Buzzardsmic clan: Difference between revisions

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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~21/16 = 475.727{{c}}
* [[WE]]: ~2 = 1199.2548{{c}}, ~21/16 = 475.5761{{c}}
* [[CWE]]: ~2 = 1200.000{{c}}, ~21/16 = 475.833{{c}}
: [[error map]]: {{val| -0.745 +0.350 +1.465 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/16 = 475.8328{{c}}
: error map: {{val| 0.000 +1.376 +3.676 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~21/16 = 475.727{{c}} -->


{{Optimal ET sequence|legend=1| 5, 33, 38, 43, 48, 53, 58 }}
{{Optimal ET sequence|legend=1| 5, 33, 38, 43, 48, 53, 58 }}
Line 33: Line 36:


= Strong extensions =
= Strong extensions =
== Septimal buzzard ==
== Septimal buzzard ==
{{Main| Buzzard }}
{{Main| Buzzard }}
Line 49: Line 51:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~2 = 1200.000{{c}}, ~21/16 = 475.555{{c}}
* [[WE]]: ~2 = 1199.3061{{c}}, ~21/16 = 475.3611{{c}}
: [[error map]]: {{val| -0.694 -0.511 +0.432 +2.315 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/16 = 475.6144{{c}}
: error map: {{val| 0.000 +0.503 +1.589 +4.331 }}
<!-- * [[CTE]]: ~2 = 1200.000{{c}}, ~21/16 = 475.555{{c}}
: [[error map]]: {{val| 0.000 +0.263 +0.333 +4.510 }}
: [[error map]]: {{val| 0.000 +0.263 +0.333 +4.510 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~21/16 = 475.636{{c}}
* [[POTE]]: ~2 = 1200.000{{c}}, ~21/16 = 475.636{{c}}
: error map: {{val| 0.000 +0.589 +2.045 +4.266 }}
: error map: {{val| 0.000 +0.589 +2.045 +4.266 }} -->


{{Optimal ET sequence|legend=1| 5, 48, 53, 111, 164d, 275d }}
{{Optimal ET sequence|legend=1| 5, 48, 53, 111, 164d, 275d }}


[[Badness]] (Smith): 0.047963
[[Badness]] (Sintel): 1.21


=== 11-limit ===
=== 11-limit ===
Line 66: Line 72:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~21/16 = 475.625{{c}}
* WE: ~2 = 1199.2516{{c}}, ~21/16 = 475.4037{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/16 = 475.700{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 475.6806{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~21/16 = 475.625{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/16 = 475.700{{c}} -->


{{Optimal ET sequence|legend=0| 53, 58, 111, 280cd }}
{{Optimal ET sequence|legend=0| 53, 58, 111, 280cd }}


Badness (Smith): 0.034484
Badness (Sintel): 1.14


==== 13-limit ====
==== 13-limit ====
Line 81: Line 89:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~21/16 = 475.615{{c}}
* WE: ~2 = 1199.2391{{c}}, ~21/16 = 475.3956{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/16 = 475.697{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 475.6760{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~21/16 = 475.615{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/16 = 475.697{{c}} -->


{{Optimal ET sequence|legend=0| 53, 58, 111, 280cdf }}
{{Optimal ET sequence|legend=0| 53, 58, 111, 280cdf }}


Badness (Smith): 0.018842
Badness (Sintel): 0.779


==== 17-limit ====
==== 17-limit ====
Line 96: Line 106:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~21/16 = 475.638{{c}}
* WE: ~2 = 1199.2723{{c}}, ~21/16 = 475.4039{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/16 = 475.692{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 475.6837{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~21/16 = 475.638{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/16 = 475.692{{c}} -->


{{Optimal ET sequence|legend=0| 53, 58, 111 }}
{{Optimal ET sequence|legend=0| 53, 58, 111 }}


