Syntonic–diatonic 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)]].
{{Technical data page}}
The '''syntonic–diatonic equivalence continuum''' is a [[equivalence continuum|continuum]] of [[regular temperament|temperaments]] which equate a number of [[81/80|syntonic commas (81/80)]] with the [[256/243|Pythagorean limma (256/243)]]. This continuum is theoretically interesting in that these are all [[5-limit]] temperaments [[support]]ed 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.  
All temperaments in the continuum satisfy {{nowrap|(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 [[tempering out|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 has the advantage of being the characteristic [[3-limit]] comma tempered out in [[5edo]]. For 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:
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:
* 25/24 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)^0 = 1/1 = 16/15.
* Superpyth ({{nowrap| ''n'' {{=}} 1 }}) is generated by a fifth;
* ''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.)
* Immunity ({{nowrap| ''n'' {{=}} 2 }}) splits its twelfth in two;
* 16/15 is the simplest ratio to be tempered in the continuum, the second simplest being 81/80 at (unsigned) infinity, which together are the two smallest 5-limit [[List of superparticular intervals|superparticular intervals]] and the only superparticular intervals in the continuum if we don't count non-integer ''k''.
* Rodan ({{nowrap| ''n'' {{=}} 3 }}) splits its fifth in three;
* 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.
 
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 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 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 out in the continuum.  


{| class="wikitable center-1 center-2"
{| class="wikitable center-1 center-2"
|+ Temperaments in the continuum
|+ style="font-size: 105%;" | Temperaments with integer ''n''
|-
|-
! rowspan="2" | ''k'' = ''n'' − 2
! rowspan="2" | ''k''
! rowspan="2" | ''n'' = ''k'' + 2
! rowspan="2" | ''n''
! rowspan="2" | Temperament
! rowspan="2" | Temperament
! colspan="2" | Comma
! colspan="2" | Comma
Line 19: Line 28:
! Monzo
! Monzo
|-
|-
| -3
| −3
| -4
| −4
| [[Laquadgu]]
| Laquadgu (5 & 28)
| [[177147/160000]]
| [[177147/160000]]
| {{monzo| -8 11 -4 }}
| {{Monzo| -8 11 -4 }}
|-
|-
| -2
| −2
| -3
| −3
| [[Laconic family#Laconic|Laconic]]
| [[Laconic]]
| [[2187/2000]]
| [[2187/2000]]
| {{monzo| -4 7 -3 }}
| {{Monzo| -4 7 -3 }}
|-
|-
| -1
| −1
| -2
| −2
| [[Bug]]
| [[Bug]]
| [[27/25]]
| [[27/25]]
| {{monzo| 0 3 -2 }}
| {{Monzo| 0 3 -2 }}
|-
|-
| 0
| 0
| -1
| −1
| [[Father]]
| [[Father]]
| [[16/15]]
| [[16/15]]
| {{monzo| 4 -1 -1 }}
| {{Monzo| 4 -1 -1 }}
|-
|-
| 1
| 1
Line 47: Line 56:
| [[Blackwood]]
| [[Blackwood]]
| [[256/243]]
| [[256/243]]
| {{monzo| 8 -5 }}
| {{Monzo| 8 -5 }}
|-
|-
| 2
| 2
Line 53: Line 62:
| [[Superpyth]]
| [[Superpyth]]
| [[20480/19683]]
| [[20480/19683]]
| {{monzo| 12 -9 1 }}
| {{Monzo| 12 -9 1 }}
|-
|-
| 3
| 3
| 2
| 2
| [[Immunity family|Immunity]]
| [[Immunity]]
| [[1638400/1594323]]
| [[1638400/1594323]]
| {{monzo| 16 -13 2 }}
| {{Monzo| 16 -13 2 }}
|-
|-
| 4
| 4
Line 65: Line 74:
| [[Rodan]]
| [[Rodan]]
| [[131072000/129140163]]
| [[131072000/129140163]]
| {{monzo| 20 -17 3 }}
| {{Monzo| 20 -17 3 }}
|-
|-
| 5
| 5
| 4
| 4
| [[Vulture]]
| [[Vulture]]
| [[10485760000/10460353203]]
| [[10485760000/10460353203|(22 digits)]]
| {{monzo| 24 -21 4 }}
| {{Monzo| 24 -21 4 }}
|-
|-
| 6
| 6
| 5
| 5
| [[Pental family|Pental]]
| [[Quintile]]
|  
| (24 digits)
| {{monzo| 28 -25 5 }}
| {{Monzo| -28 25 -5 }}
|-
|-
| 7
| 7
| 6
| 6
| 5 & 72
| [[Hemiseven]]
|  
| (28 digits)
| {{monzo| 32 -29 6 }}
| {{Monzo| -32 29 -6 }}
|-
|-
| …
| …
| …
| …
| …
Line 94: Line 104:
| [[Meantone]]
| [[Meantone]]
| [[81/80]]
| [[81/80]]
| {{monzo| -4 4 -1 }}
| {{Monzo| -4 4 -1 }}
|}
|}


