Diaschismic family: Difference between revisions

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: mapping generators: ~45/32, ~3

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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.

* Pajara ~~derives from~~ [[64/63]] and is a popular and well-known choice.

* Pajara adds [[50/49]] or [[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.

* Keen adds [[875/864]].

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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 ====

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

~~=== Srutal archagall ===~~

~~{{See also | Fiventeen }}~~

~~Subgroup: 2.3.5.17~~

~~Comma list: 136/135, 256/255~~

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

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

~~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 }}~~

~~Badness (Sintel): 0.212~~

== Septimal diaschismic ==

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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.

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]] with the 104c [[val]] 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. This mapping can also be rationalized by [[parapyth]], which makes sense due to the sharp fifth, and prime 17 is found as in srutal archagall. If you want to explore higher-limit harmonies, diaschismic is certainly one excellent way to do it; [[mos]] scales 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.

[[Subgroup]]: 2.3.5.7

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[[Badness]] (Sintel): 0.507

=== 2.3.5.7.17 subgroup ===

Subgroup: 2.3.5.7.17

Comma list: 50/49, 64/63, 85/84

Mapping: {{mapping| 2 0 11 12 5 | 0 1 -2 -2 1 }}

Optimal tunings:

* WE: ~7/5 = 599.053{{c}}, ~3/2 = 706.355{{c}}

* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 707.607{{c}}

Badness (Sintel): 0.438

=== 11-limit ===

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Badness (Sintel): 0.673

==== 2.3.5.7.11.17 subgroup ====

Subgroup: 2.3.5.7.11.17

Comma list: 50/49, 64/63, 85/84, 99/98

Mapping: {{mapping| 2 0 11 12 26 5 | 0 1 -2 -2 -6 1 }}

Optimal tunings:

* WE: ~7/5 = 599.062{{c}}, ~3/2 = 706.095{{c}}

* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 707.370{{c}}

Badness (Sintel): 0.645

==== 13-limit ====

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Badness (Sintel): 0.937

==== 2.3.5.7.11.17 subgroup ====

Subgroup: 2.3.5.7.11.13.17

Comma list: 50/49, 52/51, 55/54, 64/63, 65/63

Mapping: {{mapping| 2 0 11 12 -9 1 5 | 0 1 -2 -2 5 2 1 }}

Optimal tunings:

* WE: ~7/5 = 599.408{{c}}, ~3/2 = 708.878{{c}}

* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 709.544{{c}}

Badness (Sintel): 0.766

==== 13-limit ====

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Mapping: {{mapping| 2 0 11 12 -1 | 0 2 -4 -4 5 }}

: mapping generators: ~2, ~55/32

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: mapping generators: ~25/21, ~3

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Optimal tunings:

* WE: ~25/21 = 299.6809{{c}}, ~3/2 = 704.3367{{c}}

* CWE: ~25/21 = ~~600~~.0000{{c}}, ~3/2 = 705.2170{{c}}

* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 705.2170{{c}}

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Optimal tunings:

* WE: ~25/21 = 299.7538{{c}}, ~3/2 = 704.7222{{c}}

* CWE: ~25/21 = ~~600~~.0000{{c}}, ~3/2 = 705.3241{{c}}

* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 705.3241{{c}}

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

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Optimal tunings:

* WE: ~25/21 = 299.7883{{c}}, ~3/2 = 704.8365{{c}}

* CWE: ~25/21 = ~~600~~.0000{{c}}, ~3/2 = 705.3496{{c}}

* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 705.3496{{c}}

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Optimal tunings:

* WE: ~19/16 = 299.7967{{c}}, ~3/2 = 704.8609{{c}}

* CWE: ~19/16 = ~~600~~.0000{{c}}, ~3/2 = 705.3519{{c}}

* CWE: ~19/16 = 300.0000{{c}}, ~3/2 = 705.3519{{c}}

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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(1/7) generator can again be used as tunings.

Additionally, shrutar can employ the standard diaschismic mapping of prime 17, and most naturally represents the 2.3.5.7.11.17 subgroup temperament where 15:16:17:18 and 32:33:34:35:36 are equalized. Shrutar canonically maps primes 13, 19, and 23 as the 46 & 68 temperament; these mappings are significantly more complex and need finer tuning, however.

