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:

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* 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}}~~

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* 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}}~~

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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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== 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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= 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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==== 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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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

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== 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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Badness (Sintel): 0.944

=== 17-limit ===

==== 17-limit ====

Subgroup: 2.3.5.7.11.13.17

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Badness (Sintel): 1.14

=== 19-limit ===

==== 19-limit ====

Subgroup: 2.3.5.7.11.13.17.19

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Badness (Sintel): 1.15

=== 23-limit ===

==== 23-limit ====

Subgroup: 2.3.5.7.11.13.17.19.23

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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

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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 }}

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 {{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]].''

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

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[[Category:Temperament clans]]

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

[[Category:Buzzardsmic clan| ]]

[[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:
@@ Line 69: / Line 69: @@
 * 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}}
 {{Optimal ET sequence|legend=0| 53, 58, 111, 280cd }}
@@ Line 85: / Line 84: @@
 * 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}}
 {{Optimal ET sequence|legend=0| 53, 58, 111, 280cdf }}
@@ Line 152: / 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 199: / 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 216: / 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 237: / Line 247: @@
 ==== 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 254: / 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
@@ Line 318: / Line 324: @@
 == 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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 Badness (Sintel): 0.944
-=== 17-limit ===
+==== 17-limit ====
 Subgroup: 2.3.5.7.11.13.17
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 Badness (Sintel): 1.14
-=== 19-limit ===
+==== 19-limit ====
 Subgroup: 2.3.5.7.11.13.17.19
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 Badness (Sintel): 1.15
-=== 23-limit ===
+==== 23-limit ====
 Subgroup: 2.3.5.7.11.13.17.19.23
@@ Line 411: / Line 421: @@
 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
@@ Line 427: / Line 435: @@
 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
+Mapping: {{mapping| 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: {{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 {{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]].''
+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
@@ Line 449: / Line 550: @@
 [[Category:Temperament clans]]
-[[Category:Pages with mostly numerical content]]
 [[Category:Buzzardsmic clan| ]] <!-- main article -->
 [[Category:Rank 2]]
 [[Category:Listen]]