Syntonic–diatonic equivalence continuum: Difference between revisions
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256/243 is the characteristic [[3-limit]] comma tempered out in 5edo, and has many advantages as a target. In each case, ''n'' equals the order of [[5/1|harmonic 5]] in the corresponding comma, and equals the number of generators to obtain a harmonic 3 in the generator chain. For example: | 256/243 is the characteristic [[3-limit]] comma tempered out in 5edo, and has many advantages as a target. In each case, ''n'' equals the order of [[5/1|harmonic 5]] in the corresponding comma, and equals the number of generators to obtain a harmonic 3 in the generator chain. For example: | ||
* Superpyth {{nowrap| | * Superpyth ({{nowrap| ''n'' {{=}} 1 }}) is generated by a fifth; | ||
* Immunity {{nowrap| | * Immunity ({{nowrap| ''n'' {{=}} 2 }}) splits its twelfth in two; | ||
* Rodan {{nowrap| | * Rodan ({{nowrap| ''n'' {{=}} 3 }}) splits its fifth in three; | ||
* Etc. | * Etc. | ||
At {{nowrap|''n'' {{=}} 5}}, the corresponding temperament splits the ''octave'' into five instead, as after a stack of five syntonic commas, both the orders of 3 and 5 are multiples of 5 again. | At {{nowrap| ''n'' {{=}} 5 }}, the corresponding temperament splits the ''octave'' into five instead, as after a stack of five syntonic commas, both the orders of 3 and 5 are multiples of 5 again. | ||
If we let {{nowrap|''k'' {{=}} ''n'' + 1}} so that {{nowrap|''k'' {{=}} 0}} means {{nowrap|''n'' {{=}} | If we let {{nowrap| ''k'' {{=}} ''n'' + 1 }} so that {{nowrap| ''k'' {{=}} 0 }} means {{nowrap|''n'' {{=}} −1}}, {{nowrap| ''k'' {{=}} 1 }} means {{nowrap| ''n'' {{=}} 0 }}, etc. then the continuum corresponds to {{nowrap| (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 {{nowrap|''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 {{nowrap|(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 {{nowrap| ''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 out) because the relation becomes {{nowrap| (81/80)<sup>0</sup> ~ 1/1 ~ 16/15 }}. | ||
* {{nowrap|''k'' {{=}} 1}} and upwards (up to a point) represent temperaments with | * {{nowrap| ''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 ({{nowrap| ''k'' {{=}} 4 }}), with the only exception being meantone ({{nowrap| ''n'' {{=}} ''k'' {{=}} ∞ }}). (Temperaments corresponding to {{nowrap| ''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 out in the continuum. | ||
{| class="wikitable center-1 center-2" | {| class="wikitable center-1 center-2" | ||
| Line 32: | Line 32: | ||
| Laquadgu (5 & 28) | | Laquadgu (5 & 28) | ||
| [[177147/160000]] | | [[177147/160000]] | ||
| {{ | | {{Monzo| -8 11 -4 }} | ||
|- | |- | ||
| −2 | | −2 | ||
| Line 38: | Line 38: | ||
| [[Gamelismic clan #Gorgo|Laconic]] | | [[Gamelismic clan #Gorgo|Laconic]] | ||
| [[2187/2000]] | | [[2187/2000]] | ||
| {{ | | {{Monzo| -4 7 -3 }} | ||
|- | |- | ||
| −1 | | −1 | ||
| Line 44: | Line 44: | ||
| [[Bug]] | | [[Bug]] | ||
| [[27/25]] | | [[27/25]] | ||
| {{ | | {{Monzo| 0 3 -2 }} | ||
|- | |- | ||
| 0 | | 0 | ||
| Line 50: | Line 50: | ||
| [[Father]] | | [[Father]] | ||
| [[16/15]] | | [[16/15]] | ||
| {{ | | {{Monzo| 4 -1 -1 }} | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 56: | Line 56: | ||
| [[Blackwood]] | | [[Blackwood]] | ||
| [[256/243]] | | [[256/243]] | ||
| {{ | | {{Monzo| 8 -5 }} | ||
