=== === ============= ==== === ==== == == == == == ===== == == = == ==== === == == == == == == == = == == == == == == == == == ==== M U S I C T H E O R Y O N L I N E * * MTO ELECTRONIC DISCUSSION FORUM * * Copyright (c) 1996 Society for Music Theory ============================================================================== mto-talk is a moderated discussion forum sponsored by the Society for Music Theory, a non-profit scholarly society devoted to the promotion of quality research and teaching in music theory. All messages are checked by the list editor for format and appropriateness prior to being forwarded to subscribers. Send messages for posting to . Please include your name, affiliation (optional), and email address at the end of the message. Policies exist regulating the posting of advertisements and the reprinting of items that appear on this list. See the mto-talk Guide for details. ============================================================================= From rjudd@sas.upenn.edu Wed Oct 2 10:24:57 1996 Received: from mail1.sas.upenn.edu (root@MAIL1.SAS.UPENN.EDU [165.123.26.32]) by boethius.music.ucsb.edu (8.7.1/8.7.1) with ESMTP id KAA00711 for ; Wed, 2 Oct 1996 10:24:57 -0700 (PDT) Received: (from rjudd@localhost) by mail1.sas.upenn.edu (8.7.6/SAS 8.06) id KAA11912 for mto-talk@boethius.music.ucsb.edu; Wed, 2 Oct 1996 10:42:54 -0400 (EDT) From: rjudd@sas.upenn.edu (Robert F Judd) Posted-Date: Wed, 2 Oct 1996 10:42:54 -0400 (EDT) Message-Id: <199610021442.KAA11912@mail1.sas.upenn.edu> Subject: Re: Bent, Diatonic Ficta Revisited (MTO 2.6) To: mto-talk@boethius.music.ucsb.edu (MTO -Talk) Date: Wed, 2 Oct 1996 10:42:54 -0400 (EDT) X-Mailer: ELM [version 2.4 PL23-upenn3.1] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Sender: GHAZALI@PROMETHEUS.HOL.GR (Panos Vlagopoulos) Subject: Re: Bent, Diatonic Ficta Revisited (MTO 2.6) Just a couple of first reactions after reading Margaret Bent's most interesting "Diatonic Ficta Revisited: Josquin's Ave Maria in Context" in MTO volume 2.6. 1. Bent's 'reaction-to-what-a-singer-hears' theory (p.3 of my print-out)for soliciting an ad-hoc addition of ficta is an extreme way to express the excitement, following the realization of the difference between our notation and theirs: singers cannot be imagined as (contrapunctally) trained animals that were kept isolated from each other before thrown into the arena of common musical performance. The complexity of intervening factors makes an answer of general validity impossible, especially because ficta involves complexly conditioned decision making. To begin with, singers could certainly discuss and arrive on decisions about obscure or ambivalent passages before singing together; different singers could choose for different solutions, even the same singer(s) could do so in different performances. The presence or absence of the composer himself, as well as the singers' evaluation of the piece (repertoire or exceptional, traditional or novel, etc.) could be of importance. Against all Bent's argumentation that she does not do so, she is still committed to a quest for the most valid solutions. In this, her overall attitude reminds of Gregory Bateson's "double bind" situation: she says in a very strict way we shouldn't be strict. 2. Is the ant's path complex or the terrain? Maybe both, but, in any case, we should keep the distinction clear. Let's say the ant is the singer(s), the terrain is the notated part. It would be useful to keep in mind, when talking about their notation, modes, hexachords and counterpoint, that we should be ready to accept at least two perspectives thereof, the performers' and the composer's, valid for any single real performance that took place in their time. We can still try to find rules or make abstractions, but we shall need a more (computationally) powerful and complex mental image than the "separate tracks" of counterpoint, modes or hexachords, which relate more to a static, pre-compositional state of affairs. We shall need something like a general theory for decision making, that could be applied to Renaissance musicians, in which to embed constraints from all the above and probably other "tracks". Panos Vlagopoulos The Friends of Music Society Music Library "Lilian Voudouri" 11521 Athens Greece From rjudd@sas.upenn.edu Thu Oct 3 07:04:44 1996 Received: from mail2.sas.upenn.edu (rjudd@MAIL2.SAS.UPENN.EDU [165.123.26.33]) by boethius.music.ucsb.edu (8.7.1/8.7.1) with ESMTP id HAA09962 for ; Thu, 3 Oct 1996 07:04:41 -0700 (PDT) Received: (from rjudd@localhost) by mail2.sas.upenn.edu (8.7.6/SAS 8.06) id KAA09574 for mto-talk@boethius.music.ucsb.edu; Thu, 3 Oct 1996 10:04:38 -0400 (EDT) From: rjudd@sas.upenn.edu (Robert F Judd) Posted-Date: Thu, 3 Oct 1996 10:04:38 -0400 (EDT) Message-Id: <199610031404.KAA09574@mail2.sas.upenn.edu> Subject: Walker:Intonational Injustice, mto2.6 (fwd) To: mto-talk@boethius.music.ucsb.edu (MTO -Talk) Date: Thu, 3 Oct 1996 10:04:38 -0400 (EDT) X-Mailer: ELM [version 2.4 PL23-upenn3.1] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Sender: kollos@cavehill.dnet.co.uk (Jonathan Walker) Subject: Walker:Intonational Injustice, mto2.6 Since no discussion has so far begun, I thought I might provide a couple of addenda to "Intonational Injustice" (I also hope to send in a message on Victor Grauer's article in the next couple of days). The first point concerns the Ramos/Spataro vs. Gaffurius controversy; the second point concerns a later controversy which developed between Zarlino and Galilei. 1. Gaffurius, arguing against Ramos, alleged that the 81/80 (syntonic comma) was musically insignificant; Spataro, defending his teacher, Ramos, denied this (the only real proof is in the hearing), saying that "el duro monochordo pythagorico [deve] riducto in molle al senso de lo audito" -- upholding the Ramos substitution of, inter alia, the 5-limit major third (5/4) for the 3-limit Pythagorean major third (81/64). By 1496, Gaffurius had implicitly accepted defeat, since without protest he describes how organists temper their fifths (to produce the better 5-limit intervals of mean-tone, as Ramos had advocated). I've seen unsubstantiated claims that the mean-tone principle extends back as far as 1350, but I've yet to investigate this myself -- does anyone have arguments pro or contra? I'm aware of the fact that Nicolas Meeus has written an article "Bartolomeo Ramos de Pareia et la tessiture des instruments 'a clavier entre 1450 et 1550" in the *Revue des Arche'ologues et Historiens d'Art de Louvain* vol.5 (1972), but I haven't yet managed to obtain a copy -- I'm sure it's well worth reading. Any comments Nicolas might care to make would be most welcome. 2. The two most important voices of protest against Zarlino's advocacy of just intonation were G.B. Benedetti and Galilei. In the early 1560s, the former expressed his doubts in the course of his correspondence with Cipriano de Rore. He made the familiar argument that the pitch would descend progressively and irreversibly if the syntonon diatonic (the shaded area of my Figure 1) of Ptolemy/Zarlino were to be employed by singers. This argument was published in 1585, in the *Diversarum Speculationum*, by which time Galilei (*Dialogo*, 1581) had made much the same criticisms. Now in part, this descending of the pitch was only a problem when voices were accompanied, especially by keyboard instruments (as Margaret Bent has already argued in her excellent *Diatonic Ficta*); this might have been a problem by the close of the 16th century, but prior to this, a capella performance was the norm. And as I argued in my article, such descents _need_ not happen; while I certainly allowed for the possibility of some descent, I argued that in practice there was no need to expect the large deviations predicted by those opponents of just intonation who had an interest in presenting that system under its most rigid interpretation. What I haven't had an opportunity to examine yet is Zarlino's reply to such arguments, to see whether he thought they were irrelevant (especially if he had unaccompanied singing in mind), or if relevant, whether he succeeded in constructing any kind of refutation. Is anyone familiar with the relevant passages (in the *Supplimenti Musicali* presumably)? If so, you're welcome to have your say before I look them up myself. One final comment on my footnote 25: while I find Mark Lindley's Grove entry on Just Intonation most regrettable _as an introduction to JI_, I would not like to suggest that I think ill of his work in general: he is undoubtedly an excellent writer on keyboard tunings. I simply think that he was mis-cast by Grove when they selected him as their authority on JI, since his keyboard interests were bound to come to the fore. The entry is still worth reading though: among other things, it mentions Zarlino's attempt at constructing a JI keyboard -- as any of my readers will know, I would hardly endorse such an experiment (I argued that 1/4-comma meantone and its relatives were the true practical keyboard counterparts to just intonation). This, by the way, leads me to suspect that Zarlino might well have had trouble answering Benedetti and Galilei. -- Jonathan Walker Queen's University Belfast mailto:kollos@cavehill.dnet.co.uk http://www.music.qub.ac.uk/~walker/ From rjudd@sas.upenn.edu Thu Oct 3 19:01:50 1996 Received: from mail2.sas.upenn.edu (rjudd@MAIL2.SAS.UPENN.EDU [165.123.26.33]) by boethius.music.ucsb.edu (8.7.1/8.7.1) with ESMTP id TAA28121 for ; Thu, 3 Oct 1996 19:01:49 -0700 (PDT) Received: (from rjudd@localhost) by mail2.sas.upenn.edu (8.7.6/SAS 8.06) id WAA12904 for mto-talk@boethius.music.ucsb.edu; Thu, 3 Oct 1996 22:01:43 -0400 (EDT) From: rjudd@sas.upenn.edu (Robert F Judd) Posted-Date: Thu, 3 Oct 1996 22:01:43 -0400 (EDT) Message-Id: <199610040201.WAA12904@mail2.sas.upenn.edu> Subject: Re: Walker:Intonational Injustice, mto2.6 (fwd) To: mto-talk@boethius.music.ucsb.edu (MTO -Talk) Date: Thu, 3 Oct 1996 22:01:43 -0400 (EDT) X-Mailer: ELM [version 2.4 PL23-upenn3.1] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Sender: kollos@cavehill.dnet.co.uk (Jonathan Walker) Subject: Re: Walker:Intonational Injustice, mto2.6 Apologies for continuing this monologue, but I thought I'd better correct a misleading passage in my previous message: at the end of the second paragraph I imply that the Gaffurius/Spataro controversy had already taken place before 1496, when Gaffurius referred to the tempered fifths and good thirds of mean-tone temperament. This is wrong. The Gaffurius/Spataro controversy began in 1520, when Gaffurius published his *Apologia*, which was in part his belated contribution to an earlier dispute which Spataro conducted with Nicolo Burzio in the late 1480s/early 1490s; to complicate matters further, Aaron was also involved in the later dispute, on Spataro's side, even though he seems to endorse a Pythagorean standard elsewhere. By the time I happened to re-read the offending passage the message had already been sent out to mto-talk subscribers (fast work, Bob!), otherwise I could have made the correction behind the scenes. Apologies again to all outraged Spatarians and Gaffurians. -- Jonathan Walker Queen's University Belfast mailto:kollos@cavehill.dnet.co.uk http://www.music.qub.ac.uk/~walker/ From rjudd@sas.upenn.edu Fri Oct 4 08:26:32 1996 Received: from mail2.sas.upenn.edu (rjudd@MAIL2.SAS.UPENN.EDU [165.123.26.33]) by boethius.music.ucsb.edu (8.7.1/8.7.1) with ESMTP id IAA02836 for ; Fri, 4 Oct 1996 08:26:32 -0700 (PDT) Received: (from rjudd@localhost) by mail2.sas.upenn.edu (8.7.6/SAS 8.06) id LAA03751 for mto-talk@boethius.music.ucsb.edu; Fri, 4 Oct 1996 11:26:28 -0400 (EDT) From: rjudd@sas.upenn.edu (Robert F Judd) Posted-Date: Fri, 4 Oct 1996 11:26:28 -0400 (EDT) Message-Id: <199610041526.LAA03751@mail2.sas.upenn.edu> Subject: Early meantone (fwd) To: mto-talk@boethius.music.ucsb.edu (MTO -Talk) Date: Fri, 4 Oct 1996 11:26:28 -0400 (EDT) X-Mailer: ELM [version 2.4 PL23-upenn3.1] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Sender: meeus@musi.ucl.ac.be (Nicolas Meeus) Subject: Early meantone Jonathan Walker writes: >I've seen unsubstantiated claims that the mean-tone principle extends >back as far as 1350, but I've yet to investigate this myself -- does >anyone have arguments pro or contra? The situation is as follows: several 15th-century monchord (i.e. clavichord) treatises describe a Pythagorean tuning in which the notes corresponding to the black keys are tuned as flats. A typical such tuning extends through the cycle of fifths from B to Gb (thus B-E-A-D-G-C-F-Bb-Eb-Ab-Db-Gb). Obviously, the black key between F and G, in the 15th century, could only have been used as an F#, not a Gb. Some computation shows, however, that a Pythagoran diminished 4th, such as D-Gb, amounts to 384 cents, which is extremely close to a pure (5/4) major third (the difference, 2 cents, is called a *schisma*). The tuning just described therefore amounts to tuning D#, G#, C# and F# as pure thirds above B, E, A and D respectively. All other thirds are Pythagorean (408 cents). What is worth stressing, in this, is that the notes tuned as pure 3rds are the *ficta* ones, while *recta* notes remain Pythagorean. Mark Lindley has done a thorough study of these tunings (and published about them, I think, in the Journal of the Royal Musical Association). He also studied the 15th century keyboard repertory and shew that some pieces would better fit a Pythagorean tuning from F# to Db, or from C# to Ab, etc. This has to do with Margaret Bent's remark in MTO2.6 that "even if two monochords were tuned with true Pythagorean ratios, their resulting frequencies could be slightly different if those ratios were applied from a unison by a different route through the spiral of fifths". This is true, but as there are only two routes, the ascending and the descending ones, and as these are fully equivalent, the only difference that can arise is at the starting point and at the end of the route - i.e. on the black keys. The F# key, indeed, may have been tuned as Gb, etc., the difference amounting, of course, to a Pythagorean comma. There is no evidence that this problem ever arose for the white keys (i.e., say, that the B key might have been tuned as a Pythagorean Cb). One case of such a tuning is the one that Arnault de Zwolle attributes to an unidentified "Baudecetus". I suggested, in my article on "The Chekker" in *The Organ Yearbook* 1985, that this Baudecetus might be identifiable with Baudenet de Reims (Baude Cordier), who had been active at the Burgundian court until his death in 1397 or 1398, and whose treatise Arnault may have found in the library when he himself was active at the court. If this is so, the kind of tuning under discussion must have been practiced at the end of the 14th century at the latest. Mark Lindley might have mentioned earlier sources, which would account for the date 1350 that Jonathan mentions. Another source for the same type of tuning is the treatise of Giorgio Anselmi of Parma, but there are many more. (My 1972 article on Ramos has nothing to do with these matters; it is not a very good one.) An interesting aspect of this tuning system is that it allows a fully orthodox Pythagorean description, while actually producing pure thirds. I view of this, I believe that the controversy around Ramos was not so much of knowing whether a 3rd could be tuned or entoned pure (3rds had been tuned and sung pure since a long time), but one of Pythagorean orthodoxy, one of knowing whether the figures used in the description of intervals could include the number 5. It may also be noted that, even if the tuning mentioned above opened the way to meantone tunings, it would be more accurately be termed a just intonation system, as it involves no tempering of the