Examples of temperaments with fractional values of ''n'':
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.
* [[University temperament|University]] (''n'' = -0.5)
 
* 5 & 32p (''n'' = 0.5)
{| class="wikitable center-1"
* 5 & 56 (''n'' = 1.5)
|+ style="font-size: 105%;" | Temperaments with integer ''m''
* Counterpental (''n'' = 2.5)
|-
* 99 & 94 (''n'' = 3.5)
! rowspan="2" | ''m''
* 2513 & 559 (''n'' = 4.2)
! rowspan="2" | Temperament
* 5 & 137 (''n'' = 4.5)
! 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
| [[33554432000/31381059609]]
| {{Monzo| 28 -22 3 }}
|-
| …
| …
| …
| …
|-
| ∞
| [[Superpyth]]
| [[20480/19683]]
| {{Monzo| 12 -9 1 }}
|}
 
{| class="wikitable"
|+ style="font-size: 105%;" | Temperaments with fractional ''n'' and ''m''
|-
! ''n'' !! ''m'' !! Temperament !! Comma
|-
| −3/2 = −1.5 || 3/5 = 0.6 || [[University]] || {{Monzo| 4 2 -3 }}
|-
| −1/2 = −0.5 || 1/3 = 0.{{overline|3}} || [[Uncle]] || {{Monzo| 12 -6 -1 }}
|-
| 1/3 = 0.{{overline|3}} || −1/2 = −0.5 || [[Dirt]] || {{Monzo| 28 -19 1 }}
|-
| 5/2 = 2.5 || 5/3 = 1.{{overline|6}} || [[Counterpental]] || {{Monzo| 36 -30 5 }}
|-
| 7/2 = 3.5 || 7/5 = 1.4 || [[Septiquarter]] || {{Monzo| 44 -38 7 }}
|-
| 21/5 = 4.2 || 21/16 = 1.3125 || 559 & 2513 || {{Monzo| -124 109 -21 }}
|-
| 13/3 = 4.{{overline|3}} || 13/10 = 1.3 || [[Tokko]] || {{Monzo| -76 67 -13 }}
|-
| 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 & 137 || {{Monzo| -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 {{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
 
[[Comma list]]: 20480/19683
 
{{Mapping|legend=1| 1 0 -12 | 0 1 9 }}
 
: mapping generators: ~2, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1197.6520{{c}}, ~3/2 = 708.6882{{c}}
: [[error map]]: {{val| -2.348 +4.385 -1.076 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 709.8213{{c}}
: error map: {{val| 0.000 +7.866 +2.078 }}
 
{{Optimal ET sequence|legend=1| 5, 17, 22, 49, 120b, 169bbc }}
 
[[Badness]] (Sintel): 3.17
 
== 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 [[5edo|3\5]], so it leads to an [[5L 3s|oneirotonic]] scale, or otherwise a [[5L 2s|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 {{nowrap| ''n'' {{=}} -1/2 }} or {{nowrap| ''m'' {{=}} 1/3 }}.
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 4096/3645
 
{{Mapping|legend=1| 1 0 12 | 0 1 -6 }}
 
: mapping generators: ~2, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1189.7544{{c}}, ~3/2 = 724.6670{{c}}
: [[error map]]: {{val| -10.246 +12.466 +4.210 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 731.7318{{c}}
: error map: {{val| 0.000 +29.777 +23.296 }}
 
