Hemimean clan: Difference between revisions

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Switch to Sintel's badness, WE & CWE tunings (2/); restore the name *semihemiwürschmidt*, as there's no internal conflict
Improve intro to some entries, with source of names added
 
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{{See also| Würschmidt family }}
{{See also| Würschmidt family }}


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.
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. 130edo in particular shows how hemiwürschmidt may be extended to a higher limit, mapping 11 to 40 generators which is the same as würschmidt.
 
Hemiwürschmidt was named by [[Gene Ward Smith]] in 2004 as ''hemi-'' + ''würschmidt''.<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning-math/topicId_10541.html Yahoo! Tuning Group (archive) | ''Names for important high-complexity temperaments'']</ref>


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 120: Line 122:
Badness (Sintel): 0.953
Badness (Sintel): 0.953


==== 2.3.5.7.11.13.23 subgroup ====
===== 2.3.5.7.11.13.23 subgroup =====
As described at the page for [[würschmidt]], there is an extension for prime 23 with essentially no damage, which maps the prime to 28 generators (or 14 generators of würschmidt).
As described at the page for [[würschmidt]], there is an extension for prime 23 with essentially no damage, which maps the prime to 28 generators (or 14 generators of würschmidt).


Line 153: Line 155:


=== Hemiwur ===
=== Hemiwur ===
Hemiwur may as well be described as 31 & 37, mapping prime 11 to 9 generator steps.
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Line 213: Line 217:
* CWE: ~2 = 1200.0000{{c}}, ~220/147 = 696.9548{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~220/147 = 696.9548{{c}}


{{Optimal ET sequence|legend=1| 31, 105be, 136e, 167, 198, 427c }}
{{Optimal ET sequence|legend=0| 31, 105be, 136e, 167, 198, 427c }}


Badness (Sintel): 1.15
Badness (Sintel): 1.15
Line 259: Line 263:
* CWE: ~17/12 = 600.0000{{c}}, ~28/25 = 193.9092{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~28/25 = 193.9092{{c}}


{{Optimal ET sequence|legend=1| 62e, 68, 130, 198 }}
{{Optimal ET sequence|legend=0| 62e, 68, 130, 198 }}


Badness (Sintel): 1.48
Badness (Sintel): 1.48
Line 274: Line 278:
* CWE: ~17/12 = 600.0000{{c}}, ~19/17 = 193.9128{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~19/17 = 193.9128{{c}}


{{Optimal ET sequence|legend=1| 62e, 68, 130, 198 }}
{{Optimal ET sequence|legend=0| 62e, 68, 130, 198 }}


Badness (Sintel): 1.32
Badness (Sintel): 1.32
Line 289: Line 293:
* CWE: ~17/12 = 600.0000{{c}}, ~28/25 = 193.9238{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~28/25 = 193.9238{{c}}


{{Optimal ET sequence|legend=1| 68, 130g, 198g }}
{{Optimal ET sequence|legend=0| 68, 130g, 198g }}


Badness (Sintel): 1.51
Badness (Sintel): 1.51
Line 304: Line 308:
* CWE: ~17/12 = 600.0000{{c}}, ~28/25 = 193.9417{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~28/25 = 193.9417{{c}}


{{Optimal ET sequence|legend=1| 68, 130gh, 198gh }}
{{Optimal ET sequence|legend=0| 68, 130gh, 198gh }}


Badness (Sintel): 1.80
Badness (Sintel): 1.80
Line 315: Line 319:
[[Comma list]]: 1029/1024, 3136/3125
[[Comma list]]: 1029/1024, 3136/3125


{{Mapping|legend=1| 1 4 2 2 | 0 -15 2 5 }}
{{Mapping|legend=1| 1 -11 4 7 | 0 15 -2 -5 }}
: mapping generators: ~2, ~25/14


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.244
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.3104{{c}}, ~25/14 = 1007.0160{{c}} (~28/25 = 193.2944{{c}})
: [[error map]]: {{val| +0.310 -0.129 +0.896 -1.733 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~25/14 = 1006.7606{{c}} (~28/25 = 193.2394{{c}})
: error map: {{val| 0.000 -0.545 +0.165 -2.629 }}


[[Minimax tuning]]:
[[Minimax tuning]]:
Line 327: Line 336:
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7


{{Optimal ET sequence|legend=1| 25, 31, 87, 118 }}
{{Optimal ET sequence|legend=1| 25, 31, 87, 118, 267d, 385d }}


[[Badness]]: 0.044284
[[Badness]] (Sintel): 1.12


=== 11-limit ===
=== 11-limit ===
Line 336: Line 345:
Comma list: 385/384, 441/440, 3136/3125
Comma list: 385/384, 441/440, 3136/3125


{{Mapping|legend=0| 1 4 2 2 7 | 0 -15 2 5 -22 }}
{{Mapping|legend=0| 1 -11 4 7 -15 | 0 15 -2 -5 22 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.227
Optimal tunings:
* WE: ~2 = 1200.3576{{c}}, ~25/14 = 1007.0732{{c}} (~28/25 = 193.2844{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~25/14 = 1006.7807{{c}} (~28/25 = 193.2193{{c}})


Minimax tuning:
Minimax tuning:
Line 345: Line 356:
: Eigenmonzos (unchanged-intervals): 2, 11/7
: Eigenmonzos (unchanged-intervals): 2, 11/7


{{Optimal ET sequence|legend=1| 25e, 31, 87, 118 }}
{{Optimal ET sequence|legend=0| 25e, 31, 87, 118 }}


Badness: 0.019003
Badness (Sintel): 0.628


=== 13-limit ===
=== 13-limit ===
Line 354: Line 365:
Comma list: 196/195, 352/351, 385/384, 625/624
Comma list: 196/195, 352/351, 385/384, 625/624


{{Mapping|legend=0| 1 4 2 2 7 0 | 0 -15 2 5 -22 23 }}
{{Mapping|legend=0| 1 4 2 2 7 0 | 0 15 -2 -5 22 -23 }}


Optimal tuning (POTE): ~2 = 1\1, ~28/25 = 193.166
Optimal tunings:
* WE: ~2 = 1200.0602{{c}}, ~25/14 = 1006.8844{{c}} (~28/25 = 193.1757{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~25/14 = 1006.8343{{c}} (~28/25 = 193.1457{{c}})


{{Optimal ET sequence|legend=1| 31, 56, 87, 118, 205d }}
{{Optimal ET sequence|legend=0| 31, 56, 87, 118, 205d }}


Badness: 0.021738
Badness (Sintel): 0.898


== Spell ==
== Spell ==
Line 371: Line 384:
{{Mapping|legend=1| 1 0 2 2 | 0 10 2 5 }}
{{Mapping|legend=1| 1 0 2 2 | 0 10 2 5 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 189.927
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1202.8089{{c}}, ~28/25 = 190.3712{{c}}
: [[error map]]: {{val| +2.809 +1.757 +0.047 -11.352 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/25 = 190.1078{{c}}
: error map: {{val| 0.000 -0.877 -6.098 -18.287 }}


{{Optimal ET sequence|legend=1| 6, 19, 82dd }}
{{Optimal ET sequence|legend=1| 6, 13b, 19, 82dd }}


[[Badness]]: 0.080958
[[Badness]] (Sintel): 2.05


=== 11-limit ===
=== 11-limit ===
Line 384: Line 401:
{{Mapping|legend=0| 1 0 2 2 3 | 0 10 2 5 3 }}
{{Mapping|legend=0| 1 0 2 2 3 | 0 10 2 5 3 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 190.285
Optimal tunings:
* WE: ~2 = 1200.4163{{c}}, ~11/10 = 190.3511{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 190.3091{{c}}


{{Optimal ET sequence|legend=1| 6, 19, 44de, 63dee, 82ddee }}
{{Optimal ET sequence|legend=0| 6, 13b, 19 }}


Badness: 0.059791
Badness (Sintel): 1.98


==== 13-limit ====
==== 13-limit ====
Line 397: Line 416:
{{Mapping|legend=0| 1 0 2 2 3 4 | 0 10 2 5 3 -2 }}
{{Mapping|legend=0| 1 0 2 2 3 4 | 0 10 2 5 3 -2 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 189.928
Optimal tunings:
* WE: ~2 = 1201.8239{{c}}, ~11/10 = 190.2163{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 190.0020{{c}}


{{Optimal ET sequence|legend=1| 6, 19, 82ddeeff }}
{{Optimal ET sequence|legend=0| 6, 13b, 19 }}


Badness: 0.045591
Badness (Sintel): 1.88


==== Cantrip ====
==== Cantrip ====
Line 410: Line 431:
{{Mapping|legend=0| 1 0 2 2 3 1 | 0 10 2 5 3 17 }}
{{Mapping|legend=0| 1 0 2 2 3 1 | 0 10 2 5 3 17 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/10 = 190.360
Optimal tunings:
* WE: ~2 = 1200.4118{{c}}, ~11/10 = 190.4253{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 190.3835{{c}}


{{Optimal ET sequence|legend=1| 19, 44de, 63dee, 82ddee }}
{{Optimal ET sequence|legend=0| 6f, 19 }}


Badness: 0.041603
Badness (Sintel): 1.72


= Weak extensions =
= Weak extensions =
== Semisept ==
== Semisept ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Semisept]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Semisept]].''


The minimal generator of semisept is half a tempered septimal major sixth (12/7), hence the name. Three such generator steps minus an octave give the hemithird, and six give the classical major third. It can be described as the 31 & 80 temperament, and as one may expect, [[111edo]] makes for a great tuning.  
The generator of semisept is half a tempered septimal major sixth (12/7), hence the name. Three such generator steps minus an octave give the hemithird, and six give the classical major third. It can be described as the 31 & 80 temperament, and as one may expect, [[111edo]] makes for a great tuning.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 427: Line 449:
[[Comma list]]: 1728/1715, 3136/3125
[[Comma list]]: 1728/1715, 3136/3125


{{Mapping|legend=1| 1 12 6 12 | 0 -17 -6 -15 }}
{{Mapping|legend=1| 1 -5 0 -3 | 0 17 6 15 }}
: mapping generators: ~2, ~98/75


: mapping generators: ~2, ~75/49
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1199.2968{{c}}, ~98/75 = 464.5722{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/49 = 735.155
: [[error map]]: {{val| -0.703 -0.712 +1.119 +1.867 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~98/75 = 464.8120{{c}}
: error map: {{val| 0.000 -0.150 +2.559 +3.355 }}


{{Optimal ET sequence|legend=1| 18, 31, 80, 111 }}
{{Optimal ET sequence|legend=1| 31, 80, 111, 364cdd, 475ccddd }}


[[Badness]]: 0.050472
[[Badness]] (Sintel): 1.28


=== 11-limit ===
=== 11-limit ===
Line 442: Line 467:
Comma list: 176/175, 540/539, 1331/1323
Comma list: 176/175, 540/539, 1331/1323


{{Mapping|legend=0| 1 12 6 12 20 | 0 -17 -6 -15 -27 }}
{{Mapping|legend=0| 1 -5 0 -3 -7 | 0 17 6 15 27 }}


Optimal tuning (POTE): ~2 = 1\1, ~55/36 = 735.125
Optimal tunings:
* WE: ~2 = 1199.3073{{c}}, ~72/55 = 464.6063{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/55 = 464.8423{{c}}


{{Optimal ET sequence|legend=1| 18e, 31, 80, 111, 364cd }}
{{Optimal ET sequence|legend=0| 31, 80, 111 }}