fifths. I have an answer to Margaret Bent's and Jonathan's MTO2,6 communications almost boiling, but I thought that this note on the particular point raised by Jonathan's posting did not need waiting. Nicolas Meeus Universite de Paris Sorbonne Universite Catholique de Louvain a Louvain-la-Neuve meeus@musi.ucl.ac.be From rjudd@sas.upenn.edu Fri Oct 4 08:28:50 1996 Received: from mail2.sas.upenn.edu (rjudd@MAIL2.SAS.UPENN.EDU [165.123.26.33]) by boethius.music.ucsb.edu (8.7.1/8.7.1) with ESMTP id IAA03034 for ; Fri, 4 Oct 1996 08:28:49 -0700 (PDT) Received: (from rjudd@localhost) by mail2.sas.upenn.edu (8.7.6/SAS 8.06) id LAA04226 for mto-talk@boethius.music.ucsb.edu; Fri, 4 Oct 1996 11:28:47 -0400 (EDT) From: rjudd@sas.upenn.edu (Robert F Judd) Posted-Date: Fri, 4 Oct 1996 11:28:47 -0400 (EDT) Message-Id: <199610041528.LAA04226@mail2.sas.upenn.edu> Subject: just intonation (fwd) To: mto-talk@boethius.music.ucsb.edu (MTO -Talk) Date: Fri, 4 Oct 1996 11:28:46 -0400 (EDT) X-Mailer: ELM [version 2.4 PL23-upenn3.1] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Sender: meeus@musi.ucl.ac.be (Nicolas Meeus) Subject: just intonation Margaret Bent rightly stresses in MTO2.6 that my view of tuning is anchored to a keyboard reference. Jonathan Walker similarly complains, ibidem, against "the inappropriate imposition of a keyboard-based model of pitch on a vocal repertoire". This is a criticism that I must acknowledge, as my conception of tunings and intonation systems indeed derives from a study of keyboard instruments. I will try to take some distance from this conception in what follows, but I believe on the other hand that a more or less systematic conception of intonation systems is necessary. Pythagorean intonation is a system in which all fifths are sung or tuned pure, so that the major thirds are made quite larger than pure, larger even than equally tempered ones. When Margaret Bent speaks of "Pythagorean tuning of 5ths in practice, with pure 3rds", or of "pure intonation, Pythagorean in principle, but probably with justly tempered thirds in practice", she merely is not speaking of Pythagorean intonation. ["Justly tempered", in addition, is paradoxical : "temperament", by definition, is about altering conceptually just intervals, which become anything but just.] Any system that combines pure 5ths with pure 3rds must properly be called a just intonation system. The concept of just intonation arises from a desire to sing or play everything exactly in tune. This, however, is fully incompatible with pitch stability, as our previous discussion has shown and as what follows will confirm. Jonathan's proposed solutions reduces to shifting the pitch up a comma at selected spots, in order to compensate for the descent that would result from a strict application of just intonation. As I said before, the Josquin passage would descent five commas in just intonation. Jonathan therefore suggests five shifts up a comma (in his first solution for Margaret's version), or four (in his second), the passage ending then one comma lower than it started. I am not entirely convinced by this type of solution, but never mind : Jonathan, at least, obviously agrees to the fact that retaining the starting pitch necessarily involves special procedures. Three possible solutions emerge from our discussion : -- pitch stability is fully sacrificed to just intonation, a descent of more than one semitone being considered acceptable in the passage under discussion, in which all consonant intervals would be intoned pure. This is Margaret's solution. -- pitch stability is regained by a few localized jumps, while otherwise the piece retains an overall just intonation. The jumps involve a shift of one comma during some of the sustained notes. This is Jonathan's solution. -- pitch stability is retained at the cost of just intonation, several or all of the intervals being sung somewhat out of tune. This is the solution of keyboard temperaments. I believe that it may have a counterpart in vocal practice, where the adjustments, however, would have been unequally spread on the whole piece. The amount of the adjustment may have depended, among others, on the local level of dissonance at any instant. This was the meaning of my first intervention in this discussion : because I believe that pitch adjustments are more easily performed in dissonant contexts, I thought that some level of dissonance should be retained in the passage under discussion. Margaret's version eliminated all dissonances, which of course is consistent with her opinion that pitch stability is not at all essential. It would be silly to claim that pitch should be stable at all times. This can only be attained on instruments of fixed pitch. On the other hand it must be realized that a pitch variation of several commas, as Margaret Bent suggests, in unrealisable on ANY instrument and requires an a capella realization. Wind instruments can be blown but to a limited distance from their nominal pitch, and tend to play out of tune when pushed too far. Fretted stringed instruments can deviate by very limited amounts only (by pulling the strings aside) and even unfretted ones are limited by their open strings. I am perfectly prepared to accept a capella performances of the Josquin passage, but the question is whether Margaret is ready to EXCLUDE any instrumental participation in this and in other pieces where similar problems might arise. Jonathan, in his recent posting, states that a capella performance was the norm, but can we really be sure of this? Let's not jump to conclusions, anyway. The question we are facing is an extremely difficult one, which may never have been fully considered up to now. Nobody can tell today, I am afraid, what the situation with respect to just intonation is in Josquin's or his contemporaries' work as a whole. If Josquin's *Ave Maria* is a conundrum, then it may require a strictly a capella realization with descending pitch. Its notation, however, is such that it does not explicitly exclude other possible realizations, with pitch stability and a possible instrumental participation. (Are we facing here a case of Secret Diatonic Art in the Netherlands?) Only a wider statistical study of the situation of medieval and Renaissance polyphony with respect to intonation (just or not) could tell us what the situation really is, and the results of this study, in turn, might have consequences on our understanding of musica ficta. >From its origins in the 9th century, Occidental polyphony has been based on consonances -- that is, on an awareness of the essential difference between a consonant and a dissonant interval. This is the reason why I believe that polyphonic singing, from the start (or from very close to the start), must have been much concerned with stressing this difference (i.e. with favoring sounds with a high level of harmonicity; but let's leave acoustical considerations for another occasion). Considerations of pitch (or pitch-class), therefore, must soon have played a prominent role in our polyphony as compared to rhythm (and contrarily to the situation that may have prevailed in exotic polyphonies based on inharmonic percussion instruments, as in Indonesia or Africa). Another point that should be considered is that pitch instability needs not necessarily be descending. Margaret Bent is only partly right when she writes that "the better in tune a performance sounds (...), the more likely it is to move down". This is so, in the case of "intonational descent", only if the piece favors descending 5ths (or ascending 4ths) above ascending 5ths (or descending 4ths) or, in the case of "contrapuntal descent", only if the piece spirals the cycle of fifths downwards (i.e. towards the flat side) rather than upwards. These preferences indeed can be verified in the pieces, but they result from compositional choices which should be worth a study in themselves (among others because they have much to do with the rise of tonality). These remarks must be considered preliminary : there obviously remains a lot of work to be done on these matters, on which we will have to come back. Nicolas Meeus Universite de Paris Sorbonne Universite Catholique de Louvain a Louvain-la-Neuve meeus@musi.ucl.ac.be From rjudd@sas.upenn.edu Sat Oct 5 06:16:32 1996 Received: from mail1.sas.upenn.edu (rjudd@MAIL1.SAS.UPENN.EDU [165.123.26.32]) by boethius.music.ucsb.edu (8.7.1/8.7.1) with ESMTP id GAA00799 for ; Sat, 5 Oct 1996 06:16:31 -0700 (PDT) Received: (from rjudd@localhost) by mail1.sas.upenn.edu (8.7.6/SAS 8.06) id JAA06926 for mto-talk@boethius.music.ucsb.edu; Sat, 5 Oct 1996 09:16:30 -0400 (EDT) From: rjudd@sas.upenn.edu (Robert F Judd) Posted-Date: Sat, 5 Oct 1996 09:16:30 -0400 (EDT) Message-Id: <199610051316.JAA06926@mail1.sas.upenn.edu> Subject: Pythagorean approximations of Just Intonation (fwd) To: mto-talk@boethius.music.ucsb.edu (MTO -Talk) Date: Sat, 5 Oct 1996 09:16:29 -0400 (EDT) X-Mailer: ELM [version 2.4 PL23-upenn3.1] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Sender: kollos@cavehill.dnet.co.uk (Jonathan Walker) Subject: Pythagorean approximations of Just Intonation Many thanks to Nicolas Meeus for a valuable posting on pre- mean-tone keyboard tunings. In this message I would like to examine some of his points in more detail in order to see whether meantone tunings could indeed have emerged from the Pythagorean tuning he describes (this message is long, so to save readers from scrolling down now, my answer is no). Since some of the material which follows is diagrammatic, I would ask readers to ensure that they have their e-mail system set to a fixed-width font such as Courier or Monaco (if the settings can't be changed, you will almost certainly have a fixed-width font in operation already). I would also advise readers that some details in the argument assume some familiarity with the terminology and conceptual framework outlined in my article "Intonational Injustice" (mto2.6). Nicolas Meeus wrote: > The situation is as follows: several 15th-century ... treatises > describe a Pythagorean tuning in which the notes corresponding > to the black keys are tuned as flats. A typical such tuning > extends through the cycle of fifths from B to Gb (thus > B-E-A-D-G-C-F-Bb-Eb-Ab-Db-Gb). Obviously, the black key between F and G, in > the 15th century, could only have been used as an F#, not a Gb. Some > computation shows, however, that a Pythagoran diminished 4th, such as D-Gb, > amounts to 384 cents, which is extremely close to a pure (5/4) major third > (the difference, 2 cents, is called a *schisma*). The tuning just described > therefore amounts to tuning D#, G#, C# and F# as pure thirds above B, E, A > and D respectively. All other thirds are Pythagorean (408 cents). What is > worth stressing, in this, is that the notes tuned as pure 3rds are the > *ficta* ones, while *recta* notes remain Pythagorean. Since the commonly used (fa) flats in the 15th century were Bb, Eb and Ab, and the commonly used (mi) sharps were F#, C# and G#, a normal Pythagorean keyboard layout should tune the black notes accordingly; a choice would only arise over Ab and G# -- let's presume here that the tuner has chosen G#. The keyboard layout in ratios would then be as follows: Black notes from C# to Bb: 2187/2048, 32/27, 729/512, 6561/4096, 16/9 White notes from C to C: 1/1, 9/8, 81/64, 4/3, 3/2, 27/16, 243/128, 2/1 The notes to the sharp side of C have powers of 3 as their numerator, and powers of 2 as their denominator; notes to the flat side of C reverse this. But this is not the tuning that Nicolas has described; rather, he says that the black notes were sometimes tuned entirely as flats, so that the C#, F# and G# required in practice would have been represented in theory by Db, Gb and Ab. This would produce the following keyboard layout: Black notes from Db to Bb: 256/243, 32/27, 1024/729, 128/81, 16/9 White notes from C to C: 1/1, 9/8, 81/64, 4/3, 3/2, 27/16, 243/128, 2/1 This, as Nicolas mentions, has the happy consequence of providing very good approximations of 5/4 major thirds between D and F#(Gb), E and G#(Ab), A and C#(Db) (and B and D# if it were called upon). Thus the 3-limit diminished fourth, with the ratio 8192/6561, is for practical purposes identical with the 5-limit major third a 5/4. The difference between the two intervals is the schisma, or 32805/32768, the result of dividing 5/4 by 8192/6561 (i.e. subtracting the 3-limit dim. fourth from the 5-limit maj. third). This schisma difference amounts to only 2 cents, as against the difference between the 3-limit and the 5-limit major thirds, which is a syntonic comma, or 21.5 cents. The schisma is similar in size to the twelfth root of the Pythagorean comma, which is the amount by which the fifth is tempered in 12-note equal temperament, so the approximation to the 5/4 is indeed good. Let us now lay out this second 3-limit keyboard tuning according to the principles of Figure 1 in my article. The notes used for theoretical and tuning purposes are all therefore on the vertical axis, being members of a sequence of 3/2s. However I shall also show the 5-limit near-equivalents to the right of the notes with which they form 5/4s flattened by a schisma. As in Figure 1 of the article, I shall use letter-names on the left, and ratios on the right. Remember that the plus and minus signs respectively sharpen and flatten a pitch by a syntonic comma; the letter-names left uninflected are those covered by the basic 5-limit tuning of the Ptolemy/Zarlino syntonon diatonic 1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1 (note that C need not be 1/1 -- each modal final would serves as an initial 1/1, with a rotation of the syntonon diatonic intervallic sequence). For present purposes, I shall use the lower-case for the letter-names of the near-equivalent intervals, and leave the rest as capitals. B+ d#+ a schisma flat 243/128 1215/1024 a schisma flat E+ g#+ a schisma flat 81/64 405/256 a schisma flat A+ c#+ a schisma flat 27/16 135/128 a schisma flat D f#+ a schisma flat 9/8 45/32 a schisma flat G 3/2 C 1/1 F 4/3 Bb- 16/9 eb- 32/27 ab- 128/81 db-2 256/243 gb-2 1024/729 Thus the four members of the 3/2 sequence at the bottom of the diagram map onto the 5-limit intervals to the right of the uppermost four members. Now as Nicolas says, the upper member of each 5/4-less-a-schisma is a ficta note. But what if we wanted such thirds between recta notes, so that we would have a set of 5-limit intervals for all the modes? This would require that we start the sequence of fifths still further back: for instance, a Cb-2 would now function as a B. Nicolas doesn't believe that such possibilities were explored in the 15th century: > There is no evidence that this problem ever arose for the white keys > (i.e., say, that the B key might have been tuned as a Pythagorean Cb). Since the tuning above already requires a considerable degree of enharmonic trickery, let us nevertheless see how much more tampering is needed to produce a good representation of the syntonon diatonic on the white keys. The tuning above left us with A+, E+ and B+, whereas we want A, E and B in order to form 5/4s (a schisma flat) over F, C and G respectively. This is the system we would require for such a purpose: D f#+ a schisma flat 9/8 45/32 a schisma flat G b a schisma flat 3/2 15/8 a schisma flat C e a schisma flat 1/1 5/4 a schisma flat F a a schisma flat 4/3 5/3 a schisma flat Bb- 16/9 Eb- 32/27 Ab- 128/81 Db-2 256/243 gb-2 1024/729 cb-2 4096/2187 fb-2 8192/6561 b bb-3 32768/19683 While this is still a theoretically unimpeachable Pythagorean (i.e. 3-limit) system, its practical purposes are so blatantly at variance with the theoretical description that we need hardly wonder that the desire for better 5-limit intervals gave rise instead to a set of tuning systems