Buzzardsmic clan: Difference between revisions
→Submajor: link to 5-limit |
mNo edit summary |
||
| (7 intermediate revisions by 4 users not shown) | |||
| 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. | 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 | 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: | Full 7-limit temperaments discussed elsewhere are: | ||
| 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. | 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]]. [[ | 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 | [[Subgroup]]: 2.3.5.7 | ||
| Line 199: | Line 199: | ||
== Lemongrass == | == 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 | [[Subgroup]]: 2.3.5.7 | ||
| Line 218: | Line 218: | ||
= Weak extensions = | = Weak extensions = | ||
== | == Demibuzzard == | ||
: ''For the 5-limit version, see [[Schismic–Mercator equivalence continuum# | : ''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. | |||
It diverges into two extensions for prime 11: | 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 === | === 7-limit === | ||
| Line 261: | Line 262: | ||
Badness (Sintel): 0.847 | Badness (Sintel): 0.847 | ||
=== | === Submajor === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 375: | Line 376: | ||
Badness (Sintel): 0.944 | Badness (Sintel): 0.944 | ||
=== 17-limit === | ==== 17-limit ==== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
| Line 390: | Line 391: | ||
Badness (Sintel): 1.14 | Badness (Sintel): 1.14 | ||
=== 19-limit === | ==== 19-limit ==== | ||
Subgroup: 2.3.5.7.11.13.17.19 | Subgroup: 2.3.5.7.11.13.17.19 | ||
| Line 405: | Line 406: | ||
Badness (Sintel): 1.15 | Badness (Sintel): 1.15 | ||
=== 23-limit === | ==== 23-limit ==== | ||
Subgroup: 2.3.5.7.11.13.17.19.23 | Subgroup: 2.3.5.7.11.13.17.19.23 | ||
| Line 420: | Line 421: | ||
Badness (Sintel): 1.19 | Badness (Sintel): 1.19 | ||
=== 29-limit === | ==== 29-limit ==== | ||
Subgroup: 2.3.5.7.11.13.17.19.23.29 | Subgroup: 2.3.5.7.11.13.17.19.23.29 | ||
| Line 434: | Line 435: | ||
Badness (Sintel): 1.15 | 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 [[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: {{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 == | ||
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. | : ''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 | [[Subgroup]]: 2.3.5.7 | ||