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Buzzardsmic clan: Difference between revisions

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Extensions of buzzard to incorporate [[prime interval|prime]] [[5/1|5]] along its chain of generators (and therefore the full [[7-limit]]) include septimal buzzard ({{nowrap| 53 & 58 }}), which tempers out [[1728/1715]] (and [[5120/5103]]); subfourth ({{nowrap| 58 & 63 }}), which tempers out [[10976/10935]]; and lemongrass ({{nowrap| 63 & 68 }}), which tempers out [[245/243]]. All are considered below.
Extensions of buzzard to incorporate [[prime interval|prime]] [[5/1|5]] along its chain of generators (and therefore the full [[7-limit]]) include septimal buzzard ({{nowrap| 53 & 58 }}), which tempers out [[1728/1715]] (and [[5120/5103]]); subfourth ({{nowrap| 58 & 63 }}), which tempers out [[10976/10935]]; and lemongrass ({{nowrap| 63 & 68 }}), which tempers out [[245/243]]. All are considered below.


Weak extensions include submajor ({{nowrap| 10 & 43 }}), which tempers out [[225/224]] and splits [[32/21]] (the superfifth) in two; and thuja ({{nowrap| 15 & 43 }}), which tempers out [[126/125]] and splits [[21/8]] into three.
Weak extensions include demibuzzard ({{nowrap| 10 & 53 }}), which tempers out [[225/224]] and splits [[32/21]] (the superfifth) in two; thuja ({{nowrap| 15 & 43 }}), which tempers out [[126/125]] and splits [[21/8]] into three; subsedia ({{nowrap| 10 & 111 }}), which tempers out [[16875/16807]] and splits [[21/16]] in four; and anthoine ({{nowrap| 25 & 53 }}), which tempers out [[3125/3087]] and splits [[21/2]] in five.  


Full 7-limit temperaments discussed elsewhere are:
Full 7-limit temperaments discussed elsewhere are:
* [[Blackwood]] (+28/27) → [[Limmic temperaments #Blackwood|Limmic temperaments]]
* [[Blackwood]] (+28/27) → [[Blackwood family #Blackwood|Blackwood family]]
* ''[[Quadrasruta]]'' (+2048/2025) → [[Diaschismic family #Quadrasruta|Diaschismic family]]
* ''[[Quadrasruta]]'' (+2048/2025) → [[Diaschismic family #Quadrasruta|Diaschismic family]]
* ''[[Hemikleismic]] (+4000/3969) → [[Kleismic family #Hemikleismic|Kleismic family]]
* ''[[Hemikleismic]] (+4000/3969) → [[Kleismic family #Hemikleismic|Kleismic family]]
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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 }}


{{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 35:


= Strong extensions =
= Strong extensions =
== Septimal buzzard ==
== Septimal buzzard ==
{{Main| Buzzard }}
{{Main| Buzzard }}
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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 {{nowrap| 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 scale]]s of 5, 8, 13, 18, 23, 28, 33, 38, 43, 48 or 53 notes are available.
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 {{nowrap| 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 scale]]s 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 {[[1728/1715|S6/S7]], [[5120/5103|S8/S9]], [[847/845|S11/S13]], [[676/675|S13/S15]]}, with the structure of its 7-limit implied by the first two equivalences combined with the nontrivial [[JI]] equivalence [[36/35|S6]] = [[64/63|S8]] × [[81/80|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.
Its 13-limit [[S-expression]]-based comma list is {[[1728/1715|S6/S7]], [[5120/5103|S8/S9]], [[847/845|S11/S13]], [[676/675|S13/S15]]}, with the structure of its 7-limit implied by the first two equivalences combined with the nontrivial [[JI]] equivalence [[36/35|S6]] = [[64/63|S8]] × [[81/80|S9]]. [[Aberschismic]] 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
[[Subgroup]]: 2.3.5.7
Line 49: Line 50:


[[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| 0.000 +0.263 +0.333 +4.510 }}
: [[error map]]: {{val| -0.694 -0.511 +0.432 +2.315 }}
* [[POTE]]: ~2 = 1200.000{{c}}, ~21/16 = 475.636{{c}}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/16 = 475.6144{{c}}
: error map: {{val| 0.000 +0.589 +2.045 +4.266 }}
: error map: {{val| 0.000 +0.503 +1.589 +4.331 }}


{{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 63: Line 64:
Comma list: 176/175, 540/539, 5120/5103
Comma list: 176/175, 540/539, 5120/5103


Mapping: {{mapping| 1 0 -6 4 -12 | 0 4 21 -3 39 }}
{{Mapping|legend=0| 1 0 -6 4 -12 | 0 4 21 -3 39 }}


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}}


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


Mapping: {{mapping| 1 0 -6 4 -12 -7 | 0 4 21 -3 39 27 }}
{{Mapping|legend=0| 1 0 -6 4 -12 -7 | 0 4 21 -3 39 27 }}


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}}


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


Mapping: {{mapping| 1 0 -6 4 -12 -7 14 | 0 4 21 -3 39 27 -25 }}
{{Mapping|legend=0| 1 0 -6 4 -12 -7 14 | 0 4 21 -3 39 27 -25 }}


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}}


{{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 108: Line 109:
Comma list: 176/175, 256/255, 286/285, 324/323, 351/350, 540/539
Comma list: 176/175, 256/255, 286/285, 324/323, 351/350, 540/539


Mapping: {{mapping| 1 0 -6 4 -12 -7 14 -12 | 0 4 21 -3 39 27 -25 41 }}
{{Mapping|legend=0| 1 0 -6 4 -12 -7 14 -12 | 0 4 21 -3 39 27 -25 41 }}


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}}


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


Mapping: {{mapping| 1 0 -6 4 9 | 0 4 21 -3 -14 }}
{{Mapping|legend=0| 1 0 -6 4 9 | 0 4 21 -3 -14 }}


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}}


{{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 138: Line 139:
Comma list: 99/98, 275/273, 385/384, 572/567
Comma list: 99/98, 275/273, 385/384, 572/567


Mapping: {{mapping| 1 0 -6 4 9 -7 | 0 4 21 -3 -14 27 }}
{{Mapping|legend=0| 1 0 -6 4 9 -7 | 0 4 21 -3 -14 27 }}


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}}


{{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 ==
Subfourth tempers out [[10976/10935]] and may be described as the {{nowrap| 58 & 63 }} temperament, more notable in the higher limits than the lower as it supplies a lot of essentially tempered chords there, including everything from [[parapyth]]. Among the good tunings are [[121edo]] and [[179edo]] using the 179ef val in the 13-limit.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 155: Line 158:
{{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 }}


{{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 166: Line 173:
Comma list: 540/539, 896/891, 12005/11979
Comma list: 540/539, 896/891, 12005/11979


Mapping: {{mapping| 1 0 17 4 11 | 0 4 -37 -3 -19 }}
{{Mapping|legend=0| 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}}


{{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 179: Line 188:
Comma list: 352/351, 364/363, 540/539, 676/675
Comma list: 352/351, 364/363, 540/539, 676/675


Mapping: {{mapping| 1 0 17 4 11 16 | 0 4 -37 -3 -19 -31 }}
{{Mapping|legend=0| 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}}


{{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 ==
Named by [[Lériendil]] in 2025, lemongrass tempers out [[245/243]] and may be described as the {{nowrap| 63 & 68 }} temperament. Characterized by a sharper generator than septimal buzzard, lemongrass compresses the septimal comma so much that the syntonic comma is no longer equated with it but with twice of it, or the large septimal diesis. [[68edo]] itself is a great tuning for this, though [[63edo]] and [[73edo]] are also possible.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 194: Line 207:
{{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 }}


{{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 =
== Submajor ==
== Demibuzzard ==
: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Demibuzzard]].''
 
