Dyadic chord: Difference between revisions

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For example, the [[tetrad]]

* 1-6/5-7/5-8/5

* 1 – 6/5 – 7/5 – 8/5

is a dyadic chord in the [[7-odd-limit]] since every interval involved in it is an element of the 7-odd-limit [[tonality diamond]]. Now if we replace 7/5 with 10/7:

* 1-6/5-10/7-8/5

* 1 – 6/5 – 10/7 – 8/5

is ''not'' a dyadic chord in the 7-odd-limit. Although each note is 7-odd-limit over the bass, the interval between 10/7 and 6/5 is 25/21, and that between 10/7 and 8/5 is 28/25 – these are not 7-odd-limit.

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== Essentially tempered dyadic chord ==

In [[regular temperament theory]], we may speak of a just or tempered dyadic chord. By a ''just'' dyadic chord is meant a chord in rational intonation which is dyadic, so that each of its notes in relation to the lowest note is a rational number belonging to the set of consonances, and moreover each interval between the notes belongs to the set of consonances. By an ''essentially just'' dyadic chord is meant a chord which is considered to be an approximation of a just dyadic chord, such that each of its intervals is considered to be an approximation of the corresponding interval in the just dyadic chord. So, for instance, [[~~Ratios~~|1-5/4-3/2]] is a just dyadic chord when the consonance set is the [[5-odd-limit]] diamond with [[octave equivalence]], and 0-10-18 in 31edo with consonance set {8, 10, 13, 18, 21, 23, 31} modulo 31 is an essentially just dyadic chord approximating 1-5/4-3/2.

In [[regular temperament theory]], we may speak of a just or tempered dyadic chord. By a ''just'' dyadic chord is meant a chord in rational intonation which is dyadic, so that each of its notes in relation to the lowest note is a rational number belonging to the set of consonances, and moreover each interval between the notes belongs to the set of consonances. By an ''essentially just'' dyadic chord is meant a chord which is considered to be an approximation of a just dyadic chord, such that each of its intervals is considered to be an approximation of the corresponding interval in the just dyadic chord. So, for instance, [[4:5:6|1 – 5/4 – 3/2]] is a just dyadic chord when the consonance set is the [[5-odd-limit]] diamond with [[octave equivalence]], and 0 – 10 – 18 in 31edo with consonance set {8, 10, 13, 18, 21, 23, 31} modulo 31 is an essentially just dyadic chord approximating 1 – 5/4 – 3/2.

[[File:Essentially tempered chord.png|400px|thumb|right|A more in-depth work-through of the starling 1-6/5-10/7 essentially tempered chord example]]

By an '''essentially tempered dyadic chord''' is meant a chord defined in an [[abstract regular temperament]] such that each interval belongs to a consonance set, but there is no corresponding just dyadic chord. This means there is no just chord such that each interval, when mapped by the abstract regular temperament, belongs to the consonance set. For example, the chord 1-6/5-10/7, when mapped by starling temperament, which makes [[126/125]] [[vanish]], has each of its intervals in the set of [[7-odd-limit]] consonances which is the tempering of the 7-odd-limit diamond by 126/125 (this is because 10/7 is off from 36/25 by 126/125, and therefore 10/7 and 36/25 are tempered together in starling temperament, and since 36/25 = (6/5)², the interval from 6/5 to 10/7 in starling may be heard as a second move by 6/5). However, (10/7)/(6/5) = 25/21 is [[25-odd-limit]], and there is no other 7-odd-limit just dyadic chord which can be used instead to give the result, so it is an essentially tempered dyadic chord. Essentially tempered dyadic chords are a related notion to [[comma pump]]s, and can be used as a basis for creating pumps. Using essentially tempered chords in chord progressions breaks the harmony out of exclusively just chord relations, and serves as a sort of harmonic lubricant imparting fluidity and dynamism to the harmony, at the cost fairly often of some blurring of the sense of tonality.

