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Diaschismic family: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
The [[5-limit]] parent [[comma]] for the '''diaschismic family''' of [[regular temperament|temperaments]] is 2048/2025, the [[diaschisma]]. The [[period]] is half an [[2/1|octave]], and the [[generator]] is a fifth. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. [[34edo]] is a good tuning choice, with [[46edo]], [[56edo]], [[58edo]], or [[80edo]] being other possibilities. Both [[12edo]] and [[22edo]] support it, and retuning them to a [[mos]] of diaschismic gives two scale possibilities.
The '''diaschismic family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the diaschisma, [[2048/2025]].  


== Diaschismic ==
== Diaschismic ==
{{Main| Diaschismic }}
{{Main| Diaschismic }}
The [[period]] of diaschismic is half an [[2/1|octave]], and the [[generator]] is a fifth; the [[ploidacot]] is diploid monocot. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. [[34edo]] is a good tuning choice, with [[46edo]], [[56edo]], [[58edo]], or [[80edo]] being other possibilities. Both [[12edo]] and [[22edo]] support it, and retuning them to a [[mos]] of diaschismic gives two scale possibilities.


This temperament is also known as '''srutal''' in the 5-limit, but that name more strictly speaking refers to the [[#Srutal|34d & 46 extension]] to the [[7-limit]] that adds [[4375/4374]] to the comma list.
This temperament is also known as '''srutal''' in the 5-limit, but that name more strictly speaking refers to the [[#Srutal|34d & 46 extension]] to the [[7-limit]] that adds [[4375/4374]] to the comma list.
Line 15: Line 17:
: mapping generators: ~45/32, ~3
: mapping generators: ~45/32, ~3


[[Optimal tuning]] ([[POTE]]): ~45/32 = 600.000, ~3/2 = 704.898
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.4107{{c}}, ~3/2 = 704.2059{{c}}
: [[error map]]: {{val| -1.179 +1.072 +1.150 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 704.9585{{c}}
: error map: {{val| 0.000 +3.003 +3.769 }}


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 23: Line 29:
{{Optimal ET sequence|legend=1| 10, 12, 22, 34, 46, 80, 206c, 286bc }}
{{Optimal ET sequence|legend=1| 10, 12, 22, 34, 46, 80, 206c, 286bc }}


[[Badness]] (Smith): 0.019915
[[Badness]] (Sintel): 0.467


=== Overview to extensions ===
=== Overview to extensions ===
==== 7-limit extensions ====
==== 7-limit extensions ====
To get the 7-limit extensions, we add another comma:
To get the 7-limit extensions, we add another comma:
* Septimal diaschismic adds [[126/125]], the starling comma, to obtain 7-limit harmony by more complex methods than pajara, but with greater accuracy.  
* Septimal diaschismic adds [[126/125]], the starling comma, to obtain 7-limit harmony by more complex methods than pajara, but with greater accuracy.  
* Pajara derives from [[64/63]] and is a popular and well-known choice.  
* Pajara derives from [[64/63]] and is a popular and well-known choice.  
* Srutal adds [[4375/4374]], the ragisma, which is about as accurate as septimal diaschismic but has a much more complex mapping of 7.  
* Srutal adds [[4375/4374]], the ragisma, which is about as accurate as septimal diaschismic but has a much more complex mapping of 7.  
* Keen adds [[875/864]].  
* Keen adds [[875/864]].  
* Bidia adds [[3136/3125]], the hemimean comma.
* Echidna adds [[1728/1715]], the orwellisma.
* Shrutar adds [[245/243]], the sensamagic comma.


Pajara, diaschismic, srutal and keen keep the same half-octave period and fifth generator, but shrutar has a generator of a quarter-tone (which can be taken as [[36/35]], the septimal quarter-tone) and echidna has a generator of 9/7. Bidia has a quarter-octave period and a fifth generator.
Those all keep the same half-octave period and fifth generator.
 
Bidia adds [[3136/3125]], the hemimean comma, with a 1/4-octave period. Shrutar adds [[245/243]] and shru adds [[392/375]], with a quartertone generator. Sruti adds [[19683/19600]] and anguirus adds [[49/48]], with a neutral third or hemitwelfth generator. Those split the original generator in two. Echidna adds [[1728/1715]], the orwellisma, with a ~9/7 generator. Echidnic adds [[686/675]], the senga, with a ~8/7 generator. Those split the original generator in three. Finally, quadrasruta adds [[2401/2400]] and splits the original  generator in four.  


==== Subgroup extensions ====
==== Subgroup extensions ====
Since the diaschisma factors into ([[256/255]])<sup>2</sup>([[289/288]]) in the 17-limit, it extends naturally to the 2.3.5.17 subgroup as ''srutal archagall'', documented right below. The [[S-expression]]-based comma list of this temperament is {[[256/255|S16]], [[289/288|S17]]}.  
Since the diaschisma factors into ([[256/255]])<sup>2</sup>([[289/288]]) in the 17-limit, it extends naturally to the 2.3.5.17 subgroup as ''srutal archagall'', documented right below. The [[S-expression]]-based comma list of this temperament is {[[256/255|S16]], [[289/288|S17]]}.


=== Srutal archagall ===
=== Srutal archagall ===
Line 49: Line 53:
Comma list: 136/135, 256/255
Comma list: 136/135, 256/255


Sval mapping: {{mapping| 2 0 11 5 | 0 1 -2 1 }}
Subgroup-val mapping: {{mapping| 2 0 11 5 | 0 1 -2 1 }}


: mapping generators: ~17/12, ~3
: mapping generators: ~17/12, ~3


Optimal tuning (CTE): ~17/12 = 600.000, ~3/2 = 705.1272
Optimal tunings:
* WE: ~45/32 = 599.5585{{c}}, ~3/2 = 704.6188{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 705.1356{{c}}


{{Optimal ET sequence|legend=0| 10, 12, 22, 34, 80, 114, 194bc }}
{{Optimal ET sequence|legend=0| 10, 12, 22, 34, 80, 114, 194bc }}


Badness (Smith): 0.00575
Badness (Sintel): 0.212


== Septimal diaschismic ==
== Septimal diaschismic ==
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{{See also| Srutal vs diaschismic }}
{{See also| Srutal vs diaschismic }}


A simpler characterization than the one given by the normal comma list is that diaschismic adds [[126/125]] or [[5120/5103]] to the set of commas, and it can also be called {{nowrap| 46 & 58 }}. However described, diaschismic has a 1/2-octave period and a sharp fifth generator like pajara, but not so sharp, giving a more accurate but more complex temperament. [[58edo]] provides an excellent tuning, but an alternative is to make [[7/4]] just by making the fifth 703.897 cents, as opposed to 703.448 cents for 58edo.
A simpler characterization than the one given by the normal comma list is that septimal diaschismic adds [[126/125]] or [[5120/5103]] to the set of commas, and it can also be called {{nowrap| 46 & 58 }}. However described, septimal diaschismic has a 1/2-octave period and a sharp fifth generator like the 5-limit version, but not so sharp, giving a more accurate but more complex temperament. [[104edo]] provides an excellent tuning, which is close to tuning [[7/4]] just by making the fifth 703.897 cents.  


Diaschismic extends naturally to the 17-limit, for which the same tunings may be used, making it one of the most important of the higher-limit rank-2 temperaments. Adding the 11-limit adds the commas 176/175, 896/891 and 441/440. The 13-limit yields 196/195, 351/350, and 364/363; the 17-limit adds 136/135, 221/220, and 442/441. If you want to explore higher-limit harmonies, diaschismic is certainly one excellent way to do it; [[mos]] of 34 notes and even more the 46-note mos will encompass very great deal of it. Of course 46 or 58 equal provide alternatives which in many ways are similar, particularly in the case of 58.
Diaschismic extends naturally to the 17-limit, for which the same tunings may be used, making it one of the most important of the higher-limit rank-2 temperaments. Adding the 11-limit adds the commas 176/175, 896/891 and 441/440. The 13-limit yields 196/195, 351/350, and 364/363; the 17-limit adds 136/135, 221/220, and 442/441. If you want to explore higher-limit harmonies, diaschismic is certainly one excellent way to do it; [[mos]] of 34 notes and even more the 46-note mos will encompass very great deal of it. Of course 46 or 58 equal provide alternatives which in many ways are similar, particularly in the case of 58.
Line 73: Line 79:
{{Mapping|legend=1| 2 0 11 31 | 0 1 -2 -8 }}
{{Mapping|legend=1| 2 0 11 31 | 0 1 -2 -8 }}


[[Optimal tuning]] ([[POTE]]): ~45/32 = 600.000, ~3/2 = 703.681
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.4449{{c}}, ~3/2 = 703.0299{{c}}
: [[error map]]: {{val| -1.110 -0.035 +3.740 -1.391 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 703.7739{{c}}
: error map: {{val| 0.000 +1.819 +6.138 +0.983 }}


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 79: Line 89:
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.955, 706.843]
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.955, 706.843]


{{Optimal ET sequence|legend=1| 12, 46, 58, 104c, 162c }}
{{Optimal ET sequence|legend=1| 12, 34, 46, 58, 104c, 162c }}


[[Badness]] (Smith): 0.037914
[[Badness]] (Sintel): 0.959


=== 11-limit ===
=== 11-limit ===
Line 90: Line 100:
Mapping: {{mapping| 2 0 11 31 45 | 0 1 -2 -8 -12 }}
Mapping: {{mapping| 2 0 11 31 45 | 0 1 -2 -8 -12 }}


Optimal tuning (POTE): ~45/32 = 600.000, ~3/2 = 703.714
Optimal tunings:
* WE: ~45/32 = 599.4471{{c}}, ~3/2 = 703.0657{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 703.7996{{c}}


Tuning ranges:  
Tuning ranges:  
Line 96: Line 108:
* 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]
* 11-odd-limit diamond tradeoff: ~3/2 = [701.955, 706.843]


{{Optimal ET sequence|legend=0| 12, 46, 58, 104c, 162ce }}
{{Optimal ET sequence|legend=0| 12, 34e, 46, 58, 104c, 162ce }}


Badness (Smith): 0.025034
Badness (Sintel): 0.828


=== 13-limit ===
=== 13-limit ===
Line 107: Line 119:
Mapping: {{mapping| 2 0 11 31 45 55 | 0 1 -2 -8 -12 -15 }}
Mapping: {{mapping| 2 0 11 31 45 55 | 0 1 -2 -8 -12 -15 }}


Optimal tuning (POTE): ~45/32 = 600.000, ~3/2 = 703.704
Optimal tunings:
* WE: ~45/32 = 599.4451{{c}}, ~3/2 = 703.0528{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 703.7813{{c}}