Badness: 0.022476
Badness (Sintel): 0.743


==== 13-limit ====
==== 13-limit ====
Line 455: Line 482:
Comma list: 176/175, 351/350, 540/539, 1375/1372
Comma list: 176/175, 351/350, 540/539, 1375/1372


{{Mapping|legend=0| 1 12 6 12 20 -11 | 0 -17 -6 -15 -27 24 }}
{{Mapping|legend=0| 1 -5 0 -3 -7 13 | 0 17 6 15 27 -24 }}


Optimal tuning (POTE): ~2 = 1\1, ~55/36 = 735.126
Optimal tunings:
* WE: ~2 = 1199.3145{{c}}, ~72/55 = 464.6080{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/55 = 464.8636{{c}}


{{Optimal ET sequence|legend=1| 31, 80, 111 }}
{{Optimal ET sequence|legend=0| 31, 80, 111 }}


Badness: 0.025204
Badness (Sintel): 1.04


===== 17-limit =====
===== 17-limit =====
Line 468: Line 497:
Comma list: 176/175, 256/255, 351/350, 640/637, 715/714
Comma list: 176/175, 256/255, 351/350, 640/637, 715/714


{{Mapping|legend=0| 1 12 6 12 20 -11 -10 | 0 -17 -6 -15 -27 24 23 }}
{{Mapping|legend=0| 1 -5 0 -3 -7 13 13 | 0 17 6 15 27 -24 -23 }}


Optimal tuning (POTE): ~2 = 1\1, ~26/17 = 735.125
Optimal tunings:
* WE: ~2 = 1199.3067{{c}}, ~17/13 = 464.6061{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/13 = 464.8699{{c}}


{{Optimal ET sequence|legend=1| 31, 80, 111 }}
{{Optimal ET sequence|legend=0| 31, 80, 111 }}


Badness: 0.019919
Badness (Sintel): 1.01


===== 19-limit =====
===== 19-limit =====
Line 481: Line 512:
Comma list: 176/175, 286/285, 351/350, 476/475, 540/539, 1331/1323
Comma list: 176/175, 286/285, 351/350, 476/475, 540/539, 1331/1323


{{Mapping|legend=0| 1 12 6 12 20 -11 -10 -8 | 0 -17 -6 -15 -27 24 23 20 }}
{{Mapping|legend=0| 1 -5 0 -3 -7 13 13 12 | 0 17 6 15 27 -24 -23 -20 }}


Optimal tuning (POTE): ~2 = 1\1, ~26/17 = 735.116
Optimal tunings:
* WE: ~2 = 1199.2346{{c}}, ~17/13 = 464.5878{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/13 = 464.8816{{c}}


{{Optimal ET sequence|legend=1| 31, 80, 111 }}
{{Optimal ET sequence|legend=0| 31, 80, 111 }}


Badness: 0.016301
Badness (Sintel): 0.992


===== 23-limit =====
===== 23-limit =====
Line 494: Line 527:
Comma list: 176/175, 253/252, 286/285, 345/343, 351/350, 391/390, 460/459
Comma list: 176/175, 253/252, 286/285, 345/343, 351/350, 391/390, 460/459


{{Mapping|legend=0| 1 12 6 12 20 -11 -10 -8 18 | 0 -17 -6 -15 -27 24 23 20 -22 }}
{{Mapping|legend=0| 1 -5 0 -3 -7 13 13 12 -4 | 0 17 6 15 27 -24 -23 -20 22 }}


Optimal tuning (POTE): ~2 = 1\1, ~26/17 = 735.106
Optimal tunings:
* WE: ~2 = 1199.3227{{c}}, ~17/13 = 464.6320{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/13 = 464.8912{{c}}


{{Optimal ET sequence|legend=1| 31, 80, 111, 191cdh, 302cdgh }}
{{Optimal ET sequence|legend=0| 31, 80, 111, 191cdh }}


Badness: 0.014957
Badness (Sintel): 1.07


==== Semishly ====
==== Semishly ====
Line 507: Line 542:
Comma list: 144/143, 176/175, 196/195, 275/273
Comma list: 144/143, 176/175, 196/195, 275/273


{{Mapping|legend=0| 1 12 6 12 20 8 | 0 -17 -6 -15 -27 -7 }}
{{Mapping|legend=0| 1 -5 0 -3 -7 1 | 0 17 6 15 27 7 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/10 = 464.980
Optimal tunings:
* WE: ~2 = 1198.3103{{c}}, ~13/10 = 464.3248{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/10 = 464.9110{{c}}


{{Optimal ET sequence|legend=1| 31, 49f, 80f }}
{{Optimal ET sequence|legend=0| 31, 49f, 80f }}


Badness: 0.028408
Badness (Sintel): 1.17


== Emka ==
== Emka ==
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Emka]].''
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Emka]].''
 
Emka may be described as the 37 & 50 temperament, which tempers out the hemimean comma and 84035/82944 (quinzo-ayo). It was named by [[Petr Pařízek]] in 2011.


Emka tempers out {{monzo| -50 -8 27 }} in the 5-limit. This temperament can be described as 37 &amp; 50 temperament, which tempers out the hemimean and 84035/82944 (quinzo-ayo). Alternative extension [[Horwell temperaments #Emkay|emkay]] (87 &amp; 224) tempers out the same 5-limit comma as the emka, but with the horwell (65625/65536) rather than the hemimean tempered out.
A related temperament of emka is [[horwell temperaments #Emkay|emkay]] (87 & 224), which is more accurate but more complex, and only differs from emka in the mapping of 7.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 524: Line 563:
[[Comma list]]: 3136/3125, 84035/82944
[[Comma list]]: 3136/3125, 84035/82944


{{Mapping|legend=1| 1 14 6 12 | 0 -27 -8 -20 }}
{{Mapping|legend=1| 1 -13 -2 -8 | 0 27 8 20 }}
: mapping generators: ~2, ~35/24


: mapping generators: ~2, ~48/35
[[Optimal tuning]]s:  
 
* [[WE]]: ~2 = 1200.4701{{c}}, ~35/24 = 648.4717{{c}}
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~48/35 = 551.782
: [[error map]]: {{val| +0.470 +0.669 +0.520 -3.153 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~35/24 = 648.2307{{c}}
: error map: {{val| 0.000 +0.274 -0.468 -4.212 }}


{{Optimal ET sequence|legend=1| 37, 50, 87, 137d, 224d }}
{{Optimal ET sequence|legend=1| 37, 50, 87, 137d, 224d }}


[[Badness]]: 0.144338
[[Badness]] (Sintel): 3.65


=== 11-limit ===
=== 11-limit ===
Line 539: Line 581:
Comma list: 385/384, 2401/2376, 3136/3125
Comma list: 385/384, 2401/2376, 3136/3125


{{Mapping|legend=0| 1 14 6 12 3 | 0 -27 -8 -20 1 }}
{{Mapping|legend=0| 1 -13 -2 -8 4 | 0 27 8 20 -1 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 551.765
Optimal tunings:
* WE: ~2 = 1200.2552{{c}}, ~16/11 = 648.3727{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 648.2399{{c}}


{{Optimal ET sequence|legend=1| 37, 50, 87, 224d, 311d }}
{{Optimal ET sequence|legend=0| 37, 50, 87 }}


Badness: 0.054744
Badness (Sintel): 1.81


=== 13-limit ===
=== 13-limit ===
Line 552: Line 596:
Comma list: 196/195, 364/363, 385/384, 625/624
Comma list: 196/195, 364/363, 385/384, 625/624


{{Mapping|legend=0| 1 14 6 12 3 6 | 0 -27 -8 -20 1 -5 }}
{{Mapping|legend=0| 1 -13 -2 -8 4 1 | 0 27 8 20 -1 5 }}


Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 551.758
Optimal tunings:
* WE: ~2 = 1200.1506{{c}}, ~16/11 = 648.3230{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~16/11 = 648.2442{{c}}


{{Optimal ET sequence|legend=1| 37, 50, 87, 224d, 311d, 398d }}
{{Optimal ET sequence|legend=0| 37, 50, 87, 224d, 311d, 398d }}


Badness: 0.029741
Badness (Sintel): 1.23


== Decipentic ==
== Decipentic ==
The generator for the decipentic temperament (43 &amp; 56) is the tenth root of the [[5/1|5th harmonic (5/1)]], 5<sup>1/10</sup>, tuned between [[75/64]] and [[20/17]] (close to [[27/23]]). Aside from the hemimean comma, this temperament tempers out the [[bronzisma]], 2097152/2083725. [[99edo]] is a good tuning for decipentic, with generator 23\99, and [[mos scale]]s of 9, 13, 17, 30, 43 or 56 notes are available.
Decipentic may be described as the 43 & 56 temperament. The minimal generator for this temperament is the 10th root of the [[5/1|5th harmonic]], 5<sup>1/10</sup>, tuned between [[75/64]] and [[20/17]] (close to [[27/23]]). Aside from the hemimean comma, this temperament tempers out the [[bronzisma]], 2097152/2083725. [[99edo]] is a good tuning for decipentic, with generator 23\99, and [[mos scale]]s of 9, 13, 17, 30, 43 or 56 notes are available.
 
Decipentic was named by [[Xenllium]] in 2021 for the fact that it splits the 5th harmonic into ten equal parts.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 567: Line 615:
[[Comma list]]: 3136/3125, 2097152/2083725
[[Comma list]]: 3136/3125, 2097152/2083725


{{Mapping|legend=1| 1 6 0 -3 | 0 -19 10 25 }}
{{Mapping|legend=1| 1 -13 10 22 | 0 19 -10 -25 }}
: mapping generators: ~2, ~128/75


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/64 = 278.800
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.5804{{c}}, ~128/75 = 920.8777{{c}} (~75/64 = 278.7027{{c}})
: [[error map]]: {{val| -0.420 +0.175 +0.714 +0.001 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~128/75 = 921.2002{{c}} (~75/64 = 278.7998{{c}})
: error map: {{val| 0.000 +0.849 +1.684 +1.169 }}


{{Optimal ET sequence|legend=1| 13, 43, 56, 99 }}
{{Optimal ET sequence|legend=1| 43, 56, 99, 538cd, 637cd, …, 1033bccdd }}


[[Badness]]: 0.087325
[[Badness]] (Sintel): 2.21


=== 11-limit ===
=== 11-limit ===
Line 582: Line 635:
{{Mapping|legend=0| 1 6 0 -3 3 | 0 -19 10 25 2 }}
{{Mapping|legend=0| 1 6 0 -3 3 | 0 -19 10 25 2 }}


Optimal tuning (POTE): ~2 = 1\1, ~75/64 = 278.799
Optimal tunings:
* WE: ~2 = 1199.3002{{c}}, ~128/75 = 920.6639{{c}} (~75/64 = 278.6364{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~128/75 = 921.2019{{c}} (~75/64 = 278.7981{{c}})


{{Optimal ET sequence|legend=1| 13, 43, 56, 99e }}
{{Optimal ET sequence|legend=1| 43, 56, 99e }}


Badness: 0.061413
Badness (Sintel): 2.03


==== 13-limit ====
==== 13-limit ====
Line 593: Line 648:
Comma list: 169/168, 441/440, 832/825, 975/968
Comma list: 169/168, 441/440, 832/825, 975/968


