Buzzardsmic clan: Difference between revisions

(15 intermediate revisions by 4 users not shown)

Line 1:

The [[2.3.7 subgroup|2.3.7-subgroup]] [[comma]] for the '''buzzardsmic clan''' is the buzzardsma, [[65536/64827]], with [[monzo]] {{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{{c}} fourth of [[5edo]].

The [[2.3.7 subgroup|2.3.7-subgroup]] [[comma]] for the '''buzzardsmic clan''' is the buzzardsma, [[65536/64827]], with [[monzo]] {{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 [[7/1|7th]] [[harmonic]], and especially a sharpened [[~]]21/16 generator, which approaches the 480{{c}} 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.

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~~ (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.

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:

Line 15:

= 2.3.7 subgroup =

== Buzzard ==

Line 26:

Line 25:

[[Optimal tuning]]s:

* [[~~CTE~~]]: ~2 = ~~1200~~.~~000~~, ~21/16 = 475.~~727~~

* [[WE]]: ~2 = 1199.2548{{c}}, ~21/16 = 475.5761{{c}}

* [[CWE]]: ~2 = 1200.~~000~~, ~21/16 = 475.~~833~~

: [[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 }}

[[Badness]]:

[[Badness]] (Sintel): 0.824

* Smith: 0.0240

* Dirichlet: 0.824

= Strong extensions =

== Septimal buzzard ==

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~~ scale]]s of 3, 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.

Line 52:

Line 50:

[[Optimal tuning]]s:

* [[~~CTE~~]]: ~2 = ~~1200~~.~~000~~, ~21/16 = 475.~~555~~

* [[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~~, ~21/16 = 475.~~636~~

* [[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 }}

[[Badness]] (~~Smith~~): 0.~~047963~~

[[Badness]] (Sintel): 1.21

=== 11-limit ===

Line 69:

Line 67:

Optimal tunings:

* ~~CTE~~: ~2 = ~~1200~~.~~000~~, ~21/16 = 475.~~625~~

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

* ~~POTE~~: ~2 = 1200.~~000~~, ~21/16 = 475.~~700~~

* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 475.6806{{c}}

Badness (~~Smith~~): 0.~~034484~~

Badness (Sintel): 1.14

==== 13-limit ====

Line 84:

Line 82:

Optimal tunings:

* ~~CTE~~: ~2 = ~~1200~~.~~000~~, ~21/16 = 475.~~615~~

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

* ~~POTE~~: ~2 = 1200.~~000~~, ~21/16 = 475.~~697~~

* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 475.6760{{c}}

Badness (~~Smith~~): 0.~~018842~~

Badness (Sintel): 0.779

==== 17-limit ====

Line 99:

Line 97:

Optimal tunings:

* ~~CTE~~: ~2 = ~~1200~~.~~000~~, ~21/16 = 475.~~638~~

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

* ~~POTE~~: ~2 = 1200.~~000~~, ~21/16 = 475.~~692~~

* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 475.6837{{c}}

Badness (~~Smith~~): 0.~~018403~~

Badness (Sintel): 0.938

==== 19-limit ====

Line 114:

Line 112:

Optimal tunings:

* ~~CTE~~: ~2 = ~~1200~~.~~000~~, ~21/16 = 475.~~617~~

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

* ~~POTE~~: ~2 = 1200.~~000~~, ~21/16 = 475.~~679~~

* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 475.6690{{c}}

Badness (~~Smith~~): 0.~~015649~~

Badness (Sintel): 0.952

=== Buteo ===

Line 129:

Line 127:

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~21/16 = 475.~~454~~

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

* ~~POTE~~: ~2 = 1200.~~000~~, ~21/16 = 475.~~436~~

* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 475.4393{{c}}

Badness (~~Smith~~): 0.~~060238~~

Badness (Sintel): 1.99

==== 13-limit ====

Line 144:

Line 142:

Optimal tunings:

* ~~CTE~~: ~2 = 1200.~~000~~, ~21/16 = 475.~~495~~

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

* ~~POTE~~: ~2 = 1200.~~000~~, ~21/16 = 475.~~464~~

* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 475.4696{{c}}

Badness (~~Smith~~): 0.~~039854~~

Badness (Sintel): 1.65

== 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 158:

: ~~mapping generators~~: ~2, ~21/16

[[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 ~~tuning]] ([[POTE]]): ~2~~ = 1\1, ~~~21/16 = 475.991~~

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

[[Badness]] (Sintel): 3.56

[[Badness]]: 0.~~140722~~

=== 11-limit ===

Line 173:

Line 175:

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~21/16 = 475.~~995~~

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 }}

Badness: 0.~~045323~~

Badness (Sintel): 1.50

=== 13-limit ===

Line 186:

Line 190:

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~21/16 = 475.~~996~~

Optimal tunings:

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

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

Badness: 0.~~023800~~

Badness (Sintel): 0.983

== 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 199:

Line 207:

: ~~mapping generators~~: ~2, ~21/16

[[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 tuning~~]] (~~[[POTE]]~~): ~2 = ~~1\1~~, ~~~21/16 = 476~~.~~4448~~

[[Badness]] (Sintel): 2.90

= Weak extensions =

== Demibuzzard ==

: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Demibuzzard]].''

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

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.

[[~~Badness~~]] ~~(Sintel): 2~~.~~902~~

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]].

~~= Weak~~ extensions =

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.

~~== Submajor ==~~

=== 7-limit ===

[[Subgroup]]: 2.3.5.7

Line 215:

Line 232:

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

: mapping generators: ~2, ~80/49

[[Optimal tuning]] ([[~~POTE~~]]): ~2 = ~~1\1~~, ~49/40 = ~~362~~.~~255~~

[[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]]: 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 [[No-fives subgroup temperaments#Buzzard|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 232:

Line 252:

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

Mapping: {{mapping| 1 4 -~~1 1~~ 4 | 0 -8 11 6 -1 }}

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

Optimal ~~tuning (CTE)~~: ~2 = ~~1\1~~, ~16/13 = ~~362~~.~~242~~

Optimal tunings:

* WE: ~2 = 1199.9444{{c}}, ~13/8 = 837.7178{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 837.7569{{c}}

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~27/22 = ~~362~~.~~101~~

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~~ }}

Badness: 0.~~050582~~

Badness (Sintel): 1.67

==== 13-limit ====

Line 260:

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~16/13 = ~~362~~.~~105~~

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~~ }}

Badness: 0.~~027689~~

Badness (Sintel): 1.14

=== Interpental ===

Line 273:

Line 297:

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

Mapping: {{mapping| 1 4 ~~-1 1 -5~~ | 0 -8 11 6 28 }}

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~49/40 = ~~362~~.~~418~~

Optimal tunings:

* WE: ~2 = 1199.9381{{c}}, ~80/49 = 838.5389{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~80/49 = 837.5832{{c}}

Badness: 0.~~051806~~

Badness (Sintel): 1.71

==== 13-limit ====

Line 286:

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~16/13 = ~~362~~.~~402~~

Optimal tunings:

* WE: ~2 = 1200.1048{{c}}, ~13/8 = 837.6710{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~13/8 = 837.5964{{c}}

Badness: 0.~~029680~~

Badness (Sintel): 1.23

== Thuja ==

: ''For the 5-limit version ~~of this temperament~~, 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

Line 302:

Line 334:

: mapping generators: ~2, ~175/128

[[Optimal tuning]] ([[~~POTE~~]]): ~2 = 1\1, ~175/128 = 558.~~605~~

[[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]]: 0.~~088441~~

[[Badness]] (Sintel): 2.24

=== 11-limit ===

Line 316:

Line 353:

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~11/8 = 558.~~620~~

Optimal tunings:

* WE: ~2 = 1198.5470{{c}}, ~11/8 = 557.9433{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 558.5942{{c}}

Badness: 0.~~033078~~

Badness (Sintel): 1.09

=== 13-limit ===

Line 329:

Line 368:

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~11/8 = 558.~~589~~

Optimal tunings:

* WE: ~2 = 1198.5083{{c}}, ~11/8 = 557.8942{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 558.5565{{c}}