that abandoned any pretence of Pythagoreanism -- namely, the meantone family. Just before leaving this extreme attempt to approximate 5-limit tuning with a 3-limit system, let us see again how a keyboard octave would be laid out (this time I'll use letter-names, since some of the ratios are very cumbersome). Black keys: Db-2, Eb-, Gb-2, Ab-, Bb- White keys: C, D, Fb-2, F, G, Bbb-3, Cb-2, C This perverse tuning translates into the practical equivalent of: Black keys: C#+, Eb-, F#+, G#+, Bb- White keys: C, D, E, F, G, A, B, C Don't be misled by the + signs attached to the black-key sharps: all form 5/4s above the appropriate white-key notes. The two black-key flats, on the contrary, remain within the Pythagorean system, so for example C and Eb- form a Pythagorean minor third (32/27), and Bb- and D form a Pythagorean major third (81/64). As in the true syntonon diatonic of just intonation, the fifth from D to A is very flat, in this case by a syntonic comma less a schisma (i.e. about 19.5 cents flat). > An interesting aspect of this tuning system is that it allows a > fully orthodox Pythagorean description, while actually producing pure > thirds. In view of this, I believe that the controversy around Ramos was not > so much of knowing whether a 3rd could be tuned or entoned pure (3rds had > been tuned and sung pure since a long time), but one of Pythagorean > orthodoxy, one of knowing whether the figures used in the description of > intervals could include the number 5. Yes. I should mention at this point that the Spataro/Gaffurius dispute is discussed in *A Correspondence of Renaissance Musicians*, ed. Bonnie J. Blackburn, Edward E. Lowinsky, and Clement A. Miller (Oxford, 1991). Bonnie Blackburn has mentioned to me that the distinction Spataro drew between theoretical and practical discourse is of some importance in understanding the controversy over tuning in the late 14th/early 15th centuries; for example, the fact that Gaffurius gave an informal description of mean-tone tuning did not affect his espousal of Pythagoreanism at the theoretical level. It is interesting that so much effort was invested, from Ramos to Zarlino, in justifying a departure from Pythagoreanism by adducing support from classical sources (as it increasingly became known that Boethius had been far from comprehensive as a transmitter of classical teachings on *harmonia*); it was not enough, for purposes of debate, to justify theoretical claims simply on the grounds that they were a faithful description of contemporary practice. > It may also be noted that, even if the tuning mentioned above opened the > way to meantone tunings, it would be more accurately be termed a just > intonation system, as it involves no tempering of the fifths. As I mentioned in my article, "Pythagorean" most often refers to the 3-limit, and "just intonation" to the 5-limit system, but their use sometimes overlaps -- this is why I insisted on using the unambiguous "3-limit" and "5-limit", otherwise hopeless confusion would have resulted. So let's say then that Nicolas has described a theoretical 3-limit system which for practical purposes happens to include some excellent approximations to a few 5-limit intervals. On the point of substance here, I must say that I doubt whether such 3-limit tunings could have evolved into mean-tone tunings, since they are too distant in theory: one remains within Pythagorean orthodoxy by using a chain of pure fifths in such a way that good approximations to four 5/4s result; the other (i.e. 1/4-comma meantone) starts with pure 5/4s and tempers the fifths accordingly. The other related "mean-tone" tunings, temper the 5/4s a little, by way of compromise (the 1/3-comma system tempers both 5/4s and 3/2s by a 1/3 of a comma -- such a large deviation is balanced by the singular virtue of this system: there is no wolf "fifth" for practical purposes). I believe therefore that 1/4-comma mean-tone and its relatives resulted from a separate impulse, which simply ignored Pythagorean orthodoxy in practice (even when it was still upheld on the theoretical level). I have also argued above that the glaring disparity between the theoretical description and the practical uses of the hypothetical Pythagorean keyboard tuning which would produce the intervallic sequence of the syntonon diatonic on the white keys would have precluded its adoption (since it is only a simple extension of the principles governing the historical tuning described by Nicolas, we certainly need not assume that it was beyond the bounds of the 15th-century imagination). And if this, the only theoretically Pythagorean means of absorbing just intonation for practical purposes on the keyboard, was not acceptable, it is hardly surprising that eventually other, non-Pythagorean avenues were chosen -- hence the mean-tone family. -- Jonathan Walker Queen's University Belfast mailto:kollos@cavehill.dnet.co.uk http://www.music.qub.ac.uk/~walker/ From rjudd@sas.upenn.edu Sat Oct 5 18:54:20 1996 Received: from mail1.sas.upenn.edu (rjudd@MAIL1.SAS.UPENN.EDU [165.123.26.32]) by boethius.music.ucsb.edu (8.7.1/8.7.1) with ESMTP id SAA05208 for ; Sat, 5 Oct 1996 18:54:19 -0700 (PDT) Received: (from rjudd@localhost) by mail1.sas.upenn.edu (8.7.6/SAS 8.06) id VAA23259 for mto-talk@boethius.music.ucsb.edu; Sat, 5 Oct 1996 21:54:14 -0400 (EDT) From: rjudd@sas.upenn.edu (Robert F Judd) Posted-Date: Sat, 5 Oct 1996 21:54:14 -0400 (EDT) Message-Id: <199610060154.VAA23259@mail1.sas.upenn.edu> Subject: Re: just intonation (fwd) To: mto-talk@boethius.music.ucsb.edu (MTO -Talk) Date: Sat, 5 Oct 1996 21:54:13 -0400 (EDT) X-Mailer: ELM [version 2.4 PL23-upenn3.1] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Sender: kollos@cavehill.dnet.co.uk (Jonathan Walker) Subject: Re: just intonation I thank Nicolas Meeus for his response on JI, which considered some points made by Margaret Bent and myself in mto2.6. However, I am rather alarmed by the interpretation Nicolas attaches to my arguments, and I would urge him to look over paragraphs I.9, I.11, and especially I.12 (in conjunction with exx.1-3) once again in order to see why I was arguing along quite different lines. Together with my argument in those paragraphs (and not as a substitute for them), I shall now offer some further comments to indicate exactly how I think I have been misconstrued. Nicolas Meeus wrote: > The concept of just intonation arises from a desire to sing or play > everything exactly in tune. This, however, is fully incompatible with pitch > stability, as our previous discussion has shown and as what follows will > confirm. Jonathan's proposed solutions reduces to shifting the pitch up a > comma at selected spots, in order to compensate for the descent that would > result from a strict application of just intonation. As I said before, the > Josquin passage would descent five commas in just intonation. Jonathan > therefore suggests five shifts up a comma (in his first solution for > Margaret's version), or four (in his second), the passage ending then one > comma lower than it started. I am not entirely convinced by this type of > solution, but never mind : Jonathan, at least, obviously agrees to the fact > that retaining the starting pitch necessarily involves special procedures. > > Three possible solutions emerge from our discussion : > [1] pitch stability is fully sacrificed to just intonation, a descent of > more than one semitone being considered acceptable in the passage under > discussion, in which all consonant intervals would be intoned pure. This is > Margaret's solution. > [2] pitch stability is regained by a few localized jumps, while otherwise > the piece retains an overall just intonation. The jumps involve a shift of > one comma during some of the sustained notes. This is Jonathan's solution... (Margaret Bent will no doubt reply to this herself, but I might point out now that this is a somewhat odd gloss on her ficta arguments, which don't concern JI directly; one might tentatively arrive at Nicolas's construal by a few deductive steps from her writings, but she has always been concerned with larger intervallic shifts than the syntonic comma, so the simple statement "this is Margaret's solution" is a little misleading. I would suggest reading paragraph 11 from *Diatonic Ficta Revisited*, mto2.6, to see that her main argument was not directly concerned with just intonation -- where the syntonic comma makes a momentary appearance, as in paragraph 9, it is only ancillary to her argument.) Firstly, my article was _not_ intended as a justification of JI on the grounds that the initial pitch of a piece could be preserved by "special procedures". I certainly did say that the five-comma downwards shift that Nicolas had mentioned was only the product of the rigid conception of JI that I was arguing against, and that in practice a much smaller descent would have occured, and as a limiting case, possibly no descent at all. Since I can only establish this point properly by referring to details, I would request that any readers wishing to follow this argument in detail now refer to Ex.2 from my article. In measure 51, sixth quaver, Nicolas would insist that JI requires a syntonic comma descent in order to produce a just minor third, i.e a 6/5, between the F and Ab of the middle parts. Now both these notes are quavers, the Ab a passing note, the F a lower neighbour note; consider the likely tempo range for the Ave Maria: I would suggest minim = 72-80 would be quite reasonable, depending on the acoustic. The two decorative quavers would pass so quickly that it would be absurd for us to expect singers to tune this minor third with just as much care as we would expect for a minor third in semibreves. I would suggest that the tempo Nicolas would require before singers would have concerned themselves much with the precise tuning of this minor third would be about minim = 20, which I need hardly say is unacceptable. Looking again at Ex.2, measure 51, you will see that I have indicated a Pythagorean minor third here, i.e. a 32/37, between F+ and Ab; this is simply to retain the F+ pitch that the tenor had sung at the beginning of the same measure -- I believe that the tenor's melodic memory of this pitch would have had far more influence on the tuning of his quaver than the fleeting Ab in the altus part. I am therefore quite unconcerned by the Pythagorean minor third I allow here. Let me stress once again that I was giving a _theoretical_ account of _practical_ possibilities; each of Exx.1-3 was only a collection of such possibilities (although where ficta is concerned, I stated my strong preference for Margaret Bent's reading). Uninflected notes in these examples were not a vague default, but carried just as precise a meaning as the inflected note; within these notational constraints I had to make a decision for each note of the passage, which was why I warned readers against any impression of superhuman precision these examples might give if misinterpreted. As I also pointed out in the article, my representation of these sets of practical possibilities in modern notation (with syntonic-comma symbols) was certainly not intended to reflect the moment-to-moment thinking of singers but only the possible results of their singing; otherwise, as I mentioned in para. I.11, I would have come under suspicion of advocating a "Secret Microtonal Art". Gaffurius was not entirely wrong in saying that the syntonic comma was insignificant: in Renaissance polyphony it is indeed melodically insignificant, but it is by no means harmonically insignificant, as Spataro rightly argued. This was why I employed upward as well as downward comma inflections in Exx.1-3: they need not behave in the same manner as ficta inflections because ficta _is_ melodically significant. I think that Nicolas might nevertheless have imagined rather more inflection than there actually was: I cannot emphasise too strongly that Exx.1-3 simply cannot be understood without careful reference to Fig.1: they are not self-sufficient. For instance, the 3/2 above Bb must be an F+, but the 3/2 above Bb- is F. Of course it would have been simpler if I had simply assigned the uninflected note names to the chain of fifths directly above and below C; the reason I did not do this was that a Pythagorean notational default would thereby have been created. I would remind readers also that I am no happier with the word "inflection" than Margaret Bent was in *Diatonic Ficta*: again, it stems from the notation of the examples, but is not applicable to the thought-processes of singers: they would not describe their activities in terms such as "here's a Bb, so if I want to make a 3/2 with this I'd better inflect my F upwards by a syntonic comma", but rather "I always try to sing carefully in tune with the other parts". If singers in the 15th-century were asked to make any theoretical pronouncements on intonation, they would undoubtedly have spoken in orthodox Pythagorean terms; part of my argument in the article was that the explicit theorising of JI, such as we find in Zarlino, was very much retrospective as far as practice was concerned. I hope this makes my position clear. I opposed the notion that JI necessarily led to a very steep downwards descent, arguing that this was a product of an impractically rigid doctrine of JI (the variety understandably favoured by those opposed to JI). Aside from the local pitch fluctuations which JI requires, I also allowed for the possibility of some pitch descent by syntonic comma steps -- I will go further than this: I would _expect_ some irreversible descent from the initial pitch to take place, but most certainly not at the rate Nicolas demands, for the kind of practical reason I mention above. I am also happy to add to this the possibility of descent by intervals larger than the syntonic comma, as Margaret Bent argues; this leads us on to Nicolas's next point: > It would be silly to claim that pitch should be stable at all times. This > can only be attained on instruments of fixed pitch. On the other hand it > must be realized that a pitch variation of several commas, as Margaret Bent > suggests, is unrealisable on ANY instrument and requires an a capella > realization. Wind instruments can be blown but to a limited distance from > their nominal pitch, and tend to play out of tune when pushed too far. > Fretted stringed instruments can deviate by very limited amounts only (by > pulling the strings aside) and even unfretted ones are limited by their > open strings. I am perfectly prepared to accept a capella performances of > the Josquin passage, but the question is whether Margaret is ready to > EXCLUDE any instrumental participation in this and in other pieces where > similar problems might arise. Jonathan, in his recent posting, states that > a capella performance was the norm, but can we really be sure of this? I'll try to compile a short bibliography on the issue of instrumental accompaniment, which I'll send to mto-talk in a few days; if Margaret can find time to deal with this point, so much the better. For the moment I'll merely say that the informed consensus seems to have swung in favour of an a capella norm for late 14th/early 15th-century sacred vocal polyphony; the problems of instrumental accompaniment and intonational compromises became a source of much debate during the course of the 15th century, as Nicolas will know from his historical work on keyboard tunings. Instrumental accompaniment would indeed have resulted in a performance style which was necessarily less sensitive to intonational subtleties, and it would seem plausible to suggest that the intonation of singers who customarily performed together with instrumentalists would have been affected even on those occasions when a capella performances took place. A