Marvel temperaments: Difference between revisions
+ link to bunya and other update Tag: Reverted |
m this edit is exactly the kind of thing i saw ppl complaining about as due to unreadability of just stating a comma without knowing what it does, let alone in the context of the restrictions of some other set of commas. at least improve on or reword the info rather than erasing it Tags: Manual revert Reverted |
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Temperaments considered in families and clans are: | Temperaments considered in families and clans are: | ||
* ''[[Pelogic]]'' | * ''[[Pelogic]]'' → [[Mavila family #Pelogic|Mavila family]] (+21/20 or 135/128, generated by the fifth with 5/4 mapped to the m3) | ||
* [[Meantone]] | * [[Meantone]] → [[Meantone family #Septimal meantone|Meantone family]] (+81/80 or 126/125, generated by the fifth with 5/4 mapped to the M3) | ||
* [[Garibaldi]] | * [[Garibaldi]] → [[Schismatic family #Garibaldi|Schismatic family]] (+3125/3087, generated by the fifth with 5/4 mapped to the d4) | ||
* [[Pajara]] | * [[Pajara]] → [[Diaschismic family #Pajara|Diaschismic family]] (+50/49 or 64/63, generated by the fifth with a semioctave period) | ||
* ''[[Sharpie]]'' | * ''[[Sharpie]]'' → [[Dicot family #Sharpie|Dicot family]] (+25/24 or 28/27, fifth sliced in two) | ||
* ''[[Immune]]'' | * ''[[Immune]]'' → [[Immunity family #Immune|Immunity family]] (+781250/750141, twelfth sliced in two) | ||
* ''[[August]]'' | * ''[[August]]'' → [[Augmented family #August|Augmented family]] (+36/35 or 128/125, generated by the fifth with a 1/3-octave period) | ||
* ''[[Fog]]'' | * ''[[Fog]]'' → [[Misty family #Fog|Misty family]] (+156250/151263, generated by the fifth with a 1/3-octave period) | ||
* [[Negri]] → [[Slendro clan #Negri|Slendro clan]] (+49/48, fourth sliced in four) | |||
* [[Negri]] | * [[Magic]] → [[Magic family #Magic|Magic family]] (+245/243, twelfth sliced in five) | ||
* [[Magic]] | * ''[[Passive]]'' → [[Passion family #Passive|Passion family]] (+256/245, fourth sliced in five) | ||
* ''[[Passive]]'' | * ''[[Quintapole]]'' → [[Quintaleap family #Quintapole|Quintaleap family]] (+7812500/7411887, fourth sliced in five) | ||
* ''[[Quintapole]]'' | * ''[[Houborizic]]'' → [[Amity family #Houborizic|Amity family]] (+1250000/1240029, eleventh sliced in five) | ||
* ''[[Houborizic]]'' | * ''[[Qintosec]]'' → [[Quintosec family #Qintosec|Quintosec family]] (+2560000/2470629, generated by the classical minor second with a 1/5-octave period) | ||
* ''[[Qintosec]]'' | * [[Miracle]] → [[Gamelismic clan #Miracle|Gamelismic clan]] (+1029/1024, fifth sliced in six) | ||
* [[Miracle]] | * [[Catakleismic]] → [[Kleismic family #Catakleismic|Kleismic family]] (+4375/4374, twelfth sliced in six) | ||
* [[Catakleismic]] | * ''[[Marvo]]'' → [[Gravity family #Marvo|Gravity family]] (+78125000/78121827, two octaves and a fifth sliced in six) | ||
* ''[[Marvo]]'' | * [[Orwell]] → [[Semicomma family #Orwell|Semicomma family]] (+1728/1715, twelfth sliced in seven) | ||
* [[Orwell]] | * ''[[Snipes]]'' → [[Wesley family #Snipes|Wesley family]] (+6125/5832, two octaves and a fourth sliced in seven) | ||
* ''[[Snipes]]'' | * ''[[Submajor]]'' → [[Buzzardsmic clan #Submajor|Buzzardsmic clan]] (+65536/64827, two octaves and a fourth sliced in eight) | ||
* ''[[ | * ''[[Escapist]]'' → [[Escapade family #Escapist|Escapade family]] (+65625/65536, fourth sliced in nine) | ||
* ''[[Escapist]]'' | * ''[[Decic]]'' → [[Cloudy clan #Decic|Cloudy clan]] (+16807/16384, generated by the fifth with a 1/10-octave period) | ||
* ''[[Decic]]'' | * ''[[Amavil]]'' → [[Mabila family #Amavil|Mabila family]] (+17496/16807, four octaves and a fourth sliced in ten) | ||
* ''[[Amavil]]'' | * ''[[Betic]]'' → [[Sycamore family #Betic|Sycamore family]] (+1071875/1062882, fifth sliced in eleven) | ||
* ''[[Betic]]'' | * ''[[Hendeca]]'' → [[11th-octave temperaments #Hendeca|11th-octave temperaments]] (+122880/117649, generated by the fifth with a 1/11-octave period) | ||