Badness (Smith): 0.018403
Badness (Sintel): 0.938


==== 19-limit ====
==== 19-limit ====
Line 111: Line 123:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~21/16 = 475.617{{c}}
* WE: ~2 = 1199.2457{{c}}, ~21/16 = 475.3797{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/16 = 475.679{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 475.6690{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~21/16 = 475.617{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/16 = 475.679{{c}} -->


{{Optimal ET sequence|legend=0| 53, 58h, 111 }}
{{Optimal ET sequence|legend=0| 53, 58h, 111 }}


Badness (Smith): 0.015649
Badness (Sintel): 0.952


=== Buteo ===
=== Buteo ===
Line 126: Line 140:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~21/16 = 475.454{{c}}
* WE: ~2 = 1200.2867{{c}}, ~21/16 = 475.5498{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/16 = 475.436{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 475.4393{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~21/16 = 475.454{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/16 = 475.436{{c}} -->


{{Optimal ET sequence|legend=0| 5, 48, 53 }}
{{Optimal ET sequence|legend=0| 5, 48, 53 }}


Badness (Smith): 0.060238
Badness (Sintel): 1.99


==== 13-limit ====
==== 13-limit ====
Line 141: Line 157:


Optimal tunings:  
Optimal tunings:  
* CTE: ~2 = 1200.000{{c}}, ~21/16 = 475.495{{c}}
* WE: ~2 = 1200.3416{{c}}, ~21/16 = 475.5998{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/16 = 475.464{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 475.4696{{c}}
<!-- * CTE: ~2 = 1200.000{{c}}, ~21/16 = 475.495{{c}}
* POTE: ~2 = 1200.000{{c}}, ~21/16 = 475.464{{c}} -->


{{Optimal ET sequence|legend=0| 5, 48f, 53 }}
{{Optimal ET sequence|legend=0| 5, 48f, 53 }}


Badness (Smith): 0.039854
Badness (Sintel): 1.65


== Subfourth ==
== Subfourth ==
Line 155: Line 173:
{{Mapping|legend=1| 1 0 17 4 | 0 4 -37 -3 }}
{{Mapping|legend=1| 1 0 17 4 | 0 4 -37 -3 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~21/16 = 475.991{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.1804{{c}}, ~21/16 = 475.6659{{c}}
: [[error map]]: {{val| -0.820 +0.709 +0.113 +0.898 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/16 = 476.0019{{c}}
: error map: {{val| 0.000 +2.052 +1.617 +3.168 }}
<!-- * [[POTE]]: ~2 = 1200.000{{c}}, ~21/16 = 475.991{{c}} -->


{{Optimal ET sequence|legend=1| 58, 121, 179, 300bd, 479bcd }}
{{Optimal ET sequence|legend=1| 58, 121, 179, 300bd, 479bcdd }}


[[Badness]] (Smith): 0.140722
[[Badness]] (Sintel): 3.56


=== 11-limit ===
=== 11-limit ===
Line 168: Line 191:
Mapping: {{mapping| 1 0 17 4 11 | 0 4 -37 -3 -19 }}
Mapping: {{mapping| 1 0 17 4 11 | 0 4 -37 -3 -19 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/16 = 475.995{{c}}
Optimal tunings:
* WE: ~2 = 1199.0801{{c}}, ~21/16 = 475.6303{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 476.0088{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~21/16 = 475.995{{c}} -->


{{Optimal ET sequence|legend=1| 58, 121, 179e, 300bde }}
{{Optimal ET sequence|legend=0| 58, 121, 179e, 300bdee, 479bcddeee }}


Badness (Smith): 0.045323
Badness (Sintel): 1.50


=== 13-limit ===
=== 13-limit ===
Line 181: Line 207:
Mapping: {{mapping| 1 0 17 4 11 16 | 0 4 -37 -3 -19 -31 }}
Mapping: {{mapping| 1 0 17 4 11 16 | 0 4 -37 -3 -19 -31 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~21/16 = 475.996{{c}}
Optimal tunings:
* WE: ~2 = 1199.0747{{c}}, ~21/16 = 475.6291{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 476.0113{{c}}
<!-- * POTE: ~2 = 1200.000{{c}}, ~21/16 = 475.996{{c}} -->