{{Mapping|legend=0| 1 6 0 -3 3 3 | 0 -19 10 25 2 3 }}
{{Mapping|legend=0| 1 -13 10 22 5 6 | 0 19 -10 -25 -2 -3 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 278.802
Optimal tunings:
* WE: ~2 = 1199.5986{{c}}, ~22/13 = 920.8900{{c}} (~13/11 = 278.7085{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~22/13 = 921.1988{{c}} (~13/11 = 278.7981{{c}})


{{Optimal ET sequence|legend=1| 13, 43, 56, 99e }}
{{Optimal ET sequence|legend=0| 43, 56, 99e }}


Badness: 0.047611
Badness (Sintel): 1.97


==== 17-limit ====
==== 17-limit ====
Line 606: Line 663:
Comma list: 169/168, 221/220, 256/255, 273/272, 375/374
Comma list: 169/168, 221/220, 256/255, 273/272, 375/374


{{Mapping|legend=0| 1 6 0 -3 3 3 2 | 0 -19 10 25 2 3 9 }}
{{Mapping|legend=0| 1 -13 10 22 5 6 11 | 0 19 -10 -25 -2 -3 -9 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 278.798
Optimal tunings:
* WE: ~2 = 1199.5089{{c}}, ~17/10 = 920.8252{{c}} (~13/11 = 278.6837{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~17/10 = 921.2034{{c}} (~13/11 = 278.7966{{c}})


{{Optimal ET sequence|legend=1| 13, 43, 56, 99e }}
{{Optimal ET sequence|legend=0| 43, 56, 99e }}


Badness: 0.031191
Badness (Sintel): 1.59


==== 19-limit ====
==== 19-limit ====
Line 619: Line 678:
Comma list: 169/168, 210/209, 221/220, 256/255, 273/272, 286/285
Comma list: 169/168, 210/209, 221/220, 256/255, 273/272, 286/285


{{Mapping|legend=0| 1 6 0 -3 3 3 2 1 | 0 -19 10 25 2 3 9 14 }}
{{Mapping|legend=0| 1 -13 10 22 5 6 11 15 | 0 19 -10 -25 -2 -3 -9 -14 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 278.790
Optimal tunings:
* WE: ~2 = 1199.4107{{c}}, ~17/10 = 920.7575{{c}} (~13/11 = 278.6532{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~17/10 = 921.2020{{c}} (~13/11 = 278.7980{{c}})


{{Optimal ET sequence|legend=1| 13, 43, 56, 99e }}
{{Optimal ET sequence|legend=0| 43, 56, 99e }}


Badness: 0.023899
Badness (Sintel): 1.45


=== Quasijerome ===
=== Quasijerome ===
Line 632: Line 693:
Comma list: 3136/3125, 15488/15435, 16384/16335
Comma list: 3136/3125, 15488/15435, 16384/16335


{{Mapping|legend=0| 1 6 0 -3 3 | 0 -38 20 50 47 }}
{{Mapping|legend=0| 1 -32 20 47 45 | 0 38 -20 -50 -47 }}
: mapping generators: ~2, ~825/448


Optimal tuning (POTE): ~2 = 1\1, ~896/825 = 139.403
Optimal tunings:
* WE: ~2 = 1199.6276{{c}}, ~825/448 = 1060.2861{{c}} (~896/825 = 139.3595{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~825/448 = 1060.5982{{c}} (~896/825 = 139.4016{{c}})


{{Optimal ET sequence|legend=1| 43, 112, 155, 198, 439cd, 637cd }}
{{Optimal ET sequence|legend=0| 43, 112, 155, 198, 439cd, 637cd }}


Badness: 0.092996
Badness (Sintel): 3.07


==== 13-limit ====
==== 13-limit ====
Line 645: Line 709:
Comma list: 676/675, 1001/1000, 3136/3125, 15488/15435
Comma list: 676/675, 1001/1000, 3136/3125, 15488/15435


{{Mapping|legend=0| 1 6 0 -3 3 8 | 0 -38 20 50 47 -37 }}
{{Mapping|legend=0| 1 -32 20 47 45 -29 | 0 38 -20 -50 -47 37 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/12 = 139.403
Optimal tunings:
* WE: ~2 = 1199.6184{{c}}, ~24/13 = 1060.2595{{c}} (~13/12 = 139.3589{{c}})
* CWE: ~2 = 1200.0000{{c}}, ~24/13 = 1060.5971{{c}} (~13/12 = 139.4029{{c}})


{{Optimal ET sequence|legend=1| 43, 155, 198, 439cdf, 637cdf }}
{{Optimal ET sequence|legend=0| 43, 155, 198, 439cdf, 637cdf }}


Badness: 0.044328
Badness (Sintel): 1.83


== Sengagen ==
== Sengagen ==
Line 659: Line 725:


{{Mapping|legend=1| 1 1 2 2 | 0 29 16 40 }}
{{Mapping|legend=1| 1 1 2 2 | 0 29 16 40 }}
: mapping generators: ~2, ~686/675


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~686/675 = 24.217
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8393{{c}}, ~686/675 = 24.2133{{c}}
: [[error map]]: {{val| -0.161 +0.070 +0.778 -0.615 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~686/675 = 24.2119{{c}}
: error map: {{val| 0.000 +0.191 +1.077 -0.349 }}


{{Optimal ET sequence|legend=1| 49, 50, 99, 248, 347, 446 }}
{{Optimal ET sequence|legend=1| 49, 50, 99, 248, 347, 446 }}


[[Badness]]: 0.057978
[[Badness]] (Sintel): 1.47


=== 11-limit ===
=== 11-limit ===
Line 673: Line 744:
{{Mapping|legend=0| 1 1 2 2 3 | 0 29 16 40 23 }}
{{Mapping|legend=0| 1 1 2 2 3 | 0 29 16 40 23 }}


Optimal tuning (POTE): ~2 = 1\1, ~99/98 = 24.235
Optimal tunings:
* WE: ~2 = 1199.3714{{c}}, ~99/98 = 24.2224{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~99/98 = 24.2176{{c}}


{{Optimal ET sequence|legend=1| 49, 50, 99e }}
{{Optimal ET sequence|legend=0| 49, 50, 99e }}


Badness: 0.053828
Badness (Sintel): 1.78


==== 13-limit ====
==== 13-limit ====
Line 686: Line 759:
{{Mapping|legend=0| 1 1 2 2 3 4 | 0 29 16 40 23 -15 }}
{{Mapping|legend=0| 1 1 2 2 3 4 | 0 29 16 40 23 -15 }}


Optimal tuning (POTE): ~2 = 1\1, ~99/98 = 24.181
Optimal tunings:
* WE: ~2 = 1200.0364{{c}}, ~99/98 = 24.1817{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~99/98 = 24.1815{{c}}


{{Optimal ET sequence|legend=1| 49, 50, 99e, 149e }}
{{Optimal ET sequence|legend=0| 49, 50, 99e, 149e }}


Badness: 0.053531
Badness (Sintel): 2.21


==== Sengage ====
==== Sengage ====
Line 699: Line 774:
{{Mapping|legend=0| 1 1 2 2 3 3 | 0 29 16 40 23 35 }}
{{Mapping|legend=0| 1 1 2 2 3 3 | 0 29 16 40 23 35 }}


Optimal tuning (POTE): ~2 = 1\1, ~99/98 = 24.234
Optimal tunings:
* WE: ~2 = 1199.1402{{c}}, ~99/98 = 24.2164{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~99/98 = 24.2092{{c}}


{{Optimal ET sequence|legend=1| 49f, 50, 99ef }}
{{Optimal ET sequence|legend=0| 49f, 50, 99ef }}


Badness: 0.037416
Badness (Sintel): 1.55


== Mowglic ==
== Mowglic ==
The mowglic temperament (19 &amp; 161) is an extension of the [[Syntonic–kleismic equivalence continuum #Mowgli|mowgli temperament]] which tempers out the hemimean comma and the secanticornisma (177147/175000, laruquingu) in the 7-limit.
: ''For the 5-limit version, see [[Syntonic–kleismic equivalence continuum #Mowgli]].''
 
Mowglic may be described as the 19 & 161 temperament, named by [[Xenllium]] in 2021 as a variant of ''mowgli''.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 713: Line 792:


{{Mapping|legend=1| 1 0 0 -3 | 0 15 22 55 }}
{{Mapping|legend=1| 1 0 0 -3 | 0 15 22 55 }}
: mapping generators: ~2, ~27/25


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/25 = 126.706
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0862{{c}}, ~27/25 = 126.7147{{c}}
: [[error map]]: {{val| +0.086 -1.234 +1.411 +0.226 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~27/25 = 126.7081{{c}}
: error map: {{val| 0.000 -1.333 +1.265 +0.122 }}


{{Optimal ET sequence|legend=1| 19, 123d, 142, 161 }}
{{Optimal ET sequence|legend=1| 19, …, 123d, 142, 161 }}


[[Badness]]: 0.129915
[[Badness]] (Sintel): 3.29


=== 11-limit ===
=== 11-limit ===
Line 727: Line 811:
{{Mapping|legend=0| 1 0 0 -3 8 | 0 15 22 55 -43 }}
{{Mapping|legend=0| 1 0 0 -3 8 | 0 15 22 55 -43 }}


Optimal tuning (POTE): ~2 = 1\1, ~27/25 = 126.711
Optimal tunings:
* WE: ~2 = 1200.0198{{c}}, ~27/25 = 126.7134{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~27/25 = 126.7114{{c}}


{{Optimal ET sequence|legend=1| 19, 123de, 142, 161 }}
{{Optimal ET sequence|legend=0| 19, 142, 161 }}


Badness: 0.094032
Badness (Sintel): 3.11


=== 13-limit ===
=== 13-limit ===
Line 740: Line 826:
{{Mapping|legend=0| 1 0 0 -3 8 -2 | 0 15 22 55 -43 54 }}
{{Mapping|legend=0| 1 0 0 -3 8 -2 | 0 15 22 55 -43 54 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.705
Optimal tunings:
* WE: ~2 = 1199.9743{{c}}, ~14/13 = 126.7022{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 126.7047{{c}}


{{Optimal ET sequence|legend=1| 19, 123def, 142f, 161 }}
{{Optimal ET sequence|legend=0| 19, 142f, 161 }}


Badness: 0.051571
Badness (Sintel): 2.13


=== 17-limit ===
=== 17-limit ===
Line 753: Line 841:
{{Mapping|legend=0| 1 0 0 -3 8 -2 10 | 0 15 22 55 -43 54 -56 }}
{{Mapping|legend=0| 1 0 0 -3 8 -2 10 | 0 15 22 55 -43 54 -56 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.703
Optimal tunings:
* WE: ~2 = 1200.0016{{c}}, ~14/13 = 126.7029{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 126.7027{{c}}


{{Optimal ET sequence|legend=1| 19, 123defg, 142f, 161 }}
{{Optimal ET sequence|legend=0| 19, 123defg, 142f, 161 }}


Badness: 0.041918
Badness (Sintel): 2.14


=== 19-limit ===
=== 19-limit ===
Line 766: Line 856:
{{Mapping|legend=0| 1 0 0 -3 8 -2 10 9 | 0 15 22 55 -43 54 -56 -45 }}
{{Mapping|legend=0| 1 0 0 -3 8 -2 10 9 | 0 15 22 55 -43 54 -56 -45 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.705
Optimal tunings:
* WE: ~2 = 1199.9713{{c}}, ~14/13 = 126.7018{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 126.7048{{c}}