|- | |- | ||
| 2 | | 2 | ||
| Line 62: | Line 62: | ||
| [[Superpyth]] | | [[Superpyth]] | ||
| [[20480/19683]] | | [[20480/19683]] | ||
| {{ | | {{Monzo| 12 -9 1 }} | ||
|- | |- | ||
| 3 | | 3 | ||
| Line 68: | Line 68: | ||
| [[Immunity]] | | [[Immunity]] | ||
| [[1638400/1594323]] | | [[1638400/1594323]] | ||
| {{ | | {{Monzo| 16 -13 2 }} | ||
|- | |- | ||
| 4 | | 4 | ||
| Line 74: | Line 74: | ||
| [[Rodan]] | | [[Rodan]] | ||
| [[131072000/129140163]] | | [[131072000/129140163]] | ||
| {{ | | {{Monzo| 20 -17 3 }} | ||
|- | |- | ||
| 5 | | 5 | ||
| Line 80: | Line 80: | ||
| [[Vulture]] | | [[Vulture]] | ||
| [[10485760000/10460353203|(22 digits)]] | | [[10485760000/10460353203|(22 digits)]] | ||
| {{ | | {{Monzo| 24 -21 4 }} | ||
|- | |- | ||
| 6 | | 6 | ||
| Line 86: | Line 86: | ||
| [[Quintile]] | | [[Quintile]] | ||
| (24 digits) | | (24 digits) | ||
| {{ | | {{Monzo| -28 25 -5 }} | ||
|- | |- | ||
| 7 | | 7 | ||
| Line 92: | Line 92: | ||
| [[Hemiseven]] | | [[Hemiseven]] | ||
| (28 digits) | | (28 digits) | ||
| {{ | | {{Monzo| -32 29 -6 }} | ||
|- | |- | ||
| … | | … | ||
| Line 103: | Line 103: | ||
| [[Meantone]] | | [[Meantone]] | ||
| [[81/80]] | | [[81/80]] | ||
| {{ | | {{Monzo| -4 4 -1 }} | ||
|} | |} | ||
We may invert the continuum by setting ''m'' such that 1/''m'' + 1/''n'' = 1. This may be called the '' | We may invert the continuum by setting ''m'' such that {{nowrap| 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…. The [[superpyth comma]] is both larger and more complex than the syntonic comma. As such, this continuum does not contain as many useful temperaments, but still interesting nonetheless. | ||
{| class="wikitable center-1" | {| class="wikitable center-1" | ||
| Line 121: | Line 121: | ||
| [[Ultrapyth]] | | [[Ultrapyth]] | ||
| [[5242880/4782969]] | | [[5242880/4782969]] | ||
| {{ | | {{Monzo| 20 -14 1 }} | ||
|- | |- | ||
| 0 | | 0 | ||
| [[Blackwood]] | | [[Blackwood]] | ||
| [[256/243]] | | [[256/243]] | ||
| {{ | | {{Monzo| 8 -5 }} | ||
|- | |- | ||
| 1 | | 1 | ||
| [[Meantone]] | | [[Meantone]] | ||
| [[81/80]] | | [[81/80]] | ||
| {{ | | {{Monzo| -4 4 -1 }} | ||
|- | |- | ||
| 2 | | 2 | ||
| [[Immunity]] | | [[Immunity]] | ||
| [[1638400/1594323]] | | [[1638400/1594323]] | ||
| {{ | | {{Monzo| 16 -13 2 }} | ||
|- | |- | ||
| 3 | | 3 | ||
| 5 & 56 | | 5 & 56 | ||
| [[33554432000/31381059609]] | | [[33554432000/31381059609]] | ||
| {{ | | {{Monzo| 28 -22 3 }} | ||
|- | |- | ||
| … | | … | ||
| Line 151: | Line 151: | ||
| [[Superpyth]] | | [[Superpyth]] | ||
| [[20480/19683]] | | [[20480/19683]] | ||
| {{ | | {{Monzo| 12 -9 1 }} | ||
|} | |} | ||
| Line 159: | Line 159: | ||
! ''n'' !! ''m'' !! Temperament !! Comma | ! ''n'' !! ''m'' !! Temperament !! Comma | ||
|- | |- | ||
| −3/2 = −1.5 || 3/5 = 0.6 || [[University]] || {{ | | −3/2 = −1.5 || 3/5 = 0.6 || [[University]] || {{Monzo| 4 2 -3 }} | ||
|- | |- | ||
| −1/2 = −0.5 || 1/3 = 0.{{overline|3}} || [[Uncle]] || {{ | | −1/2 = −0.5 || 1/3 = 0.{{overline|3}} || [[Uncle]] || {{Monzo| 12 -6 -1 }} | ||
|- | |- | ||
| 1/3 = 0.{{overline|3}} || −1/2 = −0.5 || [[Dirt]] || {{ | | 1/3 = 0.{{overline|3}} || −1/2 = −0.5 || [[Dirt]] || {{Monzo| 28 -19 1 }} | ||
|- | |- | ||
| 5/2 = 2.5 || 5/3 = 1.{{overline|6}} || [[Counterpental]] || {{ | | 5/2 = 2.5 || 5/3 = 1.{{overline|6}} || [[Counterpental]] || {{Monzo| 36 -30 5 }} | ||
|- | |- | ||