{{Optimal ET sequence|legend=1| 5, 13, 18, 23bc }}
 
[[Badness]] (Sintel): 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 {{nowrap| ''m'' {{=}} -1 }} and {{nowrap| ''n'' {{=}} 1/2 }}.
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 5242880/4782969
 
{{Mapping|legend=1| 1 0 -20 | 0 1 14 }}
 
: mapping generators: ~2, ~3
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1196.4357{{c}}, ~3/2 = 711.7085{{c}}
: [[error map]]: {{val| -3.564 +6.189 -1.009 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 713.5968{{c}}
: error map: {{val| 0.000 +11.642 +4.041 }}
 
{{Optimal ET sequence|legend=1| 5, 27c, 32, 37, 79bc, 116bbc }}
 
[[Badness]] (Sintel): 18.7
 
== Dirt ==
{{Main| 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 {{nowrap| ''n'' {{=}} 1/3 }} and {{nowrap| ''m'' {{=}} -1/2 }}.
 
[[Subgroup]]: 2.3.5


== 5 & 72 ==
[[Comma list]]: {{monzo| 28 -19 1 }}


Comma: {{Monzo|32 -29 6}}
{{Mapping|legend=1| 1 0 -28 | 0 1 19 }}


POTE generator: 483.2474 cents
: mapping generators: ~2, ~3


Map: [&lt;1 4 14|, &lt;0 -6 -29|]
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1195.8566{{c}}, ~3/2 = 713.0611{{c}}
: [[error map]]: {{val| -4.143 +6.963 -0.863 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~3/2 = 715.3406{{c}}
: error map: {{val| 0.000 +13.386 +5.157 }}


EDOs: {{EDOs| 5, 10c, 67c, 72, 77, 82c, 139c, 144, 149, 154 }}
{{Optimal ET sequence|legend=1| 5, 42c, 47b, 52b, 109bbc }}


[http://x31eq.com/cgi-bin/rt.cgi?ets=72_5&limit=5 The temperament finder - 5-limit 5 & 72]
[[Badness]] (Sintel): 55.3


== 5 & 32p ==
== Rodan (5-limit) ==
: ''For extensions, see [[Gamelismic clan #Rodan]].''


Comma: {{Monzo|20 -14 1}} (5242880/4782969)
The 5-limit version of rodan tempers out the [[rodan comma]], which is the difference between a stack of three [[729/640|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, whereby the generator represents [[8/7]]. It corresponds to {{nowrap| ''n'' {{=}} 3 }}.


POTE generator: ~4/3 = 486.1713 cents
[[Subgroup]]: 2.3.5


Map: [&lt;1 2 8|, &lt;0 -1 -14|]
[[Comma list]]: 131072000/129140163


EDOs: {{EDOs| 5, 32, 37, 42 }}
{{Mapping|legend=1| 1 1 -1 | 0 3 17 }}


[http://x31eq.com/cgi-bin/rt.cgi?ets=5_32p&limit=5 The temperament finder - 5-limit 5 & 32p]
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5618{{c}}, ~729/640 = 234.4424{{c}}
: [[error map]]: {{val| -0.438 +0.934 -0.355 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~729/640 = 234.4999{{c}}
: error map: {{val| 0.000 +1.545 +0.185 }}


== 5 & 56 ==
{{Optimal ET sequence|legend=1| 5, …, 41, 46, 87, 220, 307 }}


Comma: {{Monzo|28 -22 3}} (33554432000/31381059609)
[[Badness]] (Sintel): 3.95


POTE generator: 235.8673 cents
== Laconic ==
: ''For extensions, see [[Gamelismic clan #Gorgo]].''


Map: [&lt;1 1 -2|, &lt;0 3 22|]
Laconic tempers out [[2187/2000]], which is the difference between a stack of three [[10/9|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 {{nowrap| ''n'' {{=}} -3 }}.