Badness: 0.~~022838~~

Badness (Sintel): 0.944

=== 17-limit ===

==== 17-limit ====

Subgroup: 2.3.5.7.11.13.17

Line 342:

Line 383:

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~11/8 = 558.~~509~~

Optimal tunings:

* WE: ~2 = 1198.8533{{c}}, ~11/8 = 557.9750{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 558.4979{{c}}

Badness: 0.~~022293~~

Badness (Sintel): 1.14

=== 19-limit ===

==== 19-limit ====

Subgroup: 2.3.5.7.11.13.17.19

Line 355:

Line 398:

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~11/8 = 558.~~504~~

Optimal tunings:

* WE: ~2 = 1198.6460{{c}}, ~11/8 = 557.8736{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 558.4905{{c}}

Badness: 0.~~018938~~

Badness (Sintel): 1.15

=== 23-limit ===

==== 23-limit ====

Subgroup: 2.3.5.7.11.13.17.19.23

Line 368:

Line 413:

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

Optimal ~~tuning (POTE)~~: ~2 = ~~1\1~~, ~11/8 = ~~558~~.~~522~~

Optimal tunings:

* WE: ~2 = 1198.4488{{c}}, ~11/8 = 557.7999{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 558.5086{{c}}

~~Badness:~~ 0~~.016581~~

~~=== 29-limit ===~~

Badness (Sintel): 1.19

~~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~~.

==== 29-limit ====

Subgroup: 2.3.5.7.11.13.17.19.23.29

Line 383:

Line 428:

Mapping: {{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 = 1\1, ~11/8 = ~~558~~.~~520~~

Optimal tunings:

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

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

Badness (Sintel): 1.15

=== Antemka ===

This temperament has the opposite mappings of 5 and 13 to [[emka]].

Comma list: 105/104, 126/125, 66/65, 1024/1001

Edo join: 15 & 28

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

Optimal tuning (CWE): ~2 = 1200.0000{{c}}, ~16/11 = 641.492{{c}}

== Subsedia ==

Named by [[Xenllium]] in 2022, subsedia tempers out the [[mirkwai 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

Mapping: {{mapping| 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}}

Badness (Sintel): 2.21

=== 13-limit ===

Subgroup: 2.3.5.7.11.13

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

Mapping: {{mapping| 1 0 5 4 -1 4 | 0 16 -27 -12 45 -3 }}

Badness: 0.~~013762~~

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

Mapping: {{mapping| 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}}

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: {{mapping| 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 25 & 28 temperament and as the chain of 5/4s present in 53edo~~.

: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Anthoine]].''

~~=== 7~~-~~limit ===~~

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.

Subgroup: 2.3.5.7

[[Subgroup]]: 2.3.5.7

Comma list: 3125/3087, 65536/64827

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

~~Mapping:~~ {{~~mapping~~| 1 ~~8 2~~ -2 | 0 -20 1 15 }}

: mapping generators: ~2, ~8/5

Optimal tuning ~~(CTE)~~: ~2 = ~~1\1~~, ~5/4 = ~~384~~.~~856~~

[[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~~, 290d, 343dd~~ }}

Badness (Sintel): 4.~~571~~

[[Badness]] (Sintel): 4.57

[[Category:Temperament clans]]

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

~~[[Category:Buzzard]]~~

[[Category:Buzzardsmic clan| ]]

[[Category:Rank 2]]

[[Category:Listen]]

