Otonality and utonality
Otonality and utonality are properties of chords that describe if it is simpler to treat them as part of the harmonic series or subharmonic series.
The terms otonality and utonality originated from Harry Partch, who described a chord that would fit the harmonic series an otonality, and a chord that would fit the subharmonic series a utonality. However, all chords are otonalities and utonalities under this definition. For example, the just minor chord 1–6/5–3/2 is a 5-odd-limit utonality (1/(6:5:4)), but it is also a 15-odd-limit otonality (10:12:15).
Microtonalists have since adopted a more reasonble definition that uniquely identifies a chord as otonal, utonal, or ambitonal. A chord is otonal if its largest odd number is smaller when written harmonically than subharmonically, utonal if its largest odd number is smaller when written subharmonically than harmonically, and ambitonal if the numbers are equal. In other words, if inverting a chord increases the chord's maximal odd harmonic, it is otonal, and if it reduces the chord's maximal odd harmonic, it is utonal. That way, 1–5/4–3/2 is otonal because it is more simply expressed as 4:5:6 than 1/(15:12:10), and 1–6/5–3/2 is utonal because it is more simply expressed as 1/(6:5:4) than 10:12:15.
The use of odd harmonics, based on octave equivalence, makes this definition independent of the chord's voicing. Thus 1–5/4–3/2 is otonal even if voiced as 1–4/3–5/3 (3:4:5) or 1–3/2–5/2 (2:3:5). If octave equivalence is not assumed, integer harmonics can be used.
A chord's inverse can be visualized in a 2-D drawing of the harmonic lattice as a rotation by 180 degrees around 1/1.
Precise definitions
To make that definition more precise, we can define a JI chord to be a set of positive rational numbers, the all-odd voicing of a JI chord to be a set of positive rational numbers obtained by removing all factors of two from all numerators and denominators, followed by removing any duplicate ratios, and the reduced JI chord to be the set of odd integers resulting from clearing denominators in the all-odd voicing by multiplying each member of the chord by the LCM (least common multiple) of the denominators, followed by dividing out the GCD (greatest common denominator).
For example, consider the chord {1/1 2/1 3/1 15/8}. The all-odd voicing of this is {1/1 3/1 15/1}, taking then the LCM of the denominators and simplifying (all 1 so trivial in this case) the result is 1:3:15. If we define the inverse of a chord as the chord obtained by taking the reciprocal of each member, then the inverse of our original chord is {1/1, 1/2, 1/3, 8/15}. The all odd voicing is {1/1, 1/3, 1/15}, and then multiplying by the LCM of denominators and simplifying gives us 1:5:15. If the largest member of the reduction of the original chord is smaller than the largest member of the reduction of the reciprocal, we call it otonal; if the reverse is true, we call it utonal. If they are the same, as here, we may call it ambitonal. Examples of ambitonal chords include 8:9:12 = sus2 chord (inverse 6:8:9 = sus4 chord, with the same largest-odd-number) and 8:10:15 = maj7no5 (inverse 8:12:15 = maj7no3).
If a chord can be voiced as a "palindrome", it inverts to itself, and is ambitonal. Such a voicing makes the lowest interval the same as the highest, the next lowest the same as the next highest, etc. For example, the min7 chord can be voiced as 1-m3-P5-m7 = min 3rd, maj 3rd, min 3rd, therefore it must be ambitonal. Note that some ambitonal chords, such as the maj7no5, cannot be voiced as a palindrome.
Dyads vs. intervals
By this definition all monads and dyads are ambitonal. (Dyads and intervals are not the same thing; put short an interval is the distance or a set of two pitches while a dyad is a chord of two pitch-classes, may have more than two pitches along octaves).
Therefore take note that for example while 11/8 may be the "undecimal harmonic fourth" (it is said to be rooted – of the form k / 2n), only because we are seeing it as an interval it is so, since seeing it as a dyad would convey the EFRs 8:11:16 or 11:16:22 equally: it isn't clear whether it is otonal or utonal as 16/11 is the "undecimal subharmonic fifth"; interpreting it as a dyad means that whether it is harmonic or subharmonic (or neither) depends on the voicing and/or inversion used.
Note that a dyad thus has two possible inversions (which is a distinct concept to octave complements!). For further clarity, see the section directly below.
