2. Mark Lindley and Ronald Turner-Smith, "An Algebraic Approach
to Mathematical Models of Scales," *Music Theory Online* 0.3
(1993), which is a commentary based upon their book,
*Mathematical Models of Musical Scales* (Bonn: Verlag fuer
Systematische Musikwissenschaft, 1993).

3. Blackwood, *The Structure of Recognizable Diatonic Tunings*,
(Princeton: Princeton University Press, 1985).

4. See, for example, L. L. Lloyd's diagrams in his articles in
the 1954 *Grove's Dictionary of Music and Musicians* on "Just
Intonation" and "Temperaments."

5. Intervals measured by rational fractions can be converted into cents using the following approximate values: octave = 1200 cents, perfect fifth = 702 cents; Pythagorean comma = 24 cents; syntonic comma = 22 cents.

6. See Lindley and Turner-Smith, "An Algebraic Approach," paragraph 4.

7. In modern terminology, dividing the string in half gives two octaves, in thirds gives a fifth and an eleventh, and in fourths gives a fourth, octave and double octave. The other diatonic notes are then determined by calculating intervals of a fifth from these intervals. This produces two diatonic octaves in Pythagorean tuning.

8. Perfect fifths do not combine to produce a perfect octave. The nth fifth in a sequence of n fifths is defined mathematically by the expression (3/2)^(n), which can never be an exact multiple of 2 since every power of 3 is an odd number. The mathematical proof that no sequence of fourths can ever produce an octave is less obvious since every even number can be expressed as a fraction with an odd denominator, and every power of 3 can be associated with an infinite sequence of even numbers, such as 18/9, 36/9, ...; 162/81,..., etc. However, an intuitive musical proof can be deduced from the fact that fourths and fifths are inversions of one another. No sequence of fourths can generate an octave because the inverse sequence of fifths can never do so.

9. In logarithmic measure, a perfect fifth is 702 cents. Therefore twelve perfect fifths equals 8424 cents while seven octaves is 8400 cents. The difference, the Pythagorean comma, is therefore equal to 24 cents.

10. Euclid (attrib.), *Section of the Canon*, in Barker, *Greek
Musical Writings*, vol. 2 (Cambridge: Cambridge University
Press, 1989), 199. Twelve links on the chain of fifths can be
interpreted as 12 fifths, 6 whole tones, 4 minor thirds and 3
major thirds. Thus, the Pythagorean comma is also equal to the
difference between three major thirds or four minor thirds and an
octave. In Pythagorean tuning, a Pythagorean comma is the
interval between any two notes separated by 12 links on the chain
of fifths.

11. Blackwood, *Recognizable Diatonic Tunings*, 58.

12. The development of the 12-tone school of composition was a logical consequence of accepting the 12-semitone model of equal temperament in place of a chain of harmonic fifths.

13. These stacked rows appear in Helmholtz's *On the Sensations
of Tone* (London: Longman, 1885; New York: Dover, 1954; orig.,
4th ed., Braunshweig: Verlag von Fr. Vieweg u. Sohn, 1877), 312,
where it is used to illustrate the enharmonic relation between
notes in Pythagorean intonation.

14. An irrational number, such as pi or the square root of two, is one that cannot be expressed as the ratio of two integers. An irrational number includes a nonterminating decimal written to the number of decimals needed for practical accuracy.

15. Rossing, *The Science of Sound* (Reading, Mass: Addison Wesley,
1982), 161. This is a general textbook on acoustics. Rossing's
chart lists the equally-tempered, just and Pythagorean tunings for a
22-note chromatic scale in cents and decimals. The just scale
corresponds to the "C" scale in Figure 10.

16. It is possible to generate an infinite number of chromatic just scales by allowing the four adjacent links in Pythagorean tuning to freely slide along the chain of fifths. Whenever C is not part of the Pythagorean notes, it will necessarily be raised or lowered by one or more syntonic commas and no note in the chromatic scale will be tuned to 1. However, changing the absolute tuning of the notes in this manner does not change their relative tuning. Therefore, all of the just chromatic scales generated in this manner can be transposed into one of the four modes listed in Figure 11. The major and minor scales available in each of the four modes are the following:

G-mode: Cb, Eb, G, B, D# E-minor mode: Fb, Ab, C, E, G# C-mode: Fb, Ab, C, E, G# A-minor mode: Bbb, Db, F, A, C# F-mode: Bbb, Db, F, A C# D-minor mode: Gb, Bb, D, F#, A# Bb-mode: Ebb, Gb, Bb, D, F# G-minor mode: Cb, Eb, G, B, D#

17. The algorithm used to construct just chromatic scales can be used to illustrate why intervals generated by the prime 7 cannot be systematically included in diatonic keyboard tunings. Pleasant sounding intervals can include sevens. Examples which are commonly cited are a 7/4 minor seventh, a 7/5 diminished fifth, both of which are part of a diminished seventh chord in which the notes C, E, G, Bb are in the proportion of 4:5:6:7, or 1:5/4:3/2:7/4. A single scale in which these intervals are used can be contrived.

