Otonality and utonality: Difference between revisions

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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).

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.

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 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).
+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.