Tuning ranges:  
Tuning ranges:  
Line 114: Line 128:
* 15-odd-limit diamond tradeoff: ~3/2 = [701.955, 711.731]
* 15-odd-limit diamond tradeoff: ~3/2 = [701.955, 711.731]


{{Optimal ET sequence|legend=0| 46, 58, 104c, 162cef }}
{{Optimal ET sequence|legend=0| 12f, 34ef, 46, 58, 104c, 162cef }}


Badness (Smith): 0.018926
Badness (Sintel): 0.782


=== 17-limit ===
=== 17-limit ===
Line 125: Line 139:
Mapping: {{mapping| 2 0 11 31 45 55 5 | 0 1 -2 -8 -12 -15 1 }}
Mapping: {{mapping| 2 0 11 31 45 55 5 | 0 1 -2 -8 -12 -15 1 }}


Optimal tuning (POTE): ~17/12 = 600.000, ~3/2 = 703.812
Optimal tunings:
* WE: ~17/12 = 599.6253{{c}}, ~3/2 = 703.3726{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 703.8520{{c}}


Tuning ranges:  
Tuning ranges:  
Line 131: Line 147:
* 17-odd-limit diamond tradeoff: ~3/2 = [698.955, 711.731]
* 17-odd-limit diamond tradeoff: ~3/2 = [698.955, 711.731]


{{Optimal ET sequence|legend=0| 46, 58, 104c }}
{{Optimal ET sequence|legend=0| 12f, 34ef, 46, 58, 104c }}


Badness (Smith): 0.016425
Badness (Sintel): 0.837


=== 2.3.5.7.11.13.17.23 subgroup (Na"Naa') ===
=== 2.3.5.7.11.13.17.23 subgroup (Na"Naa') ===
Line 142: Line 158:
Comma list: 126/125, 136/135, 176/175, 196/195, 231/230, 256/255
Comma list: 126/125, 136/135, 176/175, 196/195, 231/230, 256/255


Sval mapping: {{mapping| 2 0 11 31 45 55 5 63 | 0 1 -2 -8 -12 -15 1 -17 }}
Subgroup-val mapping: {{mapping| 2 0 11 31 45 55 5 63 | 0 1 -2 -8 -12 -15 1 -17 }}


Optimal tuning (POTE): ~17/12 = 600.000, ~3/2 = 703.870
Optimal tunings:
* WE: ~17/12 = 599.6272{{c}}, ~3/2 = 703.4326{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 703.9093{{c}}


{{Optimal ET sequence|legend=0| 46, 58i, 104ci }}
{{Optimal ET sequence|legend=0| 12i, 34efi, 46, 58i, 104ci }}
 
Badness (Sintel): 0.882


== Pajara ==
== Pajara ==
{{Main| Pajara }}
{{Main| Pajara }}


Pajara is closely associated with 22edo (not to mention [[Paul Erlich]]) but other tunings are possible. The 1/2-octave period serves as both a [[10/7]] and a [[7/5]]. Aside from 22edo, 34 with the val {{val| 34 54 79 96 }} and 56 with the val {{val| 56 89 130 158 }} are interesting alternatives, with more acceptable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12edo and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.
Pajara is closely associated with 22edo (not to mention [[Paul Erlich]]) but other tunings are possible. The 1/2-octave period serves as both a [[10/7]] and a [[7/5]]. Aside from 22edo, 34 with the val {{val| 34 54 79 96 }} (34d) and 56 with the val {{val| 56 89 130 158 }} (56d) are interesting alternatives, with more acceptable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12edo and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.


Pajara extends nicely to an 11-limit version, for which the 56edo tuning can be used, but a good alternative is to make the major thirds pure by setting the fifth to be 706.843 cents. Now 99/98, 100/99, 176/175 and 896/891 are being tempered out.
Pajara extends nicely to an 11-limit version, for which the 56edo tuning can be used, but a good alternative is to make the major thirds pure by setting the fifth to be 706.843 cents. Now 99/98, 100/99, 176/175 and 896/891 are being tempered out.
Line 161: Line 181:
{{Mapping|legend=1| 2 0 11 12 | 0 1 -2 -2 }}
{{Mapping|legend=1| 2 0 11 12 | 0 1 -2 -2 }}


[[Optimal tuning]] ([[POTE]]): ~7/5 = 600.000, ~3/2 = 707.048
[[Optimal tuning]]s:
* [[WE]]: ~7/5 = 598.8483{{c}}, ~3/2 = 705.6906{{c}}
: [[error map]]: {{val| -2.303 +1.432 -5.756 +10.580 }}
* [[CWE]]: ~7/5 = 600.0000{{c}}, ~3/2 = 707.3438{{c}}
: error map: {{val| 0.000 +5.389 -1.001 +16.487 }}


[[Tuning ranges]]:
[[Tuning ranges]]:
Line 169: Line 193:
{{Optimal ET sequence|legend=1| 10, 12, 22, 34d, 56d }}
{{Optimal ET sequence|legend=1| 10, 12, 22, 34d, 56d }}


[[Badness]] (Smith): 0.020033
[[Badness]] (Sintel): 0.507


=== 11-limit ===
=== 11-limit ===
Line 178: Line 202:
Mapping: {{mapping| 2 0 11 12 26 | 0 1 -2 -2 -6 }}
Mapping: {{mapping| 2 0 11 12 26 | 0 1 -2 -2 -6 }}


Optimal tuning (POTE): ~7/5 = 600.000, ~3/2 = 706.885
Optimal tunings:
* WE: ~7/5 = 598.8485{{c}}, ~3/2 = 705.5285{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 707.1826{{c}}


Tuning ranges:
Tuning ranges:
Line 186: Line 212:
{{Optimal ET sequence|legend=0| 10e, 12, 22, 34d, 56d }}
{{Optimal ET sequence|legend=0| 10e, 12, 22, 34d, 56d }}


Badness (Smith): 0.020343
Badness (Sintel): 0.673


==== 13-limit ====
==== 13-limit ====
Line 195: Line 221:
Mapping: {{mapping| 2 0 11 12 26 1 | 0 1 -2 -2 -6 2 }}
Mapping: {{mapping| 2 0 11 12 26 1 | 0 1 -2 -2 -6 2 }}


Optimal tuning (POTE): ~7/5 = 600.000, ~3/2 = 708.919
Optimal tunings:
* WE: ~7/5 = 599.9732{{c}}, ~3/2 = 708.8873{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.9227{{c}}


{{Optimal ET sequence|legend=0| 10e, 12, 22 }}
{{Optimal ET sequence|legend=0| 10e, 12, 22 }}


Badness (Smith): 0.027642
Badness (Sintel): 1.14


===== 17-limit =====
===== 17-limit =====
Line 208: Line 236:
Mapping: {{mapping| 2 0 11 12 26 1 5 | 0 1 -2 -2 -6 2 1 }}
Mapping: {{mapping| 2 0 11 12 26 1 5 | 0 1 -2 -2 -6 2 1 }}


Optimal tuning (POTE): ~7/5 = 600.000, ~3/2 = 708.806
Optimal tunings:
* WE: ~7/5 = 599.8871{{c}}, ~3/2 = 708.6725{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.8176{{c}}


{{Optimal ET sequence|legend=0| 10e, 12, 22 }}
{{Optimal ET sequence|legend=0| 10e, 12, 22 }}


Badness (Smith): 0.020899
Badness (Sintel): 1.06


==== Pajarina ====
==== Pajarina ====
Line 221: Line 251:
Mapping: {{mapping| 2 0 11 12 26 36 | 0 1 -2 -2 -6 -9 }}
Mapping: {{mapping| 2 0 11 12 26 36 | 0 1 -2 -2 -6 -9 }}


Optimal tuning (POTE): ~7/5 = 600.000, ~3/2 = 706.133
Optimal tunings:
* WE: ~7/5 = 598.7732{{c}}, ~3/2 = 704.6889{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 706.3950{{c}}


{{Optimal ET sequence|legend=0| 12f, 22, 34d }}
{{Optimal ET sequence|legend=0| 12f, 22, 34d }}


Badness (Smith): 0.022327
Badness (Sintel): 0.923


===== 17-limit =====
===== 17-limit =====
Line 234: Line 266:
Mapping: {{mapping| 2 0 11 12 26 36 5 | 0 1 -2 -2 -6 -9 1 }}
Mapping: {{mapping| 2 0 11 12 26 36 5 | 0 1 -2 -2 -6 -9 1 }}


Optimal tuning (POTE): ~7/5 = 600.000, ~3/2 = 706.410
Optimal tunings:
* WE: ~7/5 = 599.0204{{c}}, ~3/2 = 705.2572{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 706.5660{{c}}


{{Optimal ET sequence|legend=0| 12f, 22, 34d }}
{{Optimal ET sequence|legend=0| 12f, 22, 34d }}


Badness (Smith): 0.018375
Badness (Sintel): 0.936


==== Pajarita ====
==== Pajarita ====
Line 247: Line 281:
Mapping: {{mapping| 2 0 11 12 26 17 | 0 1 -2 -2 -6 -3 }}
Mapping: {{mapping| 2 0 11 12 26 17 | 0 1 -2 -2 -6 -3 }}


Optimal tuning (POTE): ~7/5 = 600.000, ~3/2 = 707.450
Optimal tunings:
* WE: ~7/5 = 598.3048{{c}}, ~3/2 = 705.4512{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 707.9238{{c}}


{{Optimal ET sequence|legend=0| 10e, 12f, 22f }}
{{Optimal ET sequence|legend=0| 10e, 12f, 22f, 34dff }}


Badness (Smith): 0.022677
Badness (Sintel): 0.937


===== 17-limit =====
===== 17-limit =====
Line 260: Line 296:
Mapping: {{mapping| 2 0 11 12 26 17 5 | 0 1 -2 -2 -6 -3 1 }}
Mapping: {{mapping| 2 0 11 12 26 17 5 | 0 1 -2 -2 -6 -3 1 }}


Optimal tuning (POTE): ~7/5 = 600.000, ~3/2 = 707.947
Optimal tunings:
* WE: ~7/5 = 598.6103{{c}}, ~3/2 = 706.3076{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.2256{{c}}


{{Optimal ET sequence|legend=0| 10e, 12f, 22f }}
{{Optimal ET sequence|legend=0| 10e, 12f, 22f }}


Badness (Smith): 0.019007
Badness (Sintel): 0.968


=== Pajarous ===
=== Pajarous ===
Line 273: Line 311:
Mapping: {{mapping| 2 0 11 12 -9 | 0 1 -2 -2 5 }}
Mapping: {{mapping| 2 0 11 12 -9 | 0 1 -2 -2 5 }}