related issue is the contemporay performance of pieces of vocal polyphony at the keyboard. It is hardly surprising to find that intabulations do not witness to the kind of ficta readings that Margaret argues for because the various constraints of the keyboard and its tunings inevitably involved choosing the lesser of two evils: the wolf "fifth" of the mean-tone family of temperaments had to be regarded as the ultimate diabolus in musica for keyboard players. If Vincentino's 36-notes-to-the octave extension of 1/4-comma meantone had ever become popular, these matters could have been remedied. (Ehh... tuning humour) (Two slightly pedantic points: I place "fifth" in inverted commas when speaking of the wolf, because the interval is in fact tuned as a diminished sixth, either C# to Ab, or G# to Eb on twelve-notes-to-the-octave keyboards. Secondly, the 1/4-comma mean-tone temperament is, strictly speaking, the only mean-tone tuning, since the whole tone between the outer notes of each 5/4 major third is a half syntonic comma flat of 9/8, and a half syntonic comma sharp of 10/9 -- hence the "mean-tone"; by "half a syntonic comma" I mean (81/80)^(1/2), of course.) -- Jonathan Walker Queen's University Belfast mailto:kollos@cavehill.dnet.co.uk http://www.music.qub.ac.uk/~walker/ From rothfarb Tue Oct 8 16:22:40 1996 Received: (from rothfarb@localhost) by boethius.music.ucsb.edu (8.7.1/8.7.1) id QAA08319; Tue, 8 Oct 1996 16:22:39 -0700 (PDT) Date: Tue, 8 Oct 1996 16:22:39 -0700 (PDT) From: Lee Rothfarb To: mto-talk Subscribers Subject: Notice about archived mto-talk messages Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII mto-talk Subscribers: I have noticed that people interested in locating old, archived mto-talk messages are trying to find them in the mto-talk Hypermail archive, and are receiving error messages in their browsers. Please keep the following in mind when looking for archived 'talk' messages: We have two separate archives for mto-talk messages: a) a Hypermail archive, which contains HTML-formatted messages that are cross-indexed by subject, author, and date. Those who read mto-talk messages with a browser on the Web will probably use the Hypermail archive. Those who do should be aware that only the last four months of messages are archived. Hypermail files older than 4 months are deleted. The URL for the Hypermail archive is http://boethius.music.ucsb.edu/www-talk/mto/ To see the current month's messages, go to http://boethius.music.ucsb.edu/www-talk/mto/current/index.html b) an ASCII archive, which contains plain-text versions of all mto-talk messages, organized by month, going back two years. Monthly message files older than two years are available in compressed form (Unix tar.Z files), with a 12 months of messages in each file. These can be retrieved but cannot be read until they are uncompressed (with the Unix uncompress command), and unpacked (with the Unix tar -xvf command). For those who retrieve monthly mto-talk files with anonymous FTP, the directory to change to once logged into boethius is pub/mto/mto-talk If using a WWW browser, the URL is ftp://boethius.music.ucsb.edu/pub/mto/mto-talk/ Lee A. Rothfarb, Boethius System Administrator University of California, Santa Barbara sys-admin@boethius.music.ucsb.edu From rjudd@sas.upenn.edu Wed Oct 9 07:10:16 1996 Received: from mail1.sas.upenn.edu (rjudd@MAIL1.SAS.UPENN.EDU [165.123.26.32]) by boethius.music.ucsb.edu (8.7.1/8.7.1) with ESMTP id HAA18227 for ; Wed, 9 Oct 1996 07:10:15 -0700 (PDT) Received: (from rjudd@localhost) by mail1.sas.upenn.edu (8.7.6/SAS 8.06) id KAA17370 for mto-talk@boethius.music.ucsb.edu; Wed, 9 Oct 1996 10:10:17 -0400 (EDT) From: rjudd@sas.upenn.edu (Robert F Judd) Posted-Date: Wed, 9 Oct 1996 10:10:17 -0400 (EDT) Message-Id: <199610091410.KAA17370@mail1.sas.upenn.edu> Subject: Just intonation (fwd) To: mto-talk@boethius.music.ucsb.edu (MTO -Talk) Date: Wed, 9 Oct 1996 10:10:16 -0400 (EDT) X-Mailer: ELM [version 2.4 PL23-upenn3.1] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Sender: meeus@musi.ucl.ac.be (Nicolas Meeus) Subject: Just intonation I am somewhat at loss to answer Jonathan Walker's comments concerning just intonation. On the one hand I find some of the options that he took (with respect to the schematic description of just intonation or to terminology in general) unduly complex, and on the other hand I wonder about the purpose of this discussion. About the way in which Figure 1 of Jonathan's mto2.6 communication is constructed, he writes in his posting of 6 october : "it would have been simpler if I had simply assigned the uninflected note names to the chain of fifths directly above and below C; the reason I did not do this was that a Pythagorean notational default would thereby have been created". But this "Pythagorean notational default" is a standard notation system; it goes back to the 18th century, to Euler if I am not mistaken, and has made a long and interesting history since (among other things, it probably is at the origin of the chart of tonal regions in Schoenberg's *Structural Functions of Harmony*). It has many advantages, among which readability. And you need not be an Pythagorean integrist in order to understand it. Jonathan's arrangement is centered around one of several possible presentations of Zarlino's system, which makes it potentially ambiguous. [For those interested, the standard notation (Jonathan's "Pythagorean notational default") consists in notating all notes belonging to one chain of 5ths with the same sign. If "-" means "less a comma", then notes labelled with a "-" will be a comma lower than those in the first chain; a note in the "-" chain is pure major third higher than the corresponding one three degrees below in the first chain. Zarlinian just intonation can then be described as follows : C D E- F G A- B- C from which it is easily deduced which intervals form pure 5ths, namely those with the same sign, as C G, E- B-, etc., and which form pure 3rds, namely those of which the upper note is a comma less than the lower, C E-, F A- and G B-.] Similarly, I find it rather pointless to argue against the use of equal temperement as a yardstick to judge other systems. True, there might be a danger of equal-temperament integrism, but on the other hand there is no other readable way. I, at least, cannot easily read a description of a chromatic scale as being formed of the notes C(1/1), C#(2187/2048), D(9/8), Eb(32/37), E(81/64) etc., while I easily understand that the same is formed of C(0), C#(1,14), D(2,04), Eb(2,94), E(4,08), etc., where the figures give the number of equal semitones (and 100's of a semitone; multiply by 100 to get cents). To say that a Pythagorean C# corresponds to 2187/2048 and a Db to 256/243 does not readily tell me which is lower or higher, and by how much. To describe the first as 1,14, the second as 0,9 equal semitones from C, and the difference between them as a Pythagorean comma (0,24 semitones) tells me all I want to know. And yet, I believe that equal temperament is but a theoretical fiction ... (This is the expression, not merely of a general doubt which I perhaps to easily cultivate, but more precisely of a belief that so many factors are involved in systems, tunings and temperaments, that their theoretical descriptions usually are no more than that : theoretical.) I now that terms such as "Pythagorean" and "just intonation" at times are used loosely. This does not prevent us from using them strictly. Replacing "Pythagorean" by "3-limit" (this is reminiscent of Pierre-Joseph Roussier's "progression triple" of 1765, which FÇtis bitterly attacked in his treatise) is acceptable. But "5-limit" is more ambiguous, as it may refer to meantone (especially 1/4-comma meantone) as well as to just intonation. (A slightly pedantic point, by the way : the meaning of "meantone" is not that the tone is half a comma flat of 9/8, but that the tone is half a major third. In this sense, all meantone tunings are meantone tunings strictly speaking.) When I said that the subverted Pythagorean tuning with (almost) pure thirds opened the way to the meantone tunings, I merely meant that tuning pure thirds in the 15th century probably showed the way to tuning pure thirds in the 16th century. Jonathan is of course entirely right that Pythagorean and meantone tunings are very distant in theory. The second, indeed, is a temperament, which the first is not. (A temperament is a system in which a consonant interval, often the 5th, consciously is tuned otherwise than pure.) About just intonation in the passage of Josquin's *Ave Maria*, let's be clear. When I mentioned the problem of just intonation, I merely meant this : although we may suppose that singers in general tend to sing in tune, so doing in this passage would lead to an "absurd" descent of five commas. I am very glad to have been made to realize that this descent actually may not be as absurd as I had thought. Absurd or not, however, the question of how overall pitch stability may be maintained in such a passage remains of some interest. Certainly, something special must be done to attain that stability, and that something is of the order of not singing perfectly in tune. Jonathan proposes solutions which, he says, consist in a less rigid application of just intonation. Turn it as you want, but his proposal involves five shifts up a comma during held notes. There are five spots in his "Bent/Just Intonation, version 1" where a sustained note is labelled first with a "-", then with a "0", or first with a "0" and then with a "+", in all cases meaning that the note raises a comma in the course of its duration. There are four such spots only in version 2, with the result that it ends one comma lower than it started. Jonathan now adds that "[his] representation of these sets of practical possibilities in modern notation (with syntonic-comma symbols) was certainly not intended to reflect the moment-to-moment thinking of singers", and he explains that some of the intervals, on decorative quavers, are not perfectly pure in his version either. I argue that just intonation, if applied rigidly, leads to a potential absurdity. Jonathan answers that this results from an impractically rigid doctrine of just intonation, and that applying it less rigidly avoids the problem. Isn't this a circular argument? Jonathan rightly stresses that this matter of intonation in any case has little bearing on questions of musica ficta. I fully agree. Intonation, however, has *some* bearing on these questions, to the extent that (I repeat myself here) the problem of intonation will become all the more obvious that the music is more consonant : I trust that it is more difficult to sing slightly out of tune (or less rigidly in tune, if you prefer) in a consonant context than in a (slightly) dissonant one. Nicolas Meeus Universite de Paris Sorbonne Universite Catholique de Louvain a Louvain-la-Neuve meeus@musi.ucl.ac.be From rjudd@sas.upenn.edu Fri Oct 11 11:44:09 1996 Received: from mail2.sas.upenn.edu (rjudd@MAIL2.SAS.UPENN.EDU [165.123.26.33]) by boethius.music.ucsb.edu (8.7.1/8.7.1) with ESMTP id LAA03829 for ; Fri, 11 Oct 1996 11:44:08 -0700 (PDT) Received: (from rjudd@localhost) by mail2.sas.upenn.edu (8.7.6/SAS 8.06) id OAA00092 for mto-talk@boethius.music.ucsb.edu; Fri, 11 Oct 1996 14:44:02 -0400 (EDT) From: rjudd@sas.upenn.edu (Robert F Judd) Posted-Date: Fri, 11 Oct 1996 14:44:02 -0400 (EDT) Message-Id: <199610111844.OAA00092@mail2.sas.upenn.edu> Subject: Re: Just intonation - notation/conceptual framework "unduly complex" (fwd) To: mto-talk@boethius.music.ucsb.edu (MTO -Talk) Date: Fri, 11 Oct 1996 14:44:01 -0400 (EDT) X-Mailer: ELM [version 2.4 PL23-upenn3.1] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Sender: kollos@cavehill.dnet.co.uk (Jonathan Walker) Subject: Re: Just intonation - notation/conceptual framework "unduly complex" [This message was originally sent on Thursday] Nicolas Meeus has compiled a set of questions (challenges) to which I have no difficulty in formulating answers; the only difficulty arises in presenting these answers in a manner digestible on this e-mail list. I propose to divide my response over a series of messages, each devoted to one area of debate. First, the topic of notation and conceptualisation that must be addressed by anyone attempting to theorise JI. Rather than writing in alternation with Nicolas's paragraphs, I shall quote them at the foot of the message. Within this topic, Nicolas has challenged me on two fronts: why did I not use a Pythagorean default in my letter-name notation (i.e. with regard to the attachment of plus and minus signs); and why do I largely avoid representing JI in terms of cents deviation from equal temperament. He says that my explanation is accordingly unduly complex. If readers think my notation/terminology complex, then I can only assure them that it is _duly_ complex. In the article, I was laying out JI on its own terms; to make it dependent upon a conceptual framework imported from Pythagorean intonation, or from equal temperament would have been disastrous, precisely because readers would have been much more familiar with them. What I had hoped was that any reader who was sufficiently interested would accept an initial leap into the unfamiliar, rather than demand concessions that would have become a serious (if not insurmountable) obstacle to the understanding. The introductory character of the article ensured that the mathematics required of the reader was only rudimentary; in fact I received a few private letters from non-mathematically inclined readers who were pleasantly surprised to find the mathematics much more easily assimilable than it looked at first sight; I've had no letters of complaint so far. I'm perfectly familiar with the Pythagorean variant of my notation that Nicolas suggested, but as I have just said, I considered it far better to avoid its tempting simplicity, because of the danger of creating unnecessary obstacles to the readers' understanding later. Why should I want to force readers to translate from one tuning system to another, when the tuning system I was concerned with could be represented well enough itself? Perhaps I might draw a couple of analogies from mathematics: radians notation is initially harder to understand than the 360-degrees system, but since radians are based on the properties of the circle, they offer a superior means of conceptualizing various problems -- as the student's familiarity increases, the desire to translate from radians back into degrees will soon disappear; similarly, if we need to model problems concerning spiral functions, we would be well advised to adopt polar co-ordinates as our notation and means of conceptualization, rather than the more familiar Cartesian co-ordinates -- again, initial difficulties are compensated for by the later ease of conceptualizing the problem in terms that are native to it. Equal temperament would have been much worse again as a norm against which JI is to be measured. Cents notation establishes equal temperament as the norm, but it also serves to preclude any understanding of tuning on the theoretical level -- it is only of use as an empirical guide to tunings. So where a Pythagorean norm would have allowed translatation into JI terms, cents measurement makes even translation impossible. I have been dismayed to see articles in scholarly journals purporting to explain various keyboard tuning systems which on examination use only cents measurement: this may be enough, sometimes, for someone with an electronic meter to tune a keyboard, but it doesn't even begin to explain what the reasoning behind the tuning might have been, and so no progress is made on the theoretical level. The only readers who have any hope of deducing such things are