* ''[[Hendeca]]'' | * ''[[Compton]]'' → [[Compton family #Compton|Compton family]] (+250047/250000, generated by the classical major third with a 1/12-octave period) | ||
* [[Compton]] | * ''[[Raccoon]]'' → [[Vavoom family #Raccoon|Vavoom family]] (+41943040/40353607, twelfth sliced in seventeen) | ||
* ''[[Raccoon]]'' | * ''[[Maquila]]'' → [[Maquila family #Septimal maquila|Maquila family]] (+30233088/28824005, seven octaves and a fifth sliced in seventeen) | ||
* ''[[Maquila]]'' | * ''[[Gammy]]'' → [[Gammic family #Gammy|Gammic family]] (+94143178827/91913281250, fifth sliced in twenty) | ||
* ''[[Gammy]]'' | |||
Considered below are wizard, tritonic, septimin, merman, slender, triton, marvolo, enneaportent, gracecordial, alphorn, misneb, untriton, naiadical, quintannic, gwazy, and tertiosec, in the order of increasing [[badness]]. | Considered below are wizard, tritonic, septimin, merman, slender, triton, marvolo, enneaportent, gracecordial, alphorn, misneb, untriton, naiadical, quintannic, gwazy, and tertiosec, in the order of increasing [[badness]]. | ||
Revision as of 14:06, 1 May 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.
This page discusses miscellaneous rank-2 temperaments tempering out 225/224, the marvel comma or septimal kleisma.
Temperaments considered in families and clans are:
- Pelogic → Mavila family (+21/20 or 135/128, generated by the fifth with 5/4 mapped to the m3)
- Meantone → Meantone family (+81/80 or 126/125, generated by the fifth with 5/4 mapped to the M3)
- Garibaldi → Schismatic family (+3125/3087, generated by the fifth with 5/4 mapped to the d4)
- Pajara → Diaschismic family (+50/49 or 64/63, generated by the fifth with a semioctave period)
- Sharpie → Dicot family (+25/24 or 28/27, fifth sliced in two)
- Immune → Immunity family (+781250/750141, twelfth sliced in two)
- August → Augmented family (+36/35 or 128/125, generated by the fifth with a 1/3-octave period)
- Fog → Misty family (+156250/151263, generated by the fifth with a 1/3-octave period)
- Negri → Slendro clan (+49/48, fourth sliced in four)
- Magic → Magic family (+245/243, twelfth sliced in five)
- Passive → Passion family (+256/245, fourth sliced in five)
- Quintapole → Quintaleap family (+7812500/7411887, fourth sliced in five)
- Houborizic → Amity family (+1250000/1240029, eleventh sliced in five)
- Qintosec → Quintosec family (+2560000/2470629, generated by the classical minor second with a 1/5-octave period)
- Miracle → Gamelismic clan (+1029/1024, fifth sliced in six)
- Catakleismic → Kleismic family (+4375/4374, twelfth sliced in six)
- Marvo → Gravity family (+78125000/78121827, two octaves and a fifth sliced in six)
- Orwell → Semicomma family (+1728/1715, twelfth sliced in seven)
- Snipes → Wesley family (+6125/5832, two octaves and a fourth sliced in seven)
- Submajor → Buzzardsmic clan (+65536/64827, two octaves and a fourth sliced in eight)
- Escapist → Escapade family (+65625/65536, fourth sliced in nine)
- Decic → Cloudy clan (+16807/16384, generated by the fifth with a 1/10-octave period)
- Amavil → Mabila family (+17496/16807, four octaves and a fourth sliced in ten)
- Betic → Sycamore family (+1071875/1062882, fifth sliced in eleven)
- Hendeca → 11th-octave temperaments (+122880/117649, generated by the fifth with a 1/11-octave period)
- Compton → Compton family (+250047/250000, generated by the classical major third with a 1/12-octave period)
- Raccoon → Vavoom family (+41943040/40353607, twelfth sliced in seventeen)
- Maquila → Maquila family (+30233088/28824005, seven octaves and a fifth sliced in seventeen)
- Gammy → Gammic family (+94143178827/91913281250, fifth sliced in twenty)
Considered below are wizard, tritonic, septimin, merman, slender, triton, marvolo, enneaportent, gracecordial, alphorn, misneb, untriton, naiadical, quintannic, gwazy, and tertiosec, in the order of increasing badness.