Optimal tuning (POTE): ~7/5 = 600.000, ~3/2 = 709.578
Optimal tunings:
* WE: ~7/5 = 599.4055{{c}}, ~3/2 = 708.8747{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 709.5508{{c}}


Tuning ranges:
Tuning ranges:
Line 281: Line 321:
{{Optimal ET sequence|legend=0| 10, 12e, 22, 120bce, 142bce }}
{{Optimal ET sequence|legend=0| 10, 12e, 22, 120bce, 142bce }}


Badness (Smith): 0.028349
Badness (Sintel): 0.937


==== 13-limit ====
==== 13-limit ====
Line 290: Line 330:
Mapping: {{mapping| 2 0 11 12 -9 1 | 0 1 -2 -2 5 2 }}
Mapping: {{mapping| 2 0 11 12 -9 1 | 0 1 -2 -2 5 2 }}


Optimal tuning (POTE): ~7/5 = 600.000, ~3/2 = 710.240
Optimal tunings:
* WE: ~7/5 = 599.9064{{c}}, ~3/2 = 710.1289{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.2325{{c}}


{{Optimal ET sequence|legend=0| 10, 22, 54f, 76bdff }}
{{Optimal ET sequence|legend=0| 10, 22 }}


Badness (Smith): 0.025176
Badness (Sintel): 1.04


===== 17-limit =====
===== 17-limit =====
Line 303: Line 345:
Mapping: {{mapping| 2 0 11 12 -9 1 5 | 0 1 -2 -2 5 2 1 }}
Mapping: {{mapping| 2 0 11 12 -9 1 5 | 0 1 -2 -2 5 2 1 }}


Optimal tuning (POTE): ~7/5 = 600.000, ~3/2 = 710.221
Optimal tunings:
* WE: ~7/5 = 599.8239{{c}}, ~3/2 = 710.0128{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.2067{{c}}


{{Optimal ET sequence|legend=0| 10, 22, 54f, 76bdff }}
{{Optimal ET sequence|legend=0| 10, 22, 54f, 76bdff }}


Badness (Smith): 0.018249
Badness (Sintel): 0.930


==== Pajaro ====
==== Pajaro ====
Line 316: Line 360:
Mapping: {{mapping| 2 0 11 12 -9 17 | 0 1 -2 -2 5 -3 }}
Mapping: {{mapping| 2 0 11 12 -9 17 | 0 1 -2 -2 5 -3 }}


Optimal tuning (POTE): ~7/5 = 600.000, ~3/2 = 710.818
Optimal tunings:
* WE: ~7/5 = 598.8257{{c}}, ~3/2 = 709.4266{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.8414{{c}}


{{Optimal ET sequence|legend=0| 10, 22f, 32f, 54ff }}
{{Optimal ET sequence|legend=0| 10, 22f, 32f }}


Badness (Smith): 0.027355
Badness (Sintel): 1.13


===== 17-limit =====
===== 17-limit =====
Line 329: Line 375:
Mapping: {{mapping| 2 0 11 12 -9 17 5 | 0 1 -2 -2 5 -3 1 }}
Mapping: {{mapping| 2 0 11 12 -9 17 5 | 0 1 -2 -2 5 -3 1 }}


Optimal tuning (POTE): ~7/5 = 600.000, ~3/2 = 710.866
Optimal tunings:
* WE: ~7/5 = 598.8865{{c}}, ~3/2 = 709.5472{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 710.8704{{c}}


{{Optimal ET sequence|legend=0| 10, 22f, 32f, 54ff }}
{{Optimal ET sequence|legend=0| 10, 22f, 32f }}


Badness (Smith): 0.019844
Badness (Sintel): 1.01


=== Pajaric ===
=== Pajaric ===
Line 342: Line 390:
Mapping: {{mapping| 2 0 11 12 7 | 0 1 -2 -2 0 }}
Mapping: {{mapping| 2 0 11 12 7 | 0 1 -2 -2 0 }}


Optimal tuning (POTE): ~7/5 = 600.000, ~3/2 = 705.524
Optimal tunings:
* WE: ~7/5 = 597.4807{{c}}, ~3/2 = 702.5616{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 706.0542{{c}}


{{Optimal ET sequence|legend=0| 10, 12, 22e, 34dee }}
{{Optimal ET sequence|legend=0| 10, 12, 22e }}


Badness (Smith): 0.023798
Badness (Sintel): 0.787


==== 13-limit ====
==== 13-limit ====
Line 355: Line 405:
Mapping: {{mapping| 2 0 11 12 7 17 | 0 1 -2 -2 0 -3 }}
Mapping: {{mapping| 2 0 11 12 7 17 | 0 1 -2 -2 0 -3 }}


Optimal tuning (POTE): ~7/5 = 600.000, ~3/2 = 707.442
Optimal tunings:
* WE: ~7/5 = 597.1952{{c}}, ~3/2 = 704.1350{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.1989{{c}}


{{Optimal ET sequence|legend=0| 10, 12f, 22ef }}
{{Optimal ET sequence|legend=0| 10, 12f, 22ef }}


Badness (Smith): 0.020461
Badness (Sintel): 0.845


==== 17-limit ====
==== 17-limit ====
Line 368: Line 420:
Mapping: {{mapping| 2 0 11 12 7 17 5 | 0 1 -2 -2 0 -3 1 }}
Mapping: {{mapping| 2 0 11 12 7 17 5 | 0 1 -2 -2 0 -3 1 }}


Optimal tuning (POTE): ~7/5 = 600.000, ~3/2 = 708.544
Optimal tunings:
* WE: ~7/5 = 597.6509{{c}}, ~3/2 = 705.7702{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~3/2 = 708.9719{{c}}


{{Optimal ET sequence|legend=0| 10, 12f, 22ef }}
{{Optimal ET sequence|legend=0| 10, 12f, 22ef }}


Badness (Smith): 0.017592
Badness (Sintel): 0.896


=== Hemipaj ===
=== Hemipaj ===
Line 381: Line 435:
Mapping: {{mapping| 2 1 9 10 8 | 0 2 -4 -4 -1 }}
Mapping: {{mapping| 2 1 9 10 8 | 0 2 -4 -4 -1 }}


Optimal tuning (POTE): ~7/5 = 600.000, ~11/8 = 546.383
: mapping generators: ~2, ~16/11
 
Optimal tunings:
* WE: ~7/5 = 597.6509{{c}}, ~16/11 = 652.7788{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~16/11 = 653.7119{{c}}


{{Optimal ET sequence|legend=0| 20, 22, 68d, 90d }}
{{Optimal ET sequence|legend=0| 2, 20, 22 }}


Badness (Smith): 0.038890
Badness (Sintel): 1.29


=== Hemifourths ===
=== Hemifourths ===
Line 394: Line 452:
Mapping: {{mapping| 2 0 11 12 -1 | 0 2 -4 -4 5 }}
Mapping: {{mapping| 2 0 11 12 -1 | 0 2 -4 -4 5 }}


Optimal tuning (POTE): ~7/5 = 600.000, ~55/32 = 953.093
: mapping generators: ~2, ~55/32
 
Optimal tunings:
* WE: ~7/5 = 597.6509{{c}}, ~55/32 = 950.8475{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~55/32 = 953.1172{{c}}


{{Optimal ET sequence|legend=0| 10, 24d, 34d }}
{{Optimal ET sequence|legend=0| 10, 24d, 34d }}


Badness (Smith): 0.048885
Badness (Sintel): 1.62


==== 13-limit ====
==== 13-limit ====
Line 407: Line 469:
Mapping: {{mapping| 2 0 11 12 -1 9 | 0 2 -4 -4 5 -1 }}
Mapping: {{mapping| 2 0 11 12 -1 9 | 0 2 -4 -4 5 -1 }}


Optimal tuning (POTE): ~7/5 = 600.000, ~26/15 = 953.074
Optimal tunings:
* WE: ~7/5 = 598.6748{{c}}, ~26/15 = 950.9691{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~26/15 = 953.1052{{c}}


{{Optimal ET sequence|legend=0| 10, 24d, 34d }}
{{Optimal ET sequence|legend=0| 10, 24d, 34d }}


Badness (Smith): 0.028755
Badness (Sintel): 1.19


==== 17-limit ====
==== 17-limit ====
Line 420: Line 484:
Mapping: {{mapping| 2 0 11 12 -1 9 5 | 0 2 -4 -4 5 -1 2 }}
Mapping: {{mapping| 2 0 11 12 -1 9 5 | 0 2 -4 -4 5 -1 2 }}


Optimal tuning (POTE): ~7/5 = 600.000, ~26/15 = 953.210
Optimal tunings:
* WE: ~7/5 = 598.8411{{c}}, ~26/15 = 951.3687{{c}}
* CWE: ~7/5 = 600.0000{{c}}, ~26/15 = 953.2169{{c}}


{{Optimal ET sequence|legend=0| 10, 24d, 34d }}
{{Optimal ET sequence|legend=0| 10, 24d, 34d }}


Badness (Smith): 0.021790
Badness (Sintel): 1.11


== Srutal ==
== Srutal ==
{{See also| Srutal vs diaschismic }}
{{See also| Srutal vs diaschismic }}


Srutal can be described as the 34d & 46 temperament, where 7/4 is located at 15 generator steps, or the double-augmented fifth (C–Gx). 80edo and 126edo are among the possible tunings. Srutal, shrutar and bidia have similar 19-limit properties, tempering out 190/189, related to rank-3 [[julius]].
Srutal can be described as the {{nowrap| 34d & 46 }} temperament, where 7/4 is located at 15 generator steps, or the double-augmented fifth (C–Gx). As such, it weakly extends [[leapfrog]]. 80edo and [[126edo]] are among the possible tunings. Srutal, shrutar and bidia have similar 19-limit properties, tempering out 190/189, related to rank-3 [[julius]].