those who are already familiar with ratio descriptions of similar systems, and know how to calculate the approximate cents equivalent of ratios (remember that cents are a logarithmic measure which will always yield an approximation for any ratios, except for the octave, which they subdivide). I know, for example that a 2 cents divergence could mean that a schisma is involved, but 2 cents is also the approximation of the twelfth root of the Pythagorean comma -- similar cents measurements can mask very different theoretical derivations of intervals. (I can give a few references to such articles to any reader who might wish to check I'm not attacking a straw man, but I don't want to cause needless offence by naming any writers on the list.) Nicolas should probably be familiar with J. Murray Barbour's "Tuning and Temperament" (1951, reprinted New York, 1972) which remains the standard book-length English-language text on tuning systems. While Barbour offers much useful historical material, his theoretical descriptions use equal temperament as a norm. But in Barbour's hands, this is not merely a descriptive norm: it is also an _evaluative_ criterion, which places 12-note equal temperament at the end of a long teleological process. As one example from many, since the problem is pervasive, Barbour pronounces judgement upon 1/4-comma meantone, saying that it nearly as "unsatisfactory" as just intonation, given that "our ideal is equal temperament" (p. 27, 2nd paragraph; the following paragraph then begins with a falsehood). Barbour is not hampered by any mathematical inadequacies, but the equal-tempered telos he imposes on his material is so heavy-handed that I excluded his book from my footnotes -- it threatened to hinder, rather than help readers understand the the case I was arguing. In the context of this book alone (although alas it is far from being alone in this respect), I would indeed be courting disaster if I were to take up Nicolas's suggestion of replacing ratios with cents deviation from equal temperament. There isn't a short-cut: any reader who cannot be bothered to come to an understanding of the ratios would be well advised to forget the whole article, and spend their time usefully on whatever is more congenial. I say it again: extract the prime factors 3 and 5 from the ratio. Take the ratio 27/16: you will see that 3^3 (i.e. 27) is in the numerator, so proceed upwards from 1/1 by three fifths (you can ignore the power of 2, which only adjusts the octave). Second example -- the ratio 5/3: 5^1 is the numerator, so proceed upwards by one pure major third; 3^1 is the denominator, so proceed downwards by a pure fifth. The first interval was the 3-limit major sixth, while the second was the 5-limit major sixth. Which is the wider interval? No need to worry about cents -- there are three simple ways to deal with the matter: 1. divide the first ratio by the second; a value above 1 means that the first ratio is the wider interval; a value below 1 means that the second ratio is the wider interval. The result also tells you by what interval the two are separated (in the above case, 81/80, i.e. the syntonic comma) OR 2. find the lowest common denominator and compare the numerators (in the above case, 81/48, as against 80/48) OR 3. simply divide out the ratio (in the above case 27/16 = 1.6875, while 5/3 = 1.6667 approx.) Where the higher numbers of my mto-talk posting, "Pythagorean approximations of Just Intonation", are concerned, all I required of the reader was the recognition of some powers of three. Why demand cents measurement? It was the derivation of the tuning that I was interested in; the only results of such a tuning that we needed to know was that a 3-limit diminished fourth is almost imperceptibly narrower than a 5-limit major third (the raison d'etre of the tuning). I shall deal with the choice of a default intervallic sequence in my Fig. 1 next, together with the use of "3-limit" and "5-limit" terminology. Below are the paragraphs from Nicolas's posting which I have been answering in this message; one point arising from them shall have to wait for the next message (namely, Nicolas's comment on Zarlino). The three paragraphs I have deleted shall be dealt with in succeeding messages. Nicolas Meeus wrote: > > I am somewhat at loss to answer Jonathan Walker's comments concerning just > intonation. On the one hand I find some of the options that he took (with > respect to the schematic description of just intonation or to terminology > in general) unduly complex, and on the other hand I wonder about the > purpose of this discussion. > > About the way in which Figure 1 of Jonathan's mto2.6 communication is > constructed, he writes in his posting of 6 october : "it would have been > simpler if I had simply assigned the uninflected note names to the chain of > fifths directly above and below C; the reason I did not do this was that a > Pythagorean notational default would thereby have been created". But this > "Pythagorean notational default" is a standard notation system; it goes > back to the 18th century, to Euler if I am not mistaken, and has made a > long and interesting history since (among other things, it probably is at > the origin of the chart of tonal regions in Schoenberg's *Structural > Functions of Harmony*). It has many advantages, among which readability. > And you need not be an Pythagorean integrist in order to understand it. > Jonathan's arrangement is centered around one of several possible > presentations of Zarlino's system, which makes it potentially ambiguous. > > [For those interested, the standard notation (Jonathan's "Pythagorean > notational default") consists in notating all notes belonging to one chain > of 5ths with the same sign. If "-" means "less a comma", then notes > labelled with a "-" will be a comma lower than those in the first chain; a > note in the "-" chain is pure major third higher than the corresponding one > three degrees below in the first chain. Zarlinian just intonation can then > be described as follows : > C D E- F G A- B- C > from which it is easily deduced which intervals form pure 5ths, namely > those with the same sign, as C G, E- B-, etc., and which form pure 3rds, > namely those of which the upper note is a comma less than the lower, C E-, > F A- and G B-.] > > Similarly, I find it rather pointless to argue against the use of equal > temperement as a yardstick to judge other systems. True, there might be a > danger of equal-temperament integrism, but on the other hand there is no > other readable way. I, at least, cannot easily read a description of a > chromatic scale as being formed of the notes C(1/1), C#(2187/2048), D(9/8), > Eb(32/37), E(81/64) etc., while I easily understand that the same is formed > of C(0), C#(1,14), D(2,04), Eb(2,94), E(4,08), etc., where the figures give > the number of equal semitones (and 100's of a semitone; multiply by 100 to > get cents). To say that a Pythagorean C# corresponds to 2187/2048 and a Db > to 256/243 does not readily tell me which is lower or higher, and by how > much. To describe the first as 1,14, the second as 0,9 equal semitones from > C, and the difference between them as a Pythagorean comma (0,24 semitones) > tells me all I want to know. And yet, I believe that equal temperament is > but a theoretical fiction ... (This is the expression, not merely of a > general doubt which I perhaps to easily cultivate, but more precisely of a > belief that so many factors are involved in systems, tunings and > temperaments, that their theoretical descriptions usually are no more than > that : theoretical.) -- Jonathan Walker Queen's University Belfast mailto:kollos@cavehill.dnet.co.uk http://www.music.qub.ac.uk/~walker/ -- Jonathan Walker Queen's University Belfast mailto:kollos@cavehill.dnet.co.uk http://www.music.qub.ac.uk/~walker/ From rjudd@sas.upenn.edu Mon Oct 14 04:45:26 1996 Received: from orion.sas.upenn.edu (ORION.SAS.UPENN.EDU [165.123.26.31]) by boethius.music.ucsb.edu (8.7.1/8.7.1) with ESMTP id EAA14530 for ; Mon, 14 Oct 1996 04:45:24 -0700 (PDT) Received: from [128.91.200.251] (TS7-09.UPENN.EDU [128.91.200.251]) by orion.sas.upenn.edu (8.7.6/SAS 8.06) with SMTP id HAA09852 for ; Mon, 14 Oct 1996 07:45:22 -0400 (EDT) Date: Mon, 14 Oct 1996 07:45:22 -0400 (EDT) Posted-Date: Mon, 14 Oct 1996 07:45:22 -0400 (EDT) Message-Id: <199610141145.HAA09852@orion.sas.upenn.edu> X-Sender: rjudd@postoffice.sas.upenn.edu X-Mailer: Windows Eudora Version 1.4.4 Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" To: mto-talk@boethius.music.ucsb.edu From: kollos@cavehill.dnet.co.uk (Jonathan Walker) (by way of rjudd@sas.upenn.edu (Robert Judd)) Subject: Just intonation - prime limit terminology defended Sender: kollos@cavehill.dnet.co.uk (Jonathan Walker) Subject: Just intonation - prime limit terminology defended This is the second instalment of my response to the various questions Nicolas Meeus posed in his posting of 9 October, "Just Intonation". Here I shall consider the terminology of "limits" that I used in the article, and which failed to find favour with Nicolas. Nicolas Meeus wrote: > I know that terms such as "Pythagorean" and "just intonation" at times are > used loosely. This does not prevent us from using them strictly. Replacing > "Pythagorean" by "3-limit" (this is reminiscent of Pierre-Joseph Roussier's > "progression triple" of 1765, which Fe'tis bitterly attacked in his > treatise) is acceptable. But "5-limit" is more ambiguous, as it may refer > to meantone (especially 1/4-comma meantone) as well as to just intonation. On the contrary, "5-limit", as defined in my article (paragraphs I.2 and I.3), does not yield itself to any such ambiguities. Meantone tunings, as Nicolas knows, incorporate surds in their ratios; the theoretical description of 1/4-comma meantone, for example, requires the use of the following surds: (81/80)^(1/4) read "the fourth root of 81-over-80" (81/80)^(1/2) (81/80)^(3/4) But surds are not admissable within any n-limit tuning system: such a system is describable in terms of ratios whose numerators and denominators consist only of primes up to and including n, and of the powers and multiples of those primes. Meantone tunings cannot be described in such terms, because surds are not integers, and a fortiori not primes. The 5-limit system includes only those ratios which are constructed from 2, 3 and 5, their powers and multiples, as I said in the article. (This is entirely to leave aside the fact that the "limit" terminology was devised and clearly defined by Partch in the 40s, and has now been used consistently by two generations of theorists thereafter.) Nicolas is confusing two distinct things here: on the one hand the 5-limit system itself, and on the other, a set of tempered tuning systems designed to approximate the 5-limit system (namely, the meantone tunings). Such approximations were needed because the 5-limit system includes an infinite set of pitch relationships, whereas keyboards only allowed a finite set, with usually twelve, and sometimes up to 17 keys per octave (Vincentino and others experimented with more keys to the octave). Open systems like the 3-limit or 5-limit don't pose any theoretical problem (in this respect) for singers, but they cannot be incorporated within keyboard tuning systems, short of MIDI systems with rather complex programming involved (and even then, large-number equal-tempered approximations are generally preferred for their greater simplicity -- subscribe to the TUNING list if you want to know more). This brings me to a point related to the main burden of my last message: just as I will describe the 5-limit system in 5-limit terms, and not in 3-limit, or equal-tempered terms, so I will describe temperaments in terms of the n-limit system which they are based upon, or designed to approximate. 12-note equal temperament is describable as a tuning based on the 3-limit system; the surds involved are roots of a 3-limit ratio, namely the Pythagorean comma (3^12/2^19). Mean-tone tunings, however, require a description based on the 5-limit; the surds involved are roots of a 5-limit ratio, namely the syntonic comma (81/80). In J. Murray Barbour's "Tuning and Temperament", the meantone family of tunings is described precisely in terms of the 3-limit default notation that Nicolas was urging me to use for greater simplicity. And what happens? The very raison d'e^tre of the meantone tunings vanishes from sight: for example, 1/4-comma meantone is based around a set of pure 5/4 major thirds, with other intervals tempered accordingly, but a description of this tuning in terms derived from the 3-limit conceal this fact, since in Barbour's tablular presentation, the major third appears to have been altered just like the other intervals (it a syntonic comma flat of the 3-limit major third). Similarly the 1/3-comma tuning has pure 5/3 major sixths, which do not show up in Barbour's 3-limit based presentation. Any tuning system, as I argued in my last message, is best understood in its own terms; to describe it in other terms at best obscures, and often obliterates the peculiar features of the tuning system. Descriptions in terms of cents comes into the latter category: while I have no objection to cents measurement when within its proper limits (as in careful ethnomusicological studies), it must be realised that it offers only an approximate, empirical description, and can thus never serve as a substitute for a theoretical understanding of a system in whatever mathematical description fits it (if any). In passing, I might note here that there are a few systems, such as Kirnberger II, which were originally described in terms sufficiently vague to admit of both a 3-limit based, and a 5-limit based description (I'll send the details to anyone who happens to be interested in this). To take a rather absurd example, one 18th-century theorist (Robert Smith, *Harmonics*, Camb. 1749) devised a system which sounded similar to 1/5-comma meantone, but was in fact based upon proportions derived from pi; now if anyone seriously wished to draw up a detailed description of this system, they would be doomed to failure if they perversely excluded pi from the terms of their description because simpler terms would supposedly suffice. Having dealt with the "limit" terminology, what of the terms "Pythagorean" and "Just" intonation? Nicolas, after all, said, in the paragraph quoted above: > I know that terms such as "Pythagorean" and "just intonation" at times are > used loosely. This does not prevent us from using them strictly. While I have demonstrated above, once again, that the limit terminology does not yield the ambiguities Nicolas supposed, is there any reason why we should not simply continue using the old terms in theoretical discussions? Well, the problem is mainly that there is such a long legacy of vague usage that it would often be hard to tell whether a particular writer was using them in a rigorously non-overlapping fashion; then again, "Pythagorean" and "Just" do not themselves suggest the mathematical features of the systems they name, unlike the limit terminology. I also mention in the article that "Aristoxeneans" would be expected to retain "Pythagorean" as a derogatory term for any description of tuning that sounds -- to them -- too mathematical and a priori, and not empirical enough. But has Nicolas himself employed these terms without ambiguity? In his message of October 4, "Early meantone", he said: > It may also be noted that, even if > the tuning mentioned above opened the way to meantone tunings, it would be > more accurately be termed a just intonation system, as it involves no > tempering of the fifths. This is surely not a "strict" use of "just intonation": it follows the description of a system which is theoretically