Demibuzzard may be described as the {{nowrap| 10 & 53 }} temperament. It is generated by a submajor third; note that in the data below, the generator is the [[octave complement]], a supraminor sixth, since two of it minus an octave make buzzard's generator of ~21/16. The [[ploidacot]] for this temperament is epsilon-octacot.
 
This temperament naturally comes about from a structure in edos like [[43edo|43-]], [[53edo|53-]], and [[63edo]] 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]].
 
It diverges into two extensions for prime 11: submajor ({{nowrap| 53 & 63 }}) favoring sharp fifths, and interpental ({{nowrap| 43 & 53 }}), favoring flat fifths; the two mappings meet at [[53edo]]. Note that ''submajor'' (referring to the submajor third, not the supraminor sixth) used to be the name for the 7-limit temperament.
 
=== 7-limit ===
=== 7-limit ===
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 207: Line 232:
[[Comma list]]: 225/224, 51200/50421
[[Comma list]]: 225/224, 51200/50421


{{Mapping|legend=1| 1 4 -1 1 | 0 -8 11 6 }}
{{Mapping|legend=1| 1 -4 10 7 | 0 8 -11 -6 }}
: mapping generators: ~2, ~49/40
: mapping generators: ~2, ~80/49


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~49/40 = 362.255{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7399{{c}}, ~80/49 = 837.5637{{c}}
: [[error map]]: {{val| -0.260 -0.405 -2.116 +3.971 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~80/49 = 837.7471{{c}}
: error map: {{val| 0.000 +0.022 -1.532 +4.691 }}


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


[[Badness]] (Smith): 0.060533
[[Badness]] (Sintel): 1.53


==== 2.3.5.7.13 subgroup ====
==== 2.3.5.7.13 subgroup ====
{{See also| Greater tendoneutralic }}
{{See also| Greater tendoneutralic }}
This temperament naturally comes about from a structure in edos like [[43edo|43]] and [[53edo|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
Subgroup: 2.3.5.7.13
Line 225: Line 252:
Comma list: 169/168, 225/224, 640/637
Comma list: 169/168, 225/224, 640/637


Mapping: {{mapping| 1 4 -1 1 4 | 0 -8 11 6 -1 }}
{{Mapping|legend=0| 1 -4 10 7 3 | 0 8 -11 -6 1 }}


Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~16/13 = 362.242{{c}}
Optimal tunings:
* WE: ~2 = 1199.9444{{c}}, ~13/8 = 837.7178{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 837.7569{{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


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


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


Mapping: {{mapping| 1 4 -1 1 11 | 0 -8 11 6 -25 }}
{{Mapping|legend=0| 1 -4 10 7 -14 | 0 8 -11 -6 25 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~27/22 = 362.101{{c}}
Optimal tunings:
* WE: ~2 = 1200.0666{{c}}, ~44/27 = 837.9460{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~44/27 = 837.9000{{c}}


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


Badness (Smith): 0.050582
Badness (Sintel): 1.67


==== 13-limit ====
==== 13-limit ====
Line 253: Line 282:
Comma list: 169/168, 225/224, 275/273, 385/384
Comma list: 169/168, 225/224, 275/273, 385/384


Mapping: {{mapping| 1 4 -1 1 11 4 | 0 -8 11 6 -25 -1 }}
{{Mapping|legend=0| 1 -4 10 7 -14 3 | 0 8 -11 -6 25 1 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~16/13 = 362.105{{c}}
Optimal tunings:
* WE: ~2 = 1200.1769{{c}}, ~13/8 = 838.0187{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 837.8965{{c}}


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


Badness (Smith): 0.027689
Badness (Sintel): 1.14


=== Interpental ===
=== Interpental ===
Line 266: Line 297:
Comma list: 99/98, 176/175, 51200/50421
Comma list: 99/98, 176/175, 51200/50421


