By John Hooker
Carnegie Mellon University, Pittsburgh, USA
The familiar major and minor scales have formed the basis for nearly all Western music over the last few centuries. Are they inevitable? Could alternative scales provide greater resources to composers? A mathematical model based on combinatorics and constraint programming can provide some answers.
The traditional scales have much to recommend them, both acoustically and combinatorially. The pitch frequencies of the scale tones bear mostly simple ratios to each other, which allows the tones to combine in easily recognizable chords. For the example, the interval of a fifth (such as C-G) corresponds to a ratio of 2:3, a fourth (C-F) to 3:4, and a major triad (C-E-G) to 4:5:6. This is acoustically important because a tone played on a musical instrument normally produces a series of harmonics, or integral multiples of the fundamental frequency. If tones that are played together have simple frequency ratios, the harmonics reinforce each other in a recognizable pattern. This helps the ear to make sense of the multiple voices of choral, orchestral, or keyboard music.
The scales have the combinatorial advantage that one can play them in 12 different keys (that is, beginning on 12 different tones) by using notes from a single underlying 12-tone chromatic scale. This produces errors in the pitch ratios (flat fifths, sharp thirds, etc.), but the chromatic pitches are “tempered” in such a way that the errors are slight and equal in all the scales. Specifically, the first and i-th tones of the chromatic scale have a frequency ratio 1:2(i –1)/12. The tempered chromatic scale was originally developed to allow instruments like a piano or organ to play in all keys without retuning. Yet it also allows the composer to exploit a wide range of combinatorial relationships when moving from one key to another.
Despite the appeal of the classical scales, several alternative scales have been investigated over the years. A number of composers have experimented with quarter-tone scales based on a 24-note chromatic, including Bartók, Berg, Bloch, Boulez, Enescu, Copeland, Ives, and Mancini. The Bohlen-Pierce scale [1,2] consists of 9 notes on a 13-note chromatic that spans a twelfth rather than an octave. Other composers have explored “just” scales of 12, 16, 19, 24 and 43 tones with perfect rather than tempered ratios . Combinatorial properties of scales have been studied by Balzano , Noll [5,6], and others.
We seek nontraditional scales that have the same desirable properties as the traditional ones – simple pitch ratios and multiple keys based on tempered tuning – but perhaps more so. The first step in our quest is to examine different chromatics of equally-spaced tones and determine which ones offer the most simple ratios within a tuning tolerance of ±0.9% (the tolerance permitted by traditional equal temperament). The winning scale, among those with 24 or fewer tones, is the 19-tone chromatic. It contains all the simple ratios with the tonic of the traditional major scale (2:3, 3:4, 3:5, 4:5), plus 5:6, 5:7, 5:8, and 5:9. Interestingly, this scale was investigated by Renaissance composers Guillaume Costeley and Francisco de Salinas, and it has been occasionally discussed since that time [7,8].
The next step is to investigate which tones from the 19-note chromatic should be selected to form diatonic scales (i.e., scales in which consecutive notes are separated by one or two chromatic semitones). Scales with 10 and 12 notes have too few (1) and too many (5) semitone intervals for aesthetic purposes. We therefore examine 11-note scales, which have 3 semitone intervals. In addition, the semitones should not be bunched up together. One condition that ensures this is Myhill’s property , but it rules out nearly all 11-note scales. A more practical condition is that the scale should contain no adjacent semitone intervals. There are 77 scales with this property.
The final step is to identify 11-note scales in the collection of 77 that offer a large number of intervals with simple ratios. For each possible scale, a constraint programming model is solved in  to determine the simplest ratios that can be assigned to its intervals, within the tuning tolerance. The scale with largest number of consonant intervals is scale number 72, which consists of tones 1, 3, 5, 7, 9, 10, 12, 14, 15, 17, and 19 of the 19-tone chromatic. The consonant intervals with the tonic appear in the table below. Some are
classical intervals (major 3rd, 4th, 5th, minor 6th, major 6th), and some have ratios that do not appear in classical scales (7/6, 7/5, 9/5).
The table also shows the distance of each of the 19 keys from the tonic, where the distance is the number of notes that do not occur in the tonic key. The classical keys have distances 1,2,3,4,5 if one follows a cycle of 4ths or 5ths. The keys for scale 72 have distances 2,4,6,8 if one follows a cycle of major 6ths for the first four keys, after which the distances are irregular. Composers may wish to consider scales in which one or more keys have a distance of 1, as in the classical scales. The best such scale (as measured by the number of consonant intervals) is scale 64, also shown in the table. By following a cycle of keys at intervals with ratio 9/7 or 14/9, one encounters key distances of 1,2, …, 8.
For comparison, the traditional major scale and scale 72 are demonstrated in synthesized mp3 files. The major scale is here, along with several chords from the scale (arpeggiated): the major triad (4:5:6), major 7 chord (8:10:12:15), minor triad (10:12:15), minor 7 chord (10:12:15:18), and jazz tensions. Scale 72 is here, with several chords from that scale: a major triad (4:5:6), a minor triad (10:12:15), a minor 7 chord (10:12:15:18), four exotic chords (5:6:7:9, 6:7:8:10, 7:8:10:12, 4:5:6:7), and some tensions. This and other alternative scales could take music in an interesting new direction.
 von Bohlen, H., Tonstufen in der Duodezine, Acustica 39 (1978) 76-86.
 Mathews, M. V., Pierce, J. R., Reeves, A., and Roberts, L. A., Theoretical and experimental exploration of the Bohlen-Pierce scale, Journal of the Acoustical Society of America 68 (1988) 1214—1222.
 Benson, D. J., Music: A Mathematical Offering, Cambridge University Press, 2006.
 Balzano, G. J., The graph-theoretic description of 12-fold and microtonal pitch systems, Computer Music Journal 4 (1980) 66-84.
 Noll, T., The topos of triads, in H. Fripertinger and L. Reich, eds., Colloquium on Mathematical Music Theory
 Noll, T., Getting involved with mathematical music theory, Journal of Mathematics and Music 8 (2014)
 Woolhouse, W. S. B., Essay on Musical Intervals, Harmonics, and the Temperament of the Musical Scale, &c., J. Souter, London, 1835.
 Mandelbaum, M. J., Multiple Division of the Octave and the Tonal Resources of 19-tone Temperament, PhD
thesis, Indiana University, 1961.
 Noll, T., Musical intervals and special linear transformations, Journal of Mathematics and Music 1 (2007) 121-137.
 Hooker, J. N., Finding alternative musical scales, in M. Rueher, ed., Principles and Practice of Constraint Programming (Proceedings), Springer (2016) 753-768.