{{Optimal ET sequence|legend=1| 19, 123defg, 142f, 161 }}
{{Optimal ET sequence|legend=0| 19, 123defg, 142f, 161 }}


Badness: 0.032168
Badness (Sintel): 1.96


=== 23-limit ===
=== 23-limit ===
Line 779: Line 871:
{{Mapping|legend=0| 1 0 0 -3 8 -2 10 9 6 | 0 15 22 55 -43 54 -56 -45 -14 }}
{{Mapping|legend=0| 1 0 0 -3 8 -2 10 9 6 | 0 15 22 55 -43 54 -56 -45 -14 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.703
Optimal tunings:
* WE: ~2 = 1200.0296{{c}}, ~14/13 = 126.7061{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 126.7030{{c}}


{{Optimal ET sequence|legend=1| 19, 123defg, 142f, 161 }}
{{Optimal ET sequence|legend=0| 19, 142f, 161 }}


Badness: 0.026117
Badness (Sintel): 1.87


=== 29-limit ===
=== 29-limit ===
Line 792: Line 886:
{{Mapping|legend=0| 1 0 0 -3 8 -2 10 9 6 0 | 0 15 22 55 -43 54 -56 -45 -14 46 }}
{{Mapping|legend=0| 1 0 0 -3 8 -2 10 9 6 0 | 0 15 22 55 -43 54 -56 -45 -14 46 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.704
Optimal tunings:
* WE: ~2 = 1200.0491{{c}}, ~14/13 = 126.7094{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 126.7043{{c}}


{{Optimal ET sequence|legend=1| 19, 123defg, 142f, 161 }}
{{Optimal ET sequence|legend=0| 19, 142f, 161 }}


Badness: 0.021398
Badness (Sintel): 1.79


=== 31-limit ===
=== 31-limit ===
Line 805: Line 901:
{{Mapping|legend=0| 1 0 0 -3 8 -2 10 9 6 0 2 | 0 15 22 55 -43 54 -56 -45 -14 46 28 }}
{{Mapping|legend=0| 1 0 0 -3 8 -2 10 9 6 0 2 | 0 15 22 55 -43 54 -56 -45 -14 46 28 }}


Optimal tuning (POTE): ~2 = 1\1, ~14/13 = 126.703
Optimal tunings:
* WE: ~2 = 1199.9986{{c}}, ~14/13 = 126.7027{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~14/13 = 126.7028{{c}}


{{Optimal ET sequence|legend=1| 19, 123defgk, 142fk, 161 }}
{{Optimal ET sequence|legend=0| 19, 142fk, 161 }}


Badness: 0.019331
Badness (Sintel): 1.85


== Tremka ==
== Tremka ==
The name ''tremka'' was initially used for the [[No-sevens subgroup temperaments|no-sevens version]] of 50 &amp; 111 (especially in the 2.3.5.11.13 subgroup), but extending to full 13-limit or higher prime limit does no significant tuning damage, so for that we keep the 2.3.5.11.13 label tremka.
Tremka may be described as the 50 & 111 temperament. The name ''tremka'' was devised by [[Petr Pařízek]] in 2011, initially used for the [[no-sevens subgroup temperaments #Tremka|no-7's version]], especially in the 2.3.5.11.13 subgroup<ref>[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group (archive) | ''Suggested names for the unclasified temperaments'']</ref>, but extending to the full 13-limit or higher does no significant tuning damage, so for that we keep the 2.3.5.11.13 label tremka.


=== 7-limit ===
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 820: Line 917:


{{Mapping|legend=1| 1 -4 -2 -8 | 0 31 24 60 }}
{{Mapping|legend=1| 1 -4 -2 -8 | 0 31 24 60 }}
: mapping generators: ~2, ~4375/3888


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~4375/3888 = 216.173
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7671{{c}}, ~4375/3888 = 216.1302{{c}}
: [[error map]]: {{val| -0.233 -0.987 +1.277 +0.849 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~4375/3888 = 216.1665{{c}}
: error map: {{val| 0.000 -0.794 +1.682 +1.162 }}


{{Optimal ET sequence|legend=1| 50, 111, 161, 272 }}
{{Optimal ET sequence|legend=1| 50, 111, 161, 272 }}


[[Badness]]: 0.179925
[[Badness]] (Sintel): 4.55


=== 11-limit ===
=== 11-limit ===
Line 834: Line 936:
{{Mapping|legend=0| 1 -4 -2 -8 4 | 0 31 24 60 -3 }}
{{Mapping|legend=0| 1 -4 -2 -8 4 | 0 31 24 60 -3 }}


Optimal tuning (POTE): ~2 = 1\1, ~112/99 = 216.168
Optimal tunings:
* WE: ~2 = 1199.8503{{c}}, ~112/99 = 216.1406{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~112/99 = 216.1652{{c}}


{{Optimal ET sequence|legend=1| 50, 111, 161, 272, 433c }}
{{Optimal ET sequence|legend=0| 50, 111, 161, 272, 433c }}


Badness: 0.068825
Badness (Sintel): 2.28


=== 13-limit ===
=== 13-limit ===
Line 847: Line 951:
{{Mapping|legend=0| 1 -4 -2 -8 4 1 | 0 31 24 60 -3 15 }}
{{Mapping|legend=0| 1 -4 -2 -8 4 1 | 0 31 24 60 -3 15 }}


Optimal tuning (POTE): ~2 = 1\1, ~112/99 = 216.172
Optimal tunings:
* WE: ~2 = 1199.7479{{c}}, ~112/99 = 216.1263{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~112/99 = 216.1681{{c}}


{{Optimal ET sequence|legend=1| 50, 111, 161, 272 }}
{{Optimal ET sequence|legend=0| 50, 111, 161, 272 }}


Badness: 0.036070
Badness (Sintel): 1.49


=== 17-limit ===
=== 17-limit ===
Line 860: Line 966:
{{Mapping|legend=0| 1 -4 -2 -8 4 1 -6 | 0 31 24 60 -3 15 56 }}
{{Mapping|legend=0| 1 -4 -2 -8 4 1 -6 | 0 31 24 60 -3 15 56 }}


Optimal tuning (POTE): ~2 = 1\1, ~17/15 = 216.172
Optimal tunings:
* WE: ~2 = 1199.7537{{c}}, ~17/15 = 216.1281{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/15 = 216.1688{{c}}


{{Optimal ET sequence|legend=1| 50, 111, 161, 272 }}
{{Optimal ET sequence|legend=0| 50, 111, 161, 272 }}


Badness: 0.022528
Badness (Sintel): 1.15


=== 19-limit ===
=== 19-limit ===
Line 873: Line 981:
{{Mapping|legend=0| 1 -4 -2 -8 4 1 -6 -8 | 0 31 24 60 -3 15 56 68 }}
{{Mapping|legend=0| 1 -4 -2 -8 4 1 -6 -8 | 0 31 24 60 -3 15 56 68 }}


Optimal tuning (POTE): ~2 = 1\1, ~17/15 = 216.170
Optimal tunings:
* WE: ~2 = 1199.7496{{c}}, ~17/15 = 216.1244{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~17/15 = 216.1658{{c}}


{{Optimal ET sequence|legend=1| 50, 111, 161, 272h, 433cfh, 705ccdffhh }}
{{Optimal ET sequence|legend=0| 50, 111, 161, 272h, 433cfh }}


Badness: 0.016900
Badness (Sintel): 1.03


== Undetrita ==
== Undetrita ==
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Undetrita (5-limit)]].''
: ''For the 5-limit version, see [[Syntonic–chromatic equivalence continuum #Undetrita (5-limit)]].''


The undetrita temperament (111 &amp; 118) tempers out the hemimean comma (3136/3125) and [[scheme comma]] (14348907/14336000) in the 7-limit; 3025/3024, 3388/3375, and 8019/8000 in the 11-limit. This temperament is related to [[11edt]], and the name ''undetrita'' is a play on the words ''undecimus'' (Latin for "eleventh") and ''[[tritave]]'' (3rd harmonic). It is also related to the [[Subgroup temperaments #No-sevens subgroup|twentcufo temperament]], which is no-sevens version of 111 &amp; 118.
Undetrita tempers out 14348907/14336000, the [[scheme comma]], in addition to the hemimean comma, and may be described as the 111 & 118 temperament. It splits 3rd harmonic into eleven parts; its [[ploidacot]] is thus alpha-hendecacot. It tempers out [[3025/3024]], [[3388/3375]], and [[8019/8000]] in the [[11-limit]], which relates itself to [[no-sevens subgroup temperaments #Twentcufo|twentcufo]], the no-7's version of 111 & 118.
 
It was named by [[Xenllium]] in 2021 as a play on the words ''undecimus'' (Latin for "eleventh") and ''[[tritave]]'' (3rd harmonic).  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 889: Line 1,001:


{{Mapping|legend=1| 1 0 -2 -8 | 0 11 30 75 }}
{{Mapping|legend=1| 1 0 -2 -8 | 0 11 30 75 }}
: mapping generators: ~2, ~448/405


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~448/405 = 172.917
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8545{{c}}, ~448/405 = 172.8961{{c}}
: error map: {{val| -0.146 -0.098 +0.859 -0.458 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~448/405 = 172.9143{{c}}
: error map: {{val| 0.000 +0.102 +1.115 -0.255 }}


{{Optimal ET sequence|legend=1| 111, 118, 229, 347, 576c }}
{{Optimal ET sequence|legend=0| 111, 118, 229, 347, 576c }}


[[Badness]]: 0.114188
[[Badness]] (Sintel): 2.89


=== 11-limit ===
=== 11-limit ===
Line 903: Line 1,020:
{{Mapping|legend=0| 1 0 -2 -8 0 | 0 11 30 75 24 }}
{{Mapping|legend=0| 1 0 -2 -8 0 | 0 11 30 75 24 }}


Optimal tuning (POTE): ~2 = 1\1, ~400/363 = 172.912
Optimal tunings:
* WE: ~2 = 1200.0112{{c}}, ~400/363 = 172.9138{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~400/363 = 172.9123{{c}}


{{Optimal ET sequence|legend=1| 111, 118, 229, 347 }}
{{Optimal ET sequence|legend=0| 111, 118, 229, 347 }}


Badness: 0.043883
Badness (Sintel): 1.45


==== 13-limit ====
==== 13-limit ====
Line 916: Line 1,035:
{{Mapping|legend=0| 1 0 -2 -8 0 5 | 0 11 30 75 24 -9 }}
{{Mapping|legend=0| 1 0 -2 -8 0 5 | 0 11 30 75 24 -9 }}


Optimal tuning (POTE): ~2 = 1\1, ~72/65 = 172.930
Optimal tunings:
* WE: ~2 = 1199.7099{{c}}, ~72/65 = 172.8882{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 172.9265{{c}}


{{Optimal ET sequence|legend=1| 111, 229f }}
{{Optimal ET sequence|legend=0| 111, 229f }}


Badness: 0.038771
Badness (Sintel): 1.60


==== Undetritoid ====
==== Undetritoid ====
Line 929: Line 1,050:
{{Mapping|legend=0| 1 0 -2 -8 0 -11 | 0 11 30 75 24 102 }}
{{Mapping|legend=0| 1 0 -2 -8 0 -11 | 0 11 30 75 24 102 }}


Optimal tuning (POTE): ~2 = 1\1, ~400/363 = 172.933
Optimal tunings:
* WE: ~2 = 1199.8901{{c}}, ~400/363 = 172.9167{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~400/363 = 172.9310{{c}}


