Figures for An Algebraic Approach to Mathematical Models of Scales (mto.93.0.3.lindley.art) by Mark Lindley and Ronald Turner-Smith File: mto.93.0.3.lindley.fig **Fig 1. A sequence of constructions abstracted from the pitch continuum: a) a positive-number line for pitch frequencies b) a number line for logarithms to base 2 of the frequencies c) equivalence classes of points mod 1 on this number-line (flogs) d) pitch-class relations between these equivalence classes (also flogs) e) two kinds of generators for groups of pitch-class relations: equal-division (1/n converted to a flog) or harmonic (see below) f) the pairs, (set, Abelian group), which ensue from these generators g) equivalence-class neighborhoods around every point-mod-1 in this set h) "ideal systems", whereby every system comprises a pair: a finite set of non-overlapping neighborhoods, operated upon by a subset of one of our groups (that is, by a "halfgroup") i) unlimited scales (repeating in every octave indefinitely) j) scales with a highest and lowest note **Fig 2. Three pitches in one pitch class: (octave) (octave) pitches: ----+-----------------------+-----------------------+--- r 2r 4r frequencies: ----+-----------------------+-----------------------+--- r r + 1 r + 2 our model: ----+-----------------------+-----------------------+--- **Fig 3. Three notes in one pitch class: octave octave ---(-+-)---------------------(-+-)---------------------(-+-)--- note note note **Fig 4. Our notation for the generating harmonic pitch-class relations: "I" = flog 2 (= 0, thus the identity element) "V" = flog 3 plus or minus a much smaller flog (t^V^) "III" = flog 5 plus or minus a much smaller flog (t^III^) "VII" = flog 7 plus or minus a much smaller flog (t^VII^) **Fig 5. A rough classifcation of orders of intervallic magnitude: (ca. 10 octaves - range of hearing) (ca. 1 octave - difference between men's and women's voices) 10ths of an octave - melodic steps and leaps 100ths of an octave - out-of-tune-ness (Such a t would mar a consonance.) 1000ths of an octave - tempering 1/10 000-octave - musically insignificant **Fig 6. A table showing how to find the most feasible equations nV = III: T^n^ = ³ n ³ ñm(flog 3) ñ flog 5 = s^n^ ³ m^n^ ³ s^n^/(m^n^ + 1) ³ ³ 1 ³ -flog 3 - flog 5 = .o931 ³ 1 ³ .o466 ³ ³ 2 ³ 3 flog 3 + flog 5 = .o768 ³ 3 ³ .o192 ³ ³ 3 ³ 4 flog 3 - flog 5 = .o179 ³ 4 ³ .oo36 ³ --> 4V = III ³ 4 ³ 8 flog 3 + flog 5 = .oo16 ³ 8 ³ .ooo2 ³ --> -8V = III ³ 5 ³ 45 flog 3 - flog 5 = .oo14 ³ 45 ³ .oooo3 ³ **Fig 7. Branches in our system-tree: a) Among systems: harmonic equal-division b) Among harmonic systems: 1-dimensional (the generators are {I, V}) 2-dimensional (the generators are {I, V, III}) 3-dimensional (the generators are {I, V, III, VII}) c) Also among harmonic systems: coherent (comprising one chain of V's) not coherent d) Among coherent systems: untempered (all t = 0) tempered e) Among temperaments: regular (all t^V^ equal, all t^III^ equal, etc.) semi-regular (all t^V^ equal, but not all t^III^ or t^VII^) irregular (t^V^ and hence t^III^ & t^VII^ varying) f) Among regular, two- (and sometimes three-) dimensional temperaments: MT (wherein 4V = III) QP (wherein -8V = III) g) Conjunctions of MT and QP: ET^1^ (12V = I; there is no III and no VII) ET^2^ (12V = I; 4V = III = -8V; there is no VII) ET^3^ (12V = I; 4V = III = -8V; -2V = VII = 10V) h) Apart from ET, some musically good possibilities for MT or for QP: meantone temperaments quasi-Pythagorean temperaments i) Physically equivalent equal-division systems, for instance, the mere division of the octave into 12 equal parts (a system which is physically equivalent to an ET but has no diatonic or chromatic semitones) j) Certain families of such equal-division systems: F^1^ (characterized by equivalence to MT systems) F^2^ (characterized by equivalence to QP systems) k) A kind of temperament, CT, which includes ET and some irregular temperaments that approximate to an ET within a certain "margin of equivalence" and meet certain other specifications l) Some historically important kinds of CT: JSB *temperament ordinaire* (with an accent over the second "e") m) A "semi-regular" temperament that is physically equivalent to a quasi-Pythagorean temperament but has two different values for III **Fig 8. Diagram of a meantone system with 14 pitch classes: ("--" between two note-names means there is a V relation between the two pitch classes; "/" means there is a III relation between them. Imagine the "--"s spiraling up on a cylinder from Ab to D#) C# -- G# -- D# / / / A -- E -- B -- F# -- (C#) / / / / F -- C -- G -- D -- (A) / / / Ab -- Eb -- Bb -- (F) **Fig 9. Some formulas distinguishing F^1^ and F^2^: Only if n = 12i + 7j (where i>j) can 1/n generate a system in F^1^. The following possibilities result when i = 1, 2, 3 or 4 and j = 0, 1 or 2: Values for i: 1 2 3 4 Values for j: 0 12 24 (36) (48) 1 31 43 55 2 50 (62) (The numbers in parentheses are multiples of smaller results in the same table and represent equal-division systems which have *not* played a very substantial role in the history of music theory.) Only if n = 12i + 7j (where i>j) can 1/n generate a system in F^2^. The following possibilities result when i = 1 or 4 and j = 0 or 1: Values for i: 1 4 Values for j: 0 12 (48) 1 53 **Fig 10. The V and III relations in Newton's system: (G) -- (D) -- (A) -- (E) -- (B) -- (F#) -- (C#) 52 30 8 39 17 48 26 / / / / / / / (Eb) -- (Bb) -- (F) -- (C) -- (G) -- (D) -- (A) -- (E) 35 13 44 22 53 31 9 40