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 437: Line 503:
{{Mapping|legend=1| 2 0 11 -42 | 0 1 -2 15 }}
{{Mapping|legend=1| 2 0 11 -42 | 0 1 -2 15 }}


[[Optimal tuning]] ([[POTE]]): ~45/32 = 600.000, ~3/2 = 704.814
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.4046{{c}}, ~3/2 = 704.1150{{c}}
: [[error map]]: {{val| -1.191 +0.969 +1.289 +0.044 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 704.7646{{c}}
: error map: {{val| 0.000 +2.810 +4.157 +2.643 }}


[[Tuning ranges]]:  
[[Tuning ranges]]:  
Line 445: Line 515:
{{Optimal ET sequence|legend=1| 34d, 46, 80, 126, 206cd, 332bcd }}
{{Optimal ET sequence|legend=1| 34d, 46, 80, 126, 206cd, 332bcd }}


[[Badness]] (Smith): 0.091504
[[Badness]] (Sintel): 2.32


=== 11-limit ===
=== 11-limit ===
Line 454: Line 524:
Mapping: {{mapping| 2 0 11 -42 -28 | 0 1 -2 15 11 }}
Mapping: {{mapping| 2 0 11 -42 -28 | 0 1 -2 15 11 }}


Optimal tuning (POTE): ~45/32 = 600.000, ~3/2 = 704.856
Optimal tunings:
* WE: ~45/32 = 599.4413{{c}}, ~3/2 = 704.1999{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 704.8017{{c}}


Tuning ranges:  
Tuning ranges:  
Line 462: Line 534:
{{Optimal ET sequence|legend=0| 34d, 46, 80, 126, 206cd }}
{{Optimal ET sequence|legend=0| 34d, 46, 80, 126, 206cd }}


Badness (Smith): 0.035315
Badness (Sintel): 1.17


=== 13-limit ===
=== 13-limit ===
Line 471: Line 543:
Mapping: {{mapping| 2 0 11 -42 -28 -18 | 0 1 -2 15 11 8 }}
Mapping: {{mapping| 2 0 11 -42 -28 -18 | 0 1 -2 15 11 8 }}


Optimal tuning (POTE): ~45/32 = 600.000, ~3/2 = 704.881
Optimal tunings:
* WE: ~45/32 = 599.5490{{c}}, ~3/2 = 704.3516{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 704.8347{{c}}


Tuning ranges:  
Tuning ranges:  
Line 478: Line 552:
* 15-odd-limit diamond tradeoff: ~3/2 = [701.955, 711.731]
* 15-odd-limit diamond tradeoff: ~3/2 = [701.955, 711.731]


{{Optimal ET sequence|legend=0| 34d, 46, 80, 206cd, 286bcde }}
{{Optimal ET sequence|legend=0| 34d, 46, 80 }}


Badness (Smith): 0.025286
Badness (Sintel): 1.04


=== 17-limit ===
=== 17-limit ===
Line 489: Line 563:
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 | 0 1 -2 15 11 8 1 }}
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 | 0 1 -2 15 11 8 1 }}


Optimal tuning (POTE): ~17/12 = 1\2, ~3/2 = 704.840
Optimal tunings:
* WE: ~17/12 = 599.6459{{c}}, ~3/2 = 704.4237{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8083{{c}}


Tuning ranges:  
Tuning ranges:  
Line 495: Line 571:
* 17-odd-limit diamond tradeoff: ~3/2 = [698.955, 711.731]
* 17-odd-limit diamond tradeoff: ~3/2 = [698.955, 711.731]


{{Optimal ET sequence|legend=0| 34d, 46, 80, 126, 206cd }}
{{Optimal ET sequence|legend=0| 34d, 46, 80, 126 }}


Badness (Smith): 0.018594
Badness (Sintel): 0.947


=== 19-limit ===
=== 19-limit ===
Line 506: Line 582:
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 | 0 1 -2 15 11 8 1 20 }}
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 | 0 1 -2 15 11 8 1 20 }}


Optimal tuning (POTE): ~17/12 = 600.000, ~3/2 = 704.905
Optimal tunings:
* WE: ~17/12 = 599.6371{{c}}, ~3/2 = 704.4790{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8745{{c}}


{{Optimal ET sequence|legend=0| 34dh, 46, 80, 206cd }}
{{Optimal ET sequence|legend=0| 34dh, 46, 80 }}


Badness (Smith): 0.017063
Badness (Sintel): 1.04


==== Srutaloo ====
==== Srutaloo ====
Line 521: Line 599:
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 -10 | 0 1 -2 15 11 8 1 20 6 }}
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 -10 | 0 1 -2 15 11 8 1 20 6 }}


Optimal tuning (POTE): ~17/12 = 600.000, ~3/2 = 704.899
Optimal tunings:
* WE: ~17/12 = 599.6690{{c}}, ~3/2 = 704.5098{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8713{{c}}


{{Optimal ET sequence|legend=0| 34dh, 46, 80, 206cd }}
{{Optimal ET sequence|legend=0| 34dh, 46, 80 }}


Badness (Smith): 0.013555
Badness (Sintel): 0.971


===== 29-limit =====
===== 29-limit =====
Line 534: Line 614:
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 -10 -76 | 0 1 -2 15 11 8 1 20 6 27 }}
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 -10 -76 | 0 1 -2 15 11 8 1 20 6 27 }}


Optimal tuning (POTE): ~17/12 = 600.000, ~3/2 = 704.906
Optimal tunings:
* WE: ~17/12 = 599.6664{{c}}, ~3/2 = 704.5138{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8807{{c}}


{{Optimal ET sequence|legend=0| 34dhj, 46, 80, 206cd }}
{{Optimal ET sequence|legend=0| 34dhj, 46, 80 }}


Badness (Smith): 0.013203
Badness (Sintel): 1.10


===== 31-limit =====
===== 31-limit =====
Line 547: Line 629:
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 -10 -76 48 | 0 1 -2 15 11 8 1 20 6 27 -12 }}
Mapping: {{mapping| 2 0 11 -42 -28 -18 5 -55 -10 -76 48 | 0 1 -2 15 11 8 1 20 6 27 -12 }}


Optimal tuning (POTE): ~17/12 = 600.000, ~3/2 = 704.817
Optimal tunings:
* WE: ~17/12 = 599.8115{{c}}, ~3/2 = 704.5958{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 704.8086{{c}}


{{Optimal ET sequence|legend=0| 46, 80, 126 }}
{{Optimal ET sequence|legend=0| 46, 80, 126 }}


Badness (Smith): 0.015073
Badness (Sintel): 1.44


== Keen ==
== Keen ==
Keen adds 875/864 as well as 2240/2187 to the set of commas. It may also be described as the {{nowrap| 22 & 56 }} temperament. [[78edo]] is a good tuning choice, and remains a good one in the 11-limit, where the temperament is really more interesting, adding 100/99 and 385/384 to the list of commas.
Keen adds 875/864 as well as 2240/2187 to the set of commas. It may also be described as the {{nowrap| 22 & 34 }} temperament. [[78edo]] is a good tuning choice, and remains a good one in the 11-limit, where the temperament is really more interesting, adding 100/99 and 385/384 to the list of commas.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 562: Line 646:
{{Mapping|legend=1| 2 0 11 -23 | 0 1 -2 9 }}
{{Mapping|legend=1| 2 0 11 -23 | 0 1 -2 9 }}


[[Optimal tuning]] ([[POTE]]): ~45/32 = 600.000, ~3/2 = 707.571
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.6603{{c}}, ~3/2 = 707.1707{{c}}
: [[error map]]: {{val| -0.679 +4.536 -3.033 -2.591 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~3/2 = 707.5294{{c}}
: error map: {{val| 0.000 +5.574 -1.373 -1.061 }}


{{Optimal ET sequence|legend=1| 22, 56, 78, 134b, 212b, 290bb }}
{{Optimal ET sequence|legend=1| 22, 56, 78, 134b }}


[[Badness]] (Smith): 0.083971
[[Badness]] (Sintel): 2.13


=== 11-limit ===
=== 11-limit ===
Line 575: Line 663:
Mapping: {{mapping| 2 0 11 -23 26 | 0 1 -2 9 -6 }}
Mapping: {{mapping| 2 0 11 -23 26 | 0 1 -2 9 -6 }}


Optimal tuning (POTE): ~45/32 = 600.000, ~3/2 = 707.609
Optimal tunings:
* WE: ~45/32 = 599.6286{{c}}, ~3/2 = 707.1712{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 707.5984{{c}}


{{Optimal ET sequence|legend=0| 22, 56, 78, 212be, 290bbe }}
{{Optimal ET sequence|legend=0| 22, 56, 78 }}


Badness (Smith): 0.045270
Badness (Sintel): 1.50


==== 13-limit ====
==== 13-limit ====
Line 588: Line 678:
Mapping: {{mapping| 2 0 11 -23 26 -18 | 0 1 -2 9 -6 8 }}
Mapping: {{mapping| 2 0 11 -23 26 -18 | 0 1 -2 9 -6 8 }}


Optimal tuning (POTE): ~45/32 = 600.000, ~3/2 = 707.167
Optimal tunings:
* WE: ~45/32 = 599.3498{{c}}, ~3/2 = 706.4009{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 707.1309{{c}}


{{Optimal ET sequence|legend=0| 22f, 34, 56f }}
{{Optimal ET sequence|legend=0| 22f, 34, 56f }}


Badness (Smith): 0.044877
Badness (Sintel): 1.85


===== 17-limit =====
===== 17-limit =====
Line 601: Line 693:
Mapping: {{mapping| 2 0 11 -23 26 -18 5 | 0 1 -2 9 -6 8 1}}
Mapping: {{mapping| 2 0 11 -23 26 -18 5 | 0 1 -2 9 -6 8 1}}


Optimal tuning (POTE): ~17/12 = 600.000, ~3/2 = 707.155
Optimal tunings:
* WE: ~17/12 = 599.4053{{c}}, ~3/2 = 706.4544{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 707.1243{{c}}


{{Optimal ET sequence|legend=0| 22f, 34, 56f }}
{{Optimal ET sequence|legend=0| 22f, 34, 56f }}


Badness (Smith): 0.030297
Badness (Sintel): 1.54


==== Keenic ====
==== Keenic ====
Line 614: Line 708:
Mapping: {{mapping| 2 0 11 -23 26 36 | 0 1 -2 9 -6 -9 }}
Mapping: {{mapping| 2 0 11 -23 26 36 | 0 1 -2 9 -6 -9 }}


Optimal tuning (POTE): ~45/32 = 600.000, ~3/2 = 707.257
Optimal tunings:
* WE: ~45/32 = 599.8547{{c}}, ~3/2 = 707.0858{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~3/2 = 707.2596{{c}}


{{Optimal ET sequence|legend=0| 22, 34, 56 }}
{{Optimal ET sequence|legend=0| 22, 34, 56 }}


Badness (Smith): 0.040351
Badness (Sintel): 1.67


===== 17-limit =====
===== 17-limit =====
Line 627: Line 723:
Mapping: {{mapping| 2 0 11 -23 26 36 5 | 0 1 -2 9 -6 -9 1 }}
Mapping: {{mapping| 2 0 11 -23 26 36 5 | 0 1 -2 9 -6 -9 1 }}


