Dyadic chord

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Not to be confused with Dyad.

A dyadic chord is 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. Such a chord may also be described as dyadically or pairwise consonant.

For example, the tetrad

  • 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

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.

The significance of dyadic chords and of the paradigm where all interval pairs are examined in the chord has the psychoacoustic basis of timbral fusion and emergence of the virtual fundamental. In the above examples, it can be shown that the lower harmonics of each note in the first chord blends better than in the second. Meanwhile, the virtual fundamental of the first chord appears 5/1 below the bass, whereas that of the second appears much lower, at 35/1 below the bass as the denominators "fight" each other. For these reasons we tend to find the first chord more consonant than the second.

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, 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.

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 pumps, 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.

Anomalous saturated suspension

An anomalous saturated suspension (ASS), introduced by Graham Breed[1], 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[2]. Below are listed two 9-odd-limit ASSes of special interest, as they avoid intervals smaller than a minor whole tone.

For a complete list of ASS chords through the 23-odd-limit see List of anomalous saturated suspensions.

List of just intonation tetrads

List of essentially tempered dyadic chords

Here are some pages on certain essentially tempered dyadic chords, sorted by odd limit. See Dyadic chord/Pattern of essentially tempered chords for some notable abstract chord patterns.

7-odd-limit

Chords Associated Temperament Associated Commas
Archytas chords Archytas 64/63
Starling chords Starling 126/125

9-odd-limit

Chords Associated Temperament Associated Commas
Didymic chords Meantone 81/80
Marvel chords Marvel 225/224
Sensamagic chords Sensamagic 245/243

11-odd-limit

Chords Associated Temperament Associated Commas
Mothwellsmic chords Mothwellsmic 99/98
Ptolemismic chords Ptolemismic 100/99
Biyatismic chords Biyatismic 121/120
Valinorsmic chords Valinorsmic 176/175
Rastmic chords Rastmic 243/242
Keenanismic chords Keenanismic 385/384
Werckismic chords Werckismic 441/440
Swetismic chords Swetismic 540/539
Pentacircle chords Pentacircle 896/891
Undecimal marvel chords Marvel 225/224, 385/384
Prodigy chords Prodigy 225/224, 441/440
Undecimal sensamagic chords Sensamagic 245/243, 385/384
Jove chords Jove 243/242, 441/440
Miracle chords Miracle 225/224, 243/242, 385/384
Magic chords Magic 100/99, 225/224, 245/243
Magical seventh chord Supermagic 100/99, 385/384
Orwell tetrad Guanyin 176/175, 540/539
Tutonic hexads Meantone 81/80, 99/98, 126/125
Baldanders hexads Andromeda 100/99, 225/224, 245/242
Porcupine heptad Porkypine 55/54, 100/99

13-odd-limit

Chords Associated Temperament Associated Commas
Buzurgismic chords Buzurgismic 169/168
Mynucumic chords Mynucumic 196/195
Gassormic chords Gassormic 275/273
Marveltwin chords Marveltwin 325/324
Ratwolfsmic chords Ratwolfsmic 351/350
Major minthmic chords Major minthmic 352/351
Minor minthmic chords Minor minthmic 364/363
Huntmic chords Huntmic 640/637
Squbemic chords Squbemic 729/728
Cuthbert chords Cuthbert 847/845
Sinbadmic chords Sinbadmic 1001/1000
Kestrel chords Kestrel 1188/1183
Catadictmic chords Catadictmic 1287/1280
Lambeth chords Lambeth 1573/1568
Petrmic chords Petrmic 2200/2197
Rastgross heptad Namo 144/143, 243/242
Parapyth chords Parapyth 352/351, 364/363
Hecate hexad Hecate 225/224, 325/324, 385/384
Woodpecker octad Woodpecker 66/65, 121/120, 126/125
Miraculous decad Miraculous, Revelation 105/104, 196/195, 512/507

15-odd-limit

Chords Associated Temperament Associated Commas
Island chords Island 676/675
Nicolic chords Nicolic 1575/1573
Myhemiwell chords Myhemiwell 3388/3375
Battaglia chord Marvel 225/224
Tetracot chords Tetracot 100/99, 243/242
Orwell ennead Guanyin 176/175, 540/539

17-odd-limit

Chords Associated Temperament Associated Commas
Augustmic chords Augustmic 154/153
Charismic chords Charismic 256/255
Prototannismic chords Prototannismic 273/272
Ursulismic chords Ursulismic 375/374
Seminaiadmic chords Seminaiadmic 442/441
Monardismic chords Monardismic 561/560
Dakotismic chords Dakotismic 595/594
September chords September 715/714
Horizmic chords Horizmic 833/832
Ainismic chords Ainismic 936/935
Twosquare chords Twosquare 1089/1088
Quadrantonismic chords Quadrantonismic 1156/1155
Cimbrismic chords Cimbrismic 1275/1274
Fidesmic chords Fidesmic 2025/2023
Heptacircle chords Heptacircle 2431/2430
Sextantonismic chords Sextantonismic 2601/2600

