Physics 36 / Music 36 Duke University Spring 2008 Handout 6

Notes on Musical Scales

from Acoustical and Mathematical Points of View

If your computer and web browser software are configured to play .WAV files, you can hear some of the examples by clicking on the colored areas of the text and tables.

Musical scales are not themselves physically fundamental. (A possible exception is an "untempered" pentatonic scale based on the 1st, 3rd, 5th, 7th, and 9th harmonics: C, G, E, A/B^b, and D, for instance, which can be rearranged into the pentatonic scale C, D, E, G, A/B^b. By "A/B^b" we mean a pitch somewhere between what we might call A and what we might call B^b.)

The underlying physical fundamental of scale construction is the consonant interval. These have been harmonic intervals simply because the usual sources of musical sounds happen to have had harmonically related upper partials. Electronic synthesis of musical sound now makes alternative schemes available.

Constructing a particular scale involves compromises, culturally conditioned preferences, and arbitrary choices, such as: which intervals to emphasize, where in a scale to make them exact (i.e. maximally consonant), when to temper an important interval slightly to make others better?

Consonance of Intervals

Even for pairs of pure tones, the consonance of a given interval depends on the pitch at which it is performed. The interval may lie within the critical band at one pitch but not at another, for instance. For pairs of complex tones consonance may be correlated with the interactions among the individual harmonics of the two tones in terms of beating, roughness, et c.

Here is one traditional way of ranking intervals in terms of consonance. Two complex tones, each containing all harmonics, are assumed. After each interval are listed (a) the fraction of harmonics in the upper tone whose frequencies exactly match those of lower tone harmonics, and (b) the fraction of harmonics in the lower tone whose frequencies exactly match those of upper tone harmonics. Choose a couple of specific fundamental frequencies and verify one or two of these relationships for yourself:

[This simple, but not wholly satisfying, method of ranking the intervals is adequate to demonstrate the basis of scale building, but does not define consonance in any usefully profound way!]

How Many Notes per Octave?

The piano's 12-keys-per-octave represents a twelve-tone chromatic scale, an arbitrary limitation made attractive by a near, but not exact coincidence called the "circle of fifths". If we begin on some low frequency and successively raise that frequency by perfect fifths [f --> (3/2)f], we find that only after 12 such operations are we close to returning to the "same note", now seven octaves above the original frequency. That is, 12 fifths [f --> (3/2)¹²f = (129.7)f] is tantalizingly close to, but not the same as, 7 octaves [f --> 2⁷f = 128f]. It is a tempting place to stop.

Four Example Scales

Several seven-tone subsets of such scales are important in Western music. We shall consider four examples of a particular seven-tone (diatonic) scale -- the "major" scale, and label the notes in each octave in the common solfeggio manner: "do, re, mi, fa, sol, la, and ti" These include two examples each of two distinct approaches to scale building: (1) specification completely in terms of exact consonant intervals (Pythagorean, Just), and (2) the use of tempered intervals (Mean-tone, Equal Temperament). Charts are included in this handout, comparing the four scales in various ways.

Pythagorean Scale

Once we admit the perfect fifth and fourth to a short list of most-consonant intervals (fifth + fourth = octave, so to admit one is to admit the other), we can construct a full diatonic scale. Only two different intervals appear between adjacent notes: 9/8 (full step) and 256/243 (half step). Within this scale there are many exact fourths and fifths but no exact thirds. The 256/243 ratio was aesthetically offensive to some as a departure from the harmonic spirit of ratios of small integers.

Quarter-Comma Mean-tone Scale

One way of getting exact major thirds into such a scale is to require mi to be a major third above the tonic do. A problem then arises about where to put re! Mean-tone scales put it in the middle, keeping the first two intervals constant as in the Pythagorean scale. Exact thirds can also be provided fa-la and sol-ti, but if we leave fa an exact fourth above do and sol a fifth above do there are problems. As an example of the problems, and of the kinds of prescription that are necessary to define a practical scale in the absence of fancy tuning instruments, consider the following: The prescription "go up four perfect fifths and down two octaves" differs from "go up an exact major third" by an interval called the syntonic comma [(3/2)⁴(1/2)² = 81/64, which is a factor of 81/80 away from 4/5. that is, (81/64)(4/5) = 81/80, the syntonic comma.] If we temper the fifth, decreasing it by 1/4 of this comma, we make the prescriptions equivalent. Hence the "Quarter-comma" scale for which the fifth is narrowed to 1.4953, and the fourth is correspondingly broadened.