EDOs: {{EDOs| 5, 56, 61 }}
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 2187/2000
 
{{Mapping|legend=1| 1 1 1 | 0 3 7 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1203.1925{{c}}, ~10/9 = 228.0305{{c}}
: [[error map]]: {{val| +3.193 -14.671 +13.092 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~10/9 = 228.0128{{c}}
: error map: {{val| 0.000 -17.917 +9.776 }}
 
{{Optimal ET sequence|legend=1| 5, 11c, 16, 21, 37b }}
 
[[Badness]] (Sintel): 3.80
 
== University ==
: ''For extensions, see [[Gamelismic clan #Gidorah]] and [[Mint temperaments #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]].
 
University widens the classical major and minor chords to [[Extraclassical tonality|tendo and arto chords.]]
 
[[Subgroup]]: 2.3.5
 
[[Comma list]]: 144/125
 
{{Mapping|legend=1| 1 1 2 | 0 3 2 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1186.1969{{c}}, ~6/5 = 232.7334{{c}}
: [[error map]]: {{val| -13.803 -17.558 +51.547 }}
* [[CWE]]: ~2 = 1200.000{{c}}, ~6/5 = 231.4822{{c}}
: error map: {{val| 0.000 -7.509 +76.651 }}
 
{{Optimal ET sequence|legend=1| 1b, …, 4bc, 5 }}
 
[[Badness]] (Sintel): 2.39
 
=== Music ===
The purely 5-limit university mapping, using the 21cc [[val]], was in mind when writing this song.
 
; [[John Moriarty]]
* [https://soundcloud.com/john-lank1/uni ''Uni''] (2013) – University[6] in approximately 21edo
<!--
== Trisatriyo (5 & 56) ==
[[Subgroup]]: 2.3.5
 
[[Comma list]]: {{monzo| 28 -22 3 }} (33554432000/31381059609)
 
{{Mapping|legend=1| 1 1 -2 | 0 3 22 }}
 
: mapping generators: ~2, ~2560/2187
 
[[Optimal tuning]]s:
* [[POTE]]: ~2 = 1200.000{{c}}, ~2560/2187 = 235.867{{c}}
 
{{Optimal ET sequence|legend=1| 5, …, 51, 56, 117b, 173b }}
 
[[Badness]] (Smith): 1.323443


[http://x31eq.com/cgi-bin/rt.cgi?ets=5_56&limit=5 The temperament finder - 5-limit 5 & 56]
[http://x31eq.com/cgi-bin/rt.cgi?ets=5_56&limit=5 The temperament finder - 5-limit 5 & 56]
-->
== Hemiseven (5-limit) ==
: ''For extensions, see [[Gamelismic clan #Hemiseven]].''
[[Subgroup]]: 2.3.5
[[Comma list]]: {{monzo| 32 -29 6 }}
{{Mapping|legend=1| 1 -2 -15 | 0 6 29 }}
: mapping generators: ~2, ~243/160
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.3725{{c}}, ~243/160 = 716.9750{{c}}
: [[error map]]: {{val| +0.373 -0.850 +0.376 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~243/160 = 716.7671{{c}}
: error map: {{val| 0.000 -1.352 -0.067 }}
{{Optimal ET sequence|legend=1| 5, …, 72, 149, 221, 370, 591b }}
[[Badness]] (Sintel): 16.9


== Counterpental ==
== Counterpental ==
: ''For extensions, see [[Orwellismic temperaments #Pentaorwell]].''
[[Subgroup]]: 2.3.5
[[Comma list]]: {{monzo| 36 -30 5 }}
{{Mapping|legend=1| 5 0 -36 | 0 1 6 }}
: mapping generators: ~729/640, ~3
[[Optimal tuning]]s:
* [[WE]]: ~729/640 = 239.8575{{c}}, ~3/2 = 704.1540{{c}}
: [[error map]]: {{val| -0.712 +1.487 -0.535 }}
* [[CWE]]: ~729/640 = 240.0000{{c}}, ~3/2 = 704.4446{{c}}
: error map: {{val| 0.000 +2.490 +0.354 }}
{{Optimal ET sequence|legend=1| 5, …, 75, 80, 155, 390b, 545bbc }}
[[Badness]] (Sintel): 35.2
== Septiquarter (5-limit) ==
: ''For extensions, see [[Hemifamity temperaments #Septiquarter]].''
[[Subgroup]]: 2.3.5
[[Comma list]]: {{monzo| 44 -38 7 }}
{{Mapping|legend=1| 1 -4 -28 | 0 7 38 }}
: mapping generators: ~2, ~177147/102400
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7741{{c}}, ~177147/102400 = 957.3630{{c}}
: [[error map]]: {{val| -0.226 +0.490 -0.194 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~177147/102400 = 957.5367{{c}}
: error map: {{val| 0.000 +0.802 +0.082 }}
{{Optimal ET sequence|legend=1| 5, …, 94, 99, 193, 292, 391, 1074b, 1465bb }}
[[Badness]] (Sintel): 22.8
== Tokko (5-limit) ==
: ''For extensions, see [[Wizmic microtemperaments #Tokko]].''


