Otonality and utonality - Revision history

Hotcrystal0 at 02:08, 26 September 2026

2026-09-26T02:08:18Z

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	== Essentially tempered chords ==		== Essentially tempered chords ==

	This kind of reduction can also be used to analyze [[Dyadic_chord\|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		This kind of reduction can also be used to analyze [[Dyadic_chord\|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.

	[[Category:Otonality and utonality\| ]] <!-- main article -->		[[Category:Otonality and utonality\| ]] <!-- main article -->
	{{Todo\|improve synopsis}}		{{Todo\|improve synopsis}}

FloraC: Rework (add the origin; use more tonality-neutral language)

2026-08-04T11:20:17Z

Rework (add the origin; use more tonality-neutral language)

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	'''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]].		'''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.		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 ==		== 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).		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.		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.

Eufalesio: Changed some examples that I deemed strangely difficult and reworded some tiny things

2026-08-03T23:36:31Z

Changed some examples that I deemed strangely difficult and reworded some tiny things

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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).		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 {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.		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 ===		=== 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~~.)		By this definition all [[monad]]s and [[dyad]]s are ambitonal. (Dyads and intervals are <u>not</u> 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 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.		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'' / 2<sup>''n''</sup>), only because we are seeing it as an ''interval'' it is so, since seeing it as a ''dyad'' would convey the [[EFR]]<nowiki/>s 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 ~~(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.		Note that a dyad 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 ===		=== 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.		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''.		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 ==		== Properties of types of chords ==

FloraC: + an actual intro, more to come

2026-08-03T18:58:10Z

+ an actual intro, more to come

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	{{~~interwiki~~		{{Interwiki
			\| en = Otonality and utonality
	\| de =		\| de =
	~~\| en = Otonality and utonality~~
	\| es =		\| es =
	\| ja = OtonalityとUtonality		\| ja = OtonalityとUtonality
	}}		}}
	{{Wikipedia\|Otonality and Utonality}}		{{Wikipedia\|Otonality and Utonality}}

			'''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 ==		== 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.		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.
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	[[Category:Otonality and utonality\| ]] <!-- main article -->		[[Category:Otonality and utonality\| ]] <!-- main article -->
			{{Todo\|improve synopsis}}

Dummy index: /* Scales */ fix

2025-10-14T12:56:33Z

Scales: fix

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	== Scales ==		== 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.		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'' = 2''k'', or tonality diamonds are necessarily ambitonal, whereas dwarf scales are always either otonal or ambitonal.

	== Essentially tempered chords ==		== Essentially tempered chords ==

Dummy index: /* Ambitonal chord theorem */ verbosely

2025-10-07T13:33:10Z

Ambitonal chord theorem: verbosely

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	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.		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.		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 = 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.

TallKite: /* Ambitonal chord theorem */

2024-09-24T05:47:40Z

Ambitonal chord theorem

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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 = 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 ==		== Scales ==

Godtone: explain difference between dyads and intervals (pending further resolution)

2024-06-11T00:36:13Z

explain difference between dyads and intervals (pending further resolution)

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	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.		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 ==		== Properties of types of chords ==

Sintel: Undo revision 145976 by Sintel (talk)

2024-06-10T10:58:28Z

Undo revision 145976 by Sintel (talk)

← Older revision		Revision as of 10:58, 10 June 2024
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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).		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)~~. By this definition all dyads are ambitonal~~.		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.		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.)

	== Properties of types of chords ==		== Properties of types of chords ==

Sintel: remove incorrect statement about dyads

2024-06-09T22:43:19Z

remove incorrect statement about dyads

← Older revision		Revision as of 22:43, 9 June 2024
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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).		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). By this definition all dyads are ambitonal.

	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.		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.)~~

	== Properties of types of chords ==		== Properties of types of chords ==
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	=== Utonal ===		=== 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 ===		=== Ambitonal ===