Aberschismic family: Difference between revisions
No edit summary |
Make counterpyth a redirect as there's not much too add in a dedicated article |
||
| (11 intermediate revisions by 3 users not shown) | |||
| Line 1: | Line 1: | ||
{{Technical data page}} | {{Technical data page}} | ||
The ''' | The '''aberschismic family''' of [[rank-3 temperament|rank-3]] [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[aberschisma]] ({{monzo|legend=1| 10 -6 1 -1 }}, [[ratio]]: [[5120/5103]]). | ||
== Aberschismic == | |||
{{Main| Aberschismic }} | |||
It is therefore very handy to adopt an additional module of accidentals such as arrows to represent the syntonic~septimal comma, in which case we have [[5/4]] at the down major third (C–vE) and [[7/4]] at the down minor seventh ( | Aberschismic (formerly ''hemifamity'') divides an exact or approximate septimal quartertone, [[36/35]] into two equal steps, each representing [[81/80]][[~]][[64/63]], the syntonic comma or the septimal comma. Therefore, classical and septimal intervals are found by the same [[chain of fifths]] inflected by the same comma to the opposite sides. In addition we may identify [[10/7]] with the augmented fourth (C–F♯) and [[50/49]] with the [[Pythagorean comma]] (C–B♯′). | ||
Aberschismic can be further tempered to [[garibaldi]], which expands the interpretations of 81/80~64/63 to include the Pythagorean comma (collapsing to a rank-2 structure), or alternatively, aberschismic can be seen as liberating the syntonic-septimal comma from garibaldi's chain of fifths. | |||
It is therefore very handy to adopt an additional module of accidentals such as arrows to represent the syntonic~septimal comma, in which case we have [[5/4]] at the down major third (C–vE) and [[7/4]] at the down minor seventh (C–vB♭). | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 20: | Line 16: | ||
{{Mapping|legend=1| 1 0 0 10 | 0 1 0 -6 | 0 0 1 1 }} | {{Mapping|legend=1| 1 0 0 10 | 0 1 0 -6 | 0 0 1 1 }} | ||
: mapping generators: ~2, ~3, ~5 | : mapping generators: ~2, ~3, ~5 | ||
| Line 48: | Line 43: | ||
[[Projection pair]]s: 7 5120/729 | [[Projection pair]]s: 7 5120/729 | ||
=== Overview to extensions === | === Overview to extensions === | ||
==== 11- and 13-limit extensions ==== | ==== 11- and 13-limit extensions ==== | ||
Strong extensions of | Strong extensions of aberschismic are [[#Pele|pele]], [[#Laka|laka]], [[#Akea|akea]], and [[#Lono|lono]]. The rest are weak extensions. Using the arrow to represent the syntonic~septimal comma, pele finds the [[11/8]] as a down diminished fifth (C–vG♭); laka, up augmented third (C–^E♯); akea, double-up fourth (C–^^F); lono, triple-down augmented fourth (C–v<sup>3</sup>F♯). All these extensions follow the trend of tuning the fifth a little sharp. Thus a successful mapping of 13 can be found by fixing the [[13/11]] at the minor third (C–Eb), tempering out [[352/351]], [[847/845]], and [[2080/2079]]. | ||
Temperaments discussed elsewhere include: | Temperaments discussed elsewhere include: | ||
| Line 61: | Line 52: | ||
==== Subgroup extensions ==== | ==== Subgroup extensions ==== | ||
A notable 2.3.5.7.19 subgroup extension, counterpyth, is | A notable 2.3.5.7.19-subgroup extension, counterpyth, is considered in [[#Subgroup extensions]]. | ||
== Pele == | == Pele == | ||
| Line 156: | Line 126: | ||
{{Main| Laka }} | {{Main| Laka }} | ||
Laka can be described as the {{nowrap| 41 & 53 & 58 }} temperament, tempering out [[540/539]], and finds the interval class of 11 at the up augmented third (C–^ | Laka can be described as the {{nowrap| 41 & 53 & 58 }} temperament, tempering out [[540/539]], and finds the interval class of 11 at the up augmented third (C–^E♯). [[Gene Ward Smith]] considered it a [[17-limit]] temperament, assigning the vanishing of [[442/441]] ({{nowrap| 41g & 53 & 58 }}) as the main extension, but {{nowrap| 41 & 53g & 58 }} also makes for a competitive extension.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101682.html#101776 Yahoo! Tuning Group | ''Laka 17-limit minimax planar temperament'']</ref> Indeed, laka makes most sense as a 2.3.5.7.11.13.19-[[subgroup]] temperament, skipping prime 17, as the 19 is accurate and easily available in a 24-tone scale. [[152edo]] makes for an excellent tuning, using the 152f val for prime 13. | ||
[[Subgroup]]: 2.3.5.7.11 | [[Subgroup]]: 2.3.5.7.11 | ||
| Line 186: | Line 149: | ||
[[Badness]] (Sintel): 0.992 | [[Badness]] (Sintel): 0.992 | ||
[[Projection pair]]s: 5120/729 11 14348907/1310720 | [[Projection pair]]s: <code>7 5120/729 11 14348907/1310720</code> | ||
=== 13-limit === | === 13-limit === | ||
| Line 226: | Line 189: | ||
Badness (Sintel): 0.647 | Badness (Sintel): 0.647 | ||
== Akea == | == Akea == | ||
| Line 303: | Line 246: | ||
== Lono == | == Lono == | ||
Lono tempers out [[176/175]] and may be described as the {{nowrap| 46 & 53 & 58 }} temperament, finding the interval class of 11 at the triple-down augmented fourth (C–v<sup>3</sup> | Lono tempers out [[176/175]] and may be described as the {{nowrap| 46 & 53 & 58 }} temperament, finding the interval class of 11 at the triple-down augmented fourth (C–v<sup>3</sup>F♯). It notably also tempers out [[8019/8000]], thus setting 11/10, 10/9, 9/8, and 8/7 a comma apart from each other. [[111edo]] is a great tuning for it. [[157edo]] is a viable alternative, which is almost as good. | ||
[[Subgroup]]: 2.3.5.7.11 | [[Subgroup]]: 2.3.5.7.11 | ||
| Line 391: | Line 334: | ||
Badness (Sintel): 0.731 | Badness (Sintel): 0.731 | ||
== Subgroup extensions == | |||
=== Counterpyth (2.3.5.7.19) === | |||
[[File:Lattice Counterpyth RTT.png|thumb|Lattice for counterpyth.]] | |||
Inspired by [[Margo Schulter]]'s [[parapyth]], counterpyth was named and first explored by [[Flora Canou]] in 2024. It is an extension of aberschismic with an even milder fifth, as it finds [[19/15]] at the major third (C–E) and [[19/10]] at the major seventh (C–B), taking advantage of the factorization {{nowrap| 5120/5103 {{=}} ([[400/399]])⋅([[1216/1215]]) }}. Other important ratios are [[21/19]] at the diminished third (C–E𝄫) and [[19/14]] at the augmented third (C–E♯). | |||
It can be further extended via the mappings of laka or akea, while working less well with pele or lono due to their much sharper fifths. | |||
Subgroup: 2.3.5.7.19 | |||
Comma list: 400/399, 1216/1215 | |||
Mapping: {{mapping| 1 0 0 10 -6 | 0 1 0 -6 5 | 0 0 1 1 1 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.6953{{c}}, ~3/2 = 702.5169{{c}}, ~5/4 = 386.2648{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6771{{c}}, ~5/4 = 386.0544{{c}} | |||
{{Optimal ET sequence|legend=0| 12, 29, 41, 53, 94, 99, 140, 152, 292h, 444dh }} | |||
Badness (Sintel): 0.347 | |||
== References == | == References == | ||
[[Category:Temperament families]] | [[Category:Temperament families]] | ||
[[Category: | [[Category:Aberschismic family| ]] <!-- main article --> | ||
[[Category:Rank 3]] | [[Category:Rank 3]] | ||
[[Category:Listen]] | [[Category:Listen]] | ||