Comma: {{Monzo|36 -30 5}}
[[Subgroup]]: 2.3.5


POTE generator: 15.4278 cents
[[Comma list]]: {{monzo| -76 67 -13 }}


Map: [&lt;5 8 12|, &lt;0 -1 -6|]
{{Mapping|legend=1| 1 -1 -11 | 0 13 67 }}
: mapping generators: ~2, ~{{monzo| -35 31 -6 }}


EDOs: {{EDOs| 5, 75, 80 }}
[[Optimal tuning]]s:  
* [[WE]]: ~2 = 1200.0377{{c}}, ~{{monzo| -35 31 -6 }} = 238.6084{{c}}
: [[error map]]: {{val| +0.038 -0.083 +0.035 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~{{monzo| -35 31 -6 }} = 238.6015{{c}}
: error map: {{val| 0.000 -0.135 -0.011 }}


[http://x31eq.com/cgi-bin/rt.cgi?ets=5_75&limit=5 The temperament finder - 5-limit 5 & 75]
{{Optimal ET sequence|legend=1| 5, …, 166, 171, 860, 1031, 1202, 1373, 1544, 3259, 4803b, 6347b }}


== 99 & 94 ==
[[Badness]] (Sintel): 20.9


Comma: {{Monzo|44 -38 7}}
<!--
== Quinla-tritrigu (5 & 118) ==
[[Subgroup]]: 2.3.5


POTE generator: 242.4567 cents
[[Comma list]]: {{monzo| -52 46 -9 }}


Map: [&lt;1 3 10|, &lt;0 -7 -38|]
{{Mapping|legend=1| 1 -2 -16 | 0 9 46 }}
: mapping generators: ~2, ~320/243


EDOs: {{EDOs| 5, 94, 99, 193, 198, 292, 297 }}
[[Optimal tuning]]s:  
* [[POTE]]: ~2 = 1200.000{{c}}, ~320/243 = 477.961{{c}}


[http://x31eq.com/cgi-bin/rt.cgi?ets=99_94&limit=5 The temperament finder - 5-limit 99 & 94]
{{Optimal ET sequence|legend=1| 5, 108c, 113, 118, 1057, 1175, 1293, 1411, 1529, 1647, 1765, 1883, 2001b, 3884b }}


== 2513 & 559 ==
[[Badness]] (Sintel): 14.5


Comma: {{Monzo|-124 109 -21}}
== Tribilalegu (5 & 137) ==
[[Subgroup]]: 2.3.5


POTE generator: 480.8595 cents
[[Comma list]]: {{Monzo| -60 54 -11 }}


Map: [&lt;1 10 46|, &lt;0 -21 -109|]
{{Mapping|legend=1| 1 -5 -30 | 0 11 54 }}
: mapping generators: ~2, ~243/160


EDOs: {{EDOs| 559, 1118, 1395, 1954, 2513, 3072, 3631, 4467, 5026, 5585 }}
[[Optimal tuning]]s:  
* [[POTE]]: ~2 = 1200.000{{c}}, ~243/160 = 718.258{{c}}


[http://x31eq.com/cgi-bin/rt.cgi?ets=2513_559&limit=5 The temperament finder - 5-limit 2153 & 559]
{{Optimal ET sequence|legend=1| 5, 127c, 132, 137, 553, 690b, 827b, 964b }}


