Hemimage temperaments: Difference between revisions

This is a collection of [[rank-2 temperament|rank-2]] [[temperament]]s [[tempering out]] the [[hemimage comma]] ({{monzo|legend=1| 5 -7 -1 3 }}, [[ratio]]: 10976/10935).

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

* ''[[Dodecacot]]'' (+3125/3087) → [[Tetracot family #Dodecacot|Tetracot family]]

* [[Parakleismic]] (+3136/3125 or 4375/4374) → [[Parakleismic family #Parakleismic|Parakleismic family]]

* ''[[Chromat]]'' (+235298/234375) → [[Amity family #Chromat|Amity family]]

* ''[[Marfifths]]'' (+15625/15552) → [[Kleismic family #Marfifths|Kleismic family]]

* ''[[Yarman I]]'' (+244140625/243045684) → [[Quartonic family]]

Considered below are bisupermajor, bicommatic, degrees, squarschmidt, and leapmonth, in the order of increasing [[badness]].  

: ''For the 5-limit version, see [[Very high accuracy temperaments #Kwazy]].''

[[Subgroup]]: 2.3.5.7

: mapping generators: ~1225/864, ~192/175

[[Optimal tuning]]s:

* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~192/175 = 162.8082{{c}}

: error map: {{val| 0.000 +0.510 -0.355 -1.087 }}

{{Optimal ET sequence|legend=1| 22, 74d, 96d, 118, 140, 258, 398, 656d }}

[[Badness]] (Sintel): 1.66

Comma list: 385/384, 3388/3375, 9801/9800

{{Mapping|legend=0| 2 1 6 1 8 | 0 8 -5 17 -4 }}

Optimal tunings:

* CWE: ~99/70 = 600.0000{{c}}, ~11/10 = 162.7788{{c}}

{{Optimal ET sequence|legend=0| 22, 74d, 96d, 118, 258e, 376de, 634dee }}

Badness (Sintel): 1.06

== Bicommatic ==

Used to be known simply as the ''commatic'' temperament, the bicommatic temperament has a period of half octave and a generator of 20.4 cents, a small interval ("commatic") which represents 81/80, 99/98, and 100/99 all tempered together.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 10976/10935, 50421/50000

: mapping generators: ~567/400, ~81/80

* [[WE]]: ~567/400 = 600.0497{{c}}, ~81/80 = 20.3790{{c}}

: [[error map]]: {{val| +0.099 +0.089 +1.085 -1.756 }}

* [[CWE]]: ~567/400 = 600.0000{{c}}, ~81/80 = 20.3837{{c}}

: error map: {{val| 0.000 -0.037 +0.976 -1.920 }}

[[Badness]] (Sintel): 2.13

Subgroup: 2.3.5.7.11

Comma list: 441/440, 3388/3375, 8019/8000

{{Mapping|legend=0| 2 3 4 5 6 | 0 5 19 18 27 }}

Optimal tunings:  

* CWE: ~99/70 = 600.0000{{c}}, ~81/80 = 20.3948{{c}}

{{Optimal ET sequence|legend=0| 58, 118, 294, 412d }}

Badness (Sintel): 1.01

=== 13-limit ===

Comma list: 196/195, 352/351, 729/728, 1001/1000

{{Mapping|legend=0| 2 3 4 5 6 7 | 0 5 19 18 27 12 }}

Optimal tunings:

* CWE: ~99/70 = 600.0000{{c}}, ~66/65 = 20.4093{{c}}

{{Optimal ET sequence|legend=0| 58, 118, 176f }}

Badness (Sintel): 1.09

=== 17-limit ===

Comma list: 170/169, 196/195, 289/288, 352/351, 561/560

{{Mapping|legend=0| 2 3 4 5 6 7 8 | 0 5 19 18 27 12 5 }}

Optimal tunings:

* CWE: ~17/12 = 600.0000{{c}}, ~66/65 = 20.3804{{c}}

{{Optimal ET sequence|legend=0| 58, 118 }}

Badness (Sintel): 1.14

== Degrees ==

{{About|the regular temperament|scale degrees|degree}}

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

Degrees temperament has a period of 1/20 octave and tempers out the hemimage (10976/10935) and the dimcomp (390625/388962). In this temperament, one period equals ~28/27, two equals ~15/14, three equals ~10/9, five equals ~25/21, six equals ~16/13, seven equals ~14/11, nine equals ~15/11, and ten equals ~99/70.  