Pythagorean (in the earlier part of his message), but which was designed to approximate some just major thirds (as I discussed at some length in my message of 5 October, "Pythagorean approximations..."). Now the use of "just" for such a system is hardly without ambiguity, but the phrase which follows renders the usage still less clear: "as it involves no tempering of the fifths". The tempering of fifths has no place in Pythagorean and just systems alike: it is not a feature that identifies just intonation alone. I have no wish to complain about Nicolas using "just intonation" in an ambiguous or casual manner -- he can use this term and "Pythagorean" as vaguely as he likes. But this only reinforces my point. For detailed theoretical discussion of tuning systems, we _do_ need a vocabulary that allows us much more clarity and precision than "Pythagorean" and "just" offer. The "limit" terminology has the advantages of being unambiguous, of making its own meaning clear, and of being firmly established for many years (even if most readers of this journal are unfamiliar with it). -- Jonathan Walker Queen's University Belfast mailto:kollos@cavehill.dnet.co.uk http://www.music.qub.ac.uk/~walker/ From rjudd@sas.upenn.edu Mon Oct 14 08:19:36 1996 Received: from orion.sas.upenn.edu (ORION.SAS.UPENN.EDU [165.123.26.31]) by boethius.music.ucsb.edu (8.7.1/8.7.1) with ESMTP id IAA18182 for ; Mon, 14 Oct 1996 08:19:35 -0700 (PDT) Received: from [128.91.201.28] (TS7-42.UPENN.EDU [128.91.201.28]) by orion.sas.upenn.edu (8.7.6/SAS 8.06) with SMTP id LAA29294 for ; Mon, 14 Oct 1996 11:19:30 -0400 (EDT) Date: Mon, 14 Oct 1996 11:19:30 -0400 (EDT) Posted-Date: Mon, 14 Oct 1996 11:19:30 -0400 (EDT) Message-Id: <199610141519.LAA29294@orion.sas.upenn.edu> X-Sender: rjudd@postoffice.sas.upenn.edu X-Mailer: Windows Eudora Version 1.4.4 Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" To: mto-talk@boethius.music.ucsb.edu From: rjudd@sas.upenn.edu (Robert Judd) Subject: Unified Theory of the Arts Sender: meeus@musi.ucl.ac.be (Nicolas Meeus) Subject: Unified Theory of the Arts Victor Grauer's "Toward a Unified Theory of the Arts" (mto 2.6) is an important text, which deserves our consideration. Victor is aware of its difficulties: "even for highly informed readers, he writes, the paper's necessary condensations can be puzzling and irritating" [0.2] (figures in brackets refer to the numbered paragraphs in mto 2.6). The theory, indeed, appears tightly integrated. Its condensation in few pages and the resulting lack of redundancy further reduces any freedom of interpretation. Such a text cannot be left without reaction, however. Several of my comments below may be viewed as questions (or as variations on one single question) to Victor himself; they do not necessarily call for an answer. Victor presents his theory as "attempting the 'unification' of two or more art forms" [0.3]. I'd view it also (if only because it originally was published in *Semiotica*) as an attempt at a general semiotic that would include forms of artistic expression. This raises the complex question whether art forms are at all justifiable of a semiotic analysis -- i.e. whether they can be considered as sign systems. Victor himself seems somewhat suspicious about that, when he writes that "the poststructuralist view [...] precipitates a crisis for theory of the arts, as it demands that any such theory be totally subservient to an (already shaky) theory of the sign function" [0.6]. A theory of the sign function needs not necessarily be shaky, however, and the terms of the debate could easily be reversed. The question may not be whether art forms can be considered semiotic systems in view of the prevailing theory of the sign function, but rather how the sign function must be (re)defined in order to adapt to the fact that art forms are considered expressive, semiotic systems. From the beginning, the project of a general semiotic has been confronted with the difficulty of accommodating non linguistic semiotics to the linguistic model. Linguists have met with problems even in accepting poetry (and the poetic aspects of literature) in their semiotic theories. Out first need, therefore, may be for a new definition of the sign. This is all the more difficult that the traditional definition ("something that stands for something else") is a very ancient and venerable one, of which Saussure's couple Signifier/Signified is but one of the last avatars. A possible new definition of the sign is latent, however, in many linguistic or semiotic writings since many years, starting perhaps with Saussure's not very well defined notion of "value" to which Victor refers [0.5 and elsewhere]: "a system of interdependent terms in which the value of each term results solely from the simultaneous presence of the others". The notion of value indeed requires the existence of a system, which Victor calls a "syntactic field". Linguists more often have called such a system a "code". A linguistic code may be viewed in two ways. Assuming the traditional definition of the sign, it may be a convention linking words to things; the study of such codes is the object of semantics. But the code may also be the system of language itself, the convention that links words to each other and assigns them a value in the sense referred to above. In both these descriptions, however, the notion of linguistic code faces the same problem: it appears as a convention, a law. A law, however, which nobody formally decreed, a convention which nobody undersigned. The question, in other terms, is whether and how a linguistic code may precede its application, how language (Saussure's *langue*) may preexist speech (*parole*). This already suggests a first question (or a first formulation of the question) addressed to Victor: to what extent can a syntactic field, in your opinion, preexist the "objects of perception" which it controls? Victor's figures 1 to 4 (variations on a "house") refer to something that, without entering too complex metaphysical considerations, may be considered as belonging to the "real world". Whether the figure is grammatical or not will be judged according to the traditional definition of the sign: can the drawing acceptably stand for a house? This, to me, is a semantic relation, i.e. one that relates to something external to the semiotic system itself. In the case of the figures 5 to 8, on the other hand, the way in which the melody relates to its tonal context does not depend on a consideration of anything worldly: the discussion never leaves the tonal system itself. The relation, in this case, may be properly syntactic in the sense defined by Victor. Whether Victor's "field" should or could be defined as "semantic" in the case of his figures 1 to 4 must be considered in view of the fact that the notion of "semantic field" is not new. It was first proposed, if I am not mistaken, by the German linguist J. Trier (*Der deutsche Wortschatz im Sinnbezirk des Verstandes*, Heidelberg, 1931), and later taken over by French semanticians (mainly P. Guiraud, *La se'mantique*, Paris, 1955), to denote a network of semantically related units within linguistic systems. (A typical semantic field would assemble, say, words having to do with the object "cat", the animal.) These relations are established in language (*langue*), thus implying the preexistence of the system. Later, the semantic school of Paris (mainly Greimas) found this situation problematic and suggested the idea of *isotopy* which, in a way, concerns the possibility for a linguistic utterance (*parole*) to build up its own field of reference. The study of isotopy in language often is concerned with rhetoric figures. Let me quote an example proposed by the Groupe Mu (a Franco-Belgian group of semioticians). In a phrase like "to drink one's shame", the word "drink" is recognized as metaphoric both because it does not fit with "shame" and because it does not normally belong to the context in which such a phrase may be found (for the phrase apparently calls for a given context). If, however, the context is about a guy who died of drinking too much champagne, then it turns out that it is "shame", rather than "drink", that is metaphoric. In any case, "drink" and "shame" are not isotopic to each other (their usage in one single phrase involves at least one metaphor). At a higher level, the usage of the phrase within the context of drinking champagne may be considered allotopic (i.e. non isotopic) to its implied context (it is not normally understood to refer to actual drinking). Victor's figures 1-4 could be analyzed in terms of isotopy: the flat rhomboid of figure 1 changes its "meaning" and becomes an object in tridimensional space when isotopically inserted in a given context in fig. 2 or 3 (the isotopy, here, has to do with the sign-complex "house"). Figure 4 is unable to create an isotopy (a coherent context) and becomes, in a sense, "meaningless". A similar analysis could be performed for figures 5 to 8, with the essential difference, however, as I noted above, that here the isotopy would no more be established semantically (i.e. with reference to the world). Several semioticians after Greimas have suggested that the notion of isotopy, although originally conceived to deal with the plane of content (the signified), could be extended to concern the plane of expression (the signifier). Greimas had defined isotopy as a redundancy of classemes which allows the coherent reading of a text. This means that the sememes (the words, etc.) forming an utterance should share one or more semes (units of meaning) which would place them within a common class of meanings. The words "drink" and "champagne" share a classeme which might be defined as *potabilitas* and which is inexistent in "shame". Such considerations could be transferred to the case illustrated by Victor's figures 5 to 8. One could argue, indeed, that the melodic line of figure 5 inherently includes something that makes it compatible with G major (even although this is not the tonality that it suggests most strongly). It also includes something that makes it even more compatible with A, major or minor. It appears, therefore, that the isotopies evidenced in figures 6 and 7 result from some reciprocal relation between the melody and its accompaniment. Victor's principle that "an object can signify only in relation with a controlling field" [1.3.2] should involve a notion of compatibility between the object and the field, or between objects belonging to the same field, and resulting more from properties of the objects themselves than from those of the field. Whatever Victor's melody may include that is compatible with G major is activated by similar compatibilities in the accompaniment, and this activation also results in a de-activation of possible compatibilities with other tonalities. There'd be a lot more to say about isotopy, but this will be sufficient for a first posting. Victor's notion of a "negative field" seems to me to require that the "positive field" be conceived in system (*langue*) rather than in utterance (*parole*). One does not see, indeed, how the syntactic field could both be asserted and negated in the same utterance. The positive field can be negated only insofar as it preexists the utterance as an expectation, an implication. It may be significant, in this respect, that the examples of negative syntax that Victor quotes are cases of "disruption" [1.6], that is, which involve a high level of expectation: disruption of the expectations of ordinary representation in the case of Cubism, of the implications of tonality in the case of serialism. Later abstract art, and later atonality, may have been less engaged in battles against former conventions and, indeed, they tend to return themselves to some sort of conventionalism. "On the lowest level, negative syntax produces the disruptions that articulate (analogous to, say, the 'phonetic' stream). Positive syntax pulls these articulations together to produce the next ('phonemic') level" [1.12.5]. This is strongly reminiscent of Martinet's theory of the double articulation, according to which the language is first articulated in a discrete (and very limited) number of phonemes which, at a second level, combine to form a large number of significant units. There also is some analogy with Saussure's and Hjelmslev's notions of substance and form. Sounds, pitches, durations, etc., which belong to Victor's "negative field", form the substance of music (much as phonemes form the substance of language). They can be organized into forms, *gestalten*, chords, themes, words, which belong to Victor's "positive field". To view the substance (the "negative field") as pertaining to aesthetics, the form (the "positive field") as belonging to language may be questionable. There is something more, in art, than the mere perception of raw stimuli. One must consider, among others, the "effect of framing". Raw materials such as flat colors, a surface of naked canvas, noises, etc., are not artistic in themselves: they need to be framed, or presented on a scene (as John Cage shew, or at least tried to show, even silence on a scene may become artistic). It may well be that this framing, this *mise en exergue* of raw materials places them in a position of negation, of denial of what art should or used to be: it this sense, I can easily follow Victor's idea of a negative field. Victor's argument aims at tracing a demarcation between Signification and Aesthesia, a demarcation which he places, if I understand him correctly, between form and substance. I would be tempted, on the contrary, to view a demarcation between Semanticism and Aesthesia: on the one hand I have difficulties in accepting informal art (I am thinking of Adorno's *musique informelle*) and, on the other hand, I believe that the difficulties met with by the project of a general semiotic are concerned mainly with the fallacy of referentialism. There'd be a lot more to say, but this will be long enough for a start. Nicolas Meeus Universite de Paris Sorbonne Universite Catholique de Louvain a Louvain-la-Neuve meeus@musi.ucl.ac.be From rjudd@sas.upenn.edu Tue Oct 15 11:52:15 1996 Received: from orion.sas.upenn.edu (ORION.SAS.UPENN.EDU [165.123.26.31]) by boethius.music.ucsb.edu (8.7.1/8.7.1) with ESMTP id LAA02022 for ; Tue, 15 Oct 1996 11:52:14 -0700 (PDT) Received: from TS5-15.UPENN.EDU (TS5-15.UPENN.EDU [128.91.200.79]) by orion.sas.upenn.edu (8.7.6/SAS 8.06) with SMTP id OAA04155 for ; Tue, 15 Oct 1996 14:52:07 -0400 (EDT) Date: Tue, 15 Oct 1996 14:52:07 -0400 (EDT) Posted-Date: Tue, 15 Oct 1996 14:52:07 -0400 (EDT) Message-Id: <199610151852.OAA04155@orion.sas.upenn.edu> X-Sender: rjudd@postoffice.sas.upenn.edu (Unverified) X-Mailer: Windows Eudora Version 1.4.4 Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" To: mto-talk@boethius.music.ucsb.edu From: rjudd@sas.upenn.edu (Robert Judd) Subject: Just intonation - the meaning of "meantone" Sender: kollos@cavehill.dnet.co.uk (Jonathan Walker) Subject: Just intonation - the meaning of "meantone" This is the third instalment of my response to Nicolas Meeus's message of 9 October, on the topic of meantone tunings, and the meaning of the name "meantone". The disagreement here arose not so much from my article, but from later mto-talk discussion (but see paragraphs II.5-9 of the article for further comments); I am including this matter in my response not merely for the sake of exhaustiveness, but because it has further ramifications in our understanding both of the meantone temperaments themselves and of the relation between these temperaments and just intonation. I said, 6 October, "Re:Just Intonation": > the 1/4-comma mean-tone temperament is, strictly speaking,the only > mean-tone tuning, since the whole tone between