== 5 & 137 ==
[[Badness]] (Sintel): 84.9


Comma: {{Monzo|60 -54 11}}
== 559 & 2513 ==
[[Subgroup]]: 2.3.5


POTE generator: 481.7421 cents
[[Comma list]]: {{monzo| -124 109 -21 }}


Map: [&lt;1 6 24|, &lt;0 -11 -54|]
{{Mapping|legend=1| 1 -11 -63 | 0 21 109 }}
: mapping generators: ~2, ~{{monzo| -29 26 -5 }}


EDOs: {{EDOs| 5, 132, 137, 142, 274, 279 }}
[[Optimal tuning]]s:  
* [[POTE]]: ~2 = 1200.0000{{c}}, ~{{monzo| -29 26 -5 }} = 719.1405{{c}}


[http://x31eq.com/cgi-bin/rt.cgi?ets=5_137&limit=5 The temperament finder - 5-limit 5 & 137]
{{Optimal ET sequence|legend=1| 5, …, 277, 559, 1395, 1954, 2513, 40767, 43280, 45793, 48306, 50819, 53332, 55845, 58358, 60871, 63384, 65897, 68410, 70923, 73436, 75949, 78462, 154411b }}


[[Badness]] (Sintel): 3.16
-->
[[Category:5edo]]
[[Category:5edo]]
[[Category:Theory]]
[[Category:Temperament]]
[[Category:Equivalence continua]]
[[Category:Equivalence continua]]

Latest revision as of 18:04, 11 May 2026

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.
Temperaments with integer n
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.

Temperaments with integer m
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⟩
Temperaments with fractional n and m
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⟩
13/3 = 4.3 13/10 = 1.3 Tokko [-76 67 -13⟩
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

Optimal tunings:

  • WE: ~2 = 1197.6520 ¢, ~3/2 = 708.6882 ¢
error map: ⟨-2.348 +4.385 -1.076]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 709.8213 ¢
error map: ⟨0.000 +7.866 +2.078]

Optimal ET sequence: 5, 17, 22, 49, 120b, 169bbc

Badness (Sintel): 3.17

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

Optimal tunings:

  • WE: ~2 = 1189.7544 ¢, ~3/2 = 724.6670 ¢
error map: ⟨-10.246 +12.466 +4.210]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 731.7318 ¢
error map: ⟨0.000 +29.777 +23.296]

Optimal ET sequence: 5, 13, 18, 23bc

Badness (Sintel): 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

Optimal tunings:

  • WE: ~2 = 1196.4357 ¢, ~3/2 = 711.7085 ¢
error map: ⟨-3.564 +6.189 -1.009]
  • CWE: ~2 = 1200.0000 ¢, ~3/2 = 713.5968 ¢
error map: ⟨0.000 +11.642 +4.041]

Optimal ET sequence: 5, 27c, 32, 37, 79bc, 116bbc

Badness (Sintel): 18.7

Dirt

Main article: 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

Optimal tunings:

  • WE: ~2 = 1195.8566 ¢, ~3/2 = 713.0611 ¢
error map: ⟨-4.143 +6.963 -0.863]
  • CWE: ~2 = 1200.000 ¢, ~3/2 = 715.3406 ¢
error map: ⟨0.000 +13.386 +5.157]

Optimal ET sequence: 5, 42c, 47b, 52b, 109bbc

Badness (Sintel): 55.3

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, whereby the generator represents 8/7. It corresponds to n = 3.

Subgroup: 2.3.5

Comma list: 131072000/129140163

Mapping: [⟨1 1 -1], ⟨0 3 17]]

Optimal tunings:

  • WE: ~2 = 1199.5618 ¢, ~729/640 = 234.4424 ¢
error map: ⟨-0.438 +0.934 -0.355]
  • CWE: ~2 = 1200.000 ¢, ~729/640 = 234.4999 ¢
error map: ⟨0.000 +1.545 +0.185]

Optimal ET sequence: 5, …, 41, 46, 87, 220, 307

Badness (Sintel): 3.95

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]]

Optimal tunings:

  • WE: ~2 = 1203.1925 ¢, ~10/9 = 228.0305 ¢
error map: ⟨+3.193 -14.671 +13.092]
  • CWE: ~2 = 1200.000 ¢, ~10/9 = 228.0128 ¢
error map: ⟨0.000 -17.917 +9.776]

Optimal ET sequence: 5, 11c, 16, 21, 37b

Badness (Sintel): 3.80

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.

