Syntonic–limmic equivalence continuum: Difference between revisions
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The '''syntonic-diatonic equivalence continuum''' is a continuum of temperaments which equate a number of [[81/80|syntonic commas (81/80)]] with the [[256/243|limma (256/243)]]. | The '''syntonic-diatonic equivalence continuum''' is a continuum of temperaments which equate a number of [[81/80|syntonic commas (81/80)]] with the [[256/243|limma (256/243)]]. | ||
All temperaments in the continuum satisfy (81/80)<sup>''n''</sup> ~ 256/243. Varying ''n'' results in different temperaments listed in the table below. It converges to [[meantone]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[5edo]] | All temperaments in the continuum satisfy (81/80)<sup>''n''</sup> ~ 256/243. Varying ''n'' results in different temperaments listed in the table below. It converges to [[meantone]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[5-limit]] temperaments supported by [[5edo]] due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them. The just value of ''n'' is 4.1952…, and temperaments near this tend to be the most accurate ones. | ||
256/243 | 256/243 is the characteristic [[3-limit]] comma tempered out in [[5edo]]. In each case, we notice that ''n'' equals the order of harmonic 5 in the corresponding comma, and equals the number of generators to obtain a harmonic 3 in the MOS scale. However, if we let ''k'' = ''n'' + 1 (meaning ''n'' = ''k'' - 1) so that ''k'' = 0 means ''n'' = -1, ''k'' = 1 means ''n'' = 0, etc. then the continuum corresponds to (81/80)<sup>''k''</sup> = 16/15, which might be a preferred way of conceptualising it because: | ||
* 16/15 is the diatonic semitone, notable in the 5-limit as the difference between 4/3 and 5/4, so this shifted continuum could also logically be termed the "syntonic-diatonic equivalence continuum". This means that at ''k'' = 0, 4/3 and 5/4 are mapped to the same interval while 81/80 becomes independent of 16/15 (meaning 81/80 may or may not be tempered) because the relation becomes (81/80) | * 16/15 is the classic diatonic semitone, notable in the 5-limit as the difference between 4/3 and 5/4, so this shifted continuum could also logically be termed the "syntonic-diatonic equivalence continuum". This means that at ''k'' = 0, 4/3 and 5/4 are mapped to the same interval while 81/80 becomes independent of 16/15 (meaning 81/80 may or may not be tempered) because the relation becomes (81/80)<sup>0</sup> ~ 1/1 ~ 16/15. | ||
* ''k'' = 1 and upwards (up to a point) represent temperaments with (the potential for) reasonably good accuracy as equating at least one 81/80 with 16/15 seems like a good lower bound for a temperament intended to model JI. A good upper bound might be rodan (''k'' = 4), with the only exception being meantone (''n'' = ''k'' = (unsigned) infinity). (Temperaments corresponding to ''k'' = 0, -1, -2 are comparatively low-accuracy to the point of developing various intriguing structures and consequences.) | * ''k'' = 1 and upwards (up to a point) represent temperaments with (the potential for) reasonably good accuracy as equating at least one 81/80 with 16/15 seems like a good lower bound for a temperament intended to model JI. A good upper bound might be rodan (''k'' = 4), with the only exception being meantone (''n'' = ''k'' = (unsigned) infinity). (Temperaments corresponding to ''k'' = 0, -1, -2 are comparatively low-accuracy to the point of developing various intriguing structures and consequences.) | ||
* 16/15 is the simplest ratio to be tempered in the continuum. | * 16/15 is the simplest ratio to be tempered in the continuum. | ||
Revision as of 16:41, 9 December 2021
The syntonic-diatonic equivalence continuum is a continuum of temperaments which equate a number of syntonic commas (81/80) with the limma (256/243).
All temperaments in the continuum satisfy (81/80)n ~ 256/243. Varying n results in different temperaments listed in the table below. It converges to meantone as n approaches infinity. If we allow non-integer and infinite n, the continuum describes the set of all 5-limit temperaments supported by 5edo due to it being the unique equal temperament that tempers both commas and thus tempers all combinations of them. The just value of n is 4.1952…, and temperaments near this tend to be the most accurate ones.
256/243 is the characteristic 3-limit comma tempered out in 5edo. In each case, we notice that n equals the order of harmonic 5 in the corresponding comma, and equals the number of generators to obtain a harmonic 3 in the MOS scale. However, if we let k = n + 1 (meaning n = k - 1) so that k = 0 means n = -1, k = 1 means n = 0, etc. then the continuum corresponds to (81/80)k = 16/15, which might be a preferred way of conceptualising it because:
- 16/15 is the classic diatonic semitone, notable in the 5-limit as the difference between 4/3 and 5/4, so this shifted continuum could also logically be termed the "syntonic-diatonic equivalence continuum". This means that at k = 0, 4/3 and 5/4 are mapped to the same interval while 81/80 becomes independent of 16/15 (meaning 81/80 may or may not be tempered) because the relation becomes (81/80)0 ~ 1/1 ~ 16/15.