{{Optimal ET sequence|legend=1| 58, 121, 179ef, 300bdef }}
{{Optimal ET sequence|legend=0| 58, 121, 179ef, 300bdeef }}


Badness (Smith): 0.023800
Badness (Sintel): 0.983


== Lemongrass ==
== Lemongrass ==
Line 194: Line 223:
{{Mapping|legend=1| 1 0 17 4 | 0 4 26 -3 }}
{{Mapping|legend=1| 1 0 17 4 | 0 4 26 -3 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.0000{{c}}, ~21/16 = 476.4448{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.0957{{c}}, ~21/16 = 476.0857{{c}}
: [[error map]]: {{val| -0.904 +2.388 -0.851 -0.700 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/16 = 476.4221{{c}}
: error map: {{val| 0.000 +3.733 +0.660 +1.908 }}
<!-- * [[POTE]]: ~2 = 1200.0000{{c}}, ~21/16 = 476.4448{{c}} -->


{{Optimal ET sequence|legend=1| 5, …, 63, 68 }}
{{Optimal ET sequence|legend=1| 5, …, 63, 68 }}


[[Badness]] (Sintel): 2.902
[[Badness]] (Sintel): 2.90


= Weak extensions =
= Weak extensions =
Line 229: Line 263:
Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~16/13 = 362.242{{c}}
Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~16/13 = 362.242{{c}}


{{Optimal ET sequence|legend=1| 10, 33, 43, 53 }}
{{Optimal ET sequence|legend=0| 10, 33, 43, 53 }}


Badness (Sintel): 0.847
Badness (Sintel): 0.847
Line 244: Line 278:
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~27/22 = 362.101{{c}}
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~27/22 = 362.101{{c}}


{{Optimal ET sequence|legend=1| 10, 43e, 53, 116, 169de, 285cde }}
{{Optimal ET sequence|legend=0| 10, 43e, 53, 116, 169de, 285cde }}


Badness (Smith): 0.050582
Badness (Smith): 0.050582
Line 257: Line 291:
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~16/13 = 362.105{{c}}
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~16/13 = 362.105{{c}}


{{Optimal ET sequence|legend=1| 10, 43e, 53, 116, 169de, 285cdef }}
{{Optimal ET sequence|legend=0| 10, 43e, 53, 116, 169de, 285cdef }}


Badness (Smith): 0.027689
Badness (Smith): 0.027689
Line 270: Line 304:
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/40 = 362.418{{c}}
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/40 = 362.418{{c}}


{{Optimal ET sequence|legend=1| 43, 53, 96, 149d }}
{{Optimal ET sequence|legend=0| 43, 53, 96, 149d }}


Badness (Smith): 0.051806
Badness (Smith): 0.051806
Line 283: Line 317:
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~16/13 = 362.402{{c}}
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~16/13 = 362.402{{c}}


{{Optimal ET sequence|legend=1| 43, 53, 96, 149d }}
{{Optimal ET sequence|legend=0| 43, 53, 96, 149d }}


Badness (Smith): 0.029680
Badness (Smith): 0.029680
Line 312: Line 346:
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/8 = 558.620{{c}}
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/8 = 558.620{{c}}


{{Optimal ET sequence|legend=1| 15, 43, 58 }}
{{Optimal ET sequence|legend=0| 15, 43, 58 }}


Badness (Smith): 0.033078
Badness (Smith): 0.033078
Line 325: Line 359:
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/8 = 558.589{{c}}
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/8 = 558.589{{c}}


{{Optimal ET sequence|legend=1| 15, 43, 58 }}
{{Optimal ET sequence|legend=0| 15, 43, 58 }}


