Dyadic chord: Difference between revisions
corrected link |
m Clarification and some cleanup |
||
| Line 1: | Line 1: | ||
== Definitions == | |||
=Definitions= | By a '''dyadic chord''' is meant a chord each of whose intervals belongs to a specified set of intervals considered to be consonant; it is therefore relative to the set of intervals in question. 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. | ||
By a '''dyadic chord''' is meant a chord each of whose intervals belongs to a specified set of intervals considered to be consonant; it is therefore relative to the set of intervals in question. 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-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. | |||
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 tempers out 126/125, has each of its intervals in the set of 7-limit consonances which is the tempering of the 7-limit diamond by 126/125. However, (10/7)/(6/5) = 25/21 is 25-limit, and there is no other 7-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 tempers out [[126/125]], 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. 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. | ||
Kite Giedraitis has proposed the term "innate comma chord" to describe the type of chord that can't be mapped to just intonation in a given prime limit and odd limit, hence a chord that won't "ring". This term is broader than the term "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's 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. | Kite Giedraitis has proposed the term "innate comma chord" to describe the type of chord that can't be mapped to just intonation in a given prime limit and odd limit, hence a chord that won't "ring". This term is broader than the term "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's 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. | ||
=Anomalous Saturated Suspensions= | == Anomalous Saturated Suspensions == | ||
An ''anomalous saturated suspension'', or ASS, is a term [http://www.webcitation.org/60VBgPSUS introduced] by [[Graham Breed]] for a q-limit just dyadic chord to which no pitch q-limit pitch class can be added while keeping it in the q-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 [http://www.webcitation.org/60VCUHe6d discovered] by [[Paul Erlich]]. Below are listed two 9-limit ASSes of special interest, as they avoid intervals smaller than a minor whole tone. | An ''anomalous saturated suspension'', or ASS, is a term [http://www.webcitation.org/60VBgPSUS introduced] by [[Graham Breed]] for a q-limit just dyadic chord to which no pitch q-limit pitch class can be added while keeping it in the q-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 [http://www.webcitation.org/60VCUHe6d discovered] by [[Paul Erlich]]. Below are listed two 9-limit ASSes of special interest, as they avoid intervals smaller than a minor whole tone. | ||
[[ | * [[Just added sixth chord]] | ||
* [[Swiss tetrad]] | |||
[[ | |||
For a complete list of ASS chords through the 23-limit see [[Anomalous saturated suspension]]. | For a complete list of ASS chords through the 23-limit see [[Anomalous saturated suspension]]. | ||
=Just intonation tetrads= | == Just intonation tetrads == | ||
* [[Seven limit tetrads]] | |||
* [[Nine limit tetrads]] | |||
* [[1-3-7-11 tetrads]] | |||
* [[Thirteen limit tetrads]] | |||
* [[Fifteen limit tetrads]] | |||
* [[Seventeen limit tetrads]] | |||
== Essentially tempered dyadic chords == | |||
=Essentially tempered dyadic chords= | |||
Here are some pages on certain essentially tempered dyadic chords. | Here are some pages on certain essentially tempered dyadic chords. | ||
==7-limit== | === 7-odd-limit === | ||
*[[Starling chords]] | * [[Starling chords]] | ||
*[[Hendrix chord]] | * [[Hendrix chord]] | ||
==9-limit== | === 9-odd-limit === | ||
*[[Meantone add6-9 pentad]] | * [[Meantone add6-9 pentad]] | ||
*[[Marvel chords]] | * [[Marvel chords]] | ||
*[[Sensamagic chords]] | * [[Sensamagic chords]] | ||
==11-limit== | === 11-odd-limit === | ||
*[[Mothwellsmic triad]] | * [[Mothwellsmic triad]] | ||
*[[Ptolemismic triad]] | * [[Ptolemismic triad]] | ||
*[[Valinorsmic chords]] | * [[Valinorsmic chords]] | ||
*[[Rastmic chords]] | * [[Rastmic chords]] | ||
*[[Keenanismic chords]] | * [[Keenanismic chords]] | ||
*[[Pentacircle chords]] | * [[Pentacircle chords]] | ||
*[[Werckismic chords]] | * [[Werckismic chords]] | ||
*[[Swetismic chords]] | * [[Swetismic chords]] | ||
*[[Undecimal marvel chords]] | * [[Undecimal marvel chords]] | ||
*[[Jove chords]] | * [[Jove chords]] | ||
*[[Prodigy chords]] | * [[Prodigy chords]] | ||
*[[Miracle chords]] | * [[Miracle chords]] | ||
*[[Magical seventh chord]] | * [[Magical seventh chord]] | ||
*[[Orwell tetrad]] | * [[Orwell tetrad]] | ||
*[[Tutonic sextad]] | * [[Tutonic sextad]] | ||
==13-limit== | === 13-odd-limit === | ||
*[[Ratwolf triad]] | * [[Ratwolf triad]] | ||
*[[Gentle chords]] | * [[Gentle chords]] | ||
*[[Minthmic chords]] | * [[Minthmic chords]] | ||
*[[Huntmic chords]] | * [[Huntmic chords]] | ||
*[[Kestrel chords]] | * [[Kestrel chords]] | ||
*[[Mynucumic chords]] | * [[Mynucumic chords]] | ||
*[[Squbemic chords]] | * [[Squbemic chords]] | ||
*[[Sinbadmic tetrad]] | * [[Sinbadmic tetrad]] | ||
*[[Marveltwin triad]] | * [[Marveltwin triad]] | ||
*[[Petrmic triad]] | * [[Petrmic triad]] | ||
*[[Cuthbert triad]] | * [[Cuthbert triad]] | ||
*[[Hecate hexad]] | * [[Hecate hexad]] | ||
==15-limit== | === 15-odd-limit === | ||
*[[ | * [[Orwell tetrad|Guanyin tetrad]] | ||
*[[Island tetrad]] | * [[Island tetrad]] | ||
*[[Nicolic tetrad]] | * [[Nicolic tetrad]] | ||
*[[Battaglia chord]] | * [[Battaglia chord]] | ||
*[[Biyatismic chords]] | * [[Biyatismic chords]] | ||
==17-limit== | === 17-odd-limit === | ||
*[[136/135-tempered chords]] | * [[136/135-tempered chords]] | ||
*[[170/169-tempered chords]] | * [[170/169-tempered chords]] | ||
*[[Ainic chords]] | * [[Ainic chords]] | ||
==19-limit== | === 19-odd-limit === | ||
*[[Hendrix chord]] | * [[Hendrix chord]] | ||
*[[Rootminor triad]] | * [[Rootminor triad]] | ||
*[[Rootsubminor triad]] | * [[Rootsubminor triad]] | ||
==21-limit== | === 21-odd-limit === | ||
*[[Slendric pentad]] | * [[Slendric pentad]] | ||
[[Category:Chords]] | [[Category:Chords]] | ||
[[Category:Dyadic]] | [[Category:Dyadic]] | ||
[[Category:Overview]] | [[Category:Overview]] | ||
{{todo|add introduction}} | |||