University widens the classical major and minor chords to tendo and arto chords.

Subgroup: 2.3.5

Comma list: 144/125

Mapping: [⟨1 1 2], ⟨0 3 2]]

Optimal tunings:

  • WE: ~2 = 1186.1969 ¢, ~6/5 = 232.7334 ¢
error map: ⟨-13.803 -17.558 +51.547]
  • CWE: ~2 = 1200.000 ¢, ~6/5 = 231.4822 ¢
error map: ⟨0.000 -7.509 +76.651]

Optimal ET sequence: 1b, …, 4bc, 5

Badness (Sintel): 2.39

Music

The purely 5-limit university mapping, using the 21cc val, was in mind when writing this song.

John Moriarty
  • Uni (2013) – University[6] in approximately 21edo

Hemiseven (5-limit)

For extensions, see Gamelismic clan #Hemiseven.

Subgroup: 2.3.5

Comma list: [32 -29 6⟩

Mapping: [⟨1 -2 -15], ⟨0 6 29]]

mapping generators: ~2, ~243/160

Optimal tunings:

  • WE: ~2 = 1200.3725 ¢, ~243/160 = 716.9750 ¢
error map: ⟨+0.373 -0.850 +0.376]
  • CWE: ~2 = 1200.0000 ¢, ~243/160 = 716.7671 ¢
error map: ⟨0.000 -1.352 -0.067]

Optimal ET sequence: 5, …, 72, 149, 221, 370, 591b

Badness (Sintel): 16.9

Counterpental

For extensions, see Orwellismic temperaments #Pentaorwell.

Subgroup: 2.3.5

Comma list: [36 -30 5⟩

Mapping: [⟨5 0 -36], ⟨0 1 6]]

mapping generators: ~729/640, ~3

Optimal tunings:

  • WE: ~729/640 = 239.8575 ¢, ~3/2 = 704.1540 ¢
error map: ⟨-0.712 +1.487 -0.535]
  • CWE: ~729/640 = 240.0000 ¢, ~3/2 = 704.4446 ¢
error map: ⟨0.000 +2.490 +0.354]

Optimal ET sequence: 5, …, 75, 80, 155, 390b, 545bbc

Badness (Sintel): 35.2

Septiquarter (5-limit)

For extensions, see Hemifamity temperaments #Septiquarter.

Subgroup: 2.3.5

Comma list: [44 -38 7⟩

Mapping: [⟨1 -4 -28], ⟨0 7 38]]

mapping generators: ~2, ~177147/102400

Optimal tunings:

  • WE: ~2 = 1199.7741 ¢, ~177147/102400 = 957.3630 ¢
error map: ⟨-0.226 +0.490 -0.194]
  • CWE: ~2 = 1200.0000 ¢, ~177147/102400 = 957.5367 ¢
error map: ⟨0.000 +0.802 +0.082]

Optimal ET sequence: 5, …, 94, 99, 193, 292, 391, 1074b, 1465bb

Badness (Sintel): 22.8

Tokko (5-limit)

For extensions, see Wizmic microtemperaments #Tokko.

Subgroup: 2.3.5

Comma list: [-76 67 -13⟩

Mapping: [⟨1 -1 -11], ⟨0 13 67]]

mapping generators: ~2, ~[-35 31 -6⟩

Optimal tunings:

  • WE: ~2 = 1200.0377 ¢, ~[-35 31 -6⟩ = 238.6084 ¢
error map: ⟨+0.038 -0.083 +0.035]
  • CWE: ~2 = 1200.0000 ¢, ~[-35 31 -6⟩ = 238.6015 ¢
error map: ⟨0.000 -0.135 -0.011]

Optimal ET sequence: 5, …, 166, 171, 860, 1031, 1202, 1373, 1544, 3259, 4803b, 6347b

Badness (Sintel): 20.9

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