Mapping: {{mapping| 1 4 -1 1 -5 | 0 -8 11 6 28 }}
{{Mapping|legend=0| 1 -4 10 7 23 | 0 8 -11 -6 -28 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~49/40 = 362.418{{c}}
Optimal tunings:
* WE: ~2 = 1199.9381{{c}}, ~80/49 = 838.5389{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~80/49 = 837.5832{{c}}


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


Badness (Smith): 0.051806
Badness (Sintel): 1.71


==== 13-limit ====
==== 13-limit ====
Line 279: Line 312:
Comma list: 99/98, 169/168, 176/175, 640/637
Comma list: 99/98, 169/168, 176/175, 640/637


Mapping: {{mapping| 1 4 -1 1 -5 4 | 0 -8 11 6 28 -1 }}
{{Mapping|legend=0| 1 -4 10 7 23 3 | 0 8 -11 -6 -28 1 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~16/13 = 362.402{{c}}
Optimal tunings:
* WE: ~2 = 1200.1048{{c}}, ~13/8 = 837.6710{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 837.5964{{c}}


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


Badness (Smith): 0.029680
Badness (Sintel): 1.23


== Thuja ==
== Thuja ==
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Thuja]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Thuja]].''
Thuja tempers out 126/125 and may be described as the {{nowrap| 15 & 43 }} temperament. The generator is a somewhat sharp fourth, which may be taken as a ~11/8 in the 11-limit, and three minus an octave make buzzard's generator of ~21/16. The ploidacot for this temperament is epsilon-dodecacot.
Thuja can be extended up to the 29-limit, with a simple and accurate approximation to 29, the 2.5.11.21.29 subgroup being of especially good accuracy and simplicity.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 297: Line 336:
: mapping generators: ~2, ~175/128
: mapping generators: ~2, ~175/128


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000{{c}}, ~175/128 = 558.605{{c}}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1198.7356{{c}}, ~175/128 = 558.0168{{c}}
: [[error map]]: {{val| -1.264 -0.696 +3.770 +0.172 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~175/128 = 558.5795{{c}}
: error map: {{val| 0.000 +0.999 +6.584 +3.959 }}


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


[[Badness]] (Smith): 0.088441
[[Badness]] (Sintel): 2.24


=== 11-limit ===
=== 11-limit ===
Line 308: Line 351:
Comma list: 126/125, 176/175, 1344/1331
Comma list: 126/125, 176/175, 1344/1331


Mapping: {{mapping| 1 -4 0 7 3 | 0 12 5 -9 1 }}
{{Mapping|legend=0| 1 -4 0 7 3 | 0 12 5 -9 1 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/8 = 558.620{{c}}
Optimal tunings:
* WE: ~2 = 1198.5470{{c}}, ~11/8 = 557.9433{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 558.5942{{c}}


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


Badness (Smith): 0.033078
Badness (Sintel): 1.09


=== 13-limit ===
=== 13-limit ===
Line 321: Line 366:
Comma list: 126/125, 144/143, 176/175, 364/363
Comma list: 126/125, 144/143, 176/175, 364/363


Mapping: {{mapping| 1 -4 0 7 3 -7 | 0 12 5 -9 1 23 }}
{{Mapping|legend=0| 1 -4 0 7 3 -7 | 0 12 5 -9 1 23 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/8 = 558.589{{c}}
Optimal tunings:
* WE: ~2 = 1198.5083{{c}}, ~11/8 = 557.8942{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 558.5565{{c}}


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


Badness (Smith): 0.022838
Badness (Sintel): 0.944


=== 17-limit ===
=== 17-limit ===
Line 334: Line 381:
Comma list: 126/125, 144/143, 176/175, 221/220, 256/255
Comma list: 126/125, 144/143, 176/175, 221/220, 256/255


Mapping: {{mapping| 1 -4 0 7 3 -7 12 | 0 12 5 -9 1 23 -17 }}
{{Mapping|legend=0| 1 -4 0 7 3 -7 12 | 0 12 5 -9 1 23 -17 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/8 = 558.509{{c}}
Optimal tunings:
* WE: ~2 = 1198.8533{{c}}, ~11/8 = 557.9750{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 558.4979{{c}}