the outer notes of > each 5/4 major third is a half syntonic comma flat of 9/8, and a half > syntonic comma sharp of 10/9 -- hence the "mean-tone"; by "half a > syntonic comma" I mean (81/80)^(1/2), of course. [I have dropped the hyphen in "mean-tone" since then, because it becomes tedious after many repetitions, whatever the dictionaries say.] Nicolas Meeus replied, 9 October, "Just Intonation": > A slightly pedantic point, by the way : the meaning of "meantone" is > not that the tone is half a comma flat of 9/8, but that the tone is > half a major third. In this sense, all meantone tunings are meantone > tunings strictly speaking. I must uphold what I originally said; Nicolas, perhaps in his reluctance to engage with the details of my ratio-based discussions, has failed to make the necessary calculations. I shall now show why (Nicolas can skip the initial stage, which is included for the sake of other readers). The "mean" in "meantone" is not the arithmetical mean of common parlance, but rather the geometrical mean, obtained by taking the nth root of the product of n values. In the present case, we want to compare the values of two geometric means: the mean between 1/1 and 5/4 (i.e. between a starting pitch and the just major third), and the mean between 10/9 and 9/8 (i.e. between the small and large whole-tones respectively of just intonation). Because of wretched ASCII restrictions in e-mail, I shall have to employ the usual cumbersome notation for roots; what we want to see is whether the following equality is true: (1/1 * 5/4)^(1/2) = (10/9 * 9/8)^(1/2) I can spare readers the effort of checking this, because the following identity follows from it: 1/1 * 5/4 = 10/9 * 9/8 and this we already know to be true; in non-numerical terms, the major third of just intonation (5/4) is identical to the interval made up of the large and small tones of just intonation (9/8 and 10/9). And since this identity holds, the two geometric means must be identical. Now this geometric mean between 1/1 and 5/4, equalling the geometric mean between 10/9 and 9/8, is the mean tone that we speak of in 1/4-comma meantone tuning. The interval between 10/9 and 9/8 is the syntonic comma, so the geometric mean between the two will be obtained either by multiplying 10/9 by the square root of the syntonic comma, or equivalently by dividing 9/8 by the square root of the syntonic comma. Here is the same structure presented in the form of Fig. 1 of my article; D (9/8)/([81/80]^(1/2) | | G (3/2)/([81/80]^(1/4) | | C - E 1/1 ---------------- 5/4 In 1/4-comma meantone, the major thirds are pure 5/4s, while the perfect fifth is flattened by the fourth root of the syntonic comma (remember that "1/4" is strictly speaking an exponent, and so denotes the fourth root, and not division by 4). Now if the fifth is 3/2 less the comma^(1/4), then the whole tone, being next in the series of 3/2s, will be a 9/8 less the comma^(2/4), which is to say less the comma^(1/2), i.e. the square root of the comma. And 9/8 less the comma^(4/4), i.e. less a whole comma is equal to 10/9. QED None of the other so-called meantone tunings have a tone which is the geometric mean of 1/1 and 5/4, and hence of 10/9 and 9/8. None preserves the distinction of just intonation between the large and the small tone; all use an interval which is between 10/9 and 9/8: in the case of 1/6, 1/5 and 2/9-comma tunings, this interval is closer to the larger 9/8, while in the 2/7 and 1/3-comma tunings the interval is closer to the smaller 10/9. Only the 1/4-comma tuning provides a tone which is the true geometric mean between the large and small tones of just intonation, and hence this tuning alone deserves the name "meantone"; the other "meantone" tunings are certainly based on the same principles as the 1/4-comma tuning, but they lack the mean tone that their name misleadingly suggests. Furthermore, only the 1/4-comma tuning even possesses any 5/4 major thirds for the construction of a mean tone: all the related tunings temper the 5/4 (although each has its own pure 5-limit intervals). I trust that readers who have remained with me to the end of this message will now also have seen in more detail why the "meantone" tunings, as I said last message, are properly understood in terms of a simple 5-limit ratio structure as a basis, which is then modified according to whichever roots of the syntonic comma the tuning uses in the construction of its own compromise between just intonation and the restrictions of the keyboard. I also hope that readers who were not previously familiar with 1/4-comma meantone and its relatives will now be better placed to see why (as I said in para. II.9 of the article) these tunings should most certainly not be regarded as initial blundering steps towards the 12-twelve note equal temperament that musicians supposedly wanted all along; these tunings, aside from being mathematically much more complex than equal temperament, possess various subtle advantages which rendered them the best possible keyboard tunings for the music from the late 15th to the early 18th centuries, the period when they were employed (remembering that in some cases, more than twelve keys per octave would have been required). -- Jonathan Walker Queen's University Belfast mailto:kollos@cavehill.dnet.co.uk http://www.music.qub.ac.uk/~walker/ -- Jonathan Walker Queen's University Belfast mailto:kollos@cavehill.dnet.co.uk http://www.music.qub.ac.uk/~walker/ From rjudd@sas.upenn.edu Tue Oct 15 12:20:36 1996 Received: from mail2.sas.upenn.edu (rjudd@MAIL2.SAS.UPENN.EDU [165.123.26.33]) by boethius.music.ucsb.edu (8.7.1/8.7.1) with ESMTP id MAA04732 for ; Tue, 15 Oct 1996 12:20:35 -0700 (PDT) Received: (from rjudd@localhost) by mail2.sas.upenn.edu (8.7.6/SAS 8.06) id PAA06240 for mto-talk@boethius.music.ucsb.edu; Tue, 15 Oct 1996 15:20:35 -0400 (EDT) From: rjudd@sas.upenn.edu (Robert F Judd) Posted-Date: Tue, 15 Oct 1996 15:20:35 -0400 (EDT) Message-Id: <199610151920.PAA06240@mail2.sas.upenn.edu> Subject: Re: Unified Theory of the Arts (fwd) To: mto-talk@boethius.music.ucsb.edu (MTO -Talk) Date: Tue, 15 Oct 1996 15:20:34 -0400 (EDT) X-Mailer: ELM [version 2.4 PL23-upenn3.1] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Sender: Victor Grauer Subject: Re: Unified Theory of the Arts I am most grateful to Professor Meeus for his serious, highly informed and insightful response to my paper, "Toward A Unified Theory of the Arts." As he has chosen to focus on the most abstract aspect of the issue at hand, i.e., the question of a "general semiotics," his concerns and my attempts to respond to them may not seem relevant to many on this list. Partly for this reason, partly because I do not wish to dwell on the purely abstract aspect of my theory, I will attempt as much as possible to steer the discussion in the direction of concrete musical and music-theoretic topics. Some of my responses with regard to particularly difficult technical issues of semiotics will, in the interests of music-theoretic relevance, be quite brief. I hope Meeus will be willing to explore some of these questions with me (and anyone else who might be interested) off-list. Before continuing, I want to take this opportunity to urge those who may feel that a "Unified Theory of the Arts" involving semiotics is not something that particularly concerns them to take a look at my paper in any case. The title is somewhat misleading. The paper contains a good deal that should be of interest to every music theoretician, especially those involved with Twentieth Century music, the structure of which is an important focus of my work. At 08:25 AM 10/14/96 -0700, Nicolas Meeus wrote: >Victor presents his theory as "attempting the 'unification' of two or more >art forms" [0.3]. I'd view it also (if only because it originally was >published in *Semiotica*) as an attempt at a general semiotic that would >include forms of artistic expression. Meeus has uncovered the "dirty little secret" of my text: it is indeed even more ambitious than I have claimed. In my defense I will insist that I certainly did not start out with the goal of producing a "general semiotic," but only wanted, initially, to understand certain aspects of modernist expression and/or structure. I soon discovered that this was not possible without developing a clearer understanding of "traditional" approaches. One thing led to another and, at a certain point, I realized that my project had become far more ambitious than I had ever intended. By that time I was, unfortunately, "hooked". >This raises the complex question >whether art forms are at all justifiable of a semiotic analysis -- i.e. >whether they can be considered as sign systems. Victor himself seems >somewhat suspicious about that, when he writes that "the poststructuralist >view [...] precipitates a crisis for theory of the arts, as it demands that >any such theory be totally subservient to an (already shaky) theory of the >sign function" [0.6]. To clarify, I feel strongly that 1. all forms of artistic expression do and indeed must lend themselves to semiotic analysis since the vital role of signification cannot be denied, even in music; 2. no such forms can be completely understood in such terms -- i.e., signification is only a part of what is going on in painting, music, etc. >A theory of the sign function needs not necessarily >be shaky, however, and the terms of the debate could easily be reversed. >The question may not be whether art forms can be considered semiotic >systems in view of the prevailing theory of the sign function, but rather >how the sign function must be (re)defined in order to adapt to the fact >that art forms are considered expressive, semiotic systems. Meeus here comes close to describing an aspect of my project. >Out first need, therefore, may be for a new definition of the sign. This is >all the more difficult that the traditional definition ("something that >stands for something else") is a very ancient and venerable one, of which >Saussure's couple Signifier/Signified is but one of the last avatars. A >possible new definition of the sign is latent, however, in many linguistic >or semiotic writings since many years, starting perhaps with Saussure's not >very well defined notion of "value" to which Victor refers [0.5 and >elsewhere]: "a system of interdependent terms in which the value of each >term results solely from the simultaneous presence of the others". The >notion of value indeed requires the existence of a system, which Victor >calls a "syntactic field". Linguists more often have called such a system a >"code". Rather than a new definition of the sign we need a new way of thinking about the conditions which make "signing" possible. My notion of a "syntactic field" is an attempt to think such conditions in very general terms, more general, really, than the notion of "code" or even "system" both of which, in my view, *depend* on syntactic fields. >A linguistic code may be viewed in two ways. Assuming the traditional >definition of the sign, it may be a convention linking words to things; the >study of such codes is the object of semantics. But the code may also be >the system of language itself, the convention that links words to each >other and assigns them a value in the sense referred to above. In both >these descriptions, however, the notion of linguistic code faces the same >problem: it appears as a convention, a law. A law, however, which nobody >formally decreed, a convention which nobody undersigned. The question, in >other terms, is whether and how a linguistic code may precede its >application, how language (Saussure's *langue*) may preexist speech >(*parole*). This already suggests a first question (or a first formulation >of the question) addressed to Victor: to what extent can a syntactic field, >in your opinion, preexist the "objects of perception" which it controls? This question goes to the heart of the matter. I must agree with those poststructuralists who argue that, in the words of Derrida (and I quote from memory, so this may be inaccurate), "nothing at all has ever been 'perceived.' " And I don't think Derrida would object if I rewrite that as "no-thing at all has ever been 'perceived.' " This is a conclusion I come to not only via study of texts but via my own experiences as a creative artist (and teacher) with a very strong interest in (and bias in favor of) "perception." (But it is very easy to misunderstand what I am saying here -- to say that no *thing* is ever perceived is not to say that nothing *whatever* is perceived or at least sensed.) In my paper I refer to an "object of perception" which can only signify in relation to a syntactic field. But as Meeus has guessed, without the syntactic field there can, in a very real sense, be *no* "object of perception" at all. Derrida refers to this *no* object as a (or even *the*) "trace." I use the term "perception" since I have found no better way of saying what I need to say at that point and I regret that it is misleading. There are other references to this term in my paper (and other writings) which can also be misconstrued. Perhaps the term should only be read, in this context, "under erasure," as Derrida would say. The closest I can come to expressing the situation I am thinking is to compare the syntactic field with the electromagnetic field of physics where the fields and the objects "within" them are mutually defining. If forced to choose, however, I would say it is the field which "pre-exists," the field which makes "perception" possible. (And it is the *breakup* of the field which makes "perception" in a very different sense possible. Sorry that this is so confusing.) >Victor's figures 1 to 4 (variations on a "house") refer to something that, >without entering too complex metaphysical considerations, may be considered >as belonging to the "real world". Whether the figure is grammatical or not >will be judged according to the traditional definition of the sign: can the >drawing acceptably stand for a house? This, to me, is a semantic relation, >i.e. one that relates to something external to the semiotic system itself. >In the case of the figures 5 to 8, on the other hand, the way in which the >melody relates to its tonal context does not depend on a consideration of >anything worldly: the discussion never leaves the tonal system itself. The >relation, in this case, may be properly syntactic in the sense defined by >Victor. This is a complex, somewhat technical matter to which I will not try to do complete justice here. For me the grammaticality of the "house" figure is determined purely by its syntactic (i.e. - in this case - spatial) properties. Just as the statement "all mimsy were the borogroves" is grammatical for purely syntactic reasons. Thus for me *both* figures 1-4 *and* 5-8 are illustrations of syntactic relations in which any semantic consideration would be secondary. >Whether Victor's "field" should or could be defined as "semantic" in the >case of his figures 1 to 4 must be considered in view of the fact that the >notion of "semantic field" is not new. I am aware of "semantic fields" but have not studied them in any depth so cannot comment with authority on this matter. My *impression* is that such fields consist essentially of hierarchically organized categories or classifications and seem rather static, whereas my notion of the syntactic field is dynamic (involving vectors). In the spirit of Derrida, however, I would not want to rule out the possibility that the opposition could be "deconstructible." Meeus continues with a very interesting discussion of "isotopy" as it applies to semantic fields. There are many ramifications which I won't deal with here, partly because I am simply not equipped. I will say only that syntactic fields, as I understand them, are definitely