{{Optimal ET sequence|legend=1| 111, 229 }}
{{Optimal ET sequence|legend=0| 111, 229 }}


Badness: 0.042744
Badness (Sintel): 1.77


= Subgroup extensions =
= Subgroup extensions =
== Undecimal didacus ==
== Undecimal didacus ==
In the no-3's [[11-limit]], there is a natural extension with prime 11 by equating [[25/16]] (which is already tuned sharp anyways) with [[11/7]] by tempering out [[176/175]], which is the same route that [[undecimal meantone]] uses, as this is essentially a no-3's restriction of undecimal meantone in the 11-limit, except that undecimal meantone finds ~[[28/25]] at 2 generators (as a flat ~[[9/8]]) while here it is the generator. This is equivalent to finding [[11/4]] as ([[7/5]])<sup>3</sup>. In the no-3's 19-limit extension "mediantone", this whole tone generator serves as the two simplest [[mediant]]s of [[9/8]] and [[10/9]], namely [[19/17]] and [[28/25]], while in undecimal didacus and its extension to the no-3's 13-limit only the latter interpretation is relevant.
In the no-3's [[11-limit]], there is a natural extension with prime 11 by equating [[25/16]] (which is already tuned sharp anyways) with [[11/7]] by tempering out [[176/175]], which is the same route that [[undecimal meantone]] uses, as this is essentially a no-3's restriction of undecimal meantone in the 11-limit, except that undecimal meantone finds ~[[28/25]] at 2 generators (as a flat ~[[9/8]]) while here it is the generator. This is equivalent to finding [[11/4]] as ([[7/5]])<sup>3</sup>. In the no-3's 19-limit extension "mediantone", this whole tone generator serves as the two simplest [[mediant]]s of [[9/8]] and [[10/9]], namely [[19/17]] and [[28/25]], while in undecimal didacus and its extension to the no-3's 13-limit only the latter interpretation is relevant.
Line 1,109: Line 1,231:


[[Tp tuning #T2 tuning|RMS error]]: 2.003 cents
[[Tp tuning #T2 tuning|RMS error]]: 2.003 cents
== References ==


[[Category:Hemimean clan| ]] <!-- main article -->
[[Category:Hemimean clan| ]] <!-- main article -->

Latest revision as of 14:01, 27 October 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The hemimean clan tempers out the hemimean comma, 3136/3125, with 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 extensions, 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. Other subgroup extensions include rectified hebrew and isra.

Temperaments considered below are hemiwürschmidt, hemithirds, spell, semisept, emka, decipentic, sengagen, subpental, mowglic, and undetrita. Discussed elsewhere are

2.5.7 subgroup

Didacus

Main article: Didacus

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

Subgroup: 2.5.7

Comma list: 3136/3125

Subgroup-val mapping: [⟨1 0 -3], ⟨0 2 5]]

Gencom mapping: [⟨1 0 0 -3], ⟨0 0 2 5]]

mapping generators: ~2, ~56/25

Optimal tunings:

  • WE: ~2 = 1199.8194 ¢, ~28/25 = 193.7425 ¢
error map: ⟨-0.181 +0.810 -0.474]
  • CWE: ~2 = 1200.0000 ¢, ~28/25 = 193.7167 ¢
error map: ⟨0.000 +1.120 -0.243]

Optimal ET sequence: 6, 19, 25, 31, 99, 130, 161, 353, 514c, 867c

Badness (Sintel): 0.0908

Strong extensions

Map to strong extensions
Extension 5-limit re-restriction Mapping of 3 Tuning range*
Temperament 5-limit generator location
Hemiwürschmidt Würschmidt +2 +16 ↓ 31
Hemithirds Luna +1 -15 ↑ 31
↓ 25
Spell Magic +2 +10 ↑ 25

* Defined by intersection with other documented extensions

Hemiwürschmidt

See also: Würschmidt family

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. 130edo in particular shows how hemiwürschmidt may be extended to a higher limit, mapping 11 to 40 generators which is the same as würschmidt.

Hemiwürschmidt was named by Gene Ward Smith in 2004 as hemi- + würschmidt.[1]

Subgroup: 2.3.5.7

Comma list: 2401/2400, 3136/3125

Mapping: [⟨1 -1 2 2], ⟨0 16 2 5]]

mapping generators: ~2, ~28/25

Optimal tunings:

  • WE: ~2 = 1199.7153 ¢, ~28/25 = 193.8522 ¢
error map: ⟨-0.285 -0.035 +0.821 -0.134]
  • CWE: ~2 = 1200.0000 ¢, ~28/25 = 193.8883 ¢
error map: ⟨0.000 +0.258 +1.463 +0.616]

Optimal ET sequence: 31, 68, 99, 229, 328, 557c, 885cc

Badness (Sintel): 0.514

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -1 2 2 -3], ⟨0 16 2 5 40]]

Optimal tunings:

  • WE: ~2 = 1199.7863 ¢, ~28/25 = 193.8058 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/25 = 193.8342 ¢

Optimal ET sequence: 31, 68e, 99e, 130

Badness (Sintel): 0.697

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -1 2 2 -3 10], ⟨0 16 2 5 40 -39]]

Optimal tunings:

  • WE: ~2 = 1199.8626 ¢, ~28/25 = 193.8063 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/25 = 193.8277 ¢

Optimal ET sequence: 31, 99e, 130, 291, 421e, 551ce

Badness (Sintel): 0.953

2.3.5.7.11.13.23 subgroup

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

Subgroup: 2.3.5.7.11.13.23

Comma list: 243/242, 276/275, 351/350, 441/440, 3584/3575

Mapping: [⟨1 -1 2 2 -3 10 0], ⟨0 16 2 5 40 -39 28]]

Optimal tunings:

  • WE: ~2 = 1199.9052 ¢, ~28/25 = 193.8170 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/25 = 193.8316 ¢

Optimal ET sequence: 31, 99e, 130

Badness (Sintel): 0.839

Hemithir

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 196/195, 243/242, 625/624

Mapping: [⟨1 -1 2 2 -3 5], ⟨0 16 2 5 40 -8]]

Optimal tunings:

  • WE: ~2 = 1199.1395 ¢, ~28/25 = 193.7788 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/25 = 193.9023 ¢

Optimal ET sequence: 31, 68e, 99ef

Badness (Sintel): 1.29

Hemiwur

Hemiwur may as well be described as 31 & 37, mapping prime 11 to 9 generator steps.

Subgroup: 2.3.5.7.11

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

Mapping: [⟨1 -1 2 2 2], ⟨0 16 2 5 9]]

Optimal tunings:

  • WE: ~2 = 1200.1742 ¢, ~28/25 = 193.9117 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/25 = 193.8896 ¢

Optimal ET sequence: 31, 68, 99, 130e

Badness (Sintel): 0.968

13-limit

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -1 2 2 2 5], ⟨0 16 2 5 9 -8]]

Optimal tunings:

  • WE: ~2 = 1199.4972 ¢, ~28/25 = 193.9231 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/25 = 193.9942 ¢

Optimal ET sequence: 31, 68

Badness (Sintel): 1.17

Hemiwar

Subgroup: 2.3.5.7.11.13

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

Mapping: [⟨1 -1 2 2 2 0], ⟨0 16 2 5 9 23]]

Optimal tunings:

  • WE: ~2 = 1199.9465 ¢, ~28/25 = 193.6889 ¢
  • CWE: ~2 = 1200.0000 ¢, ~28/25 = 193.6956 ¢

Optimal ET sequence: 6f, 25bf, 31

Badness (Sintel): 1.85

Quadrawürschmidt

This extension has been documented in Graham Breed's temperament finder as semihemiwürschmidt, but quadrawürschmidt arguably makes more sense because it splits the generator in four.

The generator of quadrawürschmidt is essentially a meantone fifth. However, it is not used to represent 3/2, as 3/2 is found at the hemiwürschmidt position, 16 whole tones up. The small comma between the generator and 3/2 represents 441/440~540/539.

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3025/3024, 3136/3125

Mapping: [⟨1 -17 0 -3 -25], ⟨0 32 4 10 49]]

mapping generators: ~2, ~220/147

Optimal tunings:

  • WE: ~2 = 1199.7360 ¢, ~220/147 = 696.8062 ¢
  • CWE: ~2 = 1200.0000 ¢, ~220/147 = 696.9548 ¢

Optimal ET sequence: 31, 105be, 136e, 167, 198, 427c

Badness (Sintel): 1.15

Semihemiwürschmidt

Subgroup: 2.3.5.7.11

Comma list: 2401/2400, 3136/3125, 9801/9800

Mapping: [⟨2 -2 4 4 15], ⟨0 16 2 5 -25]]

mapping generators: ~99/70, ~28/25

Optimal tunings:

  • WE: ~99/70 = 599.8485 ¢, ~28/25 = 193.8531 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~28/25 = 193.9002 ¢

Optimal ET sequence: 62e, 68, 130, 198, 328

Badness (Sintel): 1.48

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 1716/1715, 3136/3125

Mapping: [⟨2 -2 4 4 15 -1], ⟨0 16 2 5 -25 26]]

Optimal tunings:

  • WE: ~99/70 = 599.8514 ¢, ~28/25 = 193.8555 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~28/25 = 193.9006 ¢

Optimal ET sequence: 62e, 68, 130, 198, 328

Badness (Sintel): 0.966

Semihemiwürat

Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 442/441, 561/560, 676/675, 1632/1625

Mapping: [⟨2 -2 4 4 15 -1 3], ⟨0 16 2 5 -25 26 16]]

Optimal tunings:

  • WE: ~17/12 = 599.9069 ¢, ~28/25 = 193.8811 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~28/25 = 193.9092 ¢

Optimal ET sequence: 62e, 68, 130, 198

Badness (Sintel): 1.48

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 289/288, 442/441, 456/455, 476/475, 561/560, 627/625

Mapping: [⟨2 -2 4 4 15 -1 3 3], ⟨0 16 2 5 -25 26 16 17]]

Optimal tunings:

  • WE: ~17/12 = 599.9292 ¢, ~19/17 = 193.8916 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~19/17 = 193.9128 ¢

Optimal ET sequence: 62e, 68, 130, 198

Badness (Sintel): 1.32

Semihemiwüram

Subgroup: 2.3.5.7.11.13.17

Comma list: 256/255, 676/675, 715/714, 1001/1000, 1225/1224

Mapping: [⟨2 -2 4 4 15 -1 14], ⟨0 16 2 5 -25 26 -18]]

Optimal tunings:

  • WE: ~17/12 = 599.7759 ¢, ~28/25 = 193.8534 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~28/25 = 193.9238 ¢

Optimal ET sequence: 68, 130g, 198g

Badness (Sintel): 1.51

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 256/255, 286/285, 400/399, 476/475, 495/494, 1225/1224

Mapping: [⟨2 -2 4 4 15 -1 14 14], ⟨0 16 2 5 -25 26 -18 -17]]

Optimal tunings:

  • WE: ~17/12 = 599.7147 ¢, ~28/25 = 193.8506 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~28/25 = 193.9417 ¢

Optimal ET sequence: 68, 130gh, 198gh

Badness (Sintel): 1.80

Hemithirds

Main article: Hemithirds

Subgroup: 2.3.5.7

Comma list: 1029/1024, 3136/3125

Mapping: [⟨1 -11 4 7], ⟨0 15 -2 -5]]

mapping generators: ~2, ~25/14

Optimal tunings:

  • WE: ~2 = 1200.3104 ¢, ~25/14 = 1007.0160 ¢ (~28/25 = 193.2944 ¢)
error map: ⟨+0.310 -0.129 +0.896 -1.733]
  • CWE: ~2 = 1200.0000 ¢, ~25/14 = 1006.7606 ¢ (~28/25 = 193.2394 ¢)
error map: ⟨0.000 -0.545 +0.165 -2.629]

Minimax tuning:

[[1 0 0 0⟩, [5/2 3/4 0 -3/4⟩, [11/5 -1/10 0 1/10⟩, [5/2 -1/4 0 1/4⟩]
unchanged-interval (eigenmonzo) basis: 2.7/3
[[1 0 0 0⟩, [10/7 6/7 0 -3/7⟩, [82/35 -4/35 0 2/35⟩, [20/7 -2/7 0 1/7⟩]
unchanged-interval (eigenmonzo) basis: 2.9/7

Optimal ET sequence: 25, 31, 87, 118, 267d, 385d

Badness (Sintel): 1.12

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 441/440, 3136/3125

Mapping: [⟨1 -11 4 7 -15], ⟨0 15 -2 -5 22]]

Optimal tunings:

  • WE: ~2 = 1200.3576 ¢, ~25/14 = 1007.0732 ¢ (~28/25 = 193.2844 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~25/14 = 1006.7807 ¢ (~28/25 = 193.2193 ¢)

Minimax tuning:

  • 11-odd-limit: ~28/25 = [5/27 0 0 1/27 -1/27⟩
[[1 0 0 0 0⟩, [11/9 0 0 -5/9 5/9⟩, [64/27 0 0 2/27 -2/27⟩, [79/27 0 0 5/27 -5/27⟩, [79/27 0 0 -22/27 22/27⟩]
Eigenmonzos (unchanged-intervals): 2, 11/7

Optimal ET sequence: 25e, 31, 87, 118

Badness (Sintel): 0.628

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 352/351, 385/384, 625/624

Mapping: [⟨1 4 2 2 7 0], ⟨0 15 -2 -5 22 -23]]

Optimal tunings:

  • WE: ~2 = 1200.0602 ¢, ~25/14 = 1006.8844 ¢ (~28/25 = 193.1757 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~25/14 = 1006.8343 ¢ (~28/25 = 193.1457 ¢)

Optimal ET sequence: 31, 56, 87, 118, 205d

Badness (Sintel): 0.898

Spell

See also: Magic family

Subgroup: 2.3.5.7

Comma list: 49/48, 3125/3072

Mapping: [⟨1 0 2 2], ⟨0 10 2 5]]

Optimal tunings:

  • WE: ~2 = 1202.8089 ¢, ~28/25 = 190.3712 ¢
error map: ⟨+2.809 +1.757 +0.047 -11.352]
  • CWE: ~2 = 1200.0000 ¢, ~28/25 = 190.1078 ¢
error map: ⟨0.000 -0.877 -6.098 -18.287]

Optimal ET sequence: 6, 13b, 19, 82dd

Badness (Sintel): 2.05

11-limit

Subgroup: 2.3.5.7.11

Comma list: 49/48, 56/55, 125/121

Mapping: [⟨1 0 2 2 3], ⟨0 10 2 5 3]]

Optimal tunings:

  • WE: ~2 = 1200.4163 ¢, ~11/10 = 190.3511 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 190.3091 ¢

Optimal ET sequence: 6, 13b, 19

Badness (Sintel): 1.98

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 78/77, 125/121

Mapping: [⟨1 0 2 2 3 4], ⟨0 10 2 5 3 -2]]

Optimal tunings:

  • WE: ~2 = 1201.8239 ¢, ~11/10 = 190.2163 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 190.0020 ¢

Optimal ET sequence: 6, 13b, 19

Badness (Sintel): 1.88

Cantrip

Subgroup: 2.3.5.7.11.13

Comma list: 49/48, 56/55, 91/90, 125/121

Mapping: [⟨1 0 2 2 3 1], ⟨0 10 2 5 3 17]]

Optimal tunings:

  • WE: ~2 = 1200.4118 ¢, ~11/10 = 190.4253 ¢
  • CWE: ~2 = 1200.0000 ¢, ~11/10 = 190.3835 ¢

Optimal ET sequence: 6f, 19

Badness (Sintel): 1.72

Weak extensions

Semisept

For the 5-limit version, see Miscellaneous 5-limit temperaments #Semisept.

The generator of semisept is half a tempered septimal major sixth (12/7), hence the name. Three such generator steps minus an octave give the hemithird, and six give the classical major third. It can be described as the 31 & 80 temperament, and as one may expect, 111edo makes for a great tuning.

Subgroup: 2.3.5.7

Comma list: 1728/1715, 3136/3125

Mapping: [⟨1 -5 0 -3], ⟨0 17 6 15]]

mapping generators: ~2, ~98/75

Optimal tunings:

  • WE: ~2 = 1199.2968 ¢, ~98/75 = 464.5722 ¢
error map: ⟨-0.703 -0.712 +1.119 +1.867]
  • CWE: ~2 = 1200.0000 ¢, ~98/75 = 464.8120 ¢
error map: ⟨0.000 -0.150 +2.559 +3.355]

Optimal ET sequence: 31, 80, 111, 364cdd, 475ccddd

Badness (Sintel): 1.28

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 540/539, 1331/1323

Mapping: [⟨1 -5 0 -3 -7], ⟨0 17 6 15 27]]

Optimal tunings:

  • WE: ~2 = 1199.3073 ¢, ~72/55 = 464.6063 ¢
  • CWE: ~2 = 1200.0000 ¢, ~72/55 = 464.8423 ¢

Optimal ET sequence: 31, 80, 111

Badness (Sintel): 0.743

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 540/539, 1375/1372

Mapping: [⟨1 -5 0 -3 -7 13], ⟨0 17 6 15 27 -24]]

Optimal tunings:

  • WE: ~2 = 1199.3145 ¢, ~72/55 = 464.6080 ¢
  • CWE: ~2 = 1200.0000 ¢, ~72/55 = 464.8636 ¢

Optimal ET sequence: 31, 80, 111

Badness (Sintel): 1.04

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 176/175, 256/255, 351/350, 640/637, 715/714

Mapping: [⟨1 -5 0 -3 -7 13 13], ⟨0 17 6 15 27 -24 -23]]

Optimal tunings:

  • WE: ~2 = 1199.3067 ¢, ~17/13 = 464.6061 ¢
  • CWE: ~2 = 1200.0000 ¢, ~17/13 = 464.8699 ¢

Optimal ET sequence: 31, 80, 111

Badness (Sintel): 1.01

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 176/175, 286/285, 351/350, 476/475, 540/539, 1331/1323

Mapping: [⟨1 -5 0 -3 -7 13 13 12], ⟨0 17 6 15 27 -24 -23 -20]]

Optimal tunings:

  • WE: ~2 = 1199.2346 ¢, ~17/13 = 464.5878 ¢
  • CWE: ~2 = 1200.0000 ¢, ~17/13 = 464.8816 ¢

Optimal ET sequence: 31, 80, 111

Badness (Sintel): 0.992

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 176/175, 253/252, 286/285, 345/343, 351/350, 391/390, 460/459

Mapping: [⟨1 -5 0 -3 -7 13 13 12 -4], ⟨0 17 6 15 27 -24 -23 -20 22]]

Optimal tunings:

  • WE: ~2 = 1199.3227 ¢, ~17/13 = 464.6320 ¢
  • CWE: ~2 = 1200.0000 ¢, ~17/13 = 464.8912 ¢

Optimal ET sequence: 31, 80, 111, 191cdh

Badness (Sintel): 1.07

Semishly

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 176/175, 196/195, 275/273

Mapping: [⟨1 -5 0 -3 -7 1], ⟨0 17 6 15 27 7]]

Optimal tunings:

  • WE: ~2 = 1198.3103 ¢, ~13/10 = 464.3248 ¢
  • CWE: ~2 = 1200.0000 ¢, ~13/10 = 464.9110 ¢

Optimal ET sequence: 31, 49f, 80f

Badness (Sintel): 1.17

Emka

For the 5-limit version, see Miscellaneous 5-limit temperaments #Emka.

Emka may be described as the 37 & 50 temperament, which tempers out the hemimean comma and 84035/82944 (quinzo-ayo). It was named by Petr Pařízek in 2011.

A related temperament of emka is emkay (87 & 224), which is more accurate but more complex, and only differs from emka in the mapping of 7.

Subgroup: 2.3.5.7

Comma list: 3136/3125, 84035/82944

Mapping: [⟨1 -13 -2 -8], ⟨0 27 8 20]]

mapping generators: ~2, ~35/24

Optimal tunings:

  • WE: ~2 = 1200.4701 ¢, ~35/24 = 648.4717 ¢
error map: ⟨+0.470 +0.669 +0.520 -3.153]
  • CWE: ~2 = 1200.0000 ¢, ~35/24 = 648.2307 ¢
error map: ⟨0.000 +0.274 -0.468 -4.212]

Optimal ET sequence: 37, 50, 87, 137d, 224d

Badness (Sintel): 3.65

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 2401/2376, 3136/3125

Mapping: [⟨1 -13 -2 -8 4], ⟨0 27 8 20 -1]]

Optimal tunings:

  • WE: ~2 = 1200.2552 ¢, ~16/11 = 648.3727 ¢
  • CWE: ~2 = 1200.0000 ¢, ~16/11 = 648.2399 ¢

Optimal ET sequence: 37, 50, 87

Badness (Sintel): 1.81

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 196/195, 364/363, 385/384, 625/624

Mapping: [⟨1 -13 -2 -8 4 1], ⟨0 27 8 20 -1 5]]

Optimal tunings:

  • WE: ~2 = 1200.1506 ¢, ~16/11 = 648.3230 ¢
  • CWE: ~2 = 1200.0000 ¢, ~16/11 = 648.2442 ¢

Optimal ET sequence: 37, 50, 87, 224d, 311d, 398d

Badness (Sintel): 1.23

Decipentic

Decipentic may be described as the 43 & 56 temperament. The minimal generator for this temperament is the 10th root of the 5th harmonic, 51/10, tuned between 75/64 and 20/17 (close to 27/23). Aside from the hemimean comma, this temperament tempers out the bronzisma, 2097152/2083725. 99edo is a good tuning for decipentic, with generator 23\99, and mos scales of 9, 13, 17, 30, 43 or 56 notes are available.

Decipentic was named by Xenllium in 2021 for the fact that it splits the 5th harmonic into ten equal parts.