Badness (Smith): 0.022838
Badness (Smith): 0.022838
Line 338: Line 372:
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/8 = 558.509{{c}}
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/8 = 558.509{{c}}


{{Optimal ET sequence|legend=1| 15, 43, 58 }}
{{Optimal ET sequence|legend=0| 15, 43, 58 }}


Badness (Smith): 0.022293
Badness (Smith): 0.022293
Line 351: Line 385:
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/8 = 558.504{{c}}
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/8 = 558.504{{c}}


{{Optimal ET sequence|legend=1| 15, 43, 58h }}
{{Optimal ET sequence|legend=0| 15, 43, 58h }}


Badness (Smith): 0.018938
Badness (Smith): 0.018938
Line 364: Line 398:
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/8 = 558.522{{c}}
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/8 = 558.522{{c}}


{{Optimal ET sequence|legend=1| 15, 43, 58hi }}
{{Optimal ET sequence|legend=0| 15, 43, 58hi }}


Badness (Smith): 0.016581
Badness (Smith): 0.016581
Line 379: Line 413:
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/8 = 558.520{{c}}
Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/8 = 558.520{{c}}


{{Optimal ET sequence|legend=1| 15, 43, 58hi }}
{{Optimal ET sequence|legend=0| 15, 43, 58hi }}


Badness (Smith): 0.013762
Badness (Smith): 0.013762
Line 395: Line 429:
Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~5/4 = 384.856{{c}}
Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~5/4 = 384.856{{c}}


{{Optimal ET sequence|legend=1| 25, 53, 184, 237d, 290d, 343dd }}
{{Optimal ET sequence|legend=0| 25, 53, 184, 237d, 290d, 343dd }}


Badness (Sintel): 4.571
Badness (Sintel): 4.571

Revision as of 15:03, 27 October 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The 2.3.7-subgroup comma for the buzzardsmic clan is the buzzardsma, 65536/64827, with monzo [16 -3 0 -4⟩, which implies that the tritave, 3/1, is divided into four intervals each representing a 21/16 subfourth. Tempering out this comma implies a sharpened 7th harmonic, and especially a sharpened ~21/16 generator, which approaches the 480 ¢ fourth of 5edo.

Extensions of buzzard to incorporate prime 5 along its chain of generators (and therefore the full 7-limit) include septimal buzzard (53 & 58), which tempers out 1728/1715 (and 5120/5103); subfourth (58 & 63), which tempers out 10976/10935; and lemongrass (63 & 68), which tempers out 245/243. All are considered below.

Weak extensions include submajor (10 & 43), which tempers out 225/224 and splits 32/21 (the superfifth) in two; and thuja (15 & 43), which tempers out 126/125 and splits 21/8 into three.

Full 7-limit temperaments discussed elsewhere are:

The rest are considered below.

2.3.7 subgroup

Buzzard

Main article: Buzzard

Subgroup: 2.3.7

Comma list: 65536/64827

Mapping: [⟨1 0 4], ⟨0 4 -3]]

Optimal tunings:

  • WE: ~2 = 1199.2548 ¢, ~21/16 = 475.5761 ¢
error map: ⟨-0.745 +0.350 +1.465]
  • CWE: ~2 = 1200.0000 ¢, ~21/16 = 475.8328 ¢
error map: ⟨0.000 +1.376 +3.676]

Optimal ET sequence: 5, 33, 38, 43, 48, 53, 58

Badness (Sintel): 0.824

Strong extensions

Septimal buzzard

Main article: Buzzard
See also: Vulture family

Septimal buzzard is not only a naturally motivated extension to 2.3.7 buzzard, but the main extension to vulture of practical interest, finding prime 7 at only 3 generators down so that the generator is interpreted as a sharp ~21/16, though buzzard is powerful as a full 13-limit system in its own right. It is most naturally described as 53 & 58 (though 48edo is an interesting higher-damage tuning of it for some purposes). As one might expect, 111edo (111 = 53 + 58) is a great tuning for it. Mos scales of 5, 8, 13, 18, 23, 28, 33, 38, 43, 48 or 53 notes are available.