[[Subgroup]]: 2.3.5.7

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

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Badness (Sintel): 1.16

=== 17-limit ===

==== 17-limit ====

Subgroup: 2.3.5.7.11.13.17

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Badness (Sintel): 0.953

=== 19-limit ===

==== 19-limit ====

Subgroup: 2.3.5.7.11.13.17.19

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Badness (Sintel): 1.07

=== 23-limit ===

==== 23-limit ====

Subgroup: 2.3.5.7.11.13.17.19.23

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

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

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Line 1,072:

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

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Line 1,143:

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

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Line 1,213:

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

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Line 1,282:

: mapping generators: ~45/32, ~21/16

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Line 1,383:

Badness (Sintel): 1.13

== Subgroup extensions ==

=== Srutal archagall (2.3.5.17) ===

Subgroup: 2.3.5.17

Comma list: 136/135, 256/255

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

: mapping generators: ~17/12, ~3

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 }}

Badness (Sintel): 0.212

[[Category:Temperament families]]

~~[[Category:Pages with mostly numerical content]]~~

[[Category:Diaschismic family| ]]

~~[[Category:Diaschismic| ]] ~~

[[Category:Rank 2]]

@@ Line 14: / Line 14: @@
 {{Mapping|legend=1| 2 0 11 | 0 1 -2 }}
 : mapping generators: ~45/32, ~3
@@ Line 35: / Line 34: @@
 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.
-* Pajara derives from [[64/63]] and is a popular and well-known choice.
+* Pajara adds [[50/49]] or [[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.
 * Keen adds [[875/864]].
@@ Line 41: / Line 40: @@
 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 ====
-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'', considered in [[#Subgroup extensions]]. The [[S-expression]]-based comma list of this temperament is {[[256/255|S16]], [[289/288|S17]]}.
-=== Srutal archagall ===
-{{See also | Fiventeen }}
-Subgroup: 2.3.5.17
-Comma list: 136/135, 256/255
-Subgroup-val mapping: {{mapping| 2 0 11 5 | 0 1 -2 1 }}
-: mapping generators: ~17/12, ~3
-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 }}
-Badness (Sintel): 0.212
 == Septimal diaschismic ==
@@ Line 69: / Line 49: @@
 {{See also| Srutal vs diaschismic }}
-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.
+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]] with the 104c [[val]] 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. This mapping can also be rationalized by [[parapyth]], which makes sense due to the sharp fifth, and prime 17 is found as in srutal archagall. If you want to explore higher-limit harmonies, diaschismic is certainly one excellent way to do it; [[mos]] scales 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.
 [[Subgroup]]: 2.3.5.7
@@ Line 194: / Line 174: @@
 [[Badness]] (Sintel): 0.507
+=== 2.3.5.7.17 subgroup ===
+Subgroup: 2.3.5.7.17
+Comma list: 50/49, 64/63, 85/84
+Mapping: {{mapping| 2 0 11 12 5 | 0 1 -2 -2 1 }}
+Optimal tunings:
+* WE: ~7/5 = 599.053{{c}}, ~3/2 = 706.355{{c}}
+* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 707.607{{c}}
+{{Optimal ET sequence|legend=0| 10, 12, 22, 56d }}
+Badness (Sintel): 0.438
 === 11-limit ===
@@ Line 213: / Line 208: @@
 Badness (Sintel): 0.673
+==== 2.3.5.7.11.17 subgroup ====
+Subgroup: 2.3.5.7.11.17
+Comma list: 50/49, 64/63, 85/84, 99/98
+Mapping: {{mapping| 2 0 11 12 26 5 | 0 1 -2 -2 -6 1 }}
+Optimal tunings:
+* WE: ~7/5 = 599.062{{c}}, ~3/2 = 706.095{{c}}
+* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 707.370{{c}}
+{{Optimal ET sequence|legend=0| 10e, 12, 22, 34d, 56d }}
