Buzzardsmic clan: Difference between revisions

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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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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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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 ===

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.

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

@@ Line 7: / Line 7: @@
 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 41: / 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 376: / Line 376: @@
 Badness (Sintel): 0.944
-==== 17-limit ====
+=== 17-limit ===
 Subgroup: 2.3.5.7.11.13.17
@@ Line 391: / Line 391: @@
 Badness (Sintel): 1.14
-==== 19-limit ====
+=== 19-limit ===
 Subgroup: 2.3.5.7.11.13.17.19
@@ Line 406: / Line 406: @@
 Badness (Sintel): 1.15
-==== 23-limit ====
+=== 23-limit ===
 Subgroup: 2.3.5.7.11.13.17.19.23
@@ Line 421: / Line 421: @@
 Badness (Sintel): 1.19
-==== 29-limit ====
+=== 29-limit ===
 Subgroup: 2.3.5.7.11.13.17.19.23.29
@@ Line 435: / 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.
+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