Since (5/4)2 = (225/224)⋅(14/9), these temperaments tend to have a relatively small complexity for 5/4. They also possess a version of the augmented triad where each third approximates either 5/4 or 9/7. Since this is a chord of meantone temperament in wide use in Western common practice harmony long before 12edo established itself as the standard tuning, it is arguably more authentic to tune it as two stacked major thirds and a diminished fourth, which is what it is in meantone, than as the modern version of three stacked very sharp major thirds.
The melodic signature of marvel temperaments is that 16/15 and 15/14 are tempered to be equal. Hence 8/7 can be divided into two equal parts.
Marvel tempering allows for a tritone substitution whereby the dominant seventh chord formed by adding 16/9 above the root shares its tritone with a 4:5:6:7 tetrad. (The tritone of the dominant seventh is (16/9)/(5/4) = 64/45. Setting this equal to 10/7 gives (10/7)/(64/45) = 225/224.)
Wizard
- For the 5-limit version, see Miscellaneous 5-limit temperaments #Wizard.
Wizard has a semi-octave period and is generated by an interval that can be treated as ~17/15. The semi-octave complement of this interval is ~5/4. Wizard can be described as 22 & 72. Its ploidacot is diploid alpha-hexacot, so six generator steps plus a semi-octave period gives the perfect twelfth. 72edo, 94edo, and especially 166edo are good tunings for it.
Subgroup: 2.3.5.7
Comma list: 225/224, 118098/117649
Mapping: [⟨2 1 5 2], ⟨0 6 -1 10]]
- mapping generators: ~1225/864, ~245/216
- WE: ~1225/864 = 600.3438 ¢, ~245/216 = 216.8680 ¢
- error map: ⟨+0.688 -0.403 -1.463 +0.541]
- CWE: ~1225/864 = 600.0000 ¢, ~245/216 = 216.7977 ¢
- error map: ⟨0.000 -1.169 -3.111 -0.849]
Optimal ET sequence: 22, 50, 72, 238c, 310c, 382c, 454bccd
Badness (Sintel): 1.03
11-limit
Subgroup: 2.3.5.7.11
Comma list: 225/224, 385/384, 4000/3993
Mapping: [⟨2 1 5 2 8], ⟨0 6 -1 10 -3]]
Optimal tunings:
- WE: ~99/70 = 600.3051 ¢, ~25/22 = 216.8782 ¢
- CWE: ~99/70 = 600.0000 ¢, ~25/22 = 216.7961 ¢
Optimal ET sequence: 22, 50, 72, 166, 238c, 310c
Badness (Sintel): 0.613
Lizard
Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 351/350, 364/363, 385/384
Mapping: [⟨2 1 5 2 8 11], ⟨0 6 -1 10 -3 -10]]
Optimal tunings:
- WE: ~55/39 = 600.4824 ¢, ~25/22 = 216.7852 ¢
- CWE: ~55/39 = 600.0000 ¢, ~25/22 = 216.6247 ¢
Optimal ET sequence: 22, 50, 72
Badness (Sintel): 0.900
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 221/220, 273/272, 289/288, 351/350, 375/374
Mapping: [⟨2 1 5 2 8 11 6], ⟨0 6 -1 10 -3 -10 6]]
Optimal tunings:
- WE: ~17/12 = 600.5032 ¢, ~17/15 = 216.8002 ¢
- CWE: ~17/12 = 600.0000 ¢, ~17/15 = 216.6361 ¢
Optimal ET sequence: 22, 50, 72
Badness (Sintel): 0.741
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 153/152, 210/209, 221/220, 225/224, 273/272, 343/342
Mapping: [⟨2 1 5 2 8 11 6 2], ⟨0 6 -1 10 -3 -10 6 18]]
Optimal tunings:
- WE: ~17/12 = 600.4698 ¢, ~17/15 = 216.6925 ¢
- CWE: ~17/12 = 600.0000 ¢, ~17/15 = 216.5434 ¢
Optimal ET sequence: 22h, 50, 72, 122g, 194dfg
Badness (Sintel): 0.955
Gizzard
Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 325/324, 385/384, 1573/1568
Mapping: [⟨2 1 5 2 8 -2], ⟨0 6 -1 10 -3 26]]
