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.

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:

* [[Blackwood]] (+28/27) → [[~~Limmic temperaments~~ #Blackwood|~~Limmic temperaments~~]]

* [[Blackwood]] (+28/27) → [[Blackwood family #Blackwood|Blackwood family]]

* ''[[Quadrasruta]]'' (+2048/2025) → [[Diaschismic family #Quadrasruta|Diaschismic family]]

* ''[[Hemikleismic]] (+4000/3969) → [[Kleismic family #Hemikleismic|Kleismic family]]

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/16 = 475.8328{{c}}

: error map: {{val| 0.000 +1.376 +3.676 }}

~~~~

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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.

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

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* [[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 }}~~

* [[POTE]]: ~2 = 1200.000{{c}}, ~21/16 = 475.636{{c}}

~~: error map: {{val| 0.000 +0.589 +2.045 +4.266 }} -->~~

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

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

Optimal tunings:

* WE: ~2 = 1199.2516{{c}}, ~21/16 = 475.4037{{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}} -->

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

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

Optimal tunings:

* WE: ~2 = 1199.2391{{c}}, ~21/16 = 475.3956{{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}} -->

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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 }}

Optimal tunings:

* WE: ~2 = 1199.2723{{c}}, ~21/16 = 475.4039{{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}} -->

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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:

* WE: ~2 = 1199.2457{{c}}, ~21/16 = 475.3797{{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}} -->

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

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

Optimal tunings:

* WE: ~2 = 1200.2867{{c}}, ~21/16 = 475.5498{{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}} -->

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

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

Optimal tunings:

* WE: ~2 = 1200.3416{{c}}, ~21/16 = 475.5998{{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}} -->

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== 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

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/16 = 476.0019{{c}}

: error map: {{val| 0.000 +2.052 +1.617 +3.168 }}

~~~~

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

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

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=0| 58, 121, 179e, 300bdee, 479bcddeee }}

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

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

Optimal tunings:

* WE: ~2 = 1199.0747{{c}}, ~21/16 = 475.6291{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 476.0113{{c}}

~~~~

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== 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

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* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/16 = 476.4221{{c}}

: error map: {{val| 0.000 +3.733 +0.660 +1.908 }}

~~~~

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= 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 ===

[[Subgroup]]: 2.3.5.7

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[[Comma list]]: 225/224, 51200/50421

: 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 }}

[[Badness]] (~~Smith~~): 0.~~060533~~

[[Badness]] (Sintel): 1.53

==== 2.3.5.7.13 subgroup ====

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

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

~~Mapping:~~ {{~~mapping~~| 1 4 -~~1 1~~ 4 | 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}}

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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

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

~~Mapping:~~ {{~~mapping~~| 1 4 -~~1 1 11~~ | 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=0| 10, 43e, 53, 116~~, 169de, 285cde~~ }}

Badness (~~Smith~~): 0.~~050582~~

Badness (Sintel): 1.67

==== 13-limit ====

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

~~Mapping:~~ {{~~mapping~~| 1 4 -~~1 1 11 4~~ | 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=0| 10, 43e, 53, 116~~, 169de, 285cdef~~ }}

Badness (~~Smith~~): 0.~~027689~~

Badness (Sintel): 1.14

=== Interpental ===

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

~~Mapping:~~ {{~~mapping~~| 1 4 ~~-1 1 -5~~ | 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}}

Badness (~~Smith~~): 0.~~051806~~

Badness (Sintel): 1.71

==== 13-limit ====

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

~~Mapping:~~ {{~~mapping~~| 1 4 -~~1 1 -5~~ 4 | 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}}

Badness (~~Smith~~): 0.~~029680~~

Badness (Sintel): 1.23

== 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

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: 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 }}

[[Badness]] (~~Smith~~): 0.~~088441~~

[[Badness]] (Sintel): 2.24

=== 11-limit ===

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

~~Mapping:~~ {{~~mapping~~| 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}}