+Badness (Sintel): 0.645
 ==== 13-limit ====
@@ Line 322: / Line 332: @@
 Badness (Sintel): 0.937
+==== 2.3.5.7.11.17 subgroup ====
+Subgroup: 2.3.5.7.11.13.17
+Comma list: 50/49, 52/51, 55/54, 64/63, 65/63
+Mapping: {{mapping| 2 0 11 12 -9 1 5 | 0 1 -2 -2 5 2 1 }}
+Optimal tunings:
+* WE: ~7/5 = 599.408{{c}}, ~3/2 = 708.878{{c}}
+* CWE: ~7/5 = 600.000{{c}}, ~3/2 = 709.544{{c}}
+{{Optimal ET sequence|legend=0| 10, 12e, 22 }}
+Badness (Sintel): 0.766
 ==== 13-limit ====
@@ Line 451: / Line 476: @@
 Mapping: {{mapping| 2 0 11 12 -1 | 0 2 -4 -4 5 }}
 : mapping generators: ~2, ~55/32
@@ Line 739: / Line 763: @@
 {{Mapping|legend=1| 4 0 22 43 | 0 1 -2 -5 }}
 : mapping generators: ~25/21, ~3
@@ Line 761: / Line 784: @@
 Optimal tunings:
 * WE: ~25/21 = 299.6809{{c}}, ~3/2 = 704.3367{{c}}
-* CWE: ~25/21 = 600.0000{{c}}, ~3/2 = 705.2170{{c}}
+* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 705.2170{{c}}
 {{Optimal ET sequence|legend=0| 12, 56e, 68, 80 }}
@@ Line 776: / Line 799: @@
 Optimal tunings:
 * WE: ~25/21 = 299.7538{{c}}, ~3/2 = 704.7222{{c}}
-* CWE: ~25/21 = 600.0000{{c}}, ~3/2 = 705.3241{{c}}
+* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 705.3241{{c}}
 {{Optimal ET sequence|legend=0| 12, 68, 80, 148d, 228bcd, 376bbcddf }}
@@ Line 791: / Line 814: @@
 Optimal tunings:
 * WE: ~25/21 = 299.7883{{c}}, ~3/2 = 704.8365{{c}}
-* CWE: ~25/21 = 600.0000{{c}}, ~3/2 = 705.3496{{c}}
+* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 705.3496{{c}}
 {{Optimal ET sequence|legend=0| 12, 68, 80, 148d }}
@@ Line 806: / Line 829: @@
 Optimal tunings:
 * WE: ~19/16 = 299.7967{{c}}, ~3/2 = 704.8609{{c}}
-* CWE: ~19/16 = 600.0000{{c}}, ~3/2 = 705.3519{{c}}
+* CWE: ~19/16 = 300.0000{{c}}, ~3/2 = 705.3519{{c}}
 {{Optimal ET sequence|legend=0| 12, 68, 80, 148d }}
@@ Line 831: / Line 854: @@
 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.
+Additionally, shrutar can employ the standard diaschismic mapping of prime 17, and most naturally represents the 2.3.5.7.11.17 subgroup temperament where 15:16:17:18 and 32:33:34:35:36 are equalized. Shrutar canonically maps primes 13, 19, and 23 as the 46 & 68 temperament; these mappings are significantly more complex and need finer tuning, however.
 [[Subgroup]]: 2.3.5.7
@@ Line 837: / Line 862: @@
 {{Mapping|legend=1| 2 1 9 -2 | 0 2 -4 7 }}
 : mapping generators: ~45/32, ~35/24
@@ Line 880: / Line 904: @@
 Badness (Sintel): 1.16
-=== 17-limit ===
+==== 17-limit ====
 Subgroup: 2.3.5.7.11.13.17
@@ Line 895: / Line 919: @@
 Badness (Sintel): 0.953
-=== 19-limit ===
+==== 19-limit ====
 Subgroup: 2.3.5.7.11.13.17.19
@@ Line 910: / Line 934: @@
 Badness (Sintel): 1.07
-=== 23-limit ===
+==== 23-limit ====
 Subgroup: 2.3.5.7.11.13.17.19.23
@@ Line 933: / Line 957: @@
 {{Mapping|legend=1| 2 1 9 11 | 0 2 -4 -5 }}
 : mapping generators: ~45/32, ~10/7
@@ Line 984: / Line 1,007: @@
 {{Mapping|legend=1| 2 0 11 -15 | 0 2 -4 13 }}
 : mapping generators: ~45/32, ~140/81
@@ Line 1,050: / Line 1,072: @@
 {{Mapping|legend=1| 2 0 11 4 | 0 2 -4 1 }}
 : mapping generators: ~45/32, ~7/4
@@ Line 1,122: / Line 1,143: @@
 {{Mapping|legend=1| 2 1 9 2 | 0 3 -6 5 }}
 : mapping generators: ~45/32, ~9/7
@@ Line 1,193: / Line 1,213: @@
 {{Mapping|legend=1| 2 2 7 6 | 0 3 -6 -1 }}
 : mapping generators: ~45/32, ~8/7
@@ Line 1,263: / Line 1,282: @@
 {{Mapping|legend=1| 2 0 11 8 | 0 4 -8 -3 }}
 : mapping generators: ~45/32, ~21/16
@@ Line 1,365: / Line 1,383: @@
 Badness (Sintel): 1.13
+== Subgroup extensions ==
+=== Srutal archagall (2.3.5.17) ===
+{{See also | Fiventeen }}
+Subgroup: 2.3.5.17
+Comma list: 136/135, 256/255
+Subgroup-val mapping: {{mapping| 2 0 11 5 | 0 1 -2 1 }}
+: mapping generators: ~17/12, ~3
+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 }}
+Badness (Sintel): 0.212
 [[Category:Temperament families]]
-[[Category:Pages with mostly numerical content]]
 [[Category:Diaschismic family| ]] <!-- main article -->
-[[Category:Diaschismic| ]] <!-- key article -->
 [[Category:Rank 2]]