(Redirected from Essentially tempered chords)
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 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
Kestrel chords Kestrel 1188/1183
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

### 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
Neogrendelismic chords Neogrendelismic 5985/5984
Heartland chords Heartland 243/242, 1083/1078

### 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