Its 13-limit S-expression-based comma list is {S6/S7, S8/S9, S11/S13, S13/S15}, with the structure of its 7-limit implied by the first two equivalences combined with the nontrivial JI equivalence S6 = S8 × S9. Hemifamity leverages it by splitting 36/35 into two syntonic~septimal commas, so buzzard naturally finds an interval between 6/5 and 7/6 which in the 7-limit is 32/27 and in the 13-limit is 13/11. Then the vanishing of the orwellisma implies 49/48, the large septimal diesis, is equated with 36/35, so 49/48 is also split into two so that the system also finds an interval between 7/6 and 8/7 which in the 7-limit is 7/6 inflected down by a comma or 8/7 inflected up by a comma, and in the 13-limit is 15/13, so that it is clear this system naturally wants to be extended to and interpreted in the full 13-limit.

Subgroup: 2.3.5.7

Comma list: 1728/1715, 5120/5103

Mapping: [⟨1 0 -6 4], ⟨0 4 21 -3]]

Optimal tunings:

  • WE: ~2 = 1199.3061 ¢, ~21/16 = 475.3611 ¢
error map: ⟨-0.694 -0.511 +0.432 +2.315]
  • CWE: ~2 = 1200.0000 ¢, ~21/16 = 475.6144 ¢
error map: ⟨0.000 +0.503 +1.589 +4.331]

Optimal ET sequence: 5, 48, 53, 111, 164d, 275d

Badness (Sintel): 1.21

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 540/539, 5120/5103

Mapping: [⟨1 0 -6 4 -12], ⟨0 4 21 -3 39]]

Optimal tunings:

  • WE: ~2 = 1199.2516 ¢, ~21/16 = 475.4037 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/16 = 475.6806 ¢

Optimal ET sequence: 53, 58, 111, 280cd

Badness (Sintel): 1.14

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 540/539, 676/675

Mapping: [⟨1 0 -6 4 -12 -7], ⟨0 4 21 -3 39 27]]

Optimal tunings:

  • WE: ~2 = 1199.2391 ¢, ~21/16 = 475.3956 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/16 = 475.6760 ¢

Optimal ET sequence: 53, 58, 111, 280cdf

Badness (Sintel): 0.779

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 176/175, 256/255, 351/350, 442/441, 540/539

Mapping: [⟨1 0 -6 4 -12 -7 14], ⟨0 4 21 -3 39 27 -25]]

Optimal tunings:

  • WE: ~2 = 1199.2723 ¢, ~21/16 = 475.4039 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/16 = 475.6837 ¢

Optimal ET sequence: 53, 58, 111

Badness (Sintel): 0.938

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 176/175, 256/255, 286/285, 324/323, 351/350, 540/539

Mapping: [⟨1 0 -6 4 -12 -7 14 -12], ⟨0 4 21 -3 39 27 -25 41]]

Optimal tunings:

  • WE: ~2 = 1199.2457 ¢, ~21/16 = 475.3797 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/16 = 475.6690 ¢

Optimal ET sequence: 53, 58h, 111

Badness (Sintel): 0.952

Buteo

Subgroup: 2.3.5.7.11

Comma list: 99/98, 385/384, 2200/2187

Mapping: [⟨1 0 -6 4 9], ⟨0 4 21 -3 -14]]

Optimal tunings:

  • WE: ~2 = 1200.2867 ¢, ~21/16 = 475.5498 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/16 = 475.4393 ¢

Optimal ET sequence: 5, 48, 53

Badness (Sintel): 1.99

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 275/273, 385/384, 572/567

Mapping: [⟨1 0 -6 4 9 -7], ⟨0 4 21 -3 -14 27]]

Optimal tunings:

  • WE: ~2 = 1200.3416 ¢, ~21/16 = 475.5998 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/16 = 475.4696 ¢