By an '''essentially tempered dyadic chord''' is meant a chord defined in an [[abstract regular temperament]] such that each interval belongs to a consonance set, but there is no corresponding just dyadic chord. This means there is no just chord such that each interval, when mapped by the abstract regular temperament, belongs to the consonance set. For example, the chord 1 – 6/5 – 10/7, when mapped by starling temperament, which makes [[126/125]] [[vanish]], has each of its intervals in the set of [[7-odd-limit]] consonances which is the tempering of the 7-odd-limit diamond by 126/125 (this is because 10/7 is off from 36/25 by 126/125, and therefore 10/7 and 36/25 are tempered together in starling temperament, and since 36/25 = (6/5)², the interval from 6/5 to 10/7 in starling may be heard as a second move by 6/5). However, (10/7)/(6/5) = 25/21 is [[25-odd-limit]], and there is no other 7-odd-limit just dyadic chord which can be used instead to give the result, so it is an essentially tempered dyadic chord.

Essentially tempered dyadic chords are a related notion to [[comma pump]]s, and can be used as a basis for creating pumps. Using essentially tempered chords in chord progressions breaks the harmony out of exclusively just chord relations, and serves as a sort of harmonic lubricant imparting fluidity and dynamism to the harmony, at the cost fairly often of some blurring of the sense of tonality.

== Innate comma chord ==

An '''innate comma chord''', proposed by [[Kite Giedraitis]], is the type of chord that cannot be mapped to just intonation in a given prime limit and odd limit, hence a chord that will not "ring". This is broader than the essentially tempered chord because it includes the possibility that the chord is not tempered at all, and contains a wolf interval. For example, the augmented triad in 5-limit JI is an innate comma chord, because it is impossible to tune all three major ~~3rds~~ to 5/4. The innate comma here is 128/125 = 41¢. In practice, it might be sung or played justly, but with a large odd limit and hence a wolf interval, as 1/1 - 5/4 - 25/16 or 1/1 - 5/4 - 8/5. Or it might be tempered, e.g. in ~~12-edo~~ as 0¢ - 400¢ - 800¢ - 1200¢. In 7-limit JI, one of the ~~3rds~~ can be tuned to 9/7. The innate comma is reduced to 225/224, only 8¢. This comma can be distributed among the three ~~3rds~~, resulting in tempering each only a few cents, which may be close enough to be acceptable. In 11-limit JI, this chord ~~isn't~~ an innate comma chord, because it can be tuned justly as 7:9:11, a low enough odd limit to "ring". However, it's debatable that this chord qualifies as an augmented triad, because the upper ~~3rd~~ hardly sounds major.

An '''innate comma chord''', proposed by [[Kite Giedraitis]], is the type of chord that cannot be mapped to just intonation in a given prime limit and odd limit, hence a chord that will not "ring". This is broader than the essentially tempered chord because it includes the possibility that the chord is not tempered at all, and contains a [[wolf interval]]. For example, the [[augmented triad]] in 5-limit JI is an innate comma chord, because it is impossible to tune all three major thirds to 5/4. The innate comma here is 128/125 = 41¢. In practice, it might be sung or played justly, but with a large odd limit and hence a wolf interval, as 1/1 – 5/4 – 25/16 or 1/1 – 5/4 – 8/5. Or it might be tempered, e.g. in 12edo as 0¢ – 400¢ – 800¢ – 1200¢. In 7-limit JI, one of the thirds can be tuned to 9/7. The innate comma is reduced to 225/224, only 8¢. This comma can be distributed among the three thirds, resulting in tempering each only a few cents, which may be close enough to be acceptable. In 11-limit JI, this chord is not an innate comma chord, because it can be tuned justly as 7:9:11, a low enough odd limit to "ring". However, it is debatable that this chord qualifies as an augmented triad, because the upper third hardly sounds major.