Optimal tunings:
- WE: ~99/70 = 600.2896 ¢, ~25/22 = 216.9343 ¢
- CWE: ~99/70 = 600.0000 ¢, ~25/22 = 216.8501 ¢
Optimal ET sequence: 22f, 72, 166, 238cf
Badness (Sintel): 0.837
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 225/224, 289/288, 325/324, 375/374, 385/384
Mapping: [⟨2 1 5 2 8 -2 6], ⟨0 6 -1 10 -3 26 6]]
Optimal tunings:
- WE: ~17/12 = 600.3227 ¢, ~17/15 = 216.9414 ¢
- CWE: ~17/12 = 600.0000 ¢, ~17/15 = 216.8469 ¢
Optimal ET sequence: 22f, 72, 166g, 238cfg
Badness (Sintel): 0.694
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 225/224, 325/324, 375/374, 385/384, 400/399, 595/594
Mapping: [⟨2 1 5 2 8 -2 6 15], ⟨0 6 -1 10 -3 26 6 -18]]
Optimal tunings:
- WE: ~17/12 = 600.2637 ¢, ~17/15 = 216.9570 ¢
- CWE: ~17/12 = 600.0000 ¢, ~17/15 = 216.8687 ¢
Optimal ET sequence: 72, 94, 166g
Badness (Sintel): 0.901
Mage
Subgroup: 2.3.5.7.11
Comma list: 99/98, 176/175, 1331/1296
Mapping: [⟨2 1 5 2 4], ⟨0 6 -1 10 8]]
Optimal tunings:
- WE: ~77/54 = 600.6486 ¢, ~55/48 = 217.1099 ¢
- CWE: ~77/54 = 600.0000 ¢, ~55/48 = 216.9841 ¢
Optimal ET sequence: 22, 50e, 72ee
Badness (Sintel): 1.91
Tritonic
- For the 5-limit version, see Miscellaneous 5-limit temperaments #Tritonic.
Tritonic tempers out 50421/50000 and may be described as the 29 & 31 temperament. It splits the 6th harmonic into five generators of ~10/7 tritones, hence the name. Its ploidacot is beta-pentacot. 60edo may be used as a tuning, which in the 11-limit entails the 60e val.
Subgroup: 2.3.5.7
Comma list: 225/224, 50421/50000
Mapping: [⟨1 -1 8 9], ⟨0 5 -11 -12]]
- mapping generators: ~2, ~10/7
- WE: ~2 = 1201.3539 ¢, ~10/7 = 620.4131 ¢
- error map: ⟨+1.354 -1.243 -0.027 -1.598]
- CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.6778 ¢
- error map: ⟨0.000 -3.566 -2.769 -4.959]
Optimal ET sequence: 29, 31, 60, 91, 122, 213bcd
Badness (Sintel): 1.20
11-limit
Subgroup: 2.3.5.7.11
Comma list: 121/120, 225/224, 441/440
Mapping: [⟨1 -1 8 9 5], ⟨0 5 -11 -12 -3]]
Optimal tunings:
- WE: ~2 = 1201.7116 ¢, ~10/7 = 620.6166 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.6890 ¢
Optimal ET sequence: 29, 31, 60e, 91e, 213bcdeee
Badness (Sintel): 0.782
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 105/104, 121/120, 196/195, 275/273
Mapping: [⟨1 -1 8 9 5 13], ⟨0 5 -11 -12 -3 -18]]
Optimal tunings:
- WE: ~2 = 1201.5355 ¢, ~10/7 = 620.6855 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.8469 ¢
Optimal ET sequence: 29, 31, 60e
Badness (Sintel): 0.950
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 105/104, 121/120, 154/153, 196/195, 273/272
Mapping: [⟨1 -1 8 9 5 13 17], ⟨0 5 -11 -12 -3 -18 -25]]
Optimal tunings:
- WE: ~2 = 1201.5260 ¢, ~10/7 = 620.7330 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.8986 ¢
Optimal ET sequence: 29g, 31, 60e
Badness (Sintel): 0.973
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 77/76, 105/104, 121/120, 153/152, 196/195, 273/272
Mapping: [⟨1 -1 8 9 5 13 17 12], ⟨0 5 -11 -12 -3 -18 -25 -15]]
Optimal tunings:
- WE: ~2 = 1201.3100 ¢, ~10/7 = 620.6509 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.9328 ¢
Optimal ET sequence: 29g, 31, 60e
Badness (Sintel): 1.03
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 77/76, 105/104, 115/