Hemimean clan: Difference between revisions

The '''hemimean clan''' [[Tempering out|tempers out]] the hemimean comma, [[3136/3125]], with [[monzo]] {{monzo| 6 0 -5 2 }}, such that [[7/4]] is split into five steps, of which two make [[5/4]] and three make [[7/5]]; this defines the [[2.5.7 subgroup]] temperament [[didacus]], generated by a tempered hemithird of [[28/25]].

The second comma of the comma list determines which 7-limit family member we are looking at. These [[extension]]s, in general, split the [[syntonic comma]] into two, each for [[126/125]]~[[225/224]], as 3136/3125 = (126/125)/(225/224). Hemiwürschmidt adds [[2401/2400]]; hemithirds adds [[1029/1024]]; spell adds [[49/48]]. These all use the same nominal generator as didacus.

Septimal passion adds [[64/63]], splitting the hemithird into a further two. Septimal meantone adds [[81/80]] as well as [[126/125]] and [[225/224]], splitting an octave plus the hemithird into two perfect fifths. Sycamore adds [[686/675]], splitting the hemithird into three. Semisept adds [[1728/1715]], splitting an octave plus the hemithird into three. Mohavila adds [[135/128]], whereas cohemimabila adds [[65536/64827]], both splitting two octaves plus the hemithird into three. Emka adds [[84035/82944]], splitting two octaves plus the hemithird into four. Bidia adds [[2048/2025]] with a 1/4-octave period. Misty adds [[5120/5103]] with a 1/3-octave period. Bischismic adds [[32805/32768]] with a semioctave period. Hexe adds [[50/49]] with a 1/6-octave period. Clyde adds [[245/243]] with a generator of ~9/7, five of which make the original. Parakleismic adds [[4375/4374]] with a generator of ~6/5. Arch adds [[5250987/5242880]] with a generator of ~64/63. For these seven generators make the original. Sengagen adds [[420175/419904]] with a generator of ~686/675, splitting the hemithird into eight. Subpental adds [[19683/19600]] with a generator of ~56/45, nine of which make the original.  

Didacus has canonical subgroup extensions to primes 11 and 13, at [[#Undecimal didacus|undecimal didacus]]. Other subgroup extensions include rectified hebrew and isra.

* ''[[Passion]]'' (+64/63 or 3125/3087) → [[Passion family #Septimal passion|Passion family]]

* [[Meantone]] (+81/80, 126/125, 225/224) → [[Meantone family #Septimal meantone|Meantone family]]

* [[Misty]] (+5120/5103) → [[Misty family #Septimal misty|Misty family]]

{{main|Didacus}}

See also its canonical extension to the 2.5.7.11 subgroup, [[#Undecimal didacus]].

[[Subgroup]]: 2.5.7

[[Comma list]]: [[3136/3125]]

{{Mapping|legend=2| 1 0 -3 | 0 2 5 }}

: sval mapping generators: ~2, ~56/25

{{Mapping|legend=3| 1 0 0 -3 | 0 0 2 5 }}

: [[gencom]]: [2 56/25; 3136/3125]

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.772

{{Optimal ET sequence|legend=1| 6, 19, 25, 31, 99, 130, 161, 353, 514c, 867c }}

[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents

[[Badness]] (Sintel): 0.091

| [[#Hemithirds|Hemithirds]] || [[Luna family#Luna|Luna]] || +1 || -15 || ↑ 31 <br /> ↓ [[25edo|25]]

| [[#Spell|Spell]] || [[Magic family#Magic|Magic]] || +2 || +10 || ↑ 25

''[[#Strong extensions|Return to the map]]''

'''Hemiwürschmidt''' (sometimes spelled '''hemiwuerschmidt''') is not only one of the more accurate extensions of didacus, but also the most important extension of 5-limit [[würschmidt]], even with the rather large complexity for the fifth. It tempers out [[2401/2400]], [[3136/3125]], and [[6144/6125]]. [[68edo]], [[99edo]] and [[130edo]] can all be used as tunings, but 130 is not only the most accurate, it shows how hemiwürschmidt extends to a higher limit temperament, mapping 11 to 40 generators and 13 to -39.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 2401/2400, 3136/3125

{{Mapping|legend=1| 1 15 4 7 | 0 -16 -2 -5 }}

Mapping generators: ~2, ~25/14

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.898

{{Optimal ET sequence|legend=1| 31, 68, 99, 229, 328, 557c, 885cc }}

[[Badness]]: 0.020307

=== 2.3.5.7.23 subgroup ===

As described at the page for [[würschmidt]], there is an extension to prime 23 with essentially no damage, which maps the prime to 28 generators (or 14 generators of würschmidt).

Subgroup: 2.3.5.7.23

[[Comma list]]: 576/575, 736/735, 1127/1125

{{Mapping|legend=1| 1 15 4 7 28 | 0 -16 -2 -5 -28 }}

Mapping generators: ~2, ~25/14

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.901

{{Optimal ET sequence|legend=1| 31, 68, 99, 229, 328 }}

Badness (Sintel): 0.304

=== 11-limit ===

Subgroup: 2.3.5.7.11

Comma list: 243/242, 441/440, 3136/3125

{{Mapping|legend=0| 1 15 4 7 37 | 0 -16 -2 -5 -40 }}

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.840

{{Optimal ET sequence|legend=1| 31, 99e, 130, 811ce }}

Badness: 0.021069

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

Comma list: 243/242, 351/350, 441/440, 3584/3575

{{Mapping|legend=0| 1 15 4 7 37 -29 | 0 -16 -2 -5 -40 39 }}

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.829

{{Optimal ET sequence|legend=1| 31, 99e, 130, 291, 421e, 551ce }}

Badness: 0.023074

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 176/175, 196/195, 275/273

{{Mapping|legend=0| 1 15 4 7 37 -3 | 0 -16 -2 -5 -40 8 }}

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.918

{{Optimal ET sequence|legend=1| 31, 68e, 99ef }}

Badness: 0.031199

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 1375/1372

{{Mapping|legend=0| 1 15 4 7 11 | 0 -16 -2 -5 -9 }}

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.884

{{Optimal ET sequence|legend=1| 31, 68, 99, 130e, 229e }}

Badness: 0.029270

==== 13-limit ====

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 176/175, 196/195, 275/273

{{Mapping|legend=0| 1 15 4 7 11 -3 | 0 -16 -2 -5 -9 8 }}

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 194.004

{{Optimal ET sequence|legend=1| 31, 68, 99f, 167ef }}

Badness: 0.028432

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 105/104, 121/120, 1375/1372

{{Mapping|legend=0| 1 15 4 7 11 23 | 0 -16 -2 -5 -9 -23 }}

Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.698

{{Optimal ET sequence|legend=1| 6f, 31 }}

Badness: 0.044886

This has been documented in Graham Breed's temperament finder as ''semihemiwürschmidt'', but ''quadrawürschmidt'' arguably makes more sense.  

The generator of quadrawürschmidt is essentially a [[septimal meantone]] fifth. However, it is not used to represent [[3/2]], as 3/2 is found at the hemiwürschmidt position, 16 wholetones up. The small comma between the generator and 3/2 is taken to represent [[441/440]].

Subgroup: 2.3.5.7.11