Bird's eye view of temperaments by accuracy: Difference between revisions
m →7-limit focus: make formatting of gens shown consistent |
Gariwizmic and reworded parts of garibaldi and cassandra |
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[[#Note counts|Note count]]: 12 for {3, 5, 9, 15, 27, 45(, 81)} ([[5L 7s]] or [[12L 5s]]) | [[#Note counts|Note count]]: 12 for {3, 5, 9, 15, 27, 45(, 81)} ([[5L 7s]] or [[12L 5s]]) | ||
[[#Generator tunings|Generator tunings]]: 31\53, 69\118, 100\171, 131\224 | [[#Generator tunings|Generator tunings]]: (24\41,) 31\53, 69\118, 100\171, 131\224 | ||
Schismic is a very accurate and efficient [[5-limit]] temperament which is almost identical to [[Pythagorean tuning]] except that it tempers the perfect fifth very slightly flat so as to find [[8/5]] accurately at ([[9/8]])<sup>4</sup>, that is, as the [[Pythagorean augmented fifth]], or equivalently, finding [[5/4]] as the [[Pythagorean diminished fourth]]. Note that the smallest edo that validates its status as a microtemperament is [[118edo]], as [[53edo]], though a tone-efficient tuning, doesn't temper the fifth flat enough, | Schismic is a very accurate and efficient [[5-limit]] temperament which is almost identical to [[Pythagorean tuning]] except that it tempers the perfect fifth very slightly flat so as to find [[8/5]] accurately at ([[9/8]])<sup>4</sup>, that is, as the [[Pythagorean augmented fifth]], or equivalently, finding [[5/4]] as the [[Pythagorean diminished fourth]]. Note that the smallest edo that validates its status as a microtemperament is [[118edo]], as [[53edo]], though a tone-efficient tuning, doesn't temper the fifth flat enough, as it is practically a relabeling of the [[3-limit]]. 41edo arguably qualifies as the coarsest equal temperament to support schismic well enough, but it is way too tempered for a 5-limit tuning, as it also it is [[magic]]. | ||
In schismic, (9/8)<sup>6</sup> overshoots the octave by [[~]][[81/80]] so that the syntonic comma and the [[Pythagorean comma]] are equated. | |||
Many extensions to other primes exist, but most are not accurate enough to be microtemperaments, except for the extension to prime 41 by tempering out [[1025/1024]] = ([[41/32]])/([[32/25]]). However, as it is common to want to extend schismic, we will note common extensions here: | Many extensions to other primes exist, but most are not accurate enough to be microtemperaments, except for the extension to prime 41 by tempering out [[1025/1024]] = ([[41/32]])/([[32/25]]). However, as it is common to want to extend schismic, we will note common extensions here: | ||
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* [[#Garibaldi]] finds [[~]][[8/7]] as [[9/8]] * [[81/80]] by tempering out [[5120/5103]] = [[64/63|S8]]/[[81/80|S9]], so that it prefers a slightly-sharp or just fifth. | * [[#Garibaldi]] finds [[~]][[8/7]] as [[9/8]] * [[81/80]] by tempering out [[5120/5103]] = [[64/63|S8]]/[[81/80|S9]], so that it prefers a slightly-sharp or just fifth. | ||
* | * [[Schismatic family#Tridecaschismic (2.3.5.13)|Tridecaschismic]] (which is the 2.3.5.13 version of [[#Cassandra]]) finds 13/4 as (9/8)<sup>10</sup> and demands an approximately Pythagorean tuning. | ||
* [[#Nestoria]] equates [[~]][[19/16]] with [[32/27]] and [[~]][[19/15]] with [[81/64]]. | * [[#Nestoria]] equates [[~]][[19/16]] with [[32/27]] and [[~]][[19/15]] with [[81/64]]. | ||
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=== 11-limit focus === | === 11-limit focus === | ||
=== ~17-limit focus === | === ~17-limit focus === | ||
==== [[Gariwizmic]] ==== | |||
[[Bird's eye view of temperaments by accuracy#Note counts|Note counts]]: TBA | |||
Generator tunings: (55\94, 1\2), (79\176, 1\2), (158\270, 1\2) | |||
Gariwizmic has a 1/2-octave period representing [[99/70]], two of them being ~2/1 and tempering out [[9801/9800]]. It is generated by a slightly sharp fifth with the 2.3.7.11 mappings of [[gary]], tempering out [[19712/19683]] and [[131072/130977]], which essentially maps the pythagorean comma to 64/63, and two of those to 33/32. | |||
The kalisma allows the pythagorean comma to be split into two [[2835/2816|fwiwismas]], and this allows reaching a [[352/351]] ~ [[385/384]] minicomma by 47 fifths plus a semioctave, or alternatively put, a ~[[256/243|limma]] minus 3.5 pythcommas, tempering out [[4096/4095]] and [[1716/1715]]. Primes 5 and 13 are thus reached by a diminished fourth (96/77) + minicomma (39 fifths + 1 period), and a triply augmented fourth (44/27) - minicomma (. | |||
It is best represented in 270edo, which is well known for its astoundingly accurate 13-limit, making it one of the best fifth-based rank-2 temperaments. It can be easily thought as cassandra, but with the minicomma as a "generator" for primes 5 and 13 which is still reachable within the rank-2 structure (It isn't a true generator; were it independent, the temperament would be [[cassaschismic]]). | |||
It naturally extends into the 2.3.5.7.11.13.19 subgroup by adding [[1216/1215]] to the comma list, finding the major third to be [[19/15]] and 19/16 to be the minor third + minicomma, thus also working as [[361/360]]. Interestingly, gariwizmic tempers out the smallest superparticular of the 19- and 23-limit: the [[tredekisma]]. | |||
=== Higher-limit focus === | === Higher-limit focus === | ||
=== No-2's focus === | === No-2's focus === | ||
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[[#Generator tunings|Generator tunings]]: 24\31, 31\53, 55\94 | [[#Generator tunings|Generator tunings]]: 24\31, 31\53, 55\94 | ||
Garibaldi is | Garibaldi is aguably the simplest way of effectively bestowing prime 7 upon [[#Schismic|schismic]], at the cost of accuracy as needing a slightly sharper, not flatter fifth, tunes the 5-limit worse so that it is no longer a microtemperament. This is done by interpreting ([[9/8]])<sup>3</sup> as [[~]][[10/7]] by also tempering out [[5120/5103|S8/S9]] so that 10/9 - 9/8 - 8/7 have equidistant steps, with the step being a convenient tempered comma-sized interval that simultaneously represents [[64/63|S8]], [[81/80|S9]], and the [[Pythagorean comma]] (as per schismic). [[41edo]] and [[53edo]] are slightly overtempered and undertempered for it respectively, so that [[94edo]] is pretty close to optimal, though it has an slightly inconsistently flat [[~]][[25/16]] which is unbefitting of schismic. 94 + 53 = [[147edo]] also supports it but with yet more inconsistencies due to the finer gamut. | ||
* For | |||
* For | Extensions include: | ||
* Cassandra (41 & 53), tempering out 385/384 and 352/351, finds prime 11 at 23 fifths, and prime 13 at 20 fifths. It is the best extension, with support up to the 23-limit as seen in 94edo. | |||
* Andromeda (41f & 53), tempering out 385/384 and 351/350, finds prime 11 at 23 fifths, and prime 13 at -33 fifths. | |||
* Helenus (41ef & 53), tempering out 176/175 and 351/350, finds prime 11 at -30 fifths, and prime 13 at -33 fifths. | |||
Which of 41edo and 53edo do better in the 7-limit depends on how you measure them and who you ask; therefore, a better way of choosing is based on whether you care more about primes 7 and 11 or prime 5 and 13: | |||
* For primes 7 and 11, [[41edo]] is better, as it finds [[11/9]][[~]][[27/22]] as [[Sqrt(3/2)|half of the fifth]] and as a comma above [[~]][[6/5]] or a comma below [[~]][[5/4]]; This also corresponds to being the intersection of [[cassandra]] and [[andromeda]] (respectively). Primes 5 and 13 are worse, as these mappings prefer an ever so slightly flat fifth. | |||
* For primes 5 and 13, [[53edo]] is better, as it has a close to just fifth that benefits schismic, tempering out 325/324 too so that [[~]][[16/13]] is a comma below [[~]][[5/4]]. This corresponds to [[Schismatic family#Tridecaschismic (2.3.5.13)|tridecaschismic]]. Primes 7 and 11 are worse, as their cassandra mappings prefer a slightly sharper fifth. | |||
=== 11-limit focus === | === 11-limit focus === | ||
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=== ~17-limit focus === | === ~17-limit focus === | ||
==== [[Cassandra]] ==== | |||
See [[Bird's eye view of temperaments by accuracy#Garibaldi|garibaldi]]. | |||
==== [[Buzzard]] ==== | ==== [[Buzzard]] ==== | ||
[[#Note counts|Note counts]]: | [[#Note counts|Note counts]]: | ||