Bird's eye view of temperaments by accuracy: Difference between revisions
m →Godzilla: i personally dont think immunity is, in isolation, valuable enough to be worth adding, though someone else might disagree in which case they can add it and link to it from here |
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== 7-limit focus == | == 7-limit focus == | ||
=== [[Ennealimmal]] === | |||
[[Bird's eye view of temperaments by accuracy#Note counts|Note counts]]: | |||
45 for {3, 5, 7, 9} ([[27L 18s]]) | |||
72 for {3, 5, 7, 9, 15, 21, 25, 27, 35, 45, 49, 63} ([[27L 45s]]) | |||
[[Bird's eye view of temperaments by accuracy#Generator tunings|Generator tunings]]: 3\72, 4\99, 7\171, 11\270 | |||
Ennealimmal has a 1/9-octave period representing [[27/25]], and two of them represent [[7/6]], tempering out [[4375/4374]]. It is generated by a [[~]][[36/35]] quartertone, with 3/2 being mapped to 6 periods minus 2 generators, 5/4 mapped to 4 periods minus 3 generators, and 7/4 mapped to 8 periods minus 2 generators. It finds a neutral third representing [[49/40]]~[[60/49]] at 3 periods minus 1 generator, tempering out [[2401/2400]]. This temperament therefore tempers out the two smallest superparticular ratios in the [[7-limit]], 2401/2400 and 4375/4374. It is very accurate, with errors of around 0.2 cents in optimized tunings. However, due to the high note count, one may prefer lower accuracy temperaments. | |||
== 11-limit focus == | == 11-limit focus == | ||
== ~17-limit focus == | == ~17-limit focus == | ||
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== No-2's focus == | == No-2's focus == | ||
=== [[Bohlen-Pierce-Stearns]] === | |||
[[Bird's eye view of temperaments by accuracy#Note counts|Note count]]: | |||
* 7 for {5, 7, 25, 35, 49} ([[4L 5s (3/1-equivalent)|4L 5s<3/1>]]) | |||
Bound-violating intervals: [[35/27]], [[49/27]] and/or [[25/9]] (none if those higher harmonics are omitted) | |||
[[#Generator tunings|Generator tuning]]: 10\13edt | |||
This is arguably the most important temperament of the nonoctave [[3.5.7 subgroup]]. This temperament has a tritave period, and a generator of [[~]][[9/7]]. The tritave-reduced 7th harmonic, [[7/3]], is found at -1 generators, and the tritave reduced 5th harmonic, [[5/3]], is found at +2 generators, tempering out [[245/243]]. It is as simple as a good temperament in its subgroup can be, covering the entire no-evens 7-throdd-limit tonality diamond in 7 notes, with no redundant or missing notes, and any simpler temperament would have to equate simple consonances and have very low accuracy. Its accuracy is quite good, with a no-evens 7-throdd-limit minimax error of 4.73 cents. An excellent scale to explore this temperament is the 9-note mos, or lambda scale, which can be considered the 3.5.7 analog of the diatonic scale. | |||
== No-3's focus == | == No-3's focus == | ||
=== [[Didacus]] === | === [[Didacus]] === | ||
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Augene is the result of using 1\3 as an approximation of ~5/4 so that the period is a third of 2/1, and then using a sharp 3/2 or equivalently a flat 6/5 as the generator. Because the approximation of 5/4 is so sharp, any good tuning of augene will have a noticeably sharp ~3/2 so that ~6/5 becomes more in-tune (which is also important as it's ''also'' the generator, being equal to ~3/2 minus a period). Because the amount required to get ~6/5 reasonably in-tune is quite significant, it makes sense to lean into this and take advantage of it by tempering out the difference between the very sharp ~9/8 and a flat ~8/7, so that the minor third becomes ~7/6 as in [[#Superpyth]]. Augene merges with superpyth in [[27edo]], which is a recommendable tuning for also being the smallest edo to have all intervals of the [[7-odd-limit]] tuned distinctly (not equated), in which case you get harmonies of the no-11's [[13-odd-limit]] too. It's unclear whether a psychoacoustically optimal tuning of augene would have the fifth sharper or flatter than the 27edo tuning; if you think having the fifth more in-tune is preferable, you could try the [[39edo]] tuning {{nowrap| (23\39, 1\3) }}, which uses a very sharp mapping for ~7/4 so that 7 is barely sharper than 5; by some metrics this tuning is more optimal than 27edo. If you want 6/5 more in-tune, you could try the [[42edo]] tuning {{nowrap| (25\42, 1\3) }}, which may be preferred for having a potentially more convincing approximation of ~4:5:6 than all the other options discussed, as well as for being distinctly consistent in the 7-odd-limit like 27edo. Some even prefer the [[15edo]] tuning, as though it damages the 7-limit even more than augene does so that the 7-odd-limit is no longer tuned distinctly (because of ~8/7 = 1\5 = ~7/6), it includes an approximation of ~11/8, and can be thought of as xenmelodically/structurally interesting for a version of augene where the 5edo fifth is the generator and the 3edo major third is the period. | Augene is the result of using 1\3 as an approximation of ~5/4 so that the period is a third of 2/1, and then using a sharp 3/2 or equivalently a flat 6/5 as the generator. Because the approximation of 5/4 is so sharp, any good tuning of augene will have a noticeably sharp ~3/2 so that ~6/5 becomes more in-tune (which is also important as it's ''also'' the generator, being equal to ~3/2 minus a period). Because the amount required to get ~6/5 reasonably in-tune is quite significant, it makes sense to lean into this and take advantage of it by tempering out the difference between the very sharp ~9/8 and a flat ~8/7, so that the minor third becomes ~7/6 as in [[#Superpyth]]. Augene merges with superpyth in [[27edo]], which is a recommendable tuning for also being the smallest edo to have all intervals of the [[7-odd-limit]] tuned distinctly (not equated), in which case you get harmonies of the no-11's [[13-odd-limit]] too. It's unclear whether a psychoacoustically optimal tuning of augene would have the fifth sharper or flatter than the 27edo tuning; if you think having the fifth more in-tune is preferable, you could try the [[39edo]] tuning {{nowrap| (23\39, 1\3) }}, which uses a very sharp mapping for ~7/4 so that 7 is barely sharper than 5; by some metrics this tuning is more optimal than 27edo. If you want 6/5 more in-tune, you could try the [[42edo]] tuning {{nowrap| (25\42, 1\3) }}, which may be preferred for having a potentially more convincing approximation of ~4:5:6 than all the other options discussed, as well as for being distinctly consistent in the 7-odd-limit like 27edo. Some even prefer the [[15edo]] tuning, as though it damages the 7-limit even more than augene does so that the 7-odd-limit is no longer tuned distinctly (because of ~8/7 = 1\5 = ~7/6), it includes an approximation of ~11/8, and can be thought of as xenmelodically/structurally interesting for a version of augene where the 5edo fifth is the generator and the 3edo major third is the period. | ||
=== [[Pajara]] === | |||
[[Bird's eye view of temperaments by accuracy#Note counts|Note count]]: 10 for {3, 5, 7, 9} ([[2L 8s]]) | |||
[[Bird's eye view of temperaments by accuracy#Generator tunings|Generator tunings]]: 13\22, 20\34, 33\56 | |||
This temperament was discovered by [[Paul Erlich]], and its scales are known for having properties similar to diatonic in the 5-limit. This temperament tempers out [[50/49]], setting [[7/5]] and [[10/7]] to the half-octave. It is generated by a fifth, or alternatively a semitone that is the fifth minus the half-octave. It tempers out [[64/63]] and [[2048/2025]], meaning harmonics 5 is mapped to a half-octave minus two semitones, and harmonic 7 is an octave minus two semitones. This temperament is best used with a decatonic interval classification, so 3/2 is a Perfect 7th<sub>10</sub>, 5/4 is a Major 4th<sub>10</sub>, and 7/4 is a Major 9th<sub>10</sub>. The [[4:5:6:7]] otonal tetrad is written P1-M4-P7-M9. Changing both major intervals to minor ones gives us the [[70:84:105:120|1/(7:8:10:12)]] utonal tetrad. Since 50/49 is tempered out, [[25/24]] and [[49/48]] are equated, both to the augmented unison of the decatonic scale. It works just like in diatonic, where changing the major third of the [[4:5:6]] triad to a minor one gives the [[10:12:15|1/(4:5:6)]] triad. While this temperament has poor accuracy overall, since 12edo's accuracy is accepted by many, this temperament's accuracy can still be considered reasonable. | |||
== 11-limit focus == | == 11-limit focus == | ||