== Anomalous saturated suspension ==

An '''anomalous saturated suspension''' ('''ASS'''), introduced by [[Graham Breed]]<ref>[https://www.webcitation.org/60VBgPSUS ''Anomalous Saturated Suspensions'']</ref>, is a ''q''-odd-limit just dyadic chord to which no pitch ''q''-odd-limit pitch class can be added while keeping it in the ''q''-odd-limit, and which is neither an otonal or a utonal chord; that is, it is not contained as a subchord of either the 1:3:5:…:''q'' chord or the 1:1/3:1/5:…:1/q chord. The existence of such chords was discovered by [[Paul Erlich]]<ref>[https://www.webcitation.org/60VCUHe6d ''Anomalous Saturated Suspensions -- Paul Erlich's post'']</ref>. Below are listed two 9-odd-limit ASSes of special interest, as they avoid intervals smaller than a minor whole tone.

An '''anomalous saturated suspension''' ('''ASS'''), introduced by [[Graham Breed]]<ref>[https://www.webcitation.org/60VBgPSUS ''Anomalous Saturated Suspensions'']</ref>, is a ''q''-odd-limit just dyadic chord to which no pitch ''q''-odd-limit pitch class can be added while keeping it in the ''q''-odd-limit, and which is neither an otonal or a utonal chord; that is, it is not contained as a subchord of either the 1:3:5:…:''q'' chord or the 1:1/3:1/5:…:1/''q'' chord. The existence of such chords was discovered by [[Paul Erlich]]<ref>[https://www.webcitation.org/60VCUHe6d ''Anomalous Saturated Suspensions -- Paul Erlich's post'']</ref>. Below are listed two 9-odd-limit ASSes of special interest, as they avoid intervals smaller than a minor whole tone.

* [[12:15:18:20]]