Optimal ET sequence: 5, 48f, 53

Badness (Sintel): 1.65

Subfourth

Subgroup: 2.3.5.7

Comma list: 10976/10935, 65536/64827

Mapping: [⟨1 0 17 4], ⟨0 4 -37 -3]]

Optimal tunings:

  • WE: ~2 = 1199.1804 ¢, ~21/16 = 475.6659 ¢
error map: ⟨-0.820 +0.709 +0.113 +0.898]
  • CWE: ~2 = 1200.0000 ¢, ~21/16 = 476.0019 ¢
error map: ⟨0.000 +2.052 +1.617 +3.168]

Optimal ET sequence: 58, 121, 179, 300bd, 479bcdd

Badness (Sintel): 3.56

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 896/891, 12005/11979

Mapping: [⟨1 0 17 4 11], ⟨0 4 -37 -3 -19]]

Optimal tunings:

  • WE: ~2 = 1199.0801 ¢, ~21/16 = 475.6303 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/16 = 476.0088 ¢

Optimal ET sequence: 58, 121, 179e, 300bdee, 479bcddeee

Badness (Sintel): 1.50

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 364/363, 540/539, 676/675

Mapping: [⟨1 0 17 4 11 16], ⟨0 4 -37 -3 -19 -31]]

Optimal tunings:

  • WE: ~2 = 1199.0747 ¢, ~21/16 = 475.6291 ¢
  • CWE: ~2 = 1200.0000 ¢, ~21/16 = 476.0113 ¢

Optimal ET sequence: 58, 121, 179ef, 300bdeef

Badness (Sintel): 0.983

Lemongrass

Subgroup: 2.3.5.7

Comma list: 245/243, 65536/64827

Mapping: [⟨1 0 17 4], ⟨0 4 26 -3]]

Optimal tunings:

  • WE: ~2 = 1199.0957 ¢, ~21/16 = 476.0857 ¢
error map: ⟨-0.904 +2.388 -0.851 -0.700]
  • CWE: ~2 = 1200.0000 ¢, ~21/16 = 476.4221 ¢
error map: ⟨0.000 +3.733 +0.660 +1.908]

Optimal ET sequence: 5, …, 63, 68

Badness (Sintel): 2.90

Weak extensions

Submajor

7-limit

Subgroup: 2.3.5.7

Comma list: 225/224, 51200/50421

Mapping: [⟨1 4 -1 1], ⟨0 -8 11 6]]

mapping generators: ~2, ~49/40

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~49/40 = 362.255 ¢

Optimal ET sequence: 10, 33, 43, 53

Badness (Smith): 0.060533

2.3.5.7.13 subgroup

See also: Greater tendoneutralic

This temperament naturally comes about from a structure in edos like 43 and 53 where two flattened ~13/8 intervals reach the buzzard generator of ~21/16, two of which produce a semitritave (that can here be equated to 26/15, providing a mapping of 5 significantly less complex than the vulture mapping), and two of those finally reach 3/1.

Subgroup: 2.3.5.7.13

Comma list: 169/168, 225/224, 640/637

Mapping: [⟨1 4 -1 1 4], ⟨0 -8 11 6 -1]]

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~16/13 = 362.242 ¢

Optimal ET sequence: 10, 33, 43, 53

Badness (Sintel): 0.847

11-limit

Submajor diverges into two extensions to prime 11: this one favoring sharp fifths, and interpental, favoring flat fifths; the two mappings meet at 53edo.