19-odd-limit

Chords Associated Temperament Associated Commas
Photismic chords Photismic 324/323
Nutrismic chords Nutrismic 343/342
Dudon chords Dudon 361/360
Devichromic chords Devichromic 400/399
Abnobismic chords Abnobismic 456/455
Hedwigmic chords Hedwigmic 476/475
Eulalismic chords Eulalismic 495/494
Boethius chords Boethius 513/512
Buddhismic chords Buddhismic 969/968
Bihendrixmic chords Bihendrixmic 1083/1078
Eratosthenes chords Eratosthenes 1216/1215
Solvejgsmic chords Solvejgsmic 1331/1330
Aureusmic chords Aureusmic 1445/1444
Pinkanberry chords Pinkanberry 1521/1520
Kevolismic chords Kevolismic 1540/1539
Ramanujanismic chords Ramanujanismic 1729/1728
Blumeyer chords Blumeyer 2432/2431
Neovulturismic chords Neovulturismic 2926/2925
Neomirkwaismic chords Neomirkwaismic 3136/3135
Neosatanismic chords Neosatanismic 4200/4199

21-odd-limit

Chords Associated Temperament Associated Commas
Slendric pentad Gamelismic 1029/1024
Palingenetic chords Palingenetic 1701/1700
Xenismic chords Xenismic 2058/2057
Ibnsinmic chords Ibnsinmic 2080/2079
Heartlandismic chords Heartlandismic 3971/3969
Schisminic chords Schisminic 4096/4095
Baladismic chords Baladismic 4914/4913
Neogrendelismic chords Neogrendelismic 5985/5984
Heartland chords Heartland 243/242, 1083/1078

23-odd-limit

Chords Associated Temperament Associated Commas
Scanismic chords Scanismic 460/459
Pittsburghismic chords Pittsburghismic 484/483
Laodicismic chords Laodicismic 507/506
Preziosismic chords Preziosismic 529/528
Worcester chords Worcester 576/575
Harvardismic chords Harvardismic 736/735
Squadronismic chords Squadronismic 760/759
Lysistratismic chords Lysistratismic 897/896
Fragarismic chords Fragarismic 1105/1104
Rhodesismic chords Rhodesismic 1197/1196
Triaphonismic chords Triaphonismic 1288/1287
Turkismic chords Turkismic 1496/1495
Antinousismic chords Antinousismic 1863/1862
Artifismic chords Artifismic 2024/2023
Cupcake chords Cupcake 2025/2024
Guangdongismic chords Guangdongismic 2185/2184
Travellismic chords Travellismic 2300/2299
Biyativice chords Biyativice 2646/2645
Kotkismic chords Kotkismic 2737/2736
Vicious chords Vicious 3060/3059
Seminaiadvice chords Seminaiadvice 3381/3380
Vicedim chords Vicedim 3520/3519
Vicetride chords Vicetride 3888/3887
Viceaug chords Viceaug 4693/4692
Demiquartervice chords Demiquartervice 4761/4760
Broadviewsmic chords Broadviewsmic 5083/5082
Vicetertismic chords Vicetertismic 12168/12167

25-odd-limit

Chords Associated Temperament Associated Commas
Tunbarsmic chords Tunbarsmic 625/624
Nymphismic chords Nymphismic 875/874
Noellismic chords Noellismic 1225/1224
Trichthonismic chords Trichthonismic 2376/2375
Sperasmic chords Sperasmic 2500/2499
Lehmerismic chords Lehmerismic 3025/3024
Martebismic chords Martebismic 3250/3249
Leprechaun chords Leprechaun 4225/4224
Neovishmic chords Neovishmic 5776/5775
Neonewtismic chords Neonewtismic 6175/6174

27-odd-limit

Chords Associated Temperament Associated Commas
Ragismic chords Ragismic 4375/4374

29-odd-limit

Chords Associated Temperament Associated Commas
Large grapevine chords Large grapevine 4901/4900
Small grapevine chords Small grapevine 7425/7424

31-odd-limit

Chords Associated Temperament Associated Commas
Kilobyte chords Kilobyte 1024/1023
Tricetertismic chords Tricetertismic 29792/29791

33-odd-limit

Chords Associated Temperament Associated Commas
Wizardharry chords Wizardharry 4000/3993
Flashmic chords Flashmic 12376/12375

35-odd-limit

Chords Associated Temperament Associated Commas
Lummic chords Lummic 1716/1715

37-odd-limit

Chords Associated Temperament Associated Commas
Bullionismic chords Bullionismic 5292/5291

39-odd-limit

Chords Associated Temperament Associated Commas
Harmonismic chords Harmonismic 10648/10647

55-odd-limit

Chords Associated Temperament Associated Commas
Jacobin chords Jacobin 6656/6655

List of essentially just dyadic chords

As chords that are unambiguous counterparts to common JI chords are not of particular relevance to this page, most of the entries here will be what Kaiveran calls plurichords, where there are multiple sets of consonances that a given chord can be mapped to. Note that this can still lead to ambiguous tonality in the case of otonal and utonal intervals being identified together.

Chords Equivalent Mappings Associated Commas
Hendrix chord 8:10:14:19 ~ 12:15:21:28 57/56
Rootsubminor triad 6:7:9 ~ 16:19:24 57/56
Biosphere triads 6:7:9 ~ 26:30:39
10:13:15 ~ 14:18:21
91/90
Rootminor triad 10:12:15 ~ 16:19:24 96/95
Lynchismic plurichords 12:14:17:20 ~ 1/(20:17:14:12)
6:7:8:10 ~ 1/(20:17:15:12)
120/119
Augustmic plurichords 14:17:18:22 ~ 1/(22:18:17:14) 154/153

See also

Notes