Subgroup: 2.3.5.7

Comma list: 3136/3125, 2097152/2083725

Mapping: [⟨1 -13 10 22], ⟨0 19 -10 -25]]

mapping generators: ~2, ~128/75

Optimal tunings:

  • WE: ~2 = 1199.5804 ¢, ~128/75 = 920.8777 ¢ (~75/64 = 278.7027 ¢)
error map: ⟨-0.420 +0.175 +0.714 +0.001]
  • CWE: ~2 = 1200.0000 ¢, ~128/75 = 921.2002 ¢ (~75/64 = 278.7998 ¢)
error map: ⟨0.000 +0.849 +1.684 +1.169]

Optimal ET sequence: 43, 56, 99, 538cd, 637cd, …, 1033bccdd

Badness (Sintel): 2.21

11-limit

Subgroup: 2.3.5.7.11

Comma list: 441/440, 1344/1331, 3136/3125

Mapping: [⟨1 6 0 -3 3], ⟨0 -19 10 25 2]]

Optimal tunings:

  • WE: ~2 = 1199.3002 ¢, ~128/75 = 920.6639 ¢ (~75/64 = 278.6364 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~128/75 = 921.2019 ¢ (~75/64 = 278.7981 ¢)

Optimal ET sequence: 43, 56, 99e

Badness (Sintel): 2.03

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 441/440, 832/825, 975/968

Mapping: [⟨1 -13 10 22 5 6], ⟨0 19 -10 -25 -2 -3]]

Optimal tunings:

  • WE: ~2 = 1199.5986 ¢, ~22/13 = 920.8900 ¢ (~13/11 = 278.7085 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~22/13 = 921.1988 ¢ (~13/11 = 278.7981 ¢)

Optimal ET sequence: 43, 56, 99e

Badness (Sintel): 1.97

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 256/255, 273/272, 375/374

Mapping: [⟨1 -13 10 22 5 6 11], ⟨0 19 -10 -25 -2 -3 -9]]

Optimal tunings:

  • WE: ~2 = 1199.5089 ¢, ~17/10 = 920.8252 ¢ (~13/11 = 278.6837 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~17/10 = 921.2034 ¢ (~13/11 = 278.7966 ¢)

Optimal ET sequence: 43, 56, 99e

Badness (Sintel): 1.59

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 169/168, 210/209, 221/220, 256/255, 273/272, 286/285

Mapping: [⟨1 -13 10 22 5 6 11 15], ⟨0 19 -10 -25 -2 -3 -9 -14]]

Optimal tunings:

  • WE: ~2 = 1199.4107 ¢, ~17/10 = 920.7575 ¢ (~13/11 = 278.6532 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~17/10 = 921.2020 ¢ (~13/11 = 278.7980 ¢)

Optimal ET sequence: 43, 56, 99e

Badness (Sintel): 1.45

Quasijerome

Subgroup: 2.3.5.7.11

Comma list: 3136/3125, 15488/15435, 16384/16335

Mapping: [⟨1 -32 20 47 45], ⟨0 38 -20 -50 -47]]

mapping generators: ~2, ~825/448

Optimal tunings:

  • WE: ~2 = 1199.6276 ¢, ~825/448 = 1060.2861 ¢ (~896/825 = 139.3595 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~825/448 = 1060.5982 ¢ (~896/825 = 139.4016 ¢)

Optimal ET sequence: 43, 112, 155, 198, 439cd, 637cd

Badness (Sintel): 3.07

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 676/675, 1001/1000, 3136/3125, 15488/15435

Mapping: [⟨1 -32 20 47 45 -29], ⟨0 38 -20 -50 -47 37]]

Optimal tunings:

  • WE: ~2 = 1199.6184 ¢, ~24/13 = 1060.2595 ¢ (~13/12 = 139.3589 ¢)
  • CWE: ~2 = 1200.0000 ¢, ~24/13 = 1060.5971 ¢ (~13/12 = 139.4029 ¢)

Optimal ET sequence: 43, 155, 198, 439cdf, 637cdf

Badness (Sintel): 1.83

Sengagen

Subgroup: 2.3.5.7

Comma list: 3136/3125, 420175/419904

Mapping: [⟨1 1 2 2], ⟨0 29 16 40]]

mapping generators: ~2, ~686/675

Optimal tunings:

  • WE: ~2 = 1199.8393 ¢, ~686/675 = 24.2133 ¢
error map: ⟨-0.161 +0.070 +0.778 -0.615]
  • CWE: ~2 = 1200.0000 ¢, ~686/675 = 24.2119 ¢
error map: ⟨0.000 +0.191 +1.077 -0.349]

Optimal ET sequence: 49, 50, 99, 248, 347, 446

Badness (Sintel): 1.47

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1344/1331, 3136/3125

Mapping: [⟨1 1 2 2 3], ⟨0 29 16 40 23]]

Optimal tunings:

  • WE: ~2 = 1199.3714 ¢, ~99/98 = 24.2224 ¢
  • CWE: ~2 = 1200.0000 ¢, ~99/98 = 24.2176 ¢

Optimal ET sequence: 49, 50, 99e

Badness (Sintel): 1.78

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 975/968, 1344/1331

Mapping: [⟨1 1 2 2 3 4], ⟨0 29 16 40 23 -15]]

Optimal tunings:

  • WE: ~2 = 1200.0364 ¢, ~99/98 = 24.1817 ¢
  • CWE: ~2 = 1200.0000 ¢, ~99/98 = 24.1815 ¢

Optimal ET sequence: 49, 50, 99e, 149e

Badness (Sintel): 2.21

Sengage

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 196/195, 364/363, 625/624

Mapping: [⟨1 1 2 2 3 3], ⟨0 29 16 40 23 35]]

Optimal tunings:

  • WE: ~2 = 1199.1402 ¢, ~99/98 = 24.2164 ¢
  • CWE: ~2 = 1200.0000 ¢, ~99/98 = 24.2092 ¢

Optimal ET sequence: 49f, 50, 99ef

Badness (Sintel): 1.55

Mowglic

For the 5-limit version, see Syntonic–kleismic equivalence continuum #Mowgli.

Mowglic may be described as the 19 & 161 temperament, named by Xenllium in 2021 as a variant of mowgli.

Subgroup: 2.3.5.7

Comma list: 3136/3125, 177147/175000

Mapping: [⟨1 0 0 -3], ⟨0 15 22 55]]

mapping generators: ~2, ~27/25

Optimal tunings:

  • WE: ~2 = 1200.0862 ¢, ~27/25 = 126.7147 ¢
error map: ⟨+0.086 -1.234 +1.411 +0.226]
  • CWE: ~2 = 1200.0000 ¢, ~27/25 = 126.7081 ¢
error map: ⟨0.000 -1.333 +1.265 +0.122]

Optimal ET sequence: 19, …, 123d, 142, 161

Badness (Sintel): 3.29

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 3136/3125, 72171/71680

Mapping: [⟨1 0 0 -3 8], ⟨0 15 22 55 -43]]

Optimal tunings:

  • WE: ~2 = 1200.0198 ¢, ~27/25 = 126.7134 ¢
  • CWE: ~2 = 1200.0000 ¢, ~27/25 = 126.7114 ¢

Optimal ET sequence: 19, 142, 161

Badness (Sintel): 3.11

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 1701/1690, 3136/3125

Mapping: [⟨1 0 0 -3 8 -2], ⟨0 15 22 55 -43 54]]

Optimal tunings:

  • WE: ~2 = 1199.9743 ¢, ~14/13 = 126.7022 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/13 = 126.7047 ¢

Optimal ET sequence: 19, 142f, 161

Badness (Sintel): 2.13

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 540/539, 833/832, 1701/1690, 3136/3125

Mapping: [⟨1 0 0 -3 8 -2 10], ⟨0 15 22 55 -43 54 -56]]

Optimal tunings:

  • WE: ~2 = 1200.0016 ¢, ~14/13 = 126.7029 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/13 = 126.7027 ¢

Optimal ET sequence: 19, 123defg, 142f, 161

Badness (Sintel): 2.14

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 351/350, 476/475, 495/494, 513/512, 540/539, 1701/1690

Mapping: [⟨1 0 0 -3 8 -2 10 9], ⟨0 15 22 55 -43 54 -56 -45]]

Optimal tunings:

  • WE: ~2 = 1199.9713 ¢, ~14/13 = 126.7018 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/13 = 126.7048 ¢

Optimal ET sequence: 19, 123defg, 142f, 161

Badness (Sintel): 1.96

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 276/275, 351/350, 476/475, 495/494, 513/512, 529/528, 540/539

Mapping: [⟨1 0 0 -3 8 -2 10 9 6], ⟨0 15 22 55 -43 54 -56 -45 -14]]

Optimal tunings:

  • WE: ~2 = 1200.0296 ¢, ~14/13 = 126.7061 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/13 = 126.7030 ¢

Optimal ET sequence: 19, 142f, 161

Badness (Sintel): 1.87

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 261/260, 276/275, 351/350, 476/475, 495/494, 513/512, 529/528, 540/539

Mapping: [⟨1 0 0 -3 8 -2 10 9 6 0], ⟨0 15 22 55 -43 54 -56 -45 -14 46]]

Optimal tunings:

  • WE: ~2 = 1200.0491 ¢, ~14/13 = 126.7094 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/13 = 126.7043 ¢

Optimal ET sequence: 19, 142f, 161

Badness (Sintel): 1.79

31-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29.31

Comma list: 261/260, 276/275, 351/350, 435/434, 476/475, 495/494, 513/512, 529/528, 540/539

Mapping: [⟨1 0 0 -3 8 -2 10 9 6 0 2], ⟨0 15 22 55 -43 54 -56 -45 -14 46 28]]

Optimal tunings:

  • WE: ~2 = 1199.9986 ¢, ~14/13 = 126.7027 ¢
  • CWE: ~2 = 1200.0000 ¢, ~14/13 = 126.7028 ¢

Optimal ET sequence: 19, 142fk, 161

Badness (Sintel): 1.85

Tremka

Tremka may be described as the 50 & 111 temperament. The name tremka was devised by Petr Pařízek in 2011, initially used for the no-7's version, especially in the 2.3.5.11.13 subgroup[2], but extending to the full 13-limit or higher does no significant tuning damage, so for that we keep the 2.3.5.11.13 label tremka.

Subgroup: 2.3.5.7

Comma list: 3136/3125, 2125764/2100875

Mapping: [⟨1 -4 -2 -8], ⟨0 31 24 60]]

mapping generators: ~2, ~4375/3888

Optimal tunings:

  • WE: ~2 = 1199.7671 ¢, ~4375/3888 = 216.1302 ¢
error map: ⟨-0.233 -0.987 +1.277 +0.849]
  • CWE: ~2 = 1200.0000 ¢, ~4375/3888 = 216.1665 ¢
error map: ⟨0.000 -0.794 +1.682 +1.162]

Optimal ET sequence: 50, 111, 161, 272

Badness (Sintel): 4.55

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 3136/3125, 35937/35840

Mapping: [⟨1 -4 -2 -8 4], ⟨0 31 24 60 -3]]

Optimal tunings:

  • WE: ~2 = 1199.8503 ¢, ~112/99 = 216.1406 ¢
  • CWE: ~2 = 1200.0000 ¢, ~112/99 = 216.1652 ¢

Optimal ET sequence: 50, 111, 161, 272, 433c

Badness (Sintel): 2.28

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 540/539, 847/845, 3136/3125

Mapping: [⟨1 -4 -2 -8 4 1], ⟨0 31 24 60 -3 15]]

Optimal tunings:

  • WE: ~2 = 1199.7479 ¢, ~112/99 = 216.1263 ¢
  • CWE: ~2 = 1200.0000 ¢, ~112/99 = 216.1681 ¢

Optimal ET sequence: 50, 111, 161, 272

Badness (Sintel): 1.49

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 351/350, 540/539, 561/560, 847/845, 1089/1088

Mapping: [⟨1 -4 -2 -8 4 1 -6], ⟨0 31 24 60 -3 15 56]]

Optimal tunings:

  • WE: ~2 = 1199.7537 ¢, ~17/15 = 216.1281 ¢
  • CWE: ~2 = 1200.0000 ¢, ~17/15 = 216.1688 ¢

Optimal ET sequence: 50, 111, 161, 272

Badness (Sintel): 1.15

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 324/323, 351/350, 456/455, 476/455, 495/494, 540/539

Mapping: [⟨1 -4 -2 -8 4 1 -6 -8], ⟨0 31 24 60 -3 15 56 68]]

Optimal tunings:

  • WE: ~2 = 1199.7496 ¢, ~17/15 = 216.1244 ¢
  • CWE: ~2 = 1200.0000 ¢, ~17/15 = 216.1658 ¢

Optimal ET sequence: 50, 111, 161, 272h, 433cfh

Badness (Sintel): 1.03

Undetrita

For the 5-limit version, see Syntonic–chromatic equivalence continuum #Undetrita (5-limit).