Badness (~~Smith~~): 0.~~033078~~

Badness (Sintel): 1.09

=== 13-limit ===

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

~~Mapping:~~ {{~~mapping~~| 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}}

Badness (~~Smith~~): 0.~~022838~~

Badness (Sintel): 0.944

=== 17-limit ===

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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 }}

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}}

Badness (~~Smith~~): 0.~~022293~~

Badness (Sintel): 1.14

=== 19-limit ===

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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 }}

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}}

Badness (~~Smith~~): 0.~~018938~~

Badness (Sintel): 1.15

=== 23-limit ===

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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}}

Badness (~~Smith~~): 0.~~016581~~

Badness (Sintel): 1.19

=== 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:~~ {{~~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 ~~tuning (POTE)~~: ~2 = 1200.~~000~~{{c}}, ~11/8 = 558.~~520~~{{c}}

Optimal tunings:

* WE: ~2 = 1198.5114{{c}}, ~11/8 = 557.8276{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 558.5079{{c}}

Badness (~~Smith~~): 0.~~013762~~

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 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 }}

[[Badness]] (Sintel): 3.99

=== 11-limit ===

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 65536/64827

Optimal tunings:

* WE: ~2 = 1199.2891{{c}}, ~15/14 = 118.8978{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 118.9662{{c}}

Badness (Sintel): 2.21

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 540/539, 676/675, 1375/1372

Optimal tunings:

* WE: ~2 = 1199.2920{{c}}, ~15/14 = 118.8980{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 118.9666{{c}}

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 tunings:

* WE: ~2 = 1199.2648{{c}}, ~15/14 = 118.8946{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 118.9655{{c}}

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}}

Badness (Sintel): 1.09

== 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 generators: ~2, ~8/5

{{Optimal ET sequence|legend=0| 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~~

[[Badness]] (Sintel): 4.57

[[Category:Buzzardsmic clan| ]]

[[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~~]]