An obvious extension to the 23-limit exists by mapping [[23/20]] to 4\20 (1\5), [[69/56]] to 6\20 (3\10), and [[23/18]] to 7\20. By observing that 1\20 works as [[30/29]]~[[29/28]]~[[28/27]], with 29/28 being especially accurate, and by mapping [[29/22]] to 2\5, we get a uniquely elegant extension to the 29-limit which tempers out [[726/725]], which is the difference between [[33/25]] and [[29/22]], as well as [[784/783]] ({{S|28}}) and [[841/840]] ({{S|29}}). 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 for prime [[37/1|37]] agreeing with many [[semiconvergent]]s, tempering out [[481/480]]. By mapping [[60/41]] and [[41/28]] to 11\20 or equivalently [[56/41]] and [[41/30]] to 9\20 and by mapping [[44/41]] to 1\10 (among many other equivalences), there is a very efficient extension for prime [[41/1|41]] tempering out [[451/450]].

The 80-note generator chain is ideal, so [[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]].

[[Comma list]]: 10976/10935, 390625/388962

{{Mapping|legend=1| 20 0 -17 -39 | 0 1 2 3 }}

[[Optimal tuning]]s:  

* [[WE]]: ~28/27 = 59.9922{{c}}, ~3/2 = 702.9233{{c}} (~126/125 = 16.9828{{c}})

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

[[Badness]] (Sintel): 2.69

Comma list: 1331/1323, 1375/1372, 2200/2187

{{Mapping|legend=0| 20 0 -17 -39 -26 | 0 1 2 3 3 }}

Optimal tunings:

* WE: ~28/27 = 59.9929{{c}}, ~3/2 = 703.1478{{c}} (~100/99 = 16.7666{{c}})

* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.1556{{c}} (~100/99 = 16.8444{{c}})

{{Optimal ET sequence|legend=0| 60e, 80, 140, 360 }}

Badness (Sintel): 1.55

Comma list: 325/324, 352/351, 1001/1000, 1331/1323

{{Mapping|legend=0| 20 0 -17 -39 -26 74 | 0 1 2 3 3 0 }}

Optimal tunings:

* WE: ~28/27 = 59.9996{{c}}, ~3/2 = 703.0749{{c}} (~100/99 = 16.9197{{c}})

* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.0770{{c}} (~100/99 = 16.9230{{c}})

{{Optimal ET sequence|legend=0| 60e, 80, 140 }}

Badness (Sintel): 1.35

Comma list: 289/288, 325/324, 352/351, 561/560, 1001/1000

{{Mapping|legend=0| 20 0 -17 -39 -26 74 50 | 0 1 2 3 3 0 1 }}

Optimal tunings:

* WE: ~28/27 = 60.0058{{c}}, ~3/2 = 703.0364{{c}} (~100/99 = 17.0335{{c}})

* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.0061{{c}} (~100/99 = 16.9939{{c}})

{{Optimal ET sequence|legend=0| 60e, 80, 140 }}

Badness (Sintel): 1.17

=== 19-limit ===

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 286/285, 289/288, 325/324, 352/351, 400/399, 476/475

* WE: ~28/27 = 59.9961{{c}}, ~3/2 = 703.1523{{c}} (~100/99 = 16.8015{{c}})

* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.1777{{c}} (~100/99 = 16.8223{{c}})

{{Optimal ET sequence|legend=0| 60e, 80, 140 }}

Badness (Sintel): 1.27

=== 23-limit ===

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399

{{Mapping|legend=0| 20 0 -17 -39 -26 74 50 85 27 | 0 1 2 3 3 0 1 0 2 }}

Optimal tunings:

* WE: ~28/27 = 59.9990{{c}}, ~3/2 = 703.1804{{c}} (~100/99 = 16.8074{{c}})

* CWE: ~28/27 = 60.0000{{c}}, ~3/2 = 703.1870{{c}} (~100/99 = 16.8130{{c}})

{{Optimal ET sequence|legend=0| 60e, 80, 140 }}

Badness (Sintel): 1.21

=== 29-limit ===

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 253/252, 286/285, 289/288, 325/324, 352/351, 391/390, 400/399, 406/405

{{Mapping|legend=0| 20 0 -17 -39 -26 74 50 85 27 2 | 0 1 2 3 3 0 1 0 2 3 }}

Optimal tunings:

* WE: ~29/28 = 59.9990{{c}}, ~3/2 = 703.1829{{c}} (~100/99 = 16.8055{{c}})

* CWE: ~29/28 = 60.0000{{c}}, ~3/2 = 703.1891{{c}} (~100/99 = 16.8109{{c}})

{{Optimal ET sequence|legend=0| 60e, 80, 140 }}

Badness (Sintel): 1.13

: ''For the 5-limit version, see [[Father–3 equivalence continuum #Squarschmidt (5-limit)]].''

Squarschimidt may be described as {{nowrap| 118 & 121 }} temperament. The extension here is a less accurate 7-limit interpretation, tempering out the hemimage comma and quasiorwellisma, [[29360128/29296875]]. In the [[11-limit]], it tempers out [[3025/3024]], [[5632/5625]], and [[12005/11979]], and the generator represents [[~]][[44/35]].  

: [[error map]]: {{val| -0.099 +0.543 +0.029 -0.719 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~1125/896 = 396.6417{{c}}

: error map: {{val| 0.000 +0.653 +0.253 -0.552 }}

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

[[Badness]] (Sintel): 3.36

{{Mapping|legend=0| 1 -8 1 -20 -21 | 0 29 4 69 74 }}

 

* CWE: ~2 = 1200.0000{{c}}, ~44/35 = 396.6419{{c}}

{{Optimal ET sequence|legend=0| 118, 239, 357, 596 }}

Badness (Sintel): 1.26

Leapmonth may be described as the {{nowrap| 63 & 80 }} temperament, generated by a [[3/2|perfect fifth]] and being a strong extension of [[leapfrog]]. It was named by [[Flora Canou]] in 2025 following the pattern demonstrated by ''leapday'' and ''leapweek'', the two simpler extensions of leapfrog.  

[[Comma list]]: 10976/10935, 51200/50421

{{Mapping|legend=1| 1 0 -58 -21 | 0 1 38 15 }}

[[Optimal tuning]]s:

* [[WE]]: ~2 = 1198.8005{{c}}, ~3/2 = 704.2543{{c}}

: [[error map]]: {{val| -1.200 +1.100 -0.659 +2.186 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.9318{{c}}

: error map: {{val| 0.000 +2.977 +1.093 +5.150 }}

{{Optimal ET sequence|legend=1| 17c, 46c, 63, 80, 223bd, 303bdd, 383bcddd }}

[[Badness]] (Sintel): 4.79

Comma list: 540/539, 896/891, 1331/1323

{{Mapping|legend=0| 1 0 -58 -21 -14 | 0 1 38 15 11 }}

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.9318{{c}}

{{Optimal ET sequence|legend=0| 17c, 46c, 63, 80, 223bde, 303bdde }}

Badness (Sintel): 1.88

Comma list: 169/168, 352/351, 364/363, 540/539

{{Mapping|legend=0| 1 0 -58 -21 -14 -1 | 0 1 38 15 11 8 }}

Optimal tunings:

* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.9218{{c}}

{{Optimal ET sequence|legend=0| 17c, 46c, 63, 80, 143d }}

Badness (Sintel): 1.53

[[Category:Catalogs of rank-2 temperaments]]