{{Optimal ET sequence|legend=1| 15, 43, 58 }}
{{Optimal ET sequence|legend=0| 15, 43, 58, 101e, 159cdef }}


Badness (Smith): 0.022293
Badness (Sintel): 1.14


=== 19-limit ===
=== 19-limit ===
Line 347: Line 396:
Comma list: 96/95, 126/125, 144/143, 153/152, 176/175, 221/220
Comma list: 96/95, 126/125, 144/143, 153/152, 176/175, 221/220


Mapping: {{mapping| 1 -4 0 7 3 -7 12 1 | 0 12 5 -9 1 23 -17 7 }}
{{Mapping|legend=0| 1 -4 0 7 3 -7 12 1 | 0 12 5 -9 1 23 -17 7 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/8 = 558.504{{c}}
Optimal tunings:
* WE: ~2 = 1198.6460{{c}}, ~11/8 = 557.8736{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 558.4905{{c}}


{{Optimal ET sequence|legend=1| 15, 43, 58h }}
{{Optimal ET sequence|legend=0| 15, 43, 58h, 101eh }}


Badness (Smith): 0.018938
Badness (Sintel): 1.15


=== 23-limit ===
=== 23-limit ===
Line 360: Line 411:
Comma list: 96/95, 126/125, 144/143, 153/152, 176/175, 221/220, 231/230
Comma list: 96/95, 126/125, 144/143, 153/152, 176/175, 221/220, 231/230


Mapping: {{mapping| 1 -4 0 7 3 -7 12 1 5 | 0 12 5 -9 1 23 -17 7 -1 }}
{{Mapping|legend=0| 1 -4 0 7 3 -7 12 1 5 | 0 12 5 -9 1 23 -17 7 -1 }}


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/8 = 558.522{{c}}
Optimal tunings:
* WE: ~2 = 1198.4488{{c}}, ~11/8 = 557.7999{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 558.5086{{c}}


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


Badness (Smith): 0.016581
Badness (Sintel): 1.19


=== 29-limit ===
=== 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
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
Comma list: 96/95, 116/115, 126/125, 144/143, 153/152, 176/175, 221/220, 231/230


Mapping: {{mapping| 1 -4 0 7 3 -7 12 1 5 3 | 0 12 5 -9 1 23 -17 7 -1 4 }}
{{Mapping|legend=0| 1 -4 0 7 3 -7 12 1 5 3 | 0 12 5 -9 1 23 -17 7 -1 4 }}
 
Optimal tunings:
* WE: ~2 = 1198.5114{{c}}, ~11/8 = 557.8276{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 558.5079{{c}}
 
{{Optimal ET sequence|legend=0| 15, 43, 58hi }}
 
Badness (Sintel): 1.15
 
== Subsedia ==
Named by [[Xenllium]] in 2022, subsedia tempers out the [[canopic comma]] and may be described as the {{nowrap| 111 & 121 }} temperament. The generator for subsedia is 0.5 cents flat of [[15/14]]-wide semitone. In this temperament, three generators make ~[[16/13]], five make ~[[24/17]], twelve make ~[[16/7]], sixteen make ~[[3/1]], and 45 make ~22/1.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 16875/16807, 65536/64827
 
{{Mapping|legend=1| 1 0 5 4 | 0 16 -27 -12 }}
: mapping generators: ~2, ~15/14
 
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.2693{{c}}, ~15/14 = 118.8923{{c}}
: [[error map]]: {{val| -0.731 +0.322 -0.060 +1.543 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~15/14 = 118.9682{{c}}
: error map: {{val| 0.000 +1.536 +1.545 +3.556 }}
 
{{Optimal ET sequence|legend=1| 10, 91cd, 101, 111, 121, 232d }}
 
[[Badness]] (Sintel): 3.99
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 540/539, 1375/1372, 65536/64827
 