@@ Line 4: / Line 4: @@
 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:
-* [[Blackwood]] (+28/27) → [[Limmic temperaments #Blackwood|Limmic temperaments]]
+* [[Blackwood]] (+28/27) → [[Blackwood family #Blackwood|Blackwood family]]
 * ''[[Quadrasruta]]'' (+2048/2025) → [[Diaschismic family #Quadrasruta|Diaschismic family]]
 * ''[[Hemikleismic]] (+4000/3969) → [[Kleismic family #Hemikleismic|Kleismic family]]
@@ Line 29: / Line 29: @@
 * [[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 }}
@@ Line 42: / Line 41: @@
 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
@@ Line 55: / Line 54: @@
 * [[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 }}
-* [[POTE]]: ~2 = 1200.000{{c}}, ~21/16 = 475.636{{c}}
-: error map: {{val| 0.000 +0.589 +2.045 +4.266 }} -->
 {{Optimal ET sequence|legend=1| 5, 48, 53, 111, 164d, 275d }}
@@ Line 69: / Line 64: @@
 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:
 * WE: ~2 = 1199.2516{{c}}, ~21/16 = 475.4037{{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 }}
@@ Line 86: / Line 79: @@
 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:
 * WE: ~2 = 1199.2391{{c}}, ~21/16 = 475.3956{{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 }}
@@ Line 103: / Line 94: @@
 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:
 * WE: ~2 = 1199.2723{{c}}, ~21/16 = 475.4039{{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 }}
@@ Line 120: / Line 109: @@
 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:
 * WE: ~2 = 1199.2457{{c}}, ~21/16 = 475.3797{{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 }}
@@ Line 137: / Line 124: @@
 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:
 * WE: ~2 = 1200.2867{{c}}, ~21/16 = 475.5498{{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 }}
@@ Line 154: / Line 139: @@
 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:
 * WE: ~2 = 1200.3416{{c}}, ~21/16 = 475.5998{{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 }}
@@ Line 167: / Line 150: @@
 == 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
@@ Line 178: / Line 163: @@
 * [[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, 479bcdd }}
@@ Line 189: / Line 173: @@
 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 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=0| 58, 121, 179e, 300bdee, 479bcddeee }}
@@ Line 205: / Line 188: @@
 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 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=0| 58, 121, 179ef, 300bdeef }}
@@ Line 217: / Line 199: @@
 == 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
@@ Line 228: / Line 212: @@
 * [[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 }}
@@ Line 235: / Line 218: @@
 = 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 ===
 [[Subgroup]]: 2.3.5.7
@@ Line 241: / Line 232: @@
 [[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 }}
-[[Badness]] (Smith): 0.060533
+[[Badness]] (Sintel): 1.53
 ==== 2.3.5.7.13 subgroup ====
 {{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
@@ Line 259: / Line 252: @@
 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=0| 10, 33, 43, 53 }}
@@ Line 267: / Line 262: @@
 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
 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=0| 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 ====
@@ Line 287: / Line 282: @@
 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=0| 10, 43e, 53, 116, 169de, 285cdef }}
+{{Optimal ET sequence|legend=0| 10, 43e, 53, 116 }}
-Badness (Smith): 0.027689
+Badness (Sintel): 1.14
 === Interpental ===
@@ Line 300: / Line 297: @@
 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=0| 43, 53, 96, 149d }}
+{{Optimal ET sequence|legend=0| 43, 53, 96 }}
-Badness (Smith): 0.051806
+Badness (Sintel): 1.71
 ==== 13-limit ====
@@ Line 313: / Line 312: @@
 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=0| 43, 53, 96, 149d }}
+{{Optimal ET sequence|legend=0| 43, 53, 96 }}
-Badness (Smith): 0.029680
+Badness (Sintel): 1.23
 == 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
@@ Line 331: / Line 336: @@
 : 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 }}
-[[Badness]] (Smith): 0.088441
+[[Badness]] (Sintel): 2.24
 === 11-limit ===
@@ Line 342: / Line 351: @@
 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=0| 15, 43, 58 }}
-Badness (Smith): 0.033078
+Badness (Sintel): 1.09
 === 13-limit ===
@@ Line 355: / Line 366: @@
 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=0| 15, 43, 58 }}
-Badness (Smith): 0.022838
+Badness (Sintel): 0.944
 === 17-limit ===
@@ Line 368: / Line 381: @@
 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=0| 15, 43, 58 }}
+{{Optimal ET sequence|legend=0| 15, 43, 58, 101e, 159cdef }}
-Badness (Smith): 0.022293
+Badness (Sintel): 1.14
 === 19-limit ===
@@ Line 381: / Line 396: @@
 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=0| 15, 43, 58h }}
+{{Optimal ET sequence|legend=0| 15, 43, 58h, 101eh }}
-Badness (Smith): 0.018938
+Badness (Sintel): 1.15
 === 23-limit ===
@@ Line 394: / Line 411: @@
 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=0| 15, 43, 58hi }}
-Badness (Smith): 0.016581
+Badness (Sintel): 1.19
 === 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: {{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 tuning (POTE): ~2 = 1200.000{{c}}, ~11/8 = 558.520{{c}}
+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 (Smith): 0.013762
+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
+{{Mapping|legend=0| 1 0 5 4 -1 4 3 | 0 16 -27 -12 45 -3 11 }}
+Optimal tunings:
+* WE: ~2 = 1199.2648{{c}}, ~15/14 = 118.8946{{c}}
+* CWE: ~2 = 1200.0000{{c}}, ~15/14 = 118.9655{{c}}
+{{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 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=0| 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 }}
+{{Optimal ET sequence|legend=1| 25, 53, 184, 237d }}
-Badness (Sintel): 4.571
+[[Badness]] (Sintel): 4.57
+[[Category:Buzzardsmic clan| ]] <!-- main article -->
 [[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]]