Just Scale

Another approach is to emphasize major triads, containing exact major and minor thirds within exact fifths. (These form the basis for currently predominant harmony in the West.) Again we can determine all seven tones in terms of exact consonant intervals. No tempering is necessary. Mathematicians (well, numerologists anyway) will be pleased that the intervals between adjacent tones not only all involve small integers, but also involve consecutive integers: 9/8, 10/9, 16/15! There are now three different intervals, however -- two different kinds of "full step". This is the Just Scale. It is very good so long as you (1) don't stray far from the one tonic key in which it is exact and (2) don't try anything it isn't suited for, like a lot of parallel fourths.

Equal-Tempered Scale

The problem of loss of consonance when moving away from the musical key in which an instrument is tuned is common to all the scales we've discussed so far. this limits the musician's ability to transpose (change key to accommodate a vocalist's range, for instance) and the composer's freedom to modulate from one key to another within a piece of music. [A keyboard instrument tuned for a just scale in the key of C (no sharps or flats), for instance, can be played in the keys of F (one flat) or G (one sharp) with little loss of consonance, but increasing numbers of sharps or flats in the key signature will be accompanied by rapidly deteriorating intonation.] The ultimate solution to this problem might be to provide enough keys to play on for all the keys one needs to play in--but that would mean 77 keys per octave for just intonation in all major and minor keys! The practical solution that has been adopted in the West is the Equal Tempered scale, in which the fifth is tempered just enough to make the circle of fifths exact [the fifth is narrowed to 1.4983 since (1.4983)¹² = 2⁷] for the twelve-tone chromatic scale, with each "half-step" interval exactly the twelfth root of two. Now one can transpose and modulate freely with no decrease in consonance, but cannot obtain just intonation in any key, or obtain any exact perfect fifth.

As an indication of the magnitude of the differences among these four scales, here are frequencies for mi, la, and ti in each, based on a 256 Hz C as do:

Here is a prescription for how to tune a diatonic Pythagorean scale in C, using nothing but exact perfect fifths. Tune middle C to a reference pitch. Then tune the following notes as a chain of ascending exact perfect fifths, also transferring new pitches down into the central octave as necessary: G, D, A, E, and B. Finally, tune F an exact perfect fifth below the original C and transfer it up into the central octave.

Similarly, a diatonic Just scale in C can be obtained simply by tuning three exact triads. Tune middle C to a reference pitch. Then tune E and G by forming the exact major triad C-E-G. Then tune B and D by forming the triad G-B-D. Finally tune F and A by forming the triad F-A-C. A more typical procedure for obtaining a full chromatic Just scale while only having to tune two exact major thirds will be discussed below.

To tune a full chromatic scale that is Just in the key of C, begin by tuning A to a reference pitch (e.g. 440 Hz). Then tune two minimally-beating major thirds with respect to that A: the C# above and the F below. Then using a chain of ascending exact perfect fifths, tune C from F, G from C, and D from G. Then E from A, B from E, and F# from B. Finally, G# from C#, D# from G#, and A# from D#. This sequence is shown graphically in the following table; where Ref marks the reference tone in each case, +M3 and -M3 mean ascending and descending major thirds, respectively, +P5 is an ascending perfect fifth, and -8 denotes transferring the result down by an octave into the central octave being tuned.

If your computer can play .WAV files, you can listen to this full set of major triads based on the tones of this Just scale.

Triads important in keys considered harmonically "close" to the key of C Major are generally good in this scale, while some others certainly are not. Here access to the same twelve triads is organized to facilitate such comparisons. The three rows of buttons correspond to the tonic, subdominant, and dominant major triads of the key sharing the tonic chord's name. The key of C is represented in the center column, while keys with increasing numbers of flats in their signatures extend out to the left and those with increasing numbers of sharps to the right.