Optimal tuning (POTE): ~17/12 = 600.000, ~3/2 = 707.252
Optimal tunings:
* WE: ~17/12 = 599.8338{{c}}, ~3/2 = 707.0558{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~3/2 = 707.2537{{c}}


{{Optimal ET sequence|legend=0| 22, 34, 56 }}
{{Optimal ET sequence|legend=0| 22, 34, 56 }}


Badness (Smith): 0.026917
Badness (Sintel): 1.37


== Bidia ==
== Bidia ==
Bidia adds [[3136/3125]] to the commas, splitting the period into 1/4 octave. It may be called the {{nowrap| 12 & 56 }} temperament.
Bidia adds [[3136/3125]] to the commas, splitting the period into 1/4 octave. It may be called the {{nowrap| 12 & 68 }} temperament; its ploidacot is tetraploid monocot.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 644: Line 742:
: mapping generators: ~25/21, ~3
: mapping generators: ~25/21, ~3


[[Optimal tuning]] ([[POTE]]): ~25/21 = 300.000, ~3/2 = 705.364
[[Optimal tuning]]s:
* [[WE]]: ~25/21 = 299.6887{{c}}, ~3/2 = 704.6318{{c}}
: [[error map]]: {{val| -1.245 +1.432 +0.064 +0.854 }}
* [[CWE]]: ~25/21 = 300.0000{{c}}, ~3/2 = 705.5070{{c}}
: error map: {{val| 0.000 +3.552 +2.672 +3.639 }}


{{Optimal ET sequence|legend=1| 12, 56, 68, 80, 148d }}
{{Optimal ET sequence|legend=1| 12, …, 56, 68, 80, 148d }}


[[Badness]] (Smith): 0.056474
[[Badness]] (Sintel): 1.43


=== 11-limit ===
=== 11-limit ===
Line 657: Line 759:
Mapping: {{mapping| 4 0 22 43 71 | 0 1 -2 -5 -9 }}
Mapping: {{mapping| 4 0 22 43 71 | 0 1 -2 -5 -9 }}


Optimal tuning (POTE): ~25/21 = 300.000, ~3/2 = 705.087
Optimal tunings:
* WE: ~25/21 = 299.6809{{c}}, ~3/2 = 704.3367{{c}}
* CWE: ~25/21 = 600.0000{{c}}, ~3/2 = 705.2170{{c}}


{{Optimal ET sequence|legend=0| 12, 68, 80 }}
{{Optimal ET sequence|legend=0| 12, 56e, 68, 80 }}


Badness (Smith): 0.040191
Badness (Sintel): 1.33


=== 13-limit ===
=== 13-limit ===
Line 670: Line 774:
Mapping: {{mapping| 4 0 22 43 71 -36 | 0 1 -2 -5 -9 8 }}
Mapping: {{mapping| 4 0 22 43 71 -36 | 0 1 -2 -5 -9 8 }}


Optimal tuning (POTE): ~25/21 = 300.000, ~3/2 = 705.301
Optimal tunings:
* WE: ~25/21 = 299.7538{{c}}, ~3/2 = 704.7222{{c}}
* CWE: ~25/21 = 600.0000{{c}}, ~3/2 = 705.3241{{c}}


{{Optimal ET sequence|legend=0| 12, 68, 80, 148d, 228bcd, 376bbcddf }}
{{Optimal ET sequence|legend=0| 12, 68, 80, 148d, 228bcd, 376bbcddf }}


Badness (Smith): 0.041137
Badness (Sintel): 1.70


=== 17-limit ===
=== 17-limit ===
Line 683: Line 789:
Mapping: {{mapping| 4 0 22 43 71 -36 10 | 0 1 -2 -5 -9 8 1 }}
Mapping: {{mapping| 4 0 22 43 71 -36 10 | 0 1 -2 -5 -9 8 1 }}


Optimal tuning (POTE): ~25/21 = 300.000, ~3/2 = 705.334
Optimal tunings:
* WE: ~25/21 = 299.7883{{c}}, ~3/2 = 704.8365{{c}}
* CWE: ~25/21 = 600.0000{{c}}, ~3/2 = 705.3496{{c}}


{{Optimal ET sequence|legend=0| 12, 68, 80, 148d, 228bcd, 376bbcddf }}
{{Optimal ET sequence|legend=0| 12, 68, 80, 148d }}


Badness (Smith): 0.028631
Badness (Sintel): 1.46


=== 19-limit ===
=== 19-limit ===
Line 696: Line 804:
Mapping: {{mapping| 4 0 22 43 71 -36 10 17 | 0 1 -2 -5 -9 8 1 0 }}
Mapping: {{mapping| 4 0 22 43 71 -36 10 17 | 0 1 -2 -5 -9 8 1 0 }}


Optimal tuning (POTE): ~19/16 = 300.000, ~3/2 = 705.339
Optimal tunings:
* WE: ~19/16 = 299.7967{{c}}, ~3/2 = 704.8609{{c}}
* CWE: ~19/16 = 600.0000{{c}}, ~3/2 = 705.3519{{c}}


{{Optimal ET sequence|legend=0| 12, 68, 80, 148d, 376bbcddfh }}
{{Optimal ET sequence|legend=0| 12, 68, 80, 148d }}


Badness (Smith): 0.020590
Badness (Sintel): 1.25


=== 23-limit ===
=== 23-limit ===
Line 710: Line 820:


Optimal tunings:
Optimal tunings:
* [[TE]]: ~19/16 = 299.797, ~3/2 = 704.860
* WE: ~19/16 = 299.7961{{c}}, ~3/2 = 704.8577{{c}}
* [[CWE]]: ~19/16 = 300.000, ~3/2 = 705.341
* CWE: ~19/16 = 300.0000{{c}}, ~3/2 = 705.3413{{c}}
* [[POTE]]: ~19/16 = 300.000, ~3/2 = 705.337


{{Optimal ET sequence|legend=0| 12, 68, 80, 148di }}
{{Optimal ET sequence|legend=0| 12, 68, 80, 148di }}


Badness (Smith): 0.017301
Badness (Sintel): 1.24


== Echidna ==
== Shrutar ==
Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It may be called the {{nowrap| 22 & 58 }} temperament. [[58edo]] or [[80edo]] make for good tunings, or their vals can be added to {{val| 138 219 321 388 }} (138cde). In most of the tunings it has a significantly sharp 7/4 which some prefer.  
Shrutar adds 245/243 to the commas, and also tempers out [[6144/6125]]. It can also be described as {{nowrap| 22 & 46 }}. Its generator can be taken as either ~36/35 or ~35/24; the latter is interesting since along with 15/14 and 21/20, it connects opposite sides of a hexany. Its ploidacot is diploid alpha-dicot. [[68edo]] makes for a good tuning, but another excellent choice is a generator of 14<sup>(1/7)</sup>, making 7's just.


Echidna becomes more interesting when extended to be an 11-limit temperament by adding 176/175, 540/539 or 896/891 to the commas, where the same tunings can be used as before. It then is able to represent the entire 11-odd-limit diamond to within about six cents of error, within a compass of 24 notes. The 22-note 2mos gives scope for this, and the 36-note mos much more. Better yet, it is related to three important 11-limit edos: 22edo, a trivial tuning, is the smallest consistent in the 11-odd-limit, corresponding to the merge of this temperament with [[hedgehog]]; [[58edo]] is the smallest tuning that is distinctly consistent in the 11-odd-limit and [[80edo]] is the third smallest distinctly consistent in the 11-odd-limit.  
By adding 121/120 or 176/175 to the commas, shrutar can be extended to the 11-limit, which loses a bit of accuracy, but picks up low-complexity 11-limit harmony, making shrutar quite an interesting 11-limit system. 68, 114 or a 14<sup>(1/7)</sup> generator can again be used as tunings.
 
The generator can be interpreted as 11/10, the period complement of 9/7, as a stack of 11/10 and 9/7 makes [[99/70]] which is extremely close to 600{{cent}} and is equal to it if we temper out [[9801/9800|S99]]. Three 11/10's then make a 4/3 (tempering out [[4000/3993|S10/S11]] thus making 10/9 and 12/11 equidistant from 11/10), implying a flat tuning of 4/3.
 
Like most srutal extensions, the 13- and 17-limit interpretations are possible by observing that since we have tempered out [[176/175]], tempering out [[351/350]] and [[352/351]] which sum to 176/175 is very elegant. In the 17-limit we can equate the half-octave with 17/12 and 24/17 and we can take advantage of the sharp fifth by combining echidna with [[srutal archagall]], leading to a particularly beautiful temperament (one that prefers a very slightly less sharp fifth than srutal archagall). This mapping of 13 and 17 is supported by the patent vals of the three main echidna edos of 22, 58 and 80, of which all except 22 are consistent in the [[17-odd-limit]].


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 1728/1715, 2048/2025
[[Comma list]]: 245/243, 2048/2025


{{Mapping|legend=1| 2 1 9 2 | 0 3 -6 5 }}
{{Mapping|legend=1| 2 1 9 -2 | 0 2 -4 7 }}


: mapping generators: ~45/32, ~9/7
: mapping generators: ~45/32, ~35/24


[[Optimal tuning]] ([[POTE]]): ~45/32 = 600.000, ~9/7 = 434.856
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.5401{{c}}, ~35/24 = 652.3108{{c}}
: [[error map]]: {{val| -0.920 +2.207 +0.304 -1.730 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~35/24 = 652.7736{{c}}
: error map: {{val| 0.000 +3.592 +2.592 +0.589 }}


{{Optimal ET sequence|legend=1| 22, 58, 80, 138cd, 218cd }}
{{Optimal ET sequence|legend=1| 22, 46, 68, 182b, 250bc }}


[[Badness]] (Smith): 0.058033
[[Badness]] (Sintel): 1.20


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 176/175, 540/539, 896/891
Comma list: 121/120, 176/175, 245/243


Mapping: {{mapping| 2 1 9 2 12 | 0 3 -6 5 -7 }}
Mapping: {{mapping| 2 1 9 -2 8 | 0 2 -4 7 -1 }}


Optimal tuning (POTE): ~45/32 = 600.000, ~9/7 = 434.852
Optimal tunings:
* WE: ~45/32 = 599.7721{{c}}, ~16/11 = 652.4321{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~16/11 = 652.6672{{c}}


Minimax tuning:
{{Optimal ET sequence|legend=0| 22, 46, 68, 114 }}
* 11-odd-limit: ~9/7 = {{monzo| 5/12 0 0 1/12 -1/12 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 7/4 0 0 1/4 -1/4 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 37/12 0 0 5/12 -5/12 }}, {{monzo| 37/12 0 0 -7/12 7/12 }}]
: unchanged-interval (eigenmonzo) basis: 2.11/7
 