@@ Line 1: / Line 1: @@
 {{Technical data page}}
-The [[2.3.7 subgroup|2.3.7-subgroup]] [[comma]] for the '''buzzardsmic clan''' is the buzzardsma, [[65536/64827]], with [[monzo]] {{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{{c}} fourth of [[5edo]].
+The [[2.3.7 subgroup|2.3.7-subgroup]] [[comma]] for the '''buzzardsmic clan''' is the buzzardsma, [[65536/64827]], with [[monzo]] {{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 [[7/1|7th]] [[harmonic]], and especially a sharpened [[~]]21/16 generator, which approaches the 480{{c}} 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.
+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 (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.
+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:
@@ Line 15: / Line 15: @@
 = 2.3.7 subgroup =
 == Buzzard ==
 {{Main| Buzzard }}
@@ Line 26: / Line 25: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~21/16 = 475.727
+* [[WE]]: ~2 = 1199.2548{{c}}, ~21/16 = 475.5761{{c}}
-* [[CWE]]: ~2 = 1200.000, ~21/16 = 475.833
+: [[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 }}
-[[Badness]]:
+[[Badness]] (Sintel): 0.824
-* Smith: 0.0240
-* Dirichlet: 0.824
 = Strong extensions =
 == Septimal buzzard ==
 {{Main| 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 scale]]s of 3, 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.
@@ Line 52: / Line 50: @@
 [[Optimal tuning]]s:
-* [[CTE]]: ~2 = 1200.000, ~21/16 = 475.555
+* [[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, ~21/16 = 475.636
+* [[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 }}
-[[Badness]] (Smith): 0.047963
+[[Badness]] (Sintel): 1.21
 === 11-limit ===
@@ Line 69: / Line 67: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~21/16 = 475.625
+* WE: ~2 = 1199.2516{{c}}, ~21/16 = 475.4037{{c}}
-* POTE: ~2 = 1200.000, ~21/16 = 475.700
+* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 475.6806{{c}}
 {{Optimal ET sequence|legend=0| 53, 58, 111, 280cd }}
-Badness (Smith): 0.034484
+Badness (Sintel): 1.14
 ==== 13-limit ====
@@ Line 84: / Line 82: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~21/16 = 475.615
+* WE: ~2 = 1199.2391{{c}}, ~21/16 = 475.3956{{c}}
-* POTE: ~2 = 1200.000, ~21/16 = 475.697
+* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 475.6760{{c}}
 {{Optimal ET sequence|legend=0| 53, 58, 111, 280cdf }}
-Badness (Smith): 0.018842
+Badness (Sintel): 0.779
 ==== 17-limit ====
@@ Line 99: / Line 97: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~21/16 = 475.638
+* WE: ~2 = 1199.2723{{c}}, ~21/16 = 475.4039{{c}}
-* POTE: ~2 = 1200.000, ~21/16 = 475.692
+* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 475.6837{{c}}
 {{Optimal ET sequence|legend=0| 53, 58, 111 }}
-Badness (Smith): 0.018403
+Badness (Sintel): 0.938
 ==== 19-limit ====
@@ Line 114: / Line 112: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~21/16 = 475.617
+* WE: ~2 = 1199.2457{{c}}, ~21/16 = 475.3797{{c}}
-* POTE: ~2 = 1200.000, ~21/16 = 475.679
+* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 475.6690{{c}}
 {{Optimal ET sequence|legend=0| 53, 58h, 111 }}
-Badness (Smith): 0.015649
+Badness (Sintel): 0.952
 === Buteo ===
@@ Line 129: / Line 127: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~21/16 = 475.454
+* WE: ~2 = 1200.2867{{c}}, ~21/16 = 475.5498{{c}}
-* POTE: ~2 = 1200.000, ~21/16 = 475.436
+* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 475.4393{{c}}
 {{Optimal ET sequence|legend=0| 5, 48, 53 }}
-Badness (Smith): 0.060238
+Badness (Sintel): 1.99
 ==== 13-limit ====
@@ Line 144: / Line 142: @@
 Optimal tunings:
-* CTE: ~2 = 1200.000, ~21/16 = 475.495
+* WE: ~2 = 1200.3416{{c}}, ~21/16 = 475.5998{{c}}
-* POTE: ~2 = 1200.000, ~21/16 = 475.464
+* CWE: ~2 = 1200.0000{{c}}, ~21/16 = 475.4696{{c}}