anisotopic, usually (or possibly invariably) organized so as to point to some sort of center. Semantic fields seem to have a much more complex structure, another indication (to me at least) that they are probably not fundamental. >Victor's figures 1-4 could be analyzed in terms of isotopy: the flat >rhomboid of figure 1 changes its "meaning" and becomes an object in >tridimensional space when isotopically inserted in a given context in fig. >2 or 3 (the isotopy, here, has to do with the sign-complex "house"). Figure >4 is unable to create an isotopy (a coherent context) and becomes, in a >sense, "meaningless". Here I am confused -- the figure "becomes" an object in 3 dimensional space because it is perceived as oriented in a particular direction. How could it be isotopic? I am probably giving away my ignorance here but this interests me, so I hope Meeus will explain farther. A similar analysis could be performed for figures 5 >to 8, with the essential difference, however, as I noted above, that here >the isotopy would no more be established semantically (i.e. with reference >to the world). Again I'm puzzled as to how orientation in the direction of a particular pitch class (key) can be regarded as "isotopic." >Greimas had defined isotopy as a redundancy of classemes which allows the >coherent reading of a text. This means that the sememes (the words, etc.) >forming an utterance should share one or more semes (units of meaning) >which would place them within a common class of meanings. The words "drink" >and "champagne" share a classeme which might be defined as *potabilitas* >and which is inexistent in "shame". This is beginning to sound familiar. As I understand it the point here is to deal with problems of ambiguity of the sort that make machine translation so difficult. If we can grasp the semantic context (semantic *field*) of an utterance we have a better hope of determining exactly what is being referred to. Thus, in the semantic field produced by the discourse of the police station, "book" would mean something different than in the semantic field produced by the discourse of the library. >Such considerations could be >transferred to the case illustrated by Victor's figures 5 to 8. One could >argue, indeed, that the melodic line of figure 5 inherently includes >something that makes it compatible with G major (even although this is not >the tonality that it suggests most strongly). It also includes something >that makes it even more compatible with A, major or minor. Meeus seems to be suggesting that the determination of key may be semantic rather than syntactic, which is interesting. It could be a problem like that of the reference of "book." Thus we can think of G major as one "semantic field" and A major as another. The melodic line in question would thus resemble the word "book" in the explanation I've provided above. What this suggests to me is the possible collapse of syntax and semantics into one another. Or, better (from my standpoint), the notion that semantic fields, like perceptions, are *produced* from syntactic fields. The notion of *key* as a semantic entity, emerging from the essentially syntactic action of musical relations could be the prototype of a kind of *archaeology* of semantics, possibly giving us an insight as to how denotative meanings arise. This seems related to a statement I buried in a footnote, to the effect that music, while lacking a clearly defined semantic dimension, has semantic "valence" (see footnote # 30). >It appears, >therefore, that the isotopies evidenced in figures 6 and 7 result from some >reciprocal relation between the melody and its accompaniment. Victor's >principle that "an object can signify only in relation with a controlling >field" [1.3.2] should involve a notion of compatibility between the object >and the field, or between objects belonging to the same field, and >resulting more from properties of the objects themselves than from those of >the field. According to my view, the objects have no (meaningful) properties separate from the field. This is especially easy to grasp with respect to music, where even the tuning of the individual notes of the scale is determined by a syntactic field. [Perhaps I have not made it sufficiently clear that my notion of "syntax" is broadly defined by *analogy* with linguistic syntax and is very general indeed. A *syntactic field* in the context of my theory should really be regarded as a kind of axiomatic construct, defined more by the way in which it is developed in the course of the argument than by any dictionary definition of the terms it contains.] Whatever Victor's melody may include that is compatible with G >major is activated by similar compatibilities in the accompaniment, and >this activation also results in a de-activation of possible compatibilities >with other tonalities. There'd be a lot more to say about isotopy, but this >will be sufficient for a first posting. The above is interesting and certainly worth considering. But it seems to me to involve far more complexity than my approach, so I will defend myself with Occam's razor. >Victor's notion of a "negative field" seems to me to require that the >"positive field" be conceived in system (*langue*) rather than in utterance >(*parole*). The negative field does indeed tend to negate system in favor of utterance, at least until the point where it can establish a system of its own. >One does not see, indeed, how the syntactic field could both be >asserted and negated in the same utterance. Where both occur in the same utterance, the negation is perceived as a distortion of the positive field rather than an out and out negation. Or else, if the negation prevails, the positive is perceived as a *remnant* (as e.g. in places in *Pierrot Lunaire* where there are disembodied references to traditional practice, such as a waltz, in an essentially atonal context.) >The positive field can be >negated only insofar as it preexists the utterance as an expectation, an >implication. This invokes, for me, the theories of Leonard Meyer which, up to a point (and ONLY up to a point), are consistent with my theory and help to explain the workings of syntactic musical fields. It may be significant, in this respect, that the examples of >negative syntax that Victor quotes are cases of "disruption" [1.6], that >is, which involve a high level of expectation: disruption of the >expectations of ordinary representation in the case of Cubism, of the >implications of tonality in the case of serialism. Later abstract art, and >later atonality, may have been less engaged in battles against former >conventions and, indeed, they tend to return themselves to some sort of >conventionalism. In response, I will quote from the postscript to my paper (paragraph 2.22): "Negative syntax, the structural principle which promotes the negative field, can be understood as possessing two dialectical "moments," which we can call (in terms of the well known and quite apt Cubist terminology) "analytic" and "synthetic." Initially negative syntax, in opposing the positive field, pulling it apart "analytically," is radically disjunctive. Ultimately, as the analytic gives way to the synthetic, disjunction with respect to the positive field gives way to unification of the negative field." In the work of Mondrian and Webern, both of whom have their "analytic" and "synthetic" phases, there is no return to conventionalism but simply an establishment of a negative field as an entity in its own right, no longer dependent on the positive field, even as a source of opposition. >"On the lowest level, negative syntax produces the disruptions that >articulate (analogous to, say, the 'phonetic' stream). Positive syntax >pulls these articulations together to produce the next ('phonemic') level" >[1.12.5]. This is strongly reminiscent of Martinet's theory of the double >articulation, according to which the language is first articulated in a >discrete (and very limited) number of phonemes which, at a second level, >combine to form a large number of significant units. This is true. Here I am speculating that certain structures which we think we already understand and to an extent take for granted may be in fact produced by an interplay of the positive and negative fields (in a manner analogous to the way in which rhetorical effects both produce and then integrate (co-opt?) disruptions). I think I may here be close to an aspect of Derrida's "differance," which both exceeds and produces traditional discourse. This is the most speculative part of my paper and must be regarded as provisional. The rest of the theory can stand (as a theory of modernism alone) even if this part proves tenuous. I feel strongly that I am on to something here, but am aware of the enormous technical difficulties entailed in any foray into such thoroughly studied and restudied terrain. >Sounds, pitches, durations, etc., which belong to Victor's "negative >field", form the substance of music (much as phonemes form the substance of >language). You seem to be forgetting both Saussure and Derrida. There is no "substance," only a play of differences, only a play of opposing (complementary) fields. They can be organized into forms, *gestalten*, chords, themes, >words, which belong to Victor's "positive field". To view the substance >(the "negative field") as pertaining to aesthetics, the form (the "positive >field") as belonging to language may be questionable. There is something >more, in art, than the mere perception of raw stimuli. Here I think you have seriously misread me. This is a very difficult but essential point. At certain places in my writings I'm afraid I do give the impression that negative syntax is equivalent to "raw perception" but such passages are misleading and I apologize for that. "Substance vs form" is a completely traditional dialectic of the sort I oppose. In my theory there is no "substance", only an opposition between the negative field and the positive field. The negative field is aesthetic not because it promotes substance but because it determines (experiential) time and/or space (see 1.8.4 and 2.19 of my paper) and, in the same action, disrupts semiosis. It is "material" in this sense only. Now it is true that this sort of structure promotes an awareness of what we usually call "raw stimuli" (and depends on what I have misleadingly called the "perceptual axiom") but what is really important is the structure (negative syntax) and the field (the negative field) it determines. We are really dealing with two opposing or complementary structures, NOT structure vs. something that could be called "direct perception of stimuli." In my view there is no such thing as such "direct perception." >One must consider, >among others, the "effect of framing". Raw materials such as flat colors, a >surface of naked canvas, noises, etc., are not artistic in themselves: they >need to be framed, or presented on a scene (as John Cage shew, or at least >tried to show, even silence on a scene may become artistic). This "frame" is essentially what I call the "field" and it can be positive or negative. >It may well be >that this framing, this *mise en exergue* of raw materials places them in a >position of negation, of denial of what art should or used to be: it this >sense, I can easily follow Victor's idea of a negative field. Here you are on the right track but only if you see the difference between positive and negative "framing". >Victor's argument aims at tracing a demarcation between Signification and >Aesthesia, a demarcation which he places, if I understand him correctly, >between form and substance. No. Between form and the disruption of form by negative syntax (which promotes sensory experience, NOT substance). >I would be tempted, on the contrary, to view a >demarcation between Semanticism and Aesthesia: I don't think it possible to oppose the semantic without first disrupting that which grounds it, which, for me, is clearly syntactic. Syntactic structures *produce* semantic effects, thus if we oppose the syntactic, the semantic will fall away of itself. If you think about it you will see that this is already a basic aspect of the gestalt principle. If we cannot perceive something, it can have no meaning for us, in terms of either sense or reference. And we can only perceive in terms of figure/ground relations which are, at least in my definition, "syntactic." >on the one hand I have >difficulties in accepting informal art (I am thinking of Adorno's *musique >informelle*) and, on the other hand, I believe that the difficulties met >with by the project of a general semiotic are concerned mainly with the >fallacy of referentialism. I am not speaking of informal art but on the contrary my notion of negative syntax involves highly structured art indeed (e.g., Mondrian, Cubism, Schoenberg, Stravinsky, Webern). I am curious as to what Meeus means by "referentialism." My impression is that Meeus and I disagree basically with respect to the status of the semantic, to which he attaches much more importance than do I. In addition to the arguments I have already presented, I would tend to reject the semantic approach simply because so much attention has been given to it over so many years, with such disappointing results (at least as far as general semiotics is concerned) and such flagrant violations of Occam's razor. I think it is now necessary to think meaning independently of semantics, taking, as a starting point, the well known distinction of Frege, between "sense" and "reference," placing the emphasis on "sense." An approach along these lines seems the only way to account for musical "meaning," which must be accounted for in any general semiotics. Meeus has implied that music may after all have a semantic (in the sense of "referential") dimension of which we have been only dimly aware and I find this interesting. I think it likely, however, that what he has noticed is simply an example of what I have called music's "semantic valence," which is produced by musical syntax. I would like to conclude this post with the following quotation from Goran Sonesson's monumental *Pictorial Concepts*: "What, then, is the general meaning of meaning? Maybe, as Levi-Strauss said, order, organization, relatedness. When we ask for the meaning of life, what friendship means to somebody, what meaning there can be in running in Central Park every morning, or even what the meaning of meaning is, we are certainly not asking for a sign." Victor Grauer No academic affiliation grauer@pps.pgh.pa.us From rjudd@sas.upenn.edu Wed Oct 16 13:04:11 1996 Received: from mail1.sas.upenn.edu (rjudd@MAIL1.SAS.UPENN.EDU [165.123.26.32]) by boethius.music.ucsb.edu (8.7.1/8.7.1) with ESMTP id NAA23194 for ; Wed, 16 Oct 1996 13:04:10 -0700 (PDT) Received: (from rjudd@localhost) by mail1.sas.upenn.edu (8.7.6/SAS 8.06) id QAA16672 for mto-talk@boethius.music.ucsb.edu; Wed, 16 Oct 1996 16:04:15 -0400 (EDT) From: rjudd@sas.upenn.edu (Robert F Judd) Posted-Date: Wed, 16 Oct 1996 16:04:15 -0400 (EDT) Message-Id: <199610162004.QAA16672@mail1.sas.upenn.edu> Subject: Re: Unified Theory of the Ar (fwd) To: mto-talk@boethius.music.ucsb.edu (MTO -Talk) Date: Wed, 16 Oct 1996 16:04:14 -0400 (EDT) X-Mailer: ELM [version 2.4 PL23-upenn3.1] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Sender: Richard Littlefield Subject: Re: Unified Theory of the Arts Reply to: RE>Unified Theory of the Arts In response to Grauer's "unified theory" in MTO, Nicolaus Meeus raises an interesting question, on the "given-ness" of art (objects) as semiotic systems. For a comprehensive, tightly argued "NO" to that question, may I recommend the following article: "The Sign and Music: A Reflection on the Theoretical Bases of Musical Semiotics." It's in the collection entitled *Musical Sig