Eb- Bb-s-F--C--G-s-D----A 7/6 7/4 4/3 1 3/2 7/4 21/16However, temperaments based on 7's or factors of 7 do not produce a usable family of chromatic keyboard scales from the chain of fifths, which would have the desired ratios for the diminished seventh chord. The "septimal" comma which produces a 7/4 minor seventh and an 8/7 major second on the second link of the chain of fifths is 64/63 (27 cents). Therefore, reiteration of the septimal comma, "s", in an algorithm which tunes every minor seventh to 7/4 produces two modes of septimal chromatic scales in which E is 64/49, not the desired 5/4.

Mode 1 (F) Mode 2 (G) G# 4096/2041 (+4s) 4096/2041 (+4s) C# 8192/7203 (+4s) 348/343 (+3s) F# 512/343 (+3s) 512/343 (+3s) B 2048/1029 (+3s) 96/49 (+2s) E 64/49 (+2s) 64/49 (+2s) A 246/147 (+2s) 12/7 (+1s) D 8/7 (+1s) 8/7 (+1s) G 32/21 (+1s) 3/2 C 1 1 F 4/3 21/16 (-1s) Bb 7/4 (-1s) 7/4 (-1s) Eb 7/6 (-1s) 147/128 (-2s) Ab 49/32 (-2s) 49/32 (-2s) Db 49/48 (-2s) 1029/1024 (-3s) Gb 343/256 (-3s) 343/256 (-3s) Cb 343/192 (-3s) 7203/4096 (-4s) Fb 2401/2048 (-4s) 2401/2048 (-4s)These septimal modes are worse than the just modes because the septimal comma is broader than the syntonic comma by five cents. The deviation from just intonation is further aggravated because septimal commas increase the pitch of notes to the right of C and decrease the pitch of notes to the left of C, contrary to the action of the syntonic comma. Thus, compared to just intonation, the pitch of notes to the right of C will be painfully sharp and notes to the left will be dismally flat. For example, the major third necessary for the diminished seventh chord will be 2s+1k (76 cents) broader than in a just scale while the minor sixth will be 2s + 1k (49 cents) narrower.

18. Barbour, *Tuning and Temperament: A Historical Survey*
(East Lansing, Michigan: Michigan State College Press, 1953),
90-102.

19. Ibid., 100.

20. Ibid., 97-98.

21. Ibid., 94.

22. Helmholtz, *On the Sensations of Tone*, 453. A similar
list of "Extended Just Tuning" is found in Blackwood,
*Recognizable Diatonic Tunings*, 116-119.

23. Blackwood, *Recognizable Diatonic Tunings*, 74, demonstrates
that in a common progression of C-major triads from II to V, just
intonation would require that D as the root of the II chord be
one syntonic comma lower in pitch than D as the fifth of the V
chord. In *The Science of Musical Sound* (*Scientific American
Books*, 1983, 67), John Pierce shows a five chord
progression, I, IV, II, V, I, in which the just tuning of C drops
by a syntonic comma from the first chord to the last.

24. It is for this reason that Lloyd in his article on Just Intonation in the 1954 Grove Dictionary adopted the position that instruments without fixed pitch and vocalists use a flexible scale in which the size of the intervals vary according to the context and part of the reason that Lindley and Turner-Smith introduced the concept of "leeway" into their algebraic tuning theory.

25. Inspection of the chain of fifths tells us that a major third can only generate one-quarter of the infinite series of chromatic notes that are generated by the fifth. This is why the tuning of the major third is a subsidiary factor in the generation of diatonic scales.

26. Barbour, *Tuning and Temperament*, 26.

27. *Grove's Dictionary of Music and Musicians* (1954),
"Temperaments" (380).

28. Dart, *The Interpretation of Music* (New York: Harper & Row,
1963), 47.

29. Barbour, *Tuning and Temperament*, 108-9. Since the first
reference to split keys found by Barbour goes back to 1484, this
device must have been used for both Pythagorean and meantone
tunings.

30. *Grove's Dictionary of Music and Musicians*,
"Temperaments" (379).

31. Barbour's table 24 gives a monochord mean-tone tuning derived by Gibelius in 1666 by arithmetic division of the comma which is the same tuning shown in Figure 12. Gibelius's monochord is divided into an octave between 216000 and 108000, in which G = 144450, D = 193200 and A = 129200. The equivalence of these monochord tunings to Figure 12 is calculated as follows:

G = 216000/144450 = 4320/2889 = 480/321 = 160/107 D = 216000/193200 = 540/483 = 180/161 A = 216000/129200 = 540/323Barbour states that these approximations "check closely with numbers obtained by taking roots," with the G being off by 0.000003. Barbour,

32. Barbour, *Tuning and Temperament*, 64.

*End of Footnotes*