 {{Optimal ET sequence|legend=0| 5, 48f, 53 }}
-Badness (Smith): 0.039854
+Badness (Sintel): 1.65
 == 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 158: / Line 158: @@
 {{Mapping|legend=1| 1 0 17 4 | 0 4 -37 -3 }}
-: mapping generators: ~2, ~21/16
+[[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 tuning]] ([[POTE]]): ~2 = 1\1, ~21/16 = 475.991
+{{Optimal ET sequence|legend=1| 58, 121, 179, 300bd, 479bcdd }}
-{{Optimal ET sequence|legend=1| 58, 121, 179, 300bd, 479bcd }}
+[[Badness]] (Sintel): 3.56
-[[Badness]]: 0.140722
 === 11-limit ===
@@ Line 173: / Line 175: @@
 Mapping: {{mapping| 1 0 17 4 11 | 0 4 -37 -3 -19 }}
-Optimal tuning (POTE): ~2 = 1\1, ~21/16 = 475.995
+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: 0.045323
+Badness (Sintel): 1.50
 === 13-limit ===
@@ Line 186: / Line 190: @@
 Mapping: {{mapping| 1 0 17 4 11 16 | 0 4 -37 -3 -19 -31 }}
-Optimal tuning (POTE): ~2 = 1\1, ~21/16 = 475.996
+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: 0.023800
+Badness (Sintel): 0.983
 == 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 199: / Line 207: @@
 {{Mapping|legend=1| 1 0 17 4 | 0 4 26 -3 }}
-: mapping generators: ~2, ~21/16
+[[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 tuning]] ([[POTE]]): ~2 = 1\1, ~21/16 = 476.4448
+[[Badness]] (Sintel): 2.90
+= Weak extensions =
+== Demibuzzard ==
+: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum #Demibuzzard]].''
-{{Optimal ET sequence|legend=1| 5, ..., 63, 68 }}
+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.
-[[Badness]] (Sintel): 2.902
+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]].
-= Weak extensions =
+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.
-== Submajor ==
 === 7-limit ===
 [[Subgroup]]: 2.3.5.7
@@ Line 215: / 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, ~80/49
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 362.255
+[[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]]: 0.060533
+[[Badness]] (Sintel): 1.53
 ==== 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 [[No-fives subgroup temperaments#Buzzard|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 232: / Line 252: @@
 Comma list: 169/168, 225/224, 640/637
-Mapping: {{mapping| 1 4 -1 1 4 | 0 -8 11 6 -1 }}
+Mapping: {{mapping| 1 -4 10 7 3 | 0 8 -11 -6 1 }}
-Optimal tuning (CTE): ~2 = 1\1, ~16/13 = 362.242
+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
-=== 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: {{mapping| 1 -4 10 7 -14 | 0 8 -11 -6 25 }}
-Optimal tuning (POTE): ~2 = 1\1, ~27/22 = 362.101
+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: 0.050582
+Badness (Sintel): 1.67
 ==== 13-limit ====
@@ Line 260: / 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: {{mapping| 1 -4 10 7 -14 3 | 0 8 -11 -6 25 1 }}
-Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 362.105
+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: 0.027689
+Badness (Sintel): 1.14
 === Interpental ===
@@ Line 273: / Line 297: @@
 Comma list: 99/98, 176/175, 51200/50421
-Mapping: {{mapping| 1 4 -1 1 -5 | 0 -8 11 6 28 }}
+Mapping: {{mapping| 1 -4 10 7 23 | 0 8 -11 -6 -28 }}
-Optimal tuning (POTE): ~2 = 1\1, ~49/40 = 362.418
+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: 0.051806
+Badness (Sintel): 1.71
 ==== 13-limit ====
@@ Line 286: / 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: {{mapping| 1 -4 10 7 23 3 | 0 8 -11 -6 -28 1 }}
-Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 362.402
+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: 0.029680
+Badness (Sintel): 1.23
 == Thuja ==
-: ''For the 5-limit version of this temperament, 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
@@ Line 302: / Line 334: @@
 {{Mapping|legend=1| 1 -4 0 7 | 0 12 5 -9 }}