{{Optimal ET sequence|legend=0| 22, 58, 80, 138cde, 218cde }}


Badness (Smith): 0.025987
Badness (Sintel): 0.876


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 176/175, 351/350, 364/363, 540/539
Comma list: 121/120, 176/175, 196/195, 245/243


Mapping: {{mapping| 2 1 9 2 12 19 | 0 3 -6 5 -7 -16 }}
Mapping: {{mapping| 2 1 9 -2 8 -10 | 0 2 -4 7 -1 16 }}


Optimal tuning (POTE): ~45/32 = 600.000, ~9/7 = 434.756
Optimal tunings:
* WE: ~45/32 = 599.7699{{c}}, ~16/11 = 652.4035{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~16/11 = 652.6374{{c}}


{{Optimal ET sequence|legend=0| 22, 58, 80, 138cde }}
{{Optimal ET sequence|legend=0| 22f, 46, 68, 114 }}


Badness (Smith): 0.023679
Badness (Sintel): 1.16


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 136/135, 176/175, 221/220, 256/255, 540/539
Comma list: 121/120, 136/135, 154/153, 176/175, 196/195
 
Mapping: {{mapping| 2 1 9 -2 8 -10 6 | 0 2 -4 7 -1 16 2 }}
 
Optimal tunings:
* WE: ~17/12 = 599.7995{{c}}, ~16/11 = 652.4287{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 652.6334{{c}}
 
{{Optimal ET sequence|legend=0| 22f, 46, 68, 114 }}
 
Badness (Sintel): 0.953
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19


Mapping: {{mapping| 2 1 9 2 12 19 6 | 0 3 -6 5 -7 -16 3 }}
Comma list: 121/120, 136/135, 154/153, 176/175, 196/195, 343/342


Optimal tuning (POTE): ~17/12 = 600.000, ~9/7 = 434.816
Mapping: {{mapping| 2 1 9 -2 8 -10 6 -10 | 0 2 -4 7 -1 16 2 17 }}


{{Optimal ET sequence|legend=0| 22, 58, 80, 138cde }}
Optimal tunings:
* WE: ~17/12 = 599.8060{{c}}, ~16/11 = 652.5190{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 652.7164{{c}}


Badness (Smith): 0.020273
{{Optimal ET sequence|legend=0| 22fh, 46, 68, 114, 182bef }}


== Echidnic ==
Badness (Sintel): 1.07
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 686/675, 1029/1024
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23


{{Mapping|legend=1| 2 2 7 6 | 0 3 -6 -1 }}
Comma list: 121/120, 136/135, 154/153, 176/175, 196/195, 253/252, 343/342


: mapping generators: ~45/32, ~8/7
Mapping: {{mapping| 2 1 9 -2 8 -10 6 -10 -4 | 0 2 -4 7 -1 16 2 17 12 }}


[[Optimal tuning]] ([[POTE]]): ~45/32 = 600.000, ~8/7 = 234.492
Optimal tunings:
* WE: ~17/12 = 599.7879{{c}}, ~16/11 = 652.4776{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~16/11 = 652.6926{{c}}


{{Optimal ET sequence|legend=1| 10, 36, 46, 194bcd, 240bcd, 286bcd, 332bccdd }}
{{Optimal ET sequence|legend=0| 22fh, 46, 68, 114 }}


[[Badness]] (Smith): 0.072246
Badness (Sintel): 1.03


=== 11-limit ===
== Shru ==
Subgroup: 2.3.5.7.11
Shru tempers out 392/375 and slices the compound semitone into two generators of ~10/7. Its ploidacot is diploid alpha-dicot, the same as shrutar.  


Comma list: 385/384, 441/440, 686/675
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 2 2 7 6 3 | 0 3 -6 -1 10 }}
[[Comma list]]: 392/375, 1323/1280


Optimal tuning (POTE): ~45/32 = 600.000, ~8/7 = 235.096
{{Mapping|legend=1| 2 1 9 11 | 0 2 -4 -5 }}


{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148, 342bcdd }}
: mapping generators: ~45/32, ~10/7


Badness (Smith): 0.045127
[[Optimal tuning]]s:  
* [[WE]]: ~45/32 = 600.2519{{c}}, ~10/7 = 650.4083{{c}}
: [[error map]]: {{val| +0.504 -0.887 +14.321 -18.096 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~10/7 = 650.1017{{c}}
: error map: {{val| 0.000 -1.752 +13.279 -19.334 }}


=== 13-limit ===
{{Optimal ET sequence|legend=1| 2, 22d, 24 }}
Subgroup: 2.3.5.7.11.13


Comma list: 91/90, 169/168, 385/384, 441/440
[[Badness]] (Sintel): 3.99


Mapping: {{mapping| 2 2 7 6 3 7 | 0 3 -6 -1 10 1 }}
=== 11-limit ===
Subgroup: 2.3.5.7.11


Optimal tuning (POTE): ~45/32 = 600.000, ~8/7 = 235.088
Comma list: 56/55, 77/75, 1323/1280


{{Optimal ET sequence|legend=0| 10, 46, 102, 148f, 194bcdf }}
Mapping: {{mapping| 2 1 9 11 8 | 0 2 -4 -5 -1 }}


Badness (Smith): 0.028874
Optimal tunings:
* WE: ~17/12 = 600.2356{{c}}, ~10/7 = 650.3856{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~10/7 = 650.1008{{c}}


=== 17-limit ===
{{Optimal ET sequence|legend=0| 2, 22d, 24 }}
Subgroup: 2.3.5.7.11.13.17


Comma list: 91/90, 136/135, 154/153, 169/168, 256/255
Badness (Sintel): 2.10


Mapping: {{mapping| 2 2 7 6 3 7 7 | 0 3 -6 -1 10 1 3 }}
=== 13-limit ===
Subgroup: 2.3.5.7.11.13


Optimal tuning (POTE): ~17/12 = 600.000, ~8/7 = 235.088
Comma list: 56/55, 77/75, 105/104, 507/500


{{Optimal ET sequence|legend=0| 10, 46, 102, 148f, 194bcdf }}
Mapping: {{mapping| 2 1 9 11 8 15 | 0 2 -4 -5 -1 -7 }}


Badness (Smith): 0.019304
Optimal tunings:
* WE: ~45/32 = 599.9067{{c}}, ~10/7 = 649.4907{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~10/7 = 649.5950{{c}}


; Music
{{Optimal ET sequence|legend=0| 2, 24 }}
* [https://untwelve.org/competition/2011 ''A Stiff Shot of Turpentine''] [https://untwelve.org/static/audio/competition/2011/Kosmorsky-A_Stiff_Shot_of_Turpentine.mp3 play] by [[Peter Kosmorsky]]
* [https://www.youtube.com/watch?v=VsBXIvBZY6A ''56edo Track (Echidnic16 Scale)''] by [[Budjarn Lambeth]] (2025)


== Shrutar ==
Badness (Sintel): 2.12
Shrutar adds 245/243 to the commas, and also tempers out 6144/6125. It can also be described as {{nowrap| 22 & 46 }}. Its generator can be taken as either 36/35 or 35/24; the latter is interesting since along with 15/14 and 21/20, it connects opposite sides of a hexany. [[68edo]] makes for a good tuning, but another excellent choice is a generator of 14<sup>(1/7)</sup>, making 7's just.


By adding 121/120 or 176/175 to the commas, shrutar can be extended to the 11-limit, which loses a bit of accuracy, but picks up low-complexity 11-limit harmony, making shrutar quite an interesting 11-limit system. 68, 114 or a 14<sup>(1/7)</sup> generator can again be used as tunings.
== Sruti ==
Sruti tempers out 19683/19600, setting itself up as a [[hemipyth]] temperament. It has the same semi-octave period as diaschismic, but the generator can be taken as a neutral third or a hemitwelfth. The temperament can be described as {{nowrap| 24 & 34d }}; its ploidacot is diploid dicot. [[58edo]] may be recommended as a tuning.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 245/243, 2048/2025
[[Comma list]]: 2048/2025, 19683/19600


{{Mapping|legend=1| 2 1 9 -2 | 0 2 -4 7 }}
{{Mapping|legend=1| 2 0 11 -15 | 0 2 -4 13 }}


: mapping generators: ~45/32, ~35/24
: mapping generators: ~45/32, ~140/81


[[Optimal tuning]] ([[POTE]]): ~45/32 = 600.000, ~35/24 = 652.811
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.2764{{c}}, ~140/81 = 950.7284{{c}}
: [[error map]]: {{val| -1.447 -0.498 +2.813 +1.497 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~140/81 = 951.8227{{c}}
: error map: {{val| 0.000 +1.690 +6.395 +4.869 }}


{{Optimal ET sequence|legend=1| 22, 46, 68, 182b, 250bc }}
{{Optimal ET sequence|legend=1| 24, 34d, 58, 150cd, 208ccdd, 266ccdd }}


[[Badness]] (Smith): 0.189510
[[Badness]] (Sintel): 2.97


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 121/120, 176/175, 245/243
Comma list: 176/175, 243/242, 896/891


Mapping: {{mapping| 2 1 9 -2 8 | 0 2 -4 7 -1 }}
Mapping: {{mapping| 2 0 11 -15 -1 | 0 2 -4 13 5 }}


Optimal tuning (POTE): ~45/32 = 600.000, ~16/11 = 652.680
Optimal tunings:
* WE: ~45/32 = 599.1951{{c}}, ~121/70 = 950.5864{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~121/70 = 951.7972{{c}}


{{Optimal ET sequence|legend=0| 22, 46, 68, 114, 296bce, 410bce }}
{{Optimal ET sequence|legend=0| 24, 34d, 58, 150cdee, 208ccddee, 266ccddeee }}


Badness (Smith): 0.084098
Badness (Sintel): 1.37


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 121/120, 176/175, 196/195, 245/243
Comma list: 144/143, 176/175, 351/350, 676/675


Mapping: {{mapping| 2 1 9 -2 8 -10 | 0 2 -4 7 -1 16 }}
Mapping: {{mapping| 2 0 11 -15 -1 9 | 0 2 -4 13 5 -1 }}


Optimal tuning (POTE): ~45/32 = 600.000, ~16/11 = 652.654
Optimal tunings:
* WE: ~45/32 = 599.1479{{c}}, ~26/15 = 950.5337{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~26/15 = 951.8314{{c}}


{{Optimal ET sequence|legend=0| 22f, 24f, 46, 68, 114 }}
{{Optimal ET sequence|legend=0| 24, 34d, 58, 150cdeef, 208ccddeeff, 266ccddeeefff }}


Badness (Smith): 0.079358
Badness (Sintel): 0.983


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 121/120, 136/135, 154/153, 176/175, 196/195
Comma list: 136/135, 144/143, 170/169, 176/175, 221/220


Mapping: {{mapping| 2 1 9 -2 8 -10 6 | 0 2 -4 7 -1 16 2 }}
Mapping: {{mapping| 2 0 11 -15 -1 9 5 | 0 2 -4 13 5 -1 2 }}


Optimal tuning (POTE): ~17/12 = 600.000, ~16/11 = 652.647
Optimal tunings:
* WE: ~17/12 = 599.3003{{c}}, ~26/15 = 950.7465{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~26/15 = 951.8142{{c}}


{{Optimal ET sequence|legend=0| 22f, 24f, 46, 68, 114 }}
{{Optimal ET sequence|legend=0| 24, 34d, 58 }}


Badness (Smith): 0.049392
Badness (Sintel): 1.05


=== 19-limit ===
== Anguirus ==
Subgroup: 2.3.5.7.11.13.17.19
As another hemipyth temperament, anguirus tempers out 49/48. It can be described as the {{nowrap| 10 & 24 }} temperament; its ploidacot is diploid dicot, the same as sruti.  