- k = 1 and upwards (up to a point) represent temperaments with (the potential for) reasonably good accuracy as equating at least one 81/80 with 16/15 seems like a good lower bound for a temperament intended to model JI. A good upper bound might be rodan (k = 4), with the only exception being meantone (n = k = (unsigned) infinity). (Temperaments corresponding to k = 0, -1, -2 are comparatively low-accuracy to the point of developing various intriguing structures and consequences.)
- 16/15 is the simplest ratio to be tempered in the continuum.
| k = n + 1 | n = k − 1 | Temperament | Comma | |
|---|---|---|---|---|
| Ratio | Monzo | |||
| -3 | -4 | Laquadgu | 177147/160000 | [-8 11 -4⟩ |
| -2 | -3 | Laconic | 2187/2000 | [-4 7 -3⟩ |
| -1 | -2 | Bug | 27/25 | [0 3 -2⟩ |
| 0 | -1 | Father | 16/15 | [4 -1 -1⟩ |
| 1 | 0 | Blackwood | 256/243 | [8 -5⟩ |
| 2 | 1 | Superpyth | 20480/19683 | [12 -9 1⟩ |
| 3 | 2 | Immunity | 1638400/1594323 | [16 -13 2⟩ |
| 4 | 3 | Rodan | 131072000/129140163 | [20 -17 3⟩ |
| 5 | 4 | Vulture | 10485760000/10460353203 | [24 -21 4⟩ |
| 6 | 5 | Pental | [-28 25 -5⟩ | |
| 7 | 6 | Hemiseven | [-32 29 -6⟩ | |
| … | … | … | … | |
| ∞ | ∞ | Meantone | 81/80 | [-4 4 -1⟩ |
Examples of temperaments with fractional values of n:
- University (n = -1.5)
- Uncle (n = -0.5)
- 5 & 32 (n = 0.5)
- 5 & 56 (n = 1.5)
- Counterpental (n = 2.5)
- Septiquarter (n = 3.5)
- 2513 & 559 (n = 4.2)
- 5 & 118 (n = 4.5)
- 5 & 137 (n = 5.5)
Hemiseven (5-limit)
Comma: [32 -29 6⟩
Mapping: [⟨1 4 14], ⟨0 -6 -29]]
POTE generator: ~320/243 = 483.2474 cents
Vals: Template:Val list
Badness: 0.720465
The temperament finder - 5-limit 5 & 72
Sasayo (5 & 32)
Comma: [20 -14 1⟩ = 5242880/4782969
Mapping: [⟨1 2 8], ⟨0 -1 -14]]
POTE generator: ~4/3 = 486.1713 cents
Vals: Template:Val list
Badness: 0.795243
The temperament finder - 5-limit 5 & 32p
Trisatriyo (5 & 56)
Comma: [28 -22 3⟩ = 33554432000/31381059609
Mapping: [⟨1 1 -2], ⟨0 3 22]]
POTE generator: ~2560/2187 = 235.8673 cents
Vals: Template:Val list
Badness: 1.323443
The temperament finder - 5-limit 5 & 56
Counterpental
Comma: [36 -30 5⟩
Mapping: [⟨5 8 12], ⟨0 -1 -6]]
POTE generator: 15.4278 cents
Vals: Template:Val list
Badness: 1.500224
The temperament finder - 5-limit 5 & 75
Septiquarter (5-limit)
Comma: [44 -38 7⟩
Mapping: [⟨1 3 10], ⟨0 -7 -38]]
POTE generator: ~204800/177147 = 242.4567 cents
Vals: Template:Val list
Badness: 0.971284
The temperament finder - 5-limit 99 & 94
559 & 2513
Comma: [-124 109 -21⟩
Mapping: [⟨1 10 46], ⟨0 -21 -109]]
POTE generator: ~3355443200000/2541865828329 = 480.8595 cents
Vals: Template:Val list
Badness: 0.134523
The temperament finder - 5-limit 2513 & 559
Quinla-tritrigu (5 & 118)
Comma: [-52 46 -9⟩
Mapping: [⟨1 -2 -16], ⟨0 9 46]]
POTE generator: ~320/243 = 477.9609 cents
Vals: Template:Val list
Badness: 0.617683
Tribilalegu (5 & 137)
Comma: [-60 54 -11⟩
Mapping: [⟨1 6 24], ⟨0 -11 -54]]
POTE generator: ~320/243 = 481.7421 cents
Vals: Template:Val list
Badness: 3.620981