| 7/2 = 3.5 || 7/5 = 1.4 || [[Septiquarter]] || {{ | | 7/2 = 3.5 || 7/5 = 1.4 || [[Septiquarter]] || {{Monzo| 44 -38 7 }} | ||
|- | |- | ||
| 21/5 = 4.2 || 21/16 = 1.3125 || 559 & | | 21/5 = 4.2 || 21/16 = 1.3125 || 559 & 2513 || {{Monzo| -124 109 -21 }} | ||
|- | |- | ||
| 9/2 = 4.5 || 9/7 = 1.{{overline|285714}} || 5 & | | 9/2 = 4.5 || 9/7 = 1.{{overline|285714}} || 5 & 118 || {{Monzo| -52 46 -9 }} | ||
|- | |- | ||
| 11/2 = 5.5 || 11/9 = 1.{{overline|2}} || 5 & | | 11/2 = 5.5 || 11/9 = 1.{{overline|2}} || 5 & 137 || {{Monzo| -60 54 -11 }} | ||
|} | |} | ||
| Line 179: | Line 179: | ||
: ''For extensions, see [[Archytas clan #Superpyth]] and [[Jubilismic clan #Bipyth]].'' | : ''For extensions, see [[Archytas clan #Superpyth]] and [[Jubilismic clan #Bipyth]].'' | ||
In the 5-limit, superpyth tempers out [[20480/19683]]. It has a fifth generator of ~3/2 = ~ | In the 5-limit, superpyth tempers out [[20480/19683]]. It has a fifth generator of {{nowrap| ~3/2 {{=}} ~710{{c}} }} and ~5/4 is found at +9 generator steps, as an augmented second (C–D#). It corresponds to {{nowrap| ''n'' {{=}} 1 }}, meaning that the syntonic comma is equated with the diatonic semitone. | ||
[[Subgroup]]: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
| Line 227: | Line 227: | ||
: ''For extensions, see [[Archytas clan #Ultrapyth]].'' | : ''For extensions, see [[Archytas clan #Ultrapyth]].'' | ||
The 5-limit version of ultrapyth tempers out the [[ultrapyth comma]]. It is generated by a perfect fifth. The interval class of 5 is found at +14 fifths as a double-augmented unison (C–Cx). It corresponds to {{nowrap|''m'' {{=}} -1}} and {{nowrap|''n'' {{=}} 1/2}}. | The 5-limit version of ultrapyth tempers out the [[ultrapyth comma]]. It is generated by a perfect fifth. The interval class of 5 is found at +14 fifths as a double-augmented unison (C–Cx). It corresponds to {{nowrap| ''m'' {{=}} -1 }} and {{nowrap| ''n'' {{=}} 1/2 }}. | ||
[[Subgroup]]: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
| Line 316: | Line 316: | ||
: ''For extensions, see [[Gamelismic clan #Gidorah]] and [[Mint temperaments #Penta]].'' | : ''For extensions, see [[Gamelismic clan #Gidorah]] and [[Mint temperaments #Penta]].'' | ||
Named by [[John Moriarty]], university is the 5 & 6b temperament, and tempers out [[144/125]], the triptolemaic diminished third. It corresponds to ''n'' = −3/2 and ''m'' = 3/5. In this temperament, two instances of [[6/5]] make a [[5/4]], and three make a [[3/2]]. Equating 6/5 with [[8/7]] (which makes sense since it is already very flat in the most accurate tunings of this temperament) leads to [[Gamelismic clan #Gidorah|gidorah]], and 6/5 with [[7/6]] leads to [[Mint temperaments #Penta|penta]]. | Named by [[John Moriarty]], university is the {{nowrap| 5 & 6b }} temperament, and tempers out [[144/125]], the triptolemaic diminished third. It corresponds to {{nowrap| ''n'' {{=}} −3/2 }} and {{nowrap| ''m'' {{=}} 3/5 }}. In this temperament, two instances of [[6/5]] make a [[5/4]], and three make a [[3/2]]. Equating 6/5 with [[8/7]] (which makes sense since it is already very flat in the most accurate tunings of this temperament) leads to [[Gamelismic clan #Gidorah|gidorah]], and 6/5 with [[7/6]] leads to [[Mint temperaments #Penta|penta]]. | ||
[[Subgroup]]: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
| Line 334: | Line 334: | ||