@@ Line 4: / Line 4: @@
 For example, the [[tetrad]]
-* 1-6/5-7/5-8/5
+* 1&thinsp;–&thinsp;6/5&thinsp;–&thinsp;7/5&thinsp;–&thinsp;8/5
 is a dyadic chord in the [[7-odd-limit]] since every interval involved in it is an element of the 7-odd-limit [[tonality diamond]]. Now if we replace 7/5 with 10/7:
-* 1-6/5-10/7-8/5
+* 1&thinsp;–&thinsp;6/5&thinsp;–&thinsp;10/7&thinsp;–&thinsp;8/5
 is ''not'' a dyadic chord in the 7-odd-limit. Although each note is 7-odd-limit over the bass, the interval between 10/7 and 6/5 is 25/21, and that between 10/7 and 8/5 is 28/25 – these are not 7-odd-limit.
@@ Line 14: / Line 14: @@
 == Essentially tempered dyadic chord ==
-In [[regular temperament theory]], we may speak of a just or tempered dyadic chord. By a ''just'' dyadic chord is meant a chord in rational intonation which is dyadic, so that each of its notes in relation to the lowest note is a rational number belonging to the set of consonances, and moreover each interval between the notes belongs to the set of consonances. By an ''essentially just'' dyadic chord is meant a chord which is considered to be an approximation of a just dyadic chord, such that each of its intervals is considered to be an approximation of the corresponding interval in the just dyadic chord. So, for instance, [[Ratios|1-5/4-3/2]] is a just dyadic chord when the consonance set is the [[5-odd-limit]] diamond with [[octave equivalence]], and 0-10-18 in 31edo with consonance set {8, 10, 13, 18, 21, 23, 31} modulo 31 is an essentially just dyadic chord approximating 1-5/4-3/2.
+In [[regular temperament theory]], we may speak of a just or tempered dyadic chord. By a ''just'' dyadic chord is meant a chord in rational intonation which is dyadic, so that each of its notes in relation to the lowest note is a rational number belonging to the set of consonances, and moreover each interval between the notes belongs to the set of consonances. By an ''essentially just'' dyadic chord is meant a chord which is considered to be an approximation of a just dyadic chord, such that each of its intervals is considered to be an approximation of the corresponding interval in the just dyadic chord. So, for instance, [[4:5:6|1&thinsp;–&thinsp;5/4&thinsp;–&thinsp;3/2]] is a just dyadic chord when the consonance set is the [[5-odd-limit]] diamond with [[octave equivalence]], and 0&thinsp;–&thinsp;10&thinsp;–&thinsp;18 in 31edo with consonance set {8, 10, 13, 18, 21, 23, 31} modulo 31 is an essentially just dyadic chord approximating 1&thinsp;–&thinsp;5/4&thinsp;–&thinsp;3/2.
 [[File:Essentially tempered chord.png|400px|thumb|right|A more in-depth work-through of the starling 1-6/5-10/7 essentially tempered chord example]]
-By an '''essentially tempered dyadic chord''' is meant a chord defined in an [[abstract regular temperament]] such that each interval belongs to a consonance set, but there is no corresponding just dyadic chord. This means there is no just chord such that each interval, when mapped by the abstract regular temperament, belongs to the consonance set. For example, the chord 1-6/5-10/7, when mapped by starling temperament, which makes [[126/125]] [[vanish]], has each of its intervals in the set of [[7-odd-limit]] consonances which is the tempering of the 7-odd-limit diamond by 126/125 (this is because 10/7 is off from 36/25 by 126/125, and therefore 10/7 and 36/25 are tempered together in starling temperament, and since 36/25 = (6/5)², the interval from 6/5 to 10/7 in starling may be heard as a second move by 6/5). However, (10/7)/(6/5) = 25/21 is [[25-odd-limit]], and there is no other 7-odd-limit just dyadic chord which can be used instead to give the result, so it is an essentially tempered dyadic chord. Essentially tempered dyadic chords are a related notion to [[comma pump]]s, and can be used as a basis for creating pumps. Using essentially tempered chords in chord progressions breaks the harmony out of exclusively just chord relations, and serves as a sort of harmonic lubricant imparting fluidity and dynamism to the harmony, at the cost fairly often of some blurring of the sense of tonality.
+By an '''essentially tempered dyadic chord''' is meant a chord defined in an [[abstract regular temperament]] such that each interval belongs to a consonance set, but there is no corresponding just dyadic chord. This means there is no just chord such that each interval, when mapped by the abstract regular temperament, belongs to the consonance set. For example, the chord 1&thinsp;–&thinsp;6/5&thinsp;–&thinsp;10/7, when mapped by starling temperament, which makes [[126/125]] [[vanish]], has each of its intervals in the set of [[7-odd-limit]] consonances which is the tempering of the 7-odd-limit diamond by 126/125 (this is because 10/7 is off from 36/25 by 126/125, and therefore 10/7 and 36/25 are tempered together in starling temperament, and since 36/25 = (6/5)², the interval from 6/5 to 10/7 in starling may be heard as a second move by 6/5). However, (10/7)/(6/5) = 25/21 is [[25-odd-limit]], and there is no other 7-odd-limit just dyadic chord which can be used instead to give the result, so it is an essentially tempered dyadic chord.
