Otonality and utonality
Introduction
Given a JI chord, how can we decide whether it is otonal or utonal? This might seem obvious at first, but it's actually surprisingly subtle. For example, the chord 10:12:15 is a 5-limit utonality (1/6:1/5:1/4), but it's also a 15-limit otonality, consisting of the 10th, 12th, and 15th harmonics of a fundamental. One reasonable definition is to say that a chord is otonal if its largest odd number is smaller than the largest odd number of its inverse, and utonal if the inverse has a smaller largest-odd-number. In other words, if inverting a chord increases its odd limit, it's otonal, and if it reduces it, it's utonal. That way 4:5:6 is otonal because it's simpler than its inverse, 10:12:15, and 10:12:15 is utonal because it is more simply expressed as 1/6:1/5:1/4. Because we're using odd limit and not integer limit, this definition is independent of the chord's voicing. Thus 4:5:6 is otonal even if voiced 3:4:5 or 2:3:5.
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 {5/6, 5/3, 5/2, 25/16}. The all-odd voicing of this is {5/3, 5/1, 25/1}, clearing denominators gives {5, 15, 75}, and dividing out the GCD results in {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 {6/5, 3/5, 2/5, 16/25}, and the reduction of this chord is {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.
By this definition all monads, dyads and intervals are ambitonal. (Dyads and intervals are not the same thing; 2:3:4 is a dyad but not an interval, and 2/1 is an interval but not a dyad.)
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]
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, 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, combination product sets, 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