+: mapping generators: ~2, ~175/128
-[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~175/128 = 558.605
+[[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]]: 0.088441
+[[Badness]] (Sintel): 2.24
 === 11-limit ===
@@ Line 316: / Line 353: @@
 Mapping: {{mapping| 1 -4 0 7 3 | 0 12 5 -9 1 }}
-Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 558.620
+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: 0.033078
+Badness (Sintel): 1.09
 === 13-limit ===
@@ Line 329: / Line 368: @@
 Mapping: {{mapping| 1 -4 0 7 3 -7 | 0 12 5 -9 1 23 }}
-Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 558.589
+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: 0.022838
+Badness (Sintel): 0.944
-=== 17-limit ===
+==== 17-limit ====
 Subgroup: 2.3.5.7.11.13.17
@@ Line 342: / Line 383: @@
 Mapping: {{mapping| 1 -4 0 7 3 -7 12 | 0 12 5 -9 1 23 -17 }}
-Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 558.509
+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: 0.022293
+Badness (Sintel): 1.14
-=== 19-limit ===
+==== 19-limit ====
 Subgroup: 2.3.5.7.11.13.17.19
@@ Line 355: / Line 398: @@
 Mapping: {{mapping| 1 -4 0 7 3 -7 12 1 | 0 12 5 -9 1 23 -17 7 }}
-Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 558.504
+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: 0.018938
+Badness (Sintel): 1.15
-=== 23-limit ===
+==== 23-limit ====
 Subgroup: 2.3.5.7.11.13.17.19.23
@@ Line 368: / Line 413: @@
 Mapping: {{mapping| 1 -4 0 7 3 -7 12 1 5 | 0 12 5 -9 1 23 -17 7 -1 }}
-Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 558.522
+Optimal tunings:
+* WE: ~2 = 1198.4488{{c}}, ~11/8 = 557.7999{{c}}
-{{Optimal ET sequence|legend=1| 15, 43, 58hi }}
+* CWE: ~2 = 1200.0000{{c}}, ~11/8 = 558.5086{{c}}
-Badness: 0.016581
+{{Optimal ET sequence|legend=0| 15, 43, 58hi }}
-=== 29-limit ===
+Badness (Sintel): 1.19
-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.
+==== 29-limit ====
 Subgroup: 2.3.5.7.11.13.17.19.23.29
@@ Line 383: / Line 428: @@
 Mapping: {{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 = 1\1, ~11/8 = 558.520
+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
+=== Antemka ===
+This temperament has the opposite mappings of 5 and 13 to [[emka]].
+Comma list: 105/104, 126/125, 66/65, 1024/1001
+Edo join: 15 & 28
+Mapping: {{mapping| 1 8 5 -2 4 8|0 -12 -5 9 -1 -8}}
+Optimal tuning (CWE): ~2 = 1200.0000{{c}}, ~16/11 = 641.492{{c}}
+== Subsedia ==
+Named by [[Xenllium]] in 2022, subsedia tempers out the [[mirkwai 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: {{mapping| 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
-{{Optimal ET sequence|legend=1| 15, 43, 58hi }}
+Mapping: {{mapping| 1 0 5 4 -1 4 | 0 16 -27 -12 45 -3 }}
-Badness: 0.013762
+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: {{mapping| 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: {{mapping| 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 25 & 28 temperament and as the chain of 5/4s present in 53edo.
+: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Anthoine]].''
-=== 7-limit ===
+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.
-Subgroup: 2.3.5.7
+[[Subgroup]]: 2.3.5.7
-Comma list: 3125/3087, 65536/64827
+[[Comma list]]: 3125/3087, 65536/64827
-Mapping: {{mapping| 1 8 2 -2 | 0 -20 1 15 }}
+{{Mapping|legend=1| 1 -12 3 13 | 0 20 -1 -15 }}
+: mapping generators: ~2, ~8/5
-Optimal tuning (CTE): ~2 = 1\1, ~5/4 = 384.856
+[[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, 290d, 343dd }}
+{{Optimal ET sequence|legend=1| 25, 53, 184, 237d }}
-Badness (Sintel): 4.571
+[[Badness]] (Sintel): 4.57
 [[Category:Temperament clans]]
-[[Category:Pages with mostly numerical content]]
-[[Category:Buzzard]]
 [[Category:Buzzardsmic clan| ]] <!-- main article -->
 [[Category:Rank 2]]
 [[Category:Listen]]