Syntonic–diatonic equivalence continuum: Difference between revisions
Cmloegcmluin (talk | contribs) "optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence |
Let the m-continuum be a thing |
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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. | 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 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 | 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 generator chain. | ||
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. Some prefer this 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)<sup>0</sup> ~ 1/1 ~ 16/15. | * 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.) | ||
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{| class="wikitable center-1 center-2" | {| class="wikitable center-1 center-2" | ||
|+ Temperaments | |+ Temperaments with integer ''n'' | ||
|- | |- | ||
! rowspan="2" | ''k'' | ! rowspan="2" | ''k'' | ||
! rowspan="2" | ''n'' | ! rowspan="2" | ''n'' | ||
! rowspan="2" | Temperament | ! rowspan="2" | Temperament | ||
! colspan="2" | Comma | ! colspan="2" | Comma | ||
| Line 21: | Line 23: | ||
| -3 | | -3 | ||
| -4 | | -4 | ||
| | | Laquadgu (5 & 28) | ||
| [[177147/160000]] | | [[177147/160000]] | ||
| {{monzo| -8 11 -4 }} | | {{monzo| -8 11 -4 }} | ||
| Line 57: | Line 59: | ||
| 3 | | 3 | ||
| 2 | | 2 | ||
| [[ | | [[Immunity]] | ||
| [[1638400/1594323]] | | [[1638400/1594323]] | ||
| {{monzo| 16 -13 2 }} | | {{monzo| 16 -13 2 }} | ||
| Line 97: | Line 99: | ||
|} | |} | ||
We may invert the continuum by setting ''m'' such that 1/''m'' + 1/''n'' = 1. This may be called the ''superpyth-diatonic equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.3130… | |||
{| class="wikitable center-1" | |||
|+ Temperaments with integer ''m'' | |||
|- | |||
! rowspan="2" | ''m'' | |||
! rowspan="2" | Temperament | |||
! colspan="2" | Comma | |||
|- | |||
! Ratio | |||
! Monzo | |||
|- | |||
| -1 | |||
| [[Ultrapyth]] | |||
| [[5242880/4782969]] | |||
| {{monzo| 20 -14 1 }} | |||
|- | |||
| 0 | |||
| [[Blackwood]] | |||
| [[256/243]] | |||
| {{monzo| 8 -5 }} | |||
|- | |||
| 1 | |||
| [[Meantone]] | |||
| [[81/80]] | |||
| {{monzo| -4 4 -1 }} | |||
|- | |||
| 2 | |||
| [[Immunity]] | |||
| [[1638400/1594323]] | |||
| {{monzo| 16 -13 2 }} | |||
|- | |||
| 3 | |||
| 5 & 56 | |||
| | |||
| {{monzo| 28 -22 3 }} | |||
|- | |||
| … | |||
| … | |||
| … | |||
| … | |||
|- | |||
| ∞ | |||
| [[Superpyth]] | |||
| [[20480/19683]] | |||
| {{monzo| 12 -9 1 }} | |||
|} | |||
{| class="wikitable" | |||
|+ Temperaments with fractional ''n'' and ''m'' | |||
|- | |||
! Temperament !! ''n'' !! ''m'' | |||
|- | |||
| [[University]] || -3/2 = -1.5 || 3/5 = 0.6 | |||
|- | |||
| [[Uncle]] || -1/2 = -0.5 || 1/3 = 0.{{overline|3}} | |||
|- | |||
| [[Counterpental]] || 5/2 = 2.5 || 5/3 = 1.{{overline|6}} | |||
|- | |||
| [[Septiquarter]] || 7/2 = 3.5 || 7/5 = 1.4 | |||
|- | |||
| 559 & 2513 || 21/5 = 4.2 || 21/16 = 1.3125 | |||
|- | |||
| 5 & 118 || 9/2 = 4.5 || 9/7 = 1.{{overline|285714}} | |||
|- | |||
| 5 & 137 || 11/2 = 5.5 || 11/9 = 1.{{overline|2}} | |||
|} | |||
== Hemiseven == | == Hemiseven == | ||
Revision as of 11:00, 20 April 2024
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 generator chain.
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. Some prefer this 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 | Temperament | Comma | |
|---|---|---|---|---|
| Ratio | Monzo | |||
| -3 | -4 | Laquadgu (5 & 28) | 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⟩ |
We may invert the continuum by setting m such that 1/m + 1/n = 1. This may be called the superpyth-diatonic equivalence continuum, which is essentially the same thing. The just value of m is 1.3130…
| m | Temperament | Comma | |
|---|---|---|---|
| Ratio | Monzo | ||
| -1 | Ultrapyth | 5242880/4782969 | [20 -14 1⟩ |
| 0 | Blackwood | 256/243 | [8 -5⟩ |
| 1 | Meantone | 81/80 | [-4 4 -1⟩ |
| 2 | Immunity | 1638400/1594323 | [16 -13 2⟩ |
| 3 | 5 & 56 | [28 -22 3⟩ | |
| … | … | … | … |
| ∞ | Superpyth | 20480/19683 | [12 -9 1⟩ |
| Temperament | n | m |
|---|---|---|
| University | -3/2 = -1.5 | 3/5 = 0.6 |
| Uncle | -1/2 = -0.5 | 1/3 = 0.3 |
| Counterpental | 5/2 = 2.5 | 5/3 = 1.6 |
| Septiquarter | 7/2 = 3.5 | 7/5 = 1.4 |
| 559 & 2513 | 21/5 = 4.2 | 21/16 = 1.3125 |
| 5 & 118 | 9/2 = 4.5 | 9/7 = 1.285714 |
| 5 & 137 | 11/2 = 5.5 | 11/9 = 1.2 |
Hemiseven
Subgroup: 2.3.5
Comma list: [32 -29 6⟩
Mapping: [⟨1 4 14], ⟨0 -6 -29]]
Mapping generators: ~2, ~320/243
POTE generator: ~320/243 = 483.2474
Optimal ET sequence: 5, 62c, 67c, 72, 149, 221, 370, 591b, 961bb
Badness: 0.720465
Ultrapyth
Subgroup: 2.3.5
Comma list: 5242880/4782969
Mapping: [⟨1 0 -20], ⟨0 -1 -14]]
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 713.8287
Optimal ET sequence: 5, 27c, 32, 37, 79bc, 116bbc
Badness: 0.795243
Trisatriyo (5 & 56)
Subgroup: 2.3.5
Comma list: [28 -22 3⟩ = 33554432000/31381059609
Mapping: [⟨1 1 -2], ⟨0 3 22]]
Mapping generators: ~2, ~2560/2187
POTE generator: ~2560/2187 = 235.8673
Optimal ET sequence: 5, 56, 61
Badness: 1.323443
The temperament finder - 5-limit 5 & 56
Counterpental
Subgroup: 2.3.5
Comma list: [36 -30 5⟩
Mapping: [⟨5 0 -36], ⟨0 1 6]]
Mapping generators: ~729/640, ~3
POTE generator: ~3/2 = 704.5722
Optimal ET sequence: 5, 75, 80
Badness: 1.500224
Septiquarter
Subgroup: 2.3.5
Comma list: [44 -38 7⟩
Mapping: [⟨1 3 10], ⟨0 -7 -38]]
Mapping generators: ~2, ~204800/177147
POTE generator: ~204800/177147 = 242.4567
Optimal ET sequence: 5, 89c, 94, 99, 193, 292, 391
Badness: 0.971284
559 & 2513
Subgroup: 2.3.5
Comma list: [-124 109 -21⟩
Mapping: [⟨1 10 46], ⟨0 -21 -109]]
Mapping generators: ~2, ~3355443200000/2541865828329
POTE generator: ~3355443200000/2541865828329 = 480.8595
Optimal ET sequence: 5, 267c, 272c, 277, 559, 1395, 1954, 2513, 40767, 43280, 45793, 48306, 50819, 53332, 55845, 58358, 60871, 63384, 65897, 68410, 70923, 73436, 75949, 78462
Badness: 0.134523
The temperament finder - 5-limit 2513 & 559
Quinla-tritrigu (5 & 118)
Subgroup: 2.3.5
Comma list: [-52 46 -9⟩
Mapping: [⟨1 -2 -16], ⟨0 9 46]]
Mapping generators: ~2, ~320/243
POTE generator: ~320/243 = 477.9609
Optimal ET sequence: 5, 108c, 113, 118, 1057, 1175, 1293, 1411, 1529, 1647, 1765, 1883, 2001b, 3884b
Badness: 0.617683
Tribilalegu (5 & 137)
Subgroup: 2.3.5
Comma list: [-60 54 -11⟩
Mapping: [⟨1 6 24], ⟨0 -11 -54]]
Mapping generators: ~2, ~320/243
POTE generator: ~320/243 = 481.7421
Optimal ET sequence: 5, 127c, 132, 137, 553, 690b, 827b, 964b
Badness: 3.620981