Undetrita tempers out 14348907/14336000, the scheme comma, in addition to the hemimean comma, and may be described as the 111 & 118 temperament. It splits 3rd harmonic into eleven parts; its ploidacot is thus alpha-hendecacot. It tempers out 3025/3024, 3388/3375, and 8019/8000 in the 11-limit, which relates itself to twentcufo, the no-7's version of 111 & 118.

It was named by Xenllium in 2021 as a play on the words undecimus (Latin for "eleventh") and tritave (3rd harmonic).

Subgroup: 2.3.5.7

Comma list: 3136/3125, 14348907/14336000

Mapping: [⟨1 0 -2 -8], ⟨0 11 30 75]]

mapping generators: ~2, ~448/405

Optimal tunings:

  • WE: ~2 = 1199.8545 ¢, ~448/405 = 172.8961 ¢
error map: ⟨-0.146 -0.098 +0.859 -0.458]
  • CWE: ~2 = 1200.0000 ¢, ~448/405 = 172.9143 ¢
error map: ⟨0.000 +0.102 +1.115 -0.255]

Optimal ET sequence: 111, 118, 229, 347, 576c

Badness (Sintel): 2.89

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 3136/3125, 8019/8000

Mapping: [⟨1 0 -2 -8 0], ⟨0 11 30 75 24]]

Optimal tunings:

  • WE: ~2 = 1200.0112 ¢, ~400/363 = 172.9138 ¢
  • CWE: ~2 = 1200.0000 ¢, ~400/363 = 172.9123 ¢

Optimal ET sequence: 111, 118, 229, 347

Badness (Sintel): 1.45

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 352/351, 729/728, 1001/1000, 3025/3024

Mapping: [⟨1 0 -2 -8 0 5], ⟨0 11 30 75 24 -9]]

Optimal tunings:

  • WE: ~2 = 1199.7099 ¢, ~72/65 = 172.8882 ¢
  • CWE: ~2 = 1200.0000 ¢, ~72/65 = 172.9265 ¢

Optimal ET sequence: 111, 229f

Badness (Sintel): 1.60

Undetritoid

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 1573/1568, 2080/2079, 3136/3125

Mapping: [⟨1 0 -2 -8 0 -11], ⟨0 11 30 75 24 102]]

Optimal tunings:

  • WE: ~2 = 1199.8901 ¢, ~400/363 = 172.9167 ¢
  • CWE: ~2 = 1200.0000 ¢, ~400/363 = 172.9310 ¢

Optimal ET sequence: 111, 229

Badness (Sintel): 1.77

Subgroup extensions

Undecimal didacus

In the no-3's 11-limit, there is a natural extension with prime 11 by equating 25/16 (which is already tuned sharp anyways) with 11/7 by tempering out 176/175, which is the same route that undecimal meantone uses, as this is essentially a no-3's restriction of undecimal meantone in the 11-limit, except that undecimal meantone finds ~28/25 at 2 generators (as a flat ~9/8) while here it is the generator. This is equivalent to finding 11/4 as (7/5)3. In the no-3's 19-limit extension "mediantone", this whole tone generator serves as the two simplest mediants of 9/8 and 10/9, namely 19/17 and 28/25, while in undecimal didacus and its extension to the no-3's 13-limit only the latter interpretation is relevant.

Subgroup: 2.5.7.11

Comma list: 176/175, 1375/1372

Sval mapping: [⟨1 0 -3 -7], ⟨0 2 5 9]]

sval mapping generators: ~2, ~56/25

Optimal tuning (CWE): 2 = 1\1, ~28/25 = 194.428

Optimal ET sequence: 6, 19e, 25, 31, 37

RMS error: 0.5567 cents

Badness (Sintel): 0.195

Tridecimal didacus

Tridecimal didacus (formerly roulette; that name has now been reassigned to the no-threes 19-limit extension 37 & 68) is equivalent to hemiwur or grosstone with no mapping for prime 3. The mapping of prime 13 is somewhat strange, because it is the only mapping that requires a negative amount of generators (and a large amount of them), but it can be rationalized in a variety of ways, such as that because ~8/7 is already tuned almost 3 ¢ flat, it makes sense to equate two of it with ~13/10 (tempering out the 8 ¢ huntma). This mapping of 13 increases the badness of the temperament, but as it does not noticeably affect the optimal generators, it is usually a safe extension to didacus if prime 3 is not included.

Subgroup: 2.5.7.11.13

Comma list: 176/175, 640/637, 1375/1372

Sval mapping: [⟨1 0 -3 -7 13], ⟨0 2 5 9 -8]]

sval mapping generators: ~2, ~56/25

Gencom mapping: [⟨1 0 2 2 2 5], ⟨0 0 2 5 9 -8]]

gencom: [2 28/25; 176/175 1375/1372 640/637]

Optimal tuning (POTE): 2 = 1\1, ~28/25 = 194.594

Optimal ET sequence: 6, 25, 31, 37

Badness (Sintel): 0.324

Mediantone

Mediantone is named after its whole tone generator serving as the mediant of 9/8 and 10/9, namely 19/17, in addition to 28/25, as well as by the observation that this temperament seems to have been repeatedly rediscovered in parts in a variety of contexts, so that it seems to exist as a "median" of all of these temperaments' logics. It is also an intentional play on "meantone", as the context one is most likely to first discover this logic is when the tone also represents ~10/9~9/8.

In the full no-3's 19-limit, this temperament is a structure common to quite a few temperaments. It is a rank-2 version of orion with a mapping for primes 11 and 13. It is a no-3's version of 19-limit grosstone which can be seen as an extension of undecimal meantone according to the "mediant-tone" logic of this temperament, and which as aforementioned effectively doubles the complexity of the temperament as a result of finding the generator of ~19/17~28/25 as (~3/2)2/2. It does not work so well as an extension for hemiwur to the full 19-limit, but if you want to try anyway (at the cost of primes 17 and 19), a notable patent-val tuning is 37edo, which finds prime 3 through the würschmidt mapping so that 6/1 is found at 16 generators.

Subgroup: 2.5.7.11.13.17.19

Comma list: 176/175, 640/637, 221/220, 476/475, 1375/1372

Sval mapping: [⟨1 0 -3 -7 13 -18 -19], ⟨0 2 5 9 -8 19 20]]

sval mapping generators: ~2, ~56/25

Optimal tuning (CWE): ~2 = 1\1, ~19/17 = 194.927

Optimal ET sequence: 6h, 31gh, 37, 80, 117d*

* 117d only appears without prime 19

Badness (Sintel): 0.618

Roulette

See also: Chromatic pairs § Roulette

Roulette is an alternative no-threes 19-limit extension of tridecimal didacus to mediantone (the two mappings converging at 37edo), equating (8/7)2 to 17/13 in addition to 13/10, tempering out 170/169 and 833/832; in doing so, it also tempers out the micro-comma 2000033/2000000 so that (50/49)3 is equated to 17/16. The generator is then equated to 19/17 in the same way as in mediantone.

Subgroup: 2.5.7.11.13.17.19

Comma list: 170/169, 176/175, 476/475, 640/637, 1375/1372

Sval mapping: [⟨1 2 2 2 5 7 7], ⟨0 2 5 9 -8 -18 -17]]

sval mapping generators: ~2, ~28/25

Optimal tuning (CWE): ~2 = 1\1, ~19/17 = 194.259

Optimal ET sequence: 6g, ... 31, 37, 68, 105

Badness (Sintel): 0.676

Rectified hebrew

Main article: Rectified hebrew

Rectified hebrew (37 & 56) is derived from the calendar by the same name. It is leap year pattern takes a stack of 18 Metonic cycle diatonic major scales and truncates the 19th one down to its generator, 11. It adds harmonic 13 through tempering out 4394/4375 and spliting the generator of didacus in three. Notably, it is the no-threes restriction of sycamore.

Subgroup: 2.5.7.13

Comma list: 3136/3125, 4394/4375

Sval mapping: [⟨1 2 2 3], ⟨0 6 15 13]]

sval mapping generators: ~2, ~26/25

Optimal tuning (POTE): ~2 = 1\1, ~26/25 = 64.6086

Optimal ET sequence: 18, 19, 37, 93, 130

Isra

Isra (iss-RAH) results from taking every other generator of septimal meantone, or from didacus if the generator is interpreted as 9/8. It is named after the Isrāʾ night journey in the Qur'an, because it is similar to luna (septimal hemithirds, a didacus extension).

Subgroup: 2.9.5.7

Comma list: 81/80, 126/125

Subgroup-val mapping: [⟨1 0 -4 -13], ⟨0 1 2 5]]

sval mapping generators: ~2, ~9

Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 192.9898

Optimal ET sequence: 6, 19, 25, 31, 56b, 87b

Tutone

Tutone is every other step of undecimal meantone, or undecimal didacus with the generator interpreted as 9/8.

Subgroup: 2.9.5.7.11

Comma list: 81/80, 99/98, 126/125

Subgroup-val mapping: [⟨1 0 -4 -13 -25], ⟨0 1 2 5 9]]

Gencom mapping: [⟨1 3/2 2 2 2], ⟨0 1/2 2 5 9]]

gencom: [2 9/8; 81/80 99/98 126/125]

Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 193.937

Optimal ET sequence: 6, 19e, 25, 31, 68b, 99b

Badness: 0.00536

Leantone

See also: Chromatic pairs § Leantone

Leantone is every other step of vincenzo.

Subgroup: 2.9.5.7.11

Comma list: 45/44, 56/55, 81/80

Subgroup-val mapping: [⟨1 0 -4 -13 -6], ⟨0 1 2 5 3]]

Gencom mapping: [⟨1 3/2 2 2 3], ⟨0 1/2 2 5 3]]

gencom: [2 9/8; 45/44 56/55 81/80]

Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 192.500

Optimal ET sequence: 6, 7, 13, 19, 25e, 31e, 56bee, 81beee

RMS error: 3.882 cents

Deutone

See also: Chromatic pairs § Deutone

Deutone is (also) every other step of vincenzo.

Subgroup: 2.9.5.7.13

Comma list: 65/64, 81/80, 91/90

Subgroup-val mapping: [⟨1 0 -4 -13 10], ⟨0 1 2 5 -2]]

Gencom mapping: [⟨1 3/2 2 2 0 4], ⟨0 1/2 2 5 0 -2]]

gencom: [2 9/8; 65/64 81/80 91/90]

Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 191.059

Optimal ET sequence: 6, 7, 13, 19, 25f, 44df

RMS error: 2.003 cents

References

  1. ↑ Yahoo! Tuning Group (archive) | Names for important high-complexity temperaments
  2. ↑ Yahoo! Tuning Group (archive) | Suggested names for the unclasified temperaments
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