Otonality and utonality: Difference between revisions

Line 9:

'''Otonality''' and '''utonality''' are properties of [[chord]]s that describe if it is simpler to treat them as part of the [[harmonic series]] or [[subharmonic series]].

~~== Introduction ==~~

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 [[10:12:15|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).

~~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.

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).

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.

@@ Line 9: / Line 9: @@
 '''Otonality''' and '''utonality''' are properties of [[chord]]s that describe if it is simpler to treat them as part of the [[harmonic series]] or [[subharmonic series]].
-== Introduction ==
+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 [[10:12:15|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).
-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.
+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).
+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.