Telling inversion of an n-ad
To determine the inversion of an (n+d)-note chord consisting of n pitches up to octave equivalence (that is, given an n-ad), go through all the pitches from lowest to highest until every pitch class is accounted for; that representation will then tell you which inversion the n-ad has.
Example: going through the pitches of the 5-note chord 5:8:10:16:20 lowest to highest, we find that the all-odd EFR is 5:1 therefore this chord is a dyad (n=2); in this case, as one of the integers in the interval is a power of 2, we can classify this inversion of the dyad as subharmonic.
Properties of types of chords
Otonal
- If we represent an otonal chord as a set of integers in the form A1:A2: ... :An, we may add any additional integers without affecting the chord's otonality.
- All chords with delta signatures that can be reduced (scaled by a positive real number) to +1+1, +1+1+1, +1+1+1+1, etc., are otonal.
Utonal
- The dyadic odd-limit of utonal chords is always smaller than the overall odd-limit. [proof][dead link]
Ambitonal
- Essentially tempered chords can be ambitonal, even though they do not have unique representations in the harmonic series.
Ambitonal chord theorem
A chord can be represented as a set of integers whose gcd is 1. (If octave equivalence is assumed, we take the largest odd factors of all of these integers.) The inverse of this chord is the set of integers LCM(original chord)/x for each integer x in the original chord.
Assume a chord is ambitonal. Then its largest integer, max(chord), is equal to the largest integer of its inverse, which is LCM(chord)/min(chord). Therefore min(chord)*max(chord) = LCM(chord). Conversely, if a set of integers has gcd 1 and also satisfies this, then it is an ambitonal chord.
Thus, for any given odd number N (where N is not prime), all ambitonal chords with LCM N can easily be found by considering subsets of the factors of N. If a subset has at least three factors (as mentioned above, the statement always holds for two or fewer), has a GCD of 1, an LCM of N, and also satisfies min(subset)*max(subset) = N, then it is an ambitonal chord. These conditions are satisfied by any subset which includes 1 and N. There are usually other valid subsets as well.
For N = 15, the factors are 1, 3, 5 and 15, and the ambitonal chords are {1, 3, 5, 15}, {1, 3, 15} and {1, 5, 15}. These octave-reduce to {1/1, 3/2, 5/4, 15/8} = maj7 chord, {1/1, 3/2, 15/8} = maj7no3 chord, and {1/1, 5/4, 15/8} = maj7no5 chord.
For N = 45, the factors are 1, 3, 5, 9, 15 and 45. One ambitonal chord is {1, 3, 5, 9, 15, 45}, which octave-reduces to {1/1, 5/4, 3/2, 15/8, 9/4, 45/16} = 16:20:24:30:36:45 = maj9(#11) chord. Any note or notes can be dropped except the root and the 11th, and the chord will still be ambitonal. The only other chord is {3, 5, 9, 15} = {1/1, 5/4, 3/2, 5/3} = maj6 chord, or its homonym the min7 chord. {3, 9, 15} is not ambitonal because the GCD isn't 1. {3, 5, 15} is not ambitonal because the LCM isn't 45.
Scales
These definitions apply equally as well to JI scales as they do to JI chords. For instance, the reduction of the Ptolemy-Zarlino just diatonic, 1/1-9/8-5/4-4/3-3/2-5/3-15/8-2, is {1, 3, 5, 9, 15, 27, 45}. The reduction of the Redfield diatonic, 1/1-10/9-5/4-4/3-3/2-5/3-15/8-2, is {3, 5, 9, 15, 27, 45, 135}. These are inversely related, so the Zarlino diatonic is otonal and the Redfield diatonic is utonal. From the manner of their construction, certain types of scales can be classed in certain ways. For instance, Euler genera, the type of combination product sets where n = 2k, or tonality diamonds are necessarily ambitonal, whereas dwarf scales are always either otonal or ambitonal.
Essentially tempered chords
This kind of reduction can also be used to analyze essentially tempered chords. Consider for example the sinbadmic tetrad, which is the 1001/1000-tempering of 1-11/10-13/10-10/7. The reduction of the JI version of this chord is {25, 35, 77, 91}; discarding the lowest number, 25, and reducing again gives {5, 11, 13}. This tells us the chord can be analyzed as an otonbal 1-11/10-13/10 chord plus a 10/7 addition requiring essential tempering.ed