Buzzardsmic clan: Difference between revisions
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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 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 & | 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. | ||
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 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. | ||
Full 7-limit temperaments discussed elsewhere are: | Full 7-limit temperaments discussed elsewhere are: | ||
* | * [[Blackwood]] (+28/27) → [[Limmic temperaments #Blackwood|Limmic temperaments]] | ||
* ''[[Quadrasruta]]'' (+2048/2025) → [[Diaschismic family #Quadrasruta|Diaschismic family]] | * ''[[Quadrasruta]]'' (+2048/2025) → [[Diaschismic family #Quadrasruta|Diaschismic family]] | ||
* ''[[Hemikleismic]] (+4000/3969) → [[Kleismic family #Hemikleismic|Kleismic family]] | * ''[[Hemikleismic]] (+4000/3969) → [[Kleismic family #Hemikleismic|Kleismic family]] | ||
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{{See also| Vulture family }} | {{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]], | 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. | ||
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]]. [[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. | ||
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{{Mapping|legend=1| 1 0 -6 4 | 0 4 21 -3 }} | {{Mapping|legend=1| 1 0 -6 4 | 0 4 21 -3 }} | ||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
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Mapping: {{mapping| 1 0 -6 4 -12 | 0 4 21 -3 39 }} | Mapping: {{mapping| 1 0 -6 4 -12 | 0 4 21 -3 39 }} | ||
Optimal tunings: | Optimal tunings: | ||
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Mapping: {{mapping| 1 0 -6 4 -12 -7 | 0 4 21 -3 39 27 }} | Mapping: {{mapping| 1 0 -6 4 -12 -7 | 0 4 21 -3 39 27 }} | ||
Optimal tunings: | Optimal tunings: | ||
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: mapping generators: ~2, ~21/16 | : mapping generators: ~2, ~21/16 | ||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/16 = 475.991 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/16 = 475.991 | ||
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{{Mapping|legend=1| 1 4 -1 1 | 0 -8 11 6 }} | {{Mapping|legend=1| 1 4 -1 1 | 0 -8 11 6 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 362.255 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~49/40 = 362.255 | ||
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Optimal tuning (CTE): ~2 = 1\1, ~16/13 = 362.242 | Optimal tuning (CTE): ~2 = 1\1, ~16/13 = 362.242 | ||
{{Optimal ET sequence|legend=1| 10, 33, 43, 53 }} | |||
Badness (Sintel): 0.847 | Badness (Sintel): 0.847 | ||
=== 11-limit === | === 11-limit === | ||
Submajor diverges into two extensions to prime 11: this one favoring sharp fifths, and interpental, favoring flat fifths; the two mappings meet at [[53edo]]. | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
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{{Mapping|legend=1| 1 -4 0 7 | 0 12 5 -9 }} | {{Mapping|legend=1| 1 -4 0 7 | 0 12 5 -9 }} | ||
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~175/128 = 558.605 | [[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~175/128 = 558.605 | ||
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[[Category:Temperament clans]] | [[Category:Temperament clans]] | ||
[[Category:Pages with mostly numerical content]] | |||
[[Category:Buzzard]] | [[Category:Buzzard]] | ||
[[Category:Buzzardsmic clan| ]] <!-- main article --> | [[Category:Buzzardsmic clan| ]] <!-- main article --> | ||
[[Category:Rank 2]] | [[Category:Rank 2]] | ||
[[Category:Listen]] | [[Category:Listen]] | ||