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.

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.)

=== Dyads vs. intervals ===

By this definition all [[monad]]s and [[dyad]]s 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.)

Therefore take note that while [[43/32]] may be the "prime harmonic fourth" (in that it is rooted/of the form ''k'' / 2''n''), it is only because we are seeing it as an ''interval'' that it is so, because seeing it as a ''dyad'' would mean seeing it as 32:43:64 so that it isn't clear whether it is otonal or utonal as [[64/43]] is the "prime subharmonic fifth", so 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 (consisting of ''two'' [[pitch class]]es) thus has ''two'' possible ''inversions'' (which is a distinct concept to [[octave complement]]s!). 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 5:8 accounts for all higher pitches (in that all higher pitches are a whole number of octaves above one of those harmonics); 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 ==

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* 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 [[~~Linear chord|isoratios~~]] that can be reduced to 1:1, 1:1:1, 1:1:1:1 etc., are otonal.

* All chords with [[delta signature]]s 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. [http://tech.groups.yahoo.com/group/tuning-math/message/20310 [proof]]

* The dyadic odd-limit of utonal chords is always smaller than the overall odd-limit. [http://tech.groups.yahoo.com/group/tuning-math/message/20310 [proof]]{{dead link}}

=== Ambitonal ===

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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.

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 [[Chord homonym|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 ==

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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.
-By this definition all monads, dyads and intervals are ambitonal. (Dyads and intervals are <u>not</u> the same thing; 2:3:4 is a dyad but not an interval, and 2/1 is an interval but not a dyad.)
+=== Dyads vs. intervals ===
+By this definition all [[monad]]s and [[dyad]]s are ambitonal. (Dyads and intervals are <u>not</u> the same thing; 2:3:4 is a dyad but not an interval, and 2/1 is an interval but not a dyad.)
+Therefore take note that while [[43/32]] may be the "prime harmonic fourth" (in  that it is rooted/of the form ''k'' / 2<sup>''n''</sup>), it is only because we are seeing it as an ''interval'' that it is so, because seeing it as a ''dyad'' would mean seeing it as 32:43:64 so that it isn't clear whether it is otonal or utonal as [[64/43]] is the "prime subharmonic fifth", so 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 (consisting of ''two'' [[pitch class]]es) thus has ''two'' possible ''inversions'' (which is a distinct concept to [[octave complement]]s!). 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 5:8 accounts for all higher pitches (in that all higher pitches are a whole number of octaves above one of those harmonics); 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 ==
@@ Line 26: / Line 36: @@
 * If we represent an otonal chord as a set of integers in the form A<sub>1</sub>:A<sub>2</sub>: ... :A<sub>n</sub>, we may add any additional integers without affecting the chord's otonality.
-* All chords with [[Linear chord|isoratios]] that can be reduced to 1:1, 1:1:1, 1:1:1:1 etc., are otonal.
+* All chords with [[delta signature]]s 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. [http://tech.groups.yahoo.com/group/tuning-math/message/20310 [proof]]
+* The dyadic odd-limit of utonal chords is always smaller than the overall odd-limit. [http://tech.groups.yahoo.com/group/tuning-math/message/20310 [proof]]{{dead link}}
 === Ambitonal ===
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 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.
+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 [[Chord homonym|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 ==