{{Mapping|legend=0| 1 0 5 4 -1 | 0 16 -27 -12 45 }}
 
Optimal tunings:
* WE: ~2 = 1199.2891{{c}}, ~15/14 = 118.8978{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 118.9662{{c}}
 
{{Optimal ET sequence|legend=0| 10, 101, 111, 121, 232d }}
 
Badness (Sintel): 2.21
 
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
 
Comma list: 352/351, 540/539, 676/675, 1375/1372
 
{{Mapping|legend=0| 1 0 5 4 -1 4 | 0 16 -27 -12 45 -3 }}
 
Optimal tunings:
* WE: ~2 = 1199.2920{{c}}, ~15/14 = 118.8980{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 118.9666{{c}}
 
{{Optimal ET sequence|legend=0| 10, 101, 111, 121, 232d }}
 
Badness (Sintel): 1.31
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 256/255, 352/351, 442/441, 540/539, 715/714


Optimal tuning (POTE): ~2 = 1200.000{{c}}, ~11/8 = 558.520{{c}}
{{Mapping|legend=0| 1 0 5 4 -1 4 3 | 0 16 -27 -12 45 -3 11 }}


{{Optimal ET sequence|legend=1| 15, 43, 58hi }}
Optimal tunings:
* WE: ~2 = 1199.2648{{c}}, ~15/14 = 118.8946{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 118.9655{{c}}


Badness (Smith): 0.013762
{{Optimal ET sequence|legend=0| 10, 101, 111, 121, 232dg }}
 
Badness (Sintel): 1.00
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 256/255, 352/351, 400/399, 442/441, 456/455, 715/714
 
{{Mapping|legend=0| 1 0 5 4 -1 4 3 10 | 0 16 -27 -12 45 -3 11 -58 }}
 
Optimal tunings:
* WE: ~2 = 1199.2847{{c}}, ~15/14 = 118.8929{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 118.9644{{c}}
 
{{Optimal ET sequence|legend=0| 10, 111, 121, 232dg }}
 
Badness (Sintel): 1.09


== Anthoine ==
== 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 {{nowrap| 25 & 28 }} temperament and as the chain of 5/4's present in 53edo.
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Anthoine]].''


Subgroup: 2.3.5.7
Named by [[Lériendil]] in 2025, anthoine is generated by [[5/4]] and tempers out [[3125/3087]] in addition to the buzzardsma; note that the data below shows the octave complement generator, ~8/5, so that buzzard's generator is found at 5 generators up. It is most notable as the {{nowrap| 25 & 28 }} temperament and as the chain of 5/4's present in 53edo. Its ploidacot is 13-sheared-20-cot.  


Comma list: 3125/3087, 65536/64827
[[Subgroup]]: 2.3.5.7


Mapping: {{mapping| 1 8 2 -2 | 0 -20 1 15 }}
[[Comma list]]: 3125/3087, 65536/64827
: mapping generators: ~2, ~5/4


Optimal tuning (CTE): ~2 = 1200.000{{c}}, ~5/4 = 384.856{{c}}
{{Mapping|legend=1| 1 -12 3 13 | 0 20 -1 -15 }}
: mapping generators: ~2, ~8/5


{{Optimal ET sequence|legend=1| 25, 53, 184, 237d, 290d, 343dd }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.6282{{c}}, ~8/5 = 814.9050{{c}}
: [[error map]]: {{val| -0.372 +0.605 -2.334 +2.767 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~8/5 = 815.1546{{c}}
: error map: {{val| 0.000 +1.138 -1.468 +3.854 }}


Badness (Sintel): 4.571
{{Optimal ET sequence|legend=1| 25, 53, 184, 237d }}


[[Badness]] (Sintel): 4.57
[[Category:Buzzardsmic clan| ]] <!-- main article -->
[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Pages with mostly numerical content]]
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Buzzardsmic clan| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Listen]]
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