Schismatic family: Difference between revisions
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Expanded the three main garibaldi extensions' descriptions |
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{{Main| Garibaldi }} | {{Main| Garibaldi }} | ||
Garibaldi tempers out the [[septischisma]], equating the [[64/63|septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double-diminished octave (C–C𝄫), or down-minor seventh (C-vB♭) with the down-arrow representing the comma step. It necessitates a very slightly sharper fifth than pure. Its [[S-expression]]-based comma list is {[[5120/5103|S8/S9]], [[225/224|S15]]}, which also makes it the unique intersection of [[marvel]] and [[aberschismic]], two very notable 7-limit temperament paradigms; however, their errors go in oposite directions as marvel tunes fifths flatter and aberschismic sharper. | Garibaldi tempers out the [[septischisma]], equating the [[64/63|septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double-diminished octave (C–C𝄫), or down-minor seventh (C-vB♭) with the down-arrow representing the comma step. | ||
It necessitates a very slightly sharper fifth than pure. Its [[S-expression]]-based comma list is {[[5120/5103|S8/S9]], [[225/224|S15]]}, which also makes it the unique intersection of [[marvel]] and [[aberschismic]], two very notable 7-limit temperament paradigms; however, their errors go in oposite directions as marvel tunes fifths flatter and aberschismic sharper. | |||
[[147edo]] (from 94+53) is a [[patent val]] tuning with 5 and 7 very close to equally out-of-tune in opposite directions s.t only 7/5 and 10/7 are inconsistent in the 9-odd-limit, so might be considered for (EG) a 41-note subset ([[12L 29s]]). | [[147edo]] (from 94+53) is a [[patent val]] tuning with 5 and 7 very close to equally out-of-tune in opposite directions s.t only 7/5 and 10/7 are inconsistent in the 9-odd-limit, so might be considered for (EG) a 41-note subset ([[12L 29s]]). | ||
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=== Cassandra === | === Cassandra === | ||
Cassandra is one of the best extensions of garibaldi to the 11- and 13-limit as well as the 2.3.5.7.11.13.19 subgroup, | Cassandra is one of the best extensions of garibaldi to the 11- and 13-limit as well as the 2.3.5.7.11.13.19 subgroup, which provides the least error but is noticeably more complex. It equates a stack of two commas to 33/32, making it an [[olympic]] temperament. As an olympic temperament, an immediately natural extension is found by tempering out 4096/4095, which indirectly tempers out 352/351, making 13/11 a minor third, and thus equating 33/32 to 1053/1024. | ||
This makes 11/8 and 13/8 be found two commas above 4/3 and 128/81 respectively. | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
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=== Andromeda === | === Andromeda === | ||
Andromeda is the second-best extension to garibaldi, which | Andromeda is the second-best extension to garibaldi, which sacrifices accuracy for structural simplicity. It requires a sharper fifth than in garibaldi. | ||
It finds 13/11 as a minor third and 14/11 as a major third, which makes it a [[parapyth]] temperament. 729/704, not 33/32, is found as a stack of 2 commas, and so is 27/26, so that 11/8 and 13/8 are found two commas below a tritone and a major sixth, making all primes (except 2 and 3) have negative fifth-spans. | |||
The [[S-expression]] is remarkably simple, tempering out {S14, S15, S19, S20, S21} in the no-17 19-limit, and in the innacurate add-17 extension, it tempers out all of S14…21 except S17. | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
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=== Helenus === | === Helenus === | ||
Helenus is related to andromeda, but it makes 729/704~27/26 be a stack of ''three'' commas, not two, which necessitates a flatter fifth. | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||