Comma list: 121/120, 136/135, 154/153, 176/175, 196/195, 343/342
Mapping: {{mapping| 2 1 9 -2 8 -10 6 -10 | 0 2 -4 7 -1 16 2 17 }}
Optimal tuning (POTE): ~17/12 = 600.000, ~16/11 = 652.730
{{Optimal ET sequence|legend=0| 22fh, 24fh, 46, 68, 114, 182bef }}
Badness (Smith): 0.044197
=== 23-limit ===
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 121/120, 136/135, 154/153, 176/175, 196/195, 253/252, 343/342
Mapping: {{mapping| 2 1 9 -2 8 -10 6 -10 -4 | 0 2 -4 7 -1 16 2 17 12 }}
Optimal tuning (POTE): ~17/12 = 600.000, ~16/11 = 652.708
{{Optimal ET sequence|legend=0| 22fh, 46, 68, 114 }}
Badness (Smith): 0.035137
== Sruti ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 2048/2025, 19683/19600
[[Comma list]]: 49/48, 2048/2025


{{Mapping|legend=1| 2 0 11 -15 | 0 2 -4 13 }}
{{Mapping|legend=1| 2 0 11 4 | 0 2 -4 1 }}


: mapping generators: ~45/32, ~140/81
: mapping generators: ~45/32, ~7/4


[[Optimal tuning]] ([[POTE]]): ~45/32 = 600.000, ~140/81 = 951.876
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 600.2758{{c}}, ~7/4 = 953.4593{{c}}
: [[error map]]: {{val| +0.552 +4.964 +2.883 -14.264 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~7/4 = 953.0188{{c}}
: error map: {{val| 0.000 +4.083 +1.611 -15.807 }}


{{Optimal ET sequence|legend=1| 24, 34d, 58, 150cd, 208ccdd, 266ccdd }}
{{Optimal ET sequence|legend=1| 10, 24, 34 }}


[[Badness]] (Smith): 0.117358
[[Badness]] (Sintel): 1.97


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 176/175, 243/242, 896/891
Comma list: 49/48, 56/55, 243/242


Mapping: {{mapping| 2 0 11 -15 -1 | 0 2 -4 13 5 }}
Mapping: {{mapping| 2 0 11 4 -1 | 0 2 -4 1 5 }}


Optimal tuning (POTE): ~45/32 = 600.000, ~121/70 = 951.863
Optimal tunings:
* WE: ~45/32 = 599.9250{{c}}, ~7/4 = 952.0646{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~7/4 = 952.1784{{c}}


{{Optimal ET sequence|legend=0| 24, 34d, 58 }}
{{Optimal ET sequence|legend=0| 10, 24, 34 }}


Badness (Smith): 0.041459
Badness (Sintel): 1.63


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 144/143, 176/175, 351/350, 676/675
Comma list: 49/48, 56/55, 91/90, 243/242


Mapping: {{mapping| 2 0 11 -15 -1 9 | 0 2 -4 13 5 -1 }}
Mapping: {{mapping| 2 0 11 4 -1 9 | 0 2 -4 1 5 -1 }}


Optimal tuning (POTE): ~45/32 = 600.000, ~26/15 = 951.886
Optimal tunings:
* WE: ~45/32 = 599.7575{{c}}, ~7/4 = 951.9241{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~7/4 = 952.2980{{c}}


{{Optimal ET sequence|legend=0| 24, 34d, 58, 150cdeef, 208ccddeeff }}
{{Optimal ET sequence|legend=0| 10, 24, 34, 58d, 92ddef }}


Badness (Smith): 0.023791
Badness (Sintel): 1.27


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 136/135, 144/143, 170/169, 176/175, 221/220
Comma list: 49/48, 56/55, 91/90, 119/117, 154/153
 
Mapping: {{mapping| 2 0 11 4 -1 9 5 | 0 2 -4 1 5 -1 2 }}
 
Optimal tunings:
* WE: ~17/12 = 599.7925{{c}}, ~7/4 = 952.0004{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~7/4 = 952.3178{{c}}
 
{{Optimal ET sequence|legend=0| 10, 24, 34 }}
 
Badness (Sintel): 1.10


Mapping: {{mapping| 2 0 11 -15 -1 9 5 | 0 2 -4 13 5 -1 2 }}
== Echidna ==
Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It may be called the {{nowrap| 22 & 58 }} temperament; its ploidacot is diploid alpha-tricot. [[58edo]] or [[80edo]] make for good tunings, or their vals can be added to {{val| 138 219 321 388 }} (138cde). In most of the tunings it has a significantly sharp 7/4 which some prefer.


Optimal tuning (POTE): ~17/12 = 600.000, ~26/15 = 951.857
Echidna becomes more interesting when extended to be an 11-limit temperament by adding 176/175, 540/539 or 896/891 to the commas, where the same tunings can be used as before. It then is able to represent the entire 11-odd-limit diamond to within about six cents of error, within a compass of 24 notes. The 22-note 2mos gives scope for this, and the 36-note mos much more. Better yet, it is related to three important 11-limit edos: 22edo, a trivial tuning, is the smallest consistent in the 11-odd-limit, corresponding to the merge of this temperament with [[hedgehog]]; [[58edo]] is the smallest tuning that is distinctly consistent in the 11-odd-limit and [[80edo]] is the third smallest distinctly consistent in the 11-odd-limit.  


{{Optimal ET sequence|legend=0| 24, 34d, 58 }}
The generator can be interpreted as 11/10, the period complement of 9/7, as a stack of 11/10 and 9/7 makes [[99/70]] which is extremely close to 600{{cent}} and is equal to it if we temper out [[9801/9800|S99]]. Three 11/10's then make a 4/3 (tempering out [[4000/3993|S10/S11]] thus making 10/9 and 12/11 equidistant from 11/10), implying a flat tuning of 4/3.


Badness (Smith): 0.020536
Like most srutal extensions, the 13- and 17-limit interpretations are possible by observing that since we have tempered out [[176/175]], tempering out [[351/350]] and [[352/351]] which sum to 176/175 is very elegant. In the 17-limit we can equate the half-octave with 17/12 and 24/17 and we can take advantage of the sharp fifth by combining echidna with [[srutal archagall]], leading to a particularly beautiful temperament (one that prefers a very slightly less sharp fifth than srutal archagall). This mapping of 13 and 17 is supported by the patent vals of the three main echidna edos of 22, 58 and 80, of which all except 22 are consistent in the [[17-odd-limit]].


== Anguirus ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 49/48, 2048/2025
[[Comma list]]: 1728/1715, 2048/2025


{{Mapping|legend=1| 2 0 11 4 | 0 2 -4 1 }}
{{Mapping|legend=1| 2 1 9 2 | 0 3 -6 5 }}


: mapping generators: ~45/32, ~7/4
: mapping generators: ~45/32, ~9/7


[[Optimal tuning]] ([[POTE]]): ~45/32 = 600.000, ~7/4 = 953.021
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.3056{{c}}, ~9/7 = 434.3524{{c}}
: [[error map]]: {{val| -1.389 +0.408 +1.322 +1.547 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8327{{c}}
: error map: {{val| 0.000 +2.543 +4.690 +5.338 }}


{{Optimal ET sequence|legend=1| 10, 24, 34 }}
{{Optimal ET sequence|legend=1| 22, 58, 80, 138cd, 218cd }}


[[Badness]] (Smith): 0.077955
[[Badness]] (Sintel): 1.47


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 49/48, 56/55, 243/242
Comma list: 176/175, 540/539, 896/891
 
Mapping: {{mapping| 2 1 9 2 12 | 0 3 -6 5 -7 }}


Mapping: {{mapping| 2 0 11 4 -1 | 0 2 -4 1 5 }}
Optimal tunings:
* WE: ~45/32 = 599.3085{{c}}, ~9/7 = 434.3511{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8647{{c}}


Optimal tuning (POTE): ~45/32 = 600.000, ~7/4 = 952.184
Minimax tuning:
* 11-odd-limit: ~9/7 = {{monzo| 5/12 0 0 1/12 -1/12 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 7/4 0 0 1/4 -1/4 }}, {{monzo| 2 0 0 -1/2 1/2 }}, {{monzo| 37/12 0 0 5/12 -5/12 }}, {{monzo| 37/12 0 0 -7/12 7/12 }}]
: unchanged-interval (eigenmonzo) basis: 2.11/7


{{Optimal ET sequence|legend=0| 10, 24, 34, 58d, 92de }}
{{Optimal ET sequence|legend=0| 22, 58, 80, 138cde, 218cde }}


Badness (Smith): 0.049253
Badness (Sintel): 0.859


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 49/48, 56/55, 91/90, 243/242
Comma list: 176/175, 351/350, 364/363, 540/539


Mapping: {{mapping| 2 0 11 4 -1 9 | 0 2 -4 1 5 -1 }}
Mapping: {{mapping| 2 1 9 2 12 19 | 0 3 -6 5 -7 -16 }}


Optimal tuning (POTE): ~45/32 = 600.000, ~7/4 = 952.309
Optimal tunings:
* WE: ~45/32 = 599.3397{{c}}, ~9/7 = 434.2772{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.7864{{c}}


{{Optimal ET sequence|legend=0| 10, 24, 34, 58d, 92ddef }}
{{Optimal ET sequence|legend=0| 22, 36f, 58, 80, 138cde }}


Badness (Smith): 0.030829
Badness (Sintel): 0.978


=== 17-limit ===
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 49/48, 56/55, 91/90, 119/117, 154/153
Comma list: 136/135, 176/175, 221/220, 256/255, 540/539
 
Mapping: {{mapping| 2 1 9 2 12 19 6 | 0 3 -6 5 -7 -16 3 }}


Mapping: {{mapping| 2 0 11 4 -1 9 5 | 0 2 -4 1 5 -1 2 }}
Optimal tunings:
* WE: ~45/32 = 599.4645{{c}}, ~9/7 = 434.4282{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~9/7 = 434.8340{{c}}


Optimal tuning (POTE): ~17/12 = 600.000, ~7/4 = 952.330
{{Optimal ET sequence|legend=0| 22, 36f, 58, 80, 138cde }}


{{Optimal ET sequence|legend=0| 10, 24, 34, 58d, 92ddef }}
Badness (Sintel): 1.03


Badness (Smith): 0.021796
== Echidnic ==
Echidnic tempers out 686/675 and [[1029/1024]]. It has the same semi-octave period as diaschismic, but slices the generator of a fifth into three ~8/7's. It can be described as the {{nowrap| 10 & 46 }} temperament; its ploidacot is diploid tricot.  