Subgroup: 2.3.5.7.11

Comma list: 225/224, 385/384, 6655/6561

Mapping: [⟨1 4 -1 1 11], ⟨0 -8 11 6 -25]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~27/22 = 362.101 ¢

Optimal ET sequence: 10, 43e, 53, 116, 169de, 285cde

Badness (Smith): 0.050582

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 225/224, 275/273, 385/384

Mapping: [⟨1 4 -1 1 11 4], ⟨0 -8 11 6 -25 -1]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~16/13 = 362.105 ¢

Optimal ET sequence: 10, 43e, 53, 116, 169de, 285cdef

Badness (Smith): 0.027689

Interpental

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 51200/50421

Mapping: [⟨1 4 -1 1 -5], ⟨0 -8 11 6 28]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~49/40 = 362.418 ¢

Optimal ET sequence: 43, 53, 96, 149d

Badness (Smith): 0.051806

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 169/168, 176/175, 640/637

Mapping: [⟨1 4 -1 1 -5 4], ⟨0 -8 11 6 28 -1]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~16/13 = 362.402 ¢

Optimal ET sequence: 43, 53, 96, 149d

Badness (Smith): 0.029680

Thuja

For the 5-limit version, see Miscellaneous 5-limit temperaments #Thuja.

Subgroup: 2.3.5.7

Comma list: 126/125, 65536/64827

Mapping: [⟨1 -4 0 7], ⟨0 12 5 -9]]

mapping generators: ~2, ~175/128

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~175/128 = 558.605 ¢

Optimal ET sequence: 15, 43, 58

Badness (Smith): 0.088441

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 176/175, 1344/1331

Mapping: [⟨1 -4 0 7 3], ⟨0 12 5 -9 1]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~11/8 = 558.620 ¢

Optimal ET sequence: 15, 43, 58

Badness (Smith): 0.033078

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 126/125, 144/143, 176/175, 364/363

Mapping: [⟨1 -4 0 7 3 -7], ⟨0 12 5 -9 1 23]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~11/8 = 558.589 ¢

Optimal ET sequence: 15, 43, 58

Badness (Smith): 0.022838

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 126/125, 144/143, 176/175, 221/220, 256/255

Mapping: [⟨1 -4 0 7 3 -7 12], ⟨0 12 5 -9 1 23 -17]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~11/8 = 558.509 ¢

Optimal ET sequence: 15, 43, 58

Badness (Smith): 0.022293

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 96/95, 126/125, 144/143, 153/152, 176/175, 221/220

Mapping: [⟨1 -4 0 7 3 -7 12 1], ⟨0 12 5 -9 1 23 -17 7]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~11/8 = 558.504 ¢

Optimal ET sequence: 15, 43, 58h

Badness (Smith): 0.018938

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 96/95, 126/125, 144/143, 153/152, 176/175, 221/220, 231/230

Mapping: [⟨1 -4 0 7 3 -7 12 1 5], ⟨0 12 5 -9 1 23 -17 7 -1]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~11/8 = 558.522 ¢

Optimal ET sequence: 15, 43, 58hi

Badness (Smith): 0.016581

29-limit

The raison d'etre of this entry is the simple and accurate approximation of factor twenty-nine, the 2.5.11.21.29 subgroup being of especially good accuracy and simplicity.

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 96/95, 116/115, 126/125, 144/143, 153/152, 176/175, 221/220, 231/230

Mapping: [⟨1 -4 0 7 3 -7 12 1 5 3], ⟨0 12 5 -9 1 23 -17 7 -1 4]]

Optimal tuning (POTE): ~2 = 1200.000 ¢, ~11/8 = 558.520 ¢

Optimal ET sequence: 15, 43, 58hi

Badness (Smith): 0.013762

Anthoine

Anthoine is generated by 5/4 and tempers out 3125/3087 in addition to the buzzardsma, so that 32/21 is found at 5 generators up. It is most notable as the 25 & 28 temperament and as the chain of 5/4's present in 53edo.

Subgroup: 2.3.5.7

Comma list: 3125/3087, 65536/64827

Mapping: [⟨1 8 2 -2], ⟨0 -20 1 15]]

mapping generators: ~2, ~5/4

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~5/4 = 384.856 ¢

Optimal ET sequence: 25, 53, 184, 237d, 290d, 343dd

Badness (Sintel): 4.571

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