Some musicians are familiar with (compared to “otonal” and “utonal”) the quasi-homophone term atonality, which refers to the lack of perceived tonal center in music. The terms “otonal” and “utonal” are related in the sense that they are musical terms describing something about harmony, but are otherwise distinct. Harry Partch used these terms in his excellent music theory book “Genesis Of A Music”, and they have fallen into common usage among those of us who are just intonation enthusiasts.
“Otonal” can be thought of as a generalization of the term “major” when describing a chord. Given the ratio 4:5:6, we can derive the frequencies of a set of pitches which are related by this ratio. Thus, if we start at 100Hz, we obtain each of the 3 pitches by multiplying that original frequency by 4/4, 5/4, and 6/4, which gives us 100Hz, 125Hz, and 150Hz. This is a major, and thus otonal, triad. An otonal chord can be any set of frequencies derived from any ratio in this manner. Thus, 4:5:6:7:8:9 is an otonal sextet, and 2:5:19 is a different otonal triad. Utonal chords are the inverse: we derive the frequencies of that 4:5:6 ratio by dividing instead of multiplying, which gives us 100Hz, 80Hz, and 66.666(repeating)Hz. A utonally-intended ratio can be written as /4:5:6. Hence, /4:5:6 is the minor triad, and the “root” of this triad is what musicians normally think of as the fifth. If our 100Hz root is a G (which it is pretty close to) then 125Hz is a B and 150Hz is a D; and 80Hz is an E-flat and 66.666(repeating)Hz is a C.
Every otonal ratio can be written as a utonal ratio, and vice versa. For example, start with that 4:5:6 ratio: it can be expanded to the set of relationships to the root, 4/4, 5/4, and 6/4. Divide each by (in this case) the largest prime, 5, which gets 4/20, 5/20, and 6/20. Reducing each gives us 1/5, 1/4, and 3/10. With common numerators, these are 3/15, 3/12, and 3/10. Thus, a 4:5:6 otonal triad is a /10:12:15 utonal triad, and a /4:5:6 utonal triad is a 10:12:15 otonal triad. This hints at the analytical power of this method of looking past our traditional “major” and “minor” labels.
Each otonal ratio maps to a utonal ratio (and vice versa) but there are some otonal and utonal ratios which not only map to each other but are the same, called ambitonal ratios. That is, some a:b:c maps to /a:b:c. When a simple ratio is ambitonal, each proportion in the ratio is the same between elements. Every two-element ratio has this property; thus, 2:3 = /2:3 and 4:5 = /4:5. Not every three-element ratio is ambitonal. Our major/minor triad does not have this property because the distance from 4/4 to 5/4 is 5/4, but the distance from 5/4 to 6/4 is 6/5. Applying /4:5:6 to 150Hz results in 150Hz, 120Hz, and 100Hz; the middle frequency changes – in this case, the difference between a major and a minor triad, 25/24, or in just intonation speak, the chromatic semitone or 5-limit dichotic comma. (For more information about commas, see Comma (music) on Wikipedia)
An example of an ambitonal ratio is 4:6:9. The distance from 4/4 to 6/4 is 3/2; the distance from 6/4 to 9/4 is also 3/2. Hence, starting again with our root 100Hz, we get 100Hz (100*4/4), 150Hz (100*6/4), and 225Hz (100*9/4) for the otonal interpretation; starting at 225Hz we get 225Hz (225*9/9), 150Hz (225*6/9) and 100Hz (225*4/9) for the utonal.
More generally, any ratio where the distance between pitches would result in intervals which are symmetric around the “middle” of the ratio will be ambitonal. A ratio of 3 pitches (a:b:c, for example) has two intervals (a:b and b:c) from which the 3-part ratio is stacked. So, 3-part ratios must have two identical intervals, which I can refer to as [i1,i1]. A 4-part ratio has 3 stacked intervals, so the intervals need be of the form [i1,i2,i1]. A 5-part ratio has 4 stacked intervals, hence [i1,i2,i2,i1]. 6-part ratios need intervals [i1,i2,i3,i2,i1], and so on.
Consider composing a piece of music which uses pedal points. We can use ambitonality to expand the concept of a pedal point to a pedal “group”, where instead of holding one note constant against shifting harmony, we’re holding one set of pitches constant against shifting harmony. For example, we could apply additional pitches to sound against the 4:6:9 chord based on both otonal and utonal relationships. The benefit with this specific approach (in addition to understanding how ratios recur at different levels of the musical harmonic series) is that when strictly applying this analysis to choir writing or any other ensemble which has pitch-flexible instruments, the voices sounding the pitches of the pedal group do not have to make comma shifts to accommodate the shifting accompanying harmony. This will make the performance of such music much easier. Much composition in common temperaments (like 12ET) involving held pedal points and pedal group-like harmonic objects ignores this mathematical insight, forcing musicians to work out how to “help” the consonance of shifting harmony by ear, without specific guidance. This makes rehearsals tougher and performances more unpredictable.
Here are a few musically common and useful ambitonal ratios:
4:6:9 is two stacked perfect fifths, a very common chord used among composers who frequently use Pythagorean or “quartal” harmonies.
9:12:16 is two stacked perfect fourths. Also common among composers who like 4:6:9, but it is also a very common tuning of the 3-note dominant sus-4 chord.
16:20:25 is two stacked major thirds, the in-tune version of the classic augmented major triad.
25:30:36 is two stacked minor thirds, one of many in-tune versions of the classic diminished triad.
64:72:81 is two stacked 9/8 whole steps, the Pythagorean tuning of Do-Re-Mi, the first three notes of a Pythagorean major scale.
8:10:12:15 is the major seventh quartad. While major and minor triads are not ambitonal, this is.
For more information: