Hemimage temperaments: Difference between revisions

This is a collection of [[Rank-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the [[hemimage comma]], {{monzo| 5 -7 -1 3 }} = 10976/10935. These include commatic, chromat, degrees, subfourth, and bisupermajor, considered below, as well as the following discussed elsewhere:  

* ''[[Quasisuper]]'' (+64/63) → [[Archytas clan #Quasisuper|Archytas clan]]

* ''[[Unicorn]]'' (+126/125) → [[Unicorn family #Septimal unicorn|Unicorn family]]

* [[Hemififths]] (+2401/2400 or 5120/5103) → [[Breedsmic temperaments #Hemififths|Breedsmic temperaments]]

* ''[[Pluto]]'' (+4000/3969) → [[Mirkwai clan #Pluto|Mirkwai clan]]

* ''[[Hendecatonic]]'' (+6144/6125) → [[Porwell temperaments #Hendecatonic|Porwell temperaments]]

 

 

[[Comma list]]: 10976/10935, 235298/234375

 

 

 

: mapping generators: ~63/50, ~28/27

 

[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~28/27 = 60.528

 

{{Optimal ET sequence|legend=1| 39d, 60, 99, 258, 357, 456 }}

 

[[Badness]]: 0.057499

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Badness: 0.033420

{{See also| Very high accuracy temperaments #Kwazy }}

{{Multival|legend=1| 16 -10 34 -53 9 107 }}

 

[[Optimal tuning]] ([[POTE]]): ~1225/864 = 1\2, ~192/175 = 162.806

[[Badness]]: 0.065492

Optimal tuning (POTE): ~99/70, ~11/10 = 162.773

{{Optimal ET sequence|legend=1| 22, 74d, 96d, 118, 258e, 376de }}

Badness: 0.032080

== Commatic ==

The commatic temperament has a period of half octave and a generator of 20.4 cents. It is so named because the generator is a small interval ("commatic") which represents 81/80, 99/98, and 100/99 all tempered together.

{{Multival|legend=1| 10 38 36 37 29 -23 }}

[[Optimal tuning]] ([[POTE]]): ~567/400 = 1\2, ~81/80 = 20.377

 

[[Badness]]: 0.084317

Optimal tuning (POTE): ~99/70 = 1\2, ~81/80 = 20.390

{{Optimal ET sequence|legend=1| 58, 118, 294, 412d }}

Badness: 0.030461

Optimal tuning (POTE): ~99/70 = 1\2, ~66/65 = 20.427

{{Optimal ET sequence|legend=1| 58, 118, 176f }}

Badness: 0.026336

Optimal tuning (POTE): ~17/12 = 1\2, ~66/65 = 20.378

{{Optimal ET sequence|legend=1| 58, 118, 294ffg, 412dffgg }}

Badness: 0.022396

{{ See also | 20th-octave temperaments }}

An obvious extension to the 23-limit exists by equating 4\20 = 1\5 with [[23/20]], 6\20 = 3\10 with [[69/56]], 7\20 with [[23/18]], etc. By observing that 1\20 works as [[30/29]]~[[29/28]]~[[28/27]], with 29/28 being especially accurate, and by equating [[29/22]] with 2\5 = 240{{cent}}, we get a uniquely elegant extension to the 29-limit which tempers out ([[33/25]])/([[29/22]]) = [[726/725]], [[784/783|S28 = 784/783]] and [[841/840|S29 = 841/840]]. An edo as large as [[220edo|220]] supports it by patent val, though it does not appear in the optimal ET sequence, and [[80edo]] and [[140edo]] are both much more recommendable tunings.

By equating 37/28 with 2\5 and more accurately 85/74 with 1\5 and 44/37 with 1\4 (among many other equivalences) we get an extension to prime 37 agreeing with many (semi)convergents. By equating 60/41~41/28 with 11\20 or equivalently 56/41~41/30 with 9\20 and by equating 44/41 with 1\10 (among many other equivalences) there is a very efficient extension to prime 41.

By looking at the mapping, we observe an 80-note [[mos scale]] is ideal, so that [[80edo]] is in some sense both a trivial and maximally efficient tuning of this temperament. We also observe an abundance of JI interpretations of [[20edo]] by combining primes so that all things require 3 generators, yielding: 37:44:54:56:58:60:69:74:82:85. Alternatively, combining primes so that all things require 2 generators yields 36:40:46:51 which except for intervals of 51 is contained implicitly in the above. The ratios therein should thus be instructive for how the structure of 20edo relates to its representation of JI in this temperament. Note that prime 47 can be added but only really makes sense in rooted form in [[140edo]].

{{Multival|legend=1| 20 40 60 17 39 27 }}

 

[[Optimal tuning]] ([[POTE]]): ~28/27 = 1\20, ~3/2 = 703.015 (~126/125 = 16.985)

 

{{Optimal ET sequence|legend=1| 20cd, 60, 80, 140, 640b, 780b }}

Badness (Dirichlet): 2.694

Optimal tuning (POTE): ~28/27 = 1\20, ~3/2 = 703.231 (~100/99 = 16.769)

 

{{Optimal ET sequence|legend=1| 20cd, 60e, 80, 140, 360 }}

Badness: 0.046770

Badness (Dirichlet): 1.546

Optimal tuning (POTE): ~28/27 = 1\20, ~3/2 = 703.080 (~100/99 = 16.920)

 

{{Optimal ET sequence|legend=1| 20cde, 60e, 80, 140 }}

Badness: 0.032718

Badness (Dirichlet): 1.352

Optimal tuning (CTE): ~28/27 = 1\20, ~3/2 = 703.107 (~100/99 = 16.893)

{{Optimal ET sequence|legend=1| 20cde, 60e, 80, 140 }}

Badness (Dirichlet): 1.171

Optimal tuning (CTE): ~28/27 = 1\20, ~3/2 = 703.107 (~100/99 = 16.893)

{{Optimal ET sequence|legend=1| 20cde, 60e, 80, 140 }}

Badness (Dirichlet): 1.273

Optimal tuning (CTE): ~28/27 = 1\20, ~3/2 = 703.169 (~100/99 = 16.831)

{{Optimal ET sequence|legend=1| 20cdei, 60e, 80, 140 }}

Badness (Dirichlet): 1.209

Optimal tuning (CTE): ~29/28 = 1\20, ~3/2 = 703.171 (~100/99 = 16.829)

 

{{Optimal ET sequence|legend=1| 20cdeij, 60e, 80, 140 }}

 

Badness (Dirichlet): 1.134

=== no-31's 37-limit ===

 

 

 

 

Badness (Dirichlet): 1.127

 

Subgroup: 2.3.5.7.11.13.17.19.23.29.37.41

 

 

 

 

 

A generator for the squarschimidt temperament is the fourth root of [[5/2]], (5/2)^1/4, tuned around 396.6 cents. The squarschimidt temperament can be described as 118&239 temperament, tempering out the hemimage comma and quasiorwellisma, 29360128/29296875 in the 7-limit. In the 11-limit, 118&239 tempers out 3025/3024, 5632/5625, and 12005/11979, and the generator represents ~44/35.

 

[[Subgroup]]: 2.3.5

 

[[Comma list]]: {{monzo| 61 4 -29 }}

 

 

 

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~98304/78125 = 396.621

 

[[Badness]]: 0.218314

{{Multival|legend=1| 29 4 69 -61 28 149 }}

 

{{Optimal ET sequence|legend=1| 118, 239, 357, 596, 1549bd }}

[[Badness]]: 0.132821

Optimal tuning (POTE): ~2 = 1\1, ~44/35 = 396.644

{{Optimal ET sequence|legend=1| 118, 239, 357, 596 }}

Badness: 0.038186

[[Comma list]]: 10976/10935, 65536/64827

 

 

{{Multival|legend=1| 4 -37 -3 -68 -16 97 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~21/16 = 475.991

{{Optimal ET sequence|legend=1| 58, 121, 179, 300bd, 479bcd }}

[[Badness]]: 0.140722

Comma list: 540/539, 896/891, 12005/11979

Mapping: {{mapping| 1 0 17 4 11 | 0 4 -37 -3 -19 }}

Optimal tuning (POTE): ~2 = 1\1, ~21/16 = 475.995

{{Optimal ET sequence|legend=1| 58, 121, 179e, 300bde }}

Badness: 0.045323

Comma list: 352/351, 364/363, 540/539, 676/675

Mapping: {{mapping| 1 0 17 4 11 16 | 0 4 -37 -3 -19 -31 }}

Optimal tuning (POTE): ~2 = 1\1, ~21/16 = 475.996

{{Optimal ET sequence|legend=1| 58, 121, 179ef, 300bdef }}

Badness: 0.023800