== Shru ==
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 392/375, 1323/1280
[[Comma list]]: 686/675, 1029/1024


{{Mapping|legend=1| 2 1 9 11 | 0 2 -4 -5 }}
{{Mapping|legend=1| 2 2 7 6 | 0 3 -6 -1 }}


: mapping generators: ~45/32, ~10/7
: mapping generators: ~45/32, ~8/7


[[Optimal tuning]] ([[POTE]]): ~45/32 = 600.000, ~10/7 = 650.135
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.7208{{c}}, ~8/7 = 234.8330{{c}}
: [[error map]]: {{val| -0.558 +1.986 +2.733 -5.334 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~8/7 = 234.9539{{c}}
: error map: {{val| 0.000 +2.907 +3.963 -3.780 }}


{{Optimal ET sequence|legend=1| 2, 22d, 24 }}
{{Optimal ET sequence|legend=1| 10, 26c, 36, 46 }}


[[Badness]] (Smith): 0.157619
[[Badness]] (Sintel): 1.83


=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 56/55, 77/75, 1323/1280
Comma list: 385/384, 441/440, 686/675


Mapping: {{mapping| 2 1 9 11 8 | 0 2 -4 -5 -1 }}
Mapping: {{mapping| 2 2 7 6 3 | 0 3 -6 -1 10 }}


Optimal tuning (POTE): ~45/32 = 600.000, ~10/7 = 650.130
Optimal tunings:
* WE: ~45/32 = 599.8022{{c}}, ~8/7 = 235.0185{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~8/7 = 235.0893{{c}}


{{Optimal ET sequence|legend=0| 2, 22d, 24 }}
{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148 }}


Badness (Smith): 0.063483
Badness (Sintel): 1.49


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 56/55, 77/75, 105/104, 507/500
Comma list: 91/90, 169/168, 385/384, 441/440
 
Mapping: {{mapping| 2 2 7 6 3 7 | 0 3 -6 -1 10 1 }}
 
Optimal tunings:
* WE: ~45/32 = 599.9570{{c}}, ~8/7 = 235.0708{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~8/7 = 235.0862{{c}}
 
{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148f }}
 
Badness (Sintel): 1.19
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 91/90, 136/135, 154/153, 169/168, 256/255
 
Mapping: {{mapping| 2 2 7 6 3 7 7 | 0 3 -6 -1 10 1 3 }}


Mapping: {{mapping| 2 1 9 11 8 15 | 0 2 -4 -5 -1 -7 }}
Optimal tunings:
* WE: ~17/12 = 599.9571{{c}}, ~8/7 = 235.0709{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~8/7 = 235.0860{{c}}


Optimal tuning (POTE): ~45/32 = 600.000, ~10/7 = 650.535
{{Optimal ET sequence|legend=0| 10, 36e, 46, 102, 148f }}


{{Optimal ET sequence|legend=0| 22df, 24 }}
Badness (Sintel): 0.983


Badness (Smith): 0.045731
; Music
* [https://untwelve.org/competition/2011 ''A Stiff Shot of Turpentine''] [https://untwelve.org/static/audio/competition/2011/Kosmorsky-A_Stiff_Shot_of_Turpentine.mp3 play] by [[Peter Kosmorsky]]
* [https://www.youtube.com/watch?v=VsBXIvBZY6A ''56edo Track (Echidnic16 Scale)''] by [[Budjarn Lambeth]] (2025)


== Quadrasruta ==
== Quadrasruta ==
Named by [[Xenllium]] in 2022, quadrasruta tempers out 2401/2400, the breedsma, and extends [[buzzard]]. It may be described as {{nowrap| 58 & 68 }}; its ploidacot is diploid alpha-tetracot. 126edo may be recommended as a tuning.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 1,086: Line 1,266:
: mapping generators: ~45/32, ~21/16
: mapping generators: ~45/32, ~21/16


[[Optimal tuning]] ([[POTE]]): ~45/32 = 600.000, ~21/16 = 476.216
[[Optimal tuning]]s:
* [[WE]]: ~45/32 = 599.4443{{c}}, ~21/16 = 475.7746{{c}}
: [[error map]]: {{val| -1.111 +1.143 +1.377 -0.595 }}
* [[CWE]]: ~45/32 = 600.0000{{c}}, ~21/16 = 476.2394{{c}}
: error map: {{val| 0.000 +3.003 +3.771 +2.456 }}


{{Optimal ET sequence|legend=1| 10, 38c, 48c, 58, 68, 126 }}
{{Optimal ET sequence|legend=1| 10, …, 58, 68, 126, 446bbccd }}


[[Badness]] (Smith): 0.073569
[[Badness]] (Sintel): 1.86


=== 11-limit ===
=== 11-limit ===
Line 1,099: Line 1,283:
Mapping: {{mapping| 2 0 11 8 22 | 0 4 -8 -3 -19 }}
Mapping: {{mapping| 2 0 11 8 22 | 0 4 -8 -3 -19 }}


Optimal tuning (POTE): ~45/32 = 600.000, ~21/16 = 476.118
Optimal tunings:
* WE: ~45/32 = 599.4648{{c}}, ~21/16 = 475.6929{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.1507{{c}}


{{Optimal ET sequence|legend=0| 58, 126, 184c, 310bccde }}
{{Optimal ET sequence|legend=0| 10e, …, 58, 126, 184c, 310bccde }}


Badness (Smith): 0.049018
Badness (Sintel): 1.62


==== 13-limit ====
==== 13-limit ====
Line 1,112: Line 1,298:
Mapping: {{mapping| 2 0 11 8 22 9 | 0 4 -8 -3 -19 -2 }}
Mapping: {{mapping| 2 0 11 8 22 9 | 0 4 -8 -3 -19 -2 }}


Optimal tuning (POTE): ~45/32 = 600.000, ~21/16 = 476.099
Optimal tunings:
* WE: ~45/32 = 599.3787{{c}}, ~21/16 = 475.6065{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.1345{{c}}


{{Optimal ET sequence|legend=0| 58, 126f, 184cff }}
{{Optimal ET sequence|legend=0| 10e, …, 58, 126f, 184cff }}


Badness (Smith): 0.028463
Badness (Sintel): 1.18


==== 17-limit ====
==== 17-limit ====
Line 1,125: Line 1,313:
Mapping: {{mapping| 2 0 11 8 22 9 5 | 0 4 -8 -3 -19 -2 4 }}
Mapping: {{mapping| 2 0 11 8 22 9 5 | 0 4 -8 -3 -19 -2 4 }}


Optimal tuning (POTE): ~17/12 = 600.000, ~21/16 = 476.162
Optimal tunings:
* WE: ~17/12 = 599.5077{{c}}, ~21/16 = 475.7713{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~21/16 = 476.1814{{c}}


{{Optimal ET sequence|legend=0| 58, 126f }}
{{Optimal ET sequence|legend=0| 10e, 58, 126f }}


Badness (Smith): 0.023820
Badness (Sintel): 1.21


=== Quadrafourths ===
=== Quadrafourths ===
Line 1,138: Line 1,328:
Mapping: {{mapping| 2 0 11 8 -1 | 0 4 -8 -3 10 }}
Mapping: {{mapping| 2 0 11 8 -1 | 0 4 -8 -3 10 }}


Optimal tuning (POTE): ~45/32 = 600.000, ~21/16 = 476.017
Optimal tunings:
* WE: ~45/32 = 599.2593{{c}}, ~21/16 = 475.4292{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.0088{{c}}


{{Optimal ET sequence|legend=0| 10, 38c, 48c, 58 }}
{{Optimal ET sequence|legend=0| 10, 48c, 58, 184cee, 242ccdeee }}


Badness (Smith): 0.049114
Badness (Sintel): 1.62


==== 13-limit ====
==== 13-limit ====
Line 1,151: Line 1,343:
Mapping: {{mapping| 2 0 11 8 -1 9 | 0 4 -8 -3 10 -2 }}
Mapping: {{mapping| 2 0 11 8 -1 9 | 0 4 -8 -3 10 -2 }}


Optimal tuning (POTE): ~45/32 = 600.000, ~21/16 = 476.028
Optimal tunings:
* WE: ~45/32 = 599.2147{{c}}, ~21/16 = 475.4052{{c}}
* CWE: ~45/32 = 600.0000{{c}}, ~21/16 = 476.0253{{c}}


{{Optimal ET sequence|legend=0| 10, 38c, 48c, 58 }}
{{Optimal ET sequence|legend=0| 10, 48c, 58, 126eef, 184ceeff, 242ccdeeeff }}


Badness (Smith): 0.026743
Badness (Sintel): 1.11


==== 17-limit ====
==== 17-limit ====
Line 1,164: Line 1,358:
Mapping: {{mapping| 2 0 11 8 -1 9 5 | 0 4 -8 -3 10 -2 4 }}
Mapping: {{mapping| 2 0 11 8 -1 9 5 | 0 4 -8 -3 10 -2 4 }}


Optimal tuning (POTE): ~17/12 = 600.000, ~21/16 = 476.077
Optimal tunings:
* WE: ~17/12 = 599.3353{{c}}, ~21/16 = 475.5495{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~21/16 = 476.0691{{c}}


{{Optimal ET sequence|legend=0| 10, 38c, 48c, 58, 126eef, 184ceeff }}
{{Optimal ET sequence|legend=0| 10, 48c, 58 }}


Badness (Smith): 0.022239
Badness (Sintel): 1.13


[[Category:Temperament families]]
[[Category:Temperament families]]
Retrieved from "https://en.xen.wiki/w/Diaschismic_family"