+Essentially tempered dyadic chords are a related notion to [[comma pump]]s, and can be used as a basis for creating pumps. Using essentially tempered chords in chord progressions breaks the harmony out of exclusively just chord relations, and serves as a sort of harmonic lubricant imparting fluidity and dynamism to the harmony, at the cost fairly often of some blurring of the sense of tonality.
 == Innate comma chord ==
-An '''innate comma chord''', proposed by [[Kite Giedraitis]], is the type of chord that cannot be mapped to just intonation in a given prime limit and odd limit, hence a chord that will not "ring". This is broader than the essentially tempered chord because it includes the possibility that the chord is not tempered at all, and contains a wolf interval. For example, the augmented triad in 5-limit JI is an innate comma chord, because it is impossible to tune all three major 3rds to 5/4. The innate comma here is 128/125 = 41¢. In practice, it might be sung or played justly, but with a large odd limit and hence a wolf interval, as 1/1 - 5/4 - 25/16 or 1/1 - 5/4 - 8/5. Or it might be tempered, e.g. in 12-edo as 0¢ - 400¢ - 800¢ - 1200¢. In 7-limit JI, one of the 3rds can be tuned to 9/7. The innate comma is reduced to 225/224, only 8¢. This comma can be distributed among the three 3rds, resulting in tempering each only a few cents, which may be close enough to be acceptable. In 11-limit JI, this chord isn't an innate comma chord, because it can be tuned justly as 7:9:11, a low enough odd limit to "ring". However, it's debatable that this chord qualifies as an augmented triad, because the upper 3rd hardly sounds major.
+An '''innate comma chord''', proposed by [[Kite Giedraitis]], is the type of chord that cannot be mapped to just intonation in a given prime limit and odd limit, hence a chord that will not "ring". This is broader than the essentially tempered chord because it includes the possibility that the chord is not tempered at all, and contains a [[wolf interval]]. For example, the [[augmented triad]] in 5-limit JI is an innate comma chord, because it is impossible to tune all three major thirds to 5/4. The innate comma here is 128/125 = 41¢. In practice, it might be sung or played justly, but with a large odd limit and hence a wolf interval, as 1/1&thinsp;–&thinsp;5/4&thinsp;–&thinsp;25/16 or 1/1&thinsp;–&thinsp;5/4&thinsp;–&thinsp;8/5. Or it might be tempered, e.g. in 12edo as 0¢&thinsp;–&thinsp;400¢&thinsp;–&thinsp;800¢&thinsp;–&thinsp;1200¢. In 7-limit JI, one of the thirds can be tuned to 9/7. The innate comma is reduced to 225/224, only 8¢. This comma can be distributed among the three thirds, resulting in tempering each only a few cents, which may be close enough to be acceptable. In 11-limit JI, this chord is not an innate comma chord, because it can be tuned justly as 7:9:11, a low enough odd limit to "ring". However, it is debatable that this chord qualifies as an augmented triad, because the upper third hardly sounds major.
 == Anomalous saturated suspension ==
-An '''anomalous saturated suspension''' ('''ASS'''), introduced by [[Graham Breed]]<ref>[https://www.webcitation.org/60VBgPSUS ''Anomalous Saturated Suspensions'']</ref>, is a ''q''-odd-limit just dyadic chord to which no pitch ''q''-odd-limit pitch class can be added while keeping it in the ''q''-odd-limit, and which is neither an otonal or a utonal chord; that is, it is not contained as a subchord of either the 1:3:5:…:''q'' chord or the 1:1/3:1/5:…:1/q chord. The existence of such chords was discovered by [[Paul Erlich]]<ref>[https://www.webcitation.org/60VCUHe6d ''Anomalous Saturated Suspensions -- Paul Erlich's post'']</ref>. Below are listed two 9-odd-limit ASSes of special interest, as they avoid intervals smaller than a minor whole tone.
+An '''anomalous saturated suspension''' ('''ASS'''), introduced by [[Graham Breed]]<ref>[https://www.webcitation.org/60VBgPSUS ''Anomalous Saturated Suspensions'']</ref>, is a ''q''-odd-limit just dyadic chord to which no pitch ''q''-odd-limit pitch class can be added while keeping it in the ''q''-odd-limit, and which is neither an otonal or a utonal chord; that is, it is not contained as a subchord of either the 1:3:5:…:''q'' chord or the 1:1/3:1/5:…:1/''q'' chord. The existence of such chords was discovered by [[Paul Erlich]]<ref>[https://www.webcitation.org/60VCUHe6d ''Anomalous Saturated Suspensions -- Paul Erlich's post'']</ref>. Below are listed two 9-odd-limit ASSes of special interest, as they avoid intervals smaller than a minor whole tone.
 * [[12:15:18:20]]