[[Badness]] (Smith): 0.101806 | [[Badness]] (Smith): 0.101806 | ||
== Trisatriyo (5 & | == Trisatriyo (5 & 56) == | ||
[[Subgroup]]: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
[[Comma list]]: {{monzo| 28 -22 3 }} | [[Comma list]]: {{monzo| 28 -22 3 }} (33554432000/31381059609) | ||
{{Mapping|legend=1| 1 1 -2 | 0 3 22 }} | {{Mapping|legend=1| 1 1 -2 | 0 3 22 }} | ||
| Line 406: | Line 406: | ||
[[Badness]] (Smith): 0.971284 | [[Badness]] (Smith): 0.971284 | ||
== Quinla-tritrigu (5 & | == Quinla-tritrigu (5 & 118) == | ||
[[Subgroup]]: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
| Line 422: | Line 422: | ||
[[Badness]] (Smith): 0.617683 | [[Badness]] (Smith): 0.617683 | ||
== Tribilalegu (5 & | == Tribilalegu (5 & 137) == | ||
[[Subgroup]]: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
| Line 440: | Line 440: | ||
[http://x31eq.com/cgi-bin/rt.cgi?ets=5_137&limit=5 The temperament finder - 5-limit 5 & 137] | [http://x31eq.com/cgi-bin/rt.cgi?ets=5_137&limit=5 The temperament finder - 5-limit 5 & 137] | ||
== 559 & | == 559 & 2513 == | ||
[[Subgroup]]: 2.3.5 | [[Subgroup]]: 2.3.5 | ||
Revision as of 10:49, 13 August 2025
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
The syntonic–diatonic equivalence continuum is a continuum of temperaments which equate a number of syntonic commas (81/80) with the Pythagorean limma (256/243). This continuum is theoretically interesting in that these are all 5-limit temperaments supported by 5edo.
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 out both commas and thus tempers out 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, and has many advantages as a target. In each case, 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. For example:
- Superpyth (n = 1) is generated by a fifth;
- Immunity (n = 2) splits its twelfth in two;
- Rodan (n = 3) splits its fifth in three;
- Etc.
At n = 5, the corresponding temperament splits the octave into five instead, as after a stack of five syntonic commas, both the orders of 3 and 5 are multiples of 5 again.
If we let k = n + 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 out) 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 = ∞). (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 out 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 | (22 digits) | [24 -21 4⟩ |
| 6 | 5 | Quintile | (24 digits) | [-28 25 -5⟩ |
| 7 | 6 | Hemiseven | (28 digits) | [-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…. The superpyth comma is both larger and more complex than the syntonic comma. As such, this continuum does not contain as many useful temperaments, but still interesting nonetheless.
| 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 | 33554432000/31381059609 | [28 -22 3⟩ |
| … | … | … | … |
| ∞ | Superpyth | 20480/19683 | [12 -9 1⟩ |
| n | m | Temperament | Comma |
|---|---|---|---|
| −3/2 = −1.5 | 3/5 = 0.6 | University | [4 2 -3⟩ |
| −1/2 = −0.5 | 1/3 = 0.3 | Uncle | [12 -6 -1⟩ |
| 1/3 = 0.3 | −1/2 = −0.5 | Dirt | [28 -19 1⟩ |
| 5/2 = 2.5 | 5/3 = 1.6 | Counterpental | [36 -30 5⟩ |
| 7/2 = 3.5 | 7/5 = 1.4 | Septiquarter | [44 -38 7⟩ |
| 21/5 = 4.2 | 21/16 = 1.3125 | 559 & 2513 | [-124 109 -21⟩ |
| 9/2 = 4.5 | 9/7 = 1.285714 | 5 & 118 | [-52 46 -9⟩ |
| 11/2 = 5.5 | 11/9 = 1.2 | 5 & 137 | [-60 54 -11⟩ |
Superpyth (5-limit)
- For extensions, see Archytas clan #Superpyth and Jubilismic clan #Bipyth.
In the 5-limit, superpyth tempers out 20480/19683. It has a fifth generator of ~3/2 = ~710 ¢ and ~5/4 is found at +9 generator steps, as an augmented second (C–D#). It corresponds to n = 1, meaning that the syntonic comma is equated with the diatonic semitone.
Subgroup: 2.3.5
Comma list: 20480/19683
Mapping: [⟨1 0 -12], ⟨0 1 9]]
- mapping generators: ~2, ~3
- CTE: ~2 = 1200.000, ~3/2 = 709.393
- error map: ⟨0.000 +7.438 -1.774]
- POTE: ~2 = 1200.000, ~3/2 = 710.078
- error map: ⟨0.000 +8.123 +4.385]
Optimal ET sequence: 5, 17, 22, 49, 120b, 169bbc
Badness (Smith): 0.135141
Uncle (5-limit)
- For extensions, see Trienstonic clan #Uncle.
The 5-limit version of uncle tempers out 4096/3645. It is generated by a fifth that is supposedly sharper than 3\5, so it leads to an oneirotonic scale, or otherwise a diatonic scale with negative small steps. The interval class of 5 is found at -6 fifths, as a major 2-step in oneirotonic, or a diminished fifth (C–Gb) in diatonic. It corresponds to n = -1/2 or m = 1/3.
Subgroup: 2.3.5
Comma list: 4096/3645
Mapping: [⟨1 0 12], ⟨0 1 -6]]
- mapping generators: ~2, ~3
- CTE: ~2 = 1200.000, ~3/2 = 733.721
- error map: ⟨0.000 +31.766 +11.362]
- CWE: ~2 = 1200.000, ~3/2 = 731.732
- error map: ⟨0.000 +29.777 +23.296]
Optimal ET sequence: 5, 13, 18, 23bc
- Smith: 0.270
- Dirichlet: 6.33
Ultrapyth (5-limit)
- For extensions, see Archytas clan #Ultrapyth.
The 5-limit version of ultrapyth tempers out the ultrapyth comma. It is generated by a perfect fifth. The interval class of 5 is found at +14 fifths as a double-augmented unison (C–Cx). It corresponds to m = -1 and n = 1/2.
Subgroup: 2.3.5
Comma list: 5242880/4782969
Mapping: [⟨1 0 -20], ⟨0 1 14]]
- mapping generators: ~2, ~3
- CTE: ~2 = 1200.000, ~3/2 = 713.185
- error map: ⟨0.000 +11.230 -1.722]
- POTE: ~2 = 1200.000, ~3/2 = 713.829
- error map: ⟨0.000 +11.874 +7.289]
Optimal ET sequence: 5, 27c, 32, 37, 79bc, 116bbc
Badness (Smith): 0.795243
Dirt
Dirt tempers out the dirt comma, 1342177280/1162261467. It is generated by a perfect fifth. The interval class of 5 is found at +19 fifths, as a double-augmented seventh (C–Bx). It corresponds to n = 1/3 and m = -1/2.
Subgroup: 2.3.5
Comma list: [28 -19 1⟩
Mapping: [⟨1 0 -28], ⟨0 1 19]]
- mapping generators: ~2, ~3
- CTE: ~2 = 1200.000, ~3/2 = 714.992
- error map: ⟨0.000 +13.037 -1.473]
- CWE: ~2 = 1200.000, ~3/2 = 715.341
- error map: ⟨0.000 +13.386 +5.157]
Optimal ET sequence: 5, 42c, 47b, 52b, 109bbc
Badness (Smith): 2.36
Rodan (5-limit)
- For extensions, see Gamelismic clan #Rodan.
The 5-limit version of rodan tempers out the rodan comma, which is the difference between a stack of three retroptolemaic whole tones (729/640) and a perfect fifth (3/2). The only 7-limit extension that makes any sense to use is to add the gamelisma to the comma list. It corresponds to n = 3.
Subgroup: 2.3.5
Comma list: 131072000/129140163
Mapping: [⟨1 1 -1], ⟨0 3 17]]
- CTE: ~2 = 1200.000, ~729/640 = 234.457
- error map: ⟨0.000 +1.417 -0.537]
- POTE: ~2 = 1200.000, ~729/640 = 234.528
- error map: ⟨0.000 +1.629 +0.663]
Optimal ET sequence: 5, …, 41, 46, 87, 220, 307
Badness: 0.168264
Laconic
- For extensions, see Gamelismic clan #Gorgo.
Laconic tempers out 2187/2000, which is the difference between a stack of three ptolemaic whole tones (10/9)'s and a perfect fifth (3/2). Although a higher-error temperament, it does pop up enough in the low-numbered edos to be useful, most notably in 16edo and 21edo. The only 7-limit extension that makes any sense to use is to add the gamelisma to the comma list. It corresponds to n = -3.
Subgroup: 2.3.5
Comma list: 2187/2000
Mapping: [⟨1 1 1], ⟨0 3 7]]
- CTE: ~2 = 1200.000, ~10/9 = 228.700
- error map: ⟨0.000 -15.856 +14.584]
- POTE: ~2 = 1200.000, ~10/9 = 227.426
- error map: ⟨0.000 -19.679 +5.664]
Optimal ET sequence: 5, 11c, 16, 21, 37b
Badness (Smith): 0.161799
University
- For extensions, see Gamelismic clan #Gidorah and Mint temperaments #Penta.
Named by John Moriarty, university is the 5 & 6b temperament, and tempers out 144/125, the triptolemaic diminished third. It corresponds to n = −3/2 and m = 3/5. In this temperament, two instances of 6/5 make a 5/4, and three make a 3/2. Equating 6/5 with 8/7 (which makes sense since it is already very flat in the most accurate tunings of this temperament) leads to gidorah, and 6/5 with 7/6 leads to penta.
Subgroup: 2.3.5
Comma list: 144/125
Mapping: [⟨1 1 2], ⟨0 3 2]]
- CTE: ~2 = 1200.000, ~6/5 = 226.980
- error map: ⟨0.000 -21.014 +67.647]
- POTE: ~2 = 1200.000, ~6/5 = 235.442
- error map: ⟨0.000 +4.370 +84.569]
Optimal ET sequence: 1b, …, 4bc, 5
Badness (Smith): 0.101806
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: ~2 = 1200.000, ~2560/2187 = 235.867
Optimal ET sequence: 5, …, 51, 56, 117b, 173b
Badness (Smith): 1.323443
The temperament finder - 5-limit 5 & 56
Hemiseven (5-limit)
- For extensions, see Gamelismic clan #Hemiseven.
Subgroup: 2.3.5
Comma list: [32 -29 6⟩
Mapping: [⟨1 4 14], ⟨0 -6 -29]]
- mapping generators: ~2, ~320/243
- POTE: ~2 = 1200.000, ~320/243 = 483.247
Optimal ET sequence: 5, 62c, 67c, 72, 149, 221, 370, 591b, 961bb
Badness (Smith): 0.720465
Counterpental
- For extensions, see Orwellismic temperaments #Pentorwell.
Subgroup: 2.3.5
Comma list: [36 -30 5⟩
Mapping: [⟨5 0 -36], ⟨0 1 6]]
- mapping generators: ~729/640, ~3
- POTE: ~729/640 = 240.000, ~3/2 = 704.572
Optimal ET sequence: 5, …, 75, 80, 155, 390b, 545bbc
Badness (Smith): 1.500224
Septiquarter (5-limit)
- For extensions, see Hemifamity temperaments #Septiquarter.
Subgroup: 2.3.5
Comma list: [44 -38 7⟩
Mapping: [⟨1 3 10], ⟨0 -7 -38]]
- mapping generators: ~2, ~204800/177147
- POTE: ~2 = 1200.000, ~204800/177147 = 242.457
Optimal ET sequence: 5, 89c, 94, 99, 193, 292, 391
Badness (Smith): 0.971284
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: ~2 = 1200.000, ~320/243 = 477.961
Optimal ET sequence: 5, 108c, 113, 118, 1057, 1175, 1293, 1411, 1529, 1647, 1765, 1883, 2001b, 3884b
Badness (Smith): 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: ~2 = 1200.000, ~320/243 = 481.742
Optimal ET sequence: 5, 127c, 132, 137, 553, 690b, 827b, 964b
Badness (Smith): 3.620981
The temperament finder - 5-limit 5 & 137
559 & 2513
Subgroup: 2.3.5
Comma list: [-124 109 -21⟩
Mapping: [⟨1 10 46], ⟨0 -21 -109]]
- mapping generators: ~2, ~3355443200000/2541865828329
- POTE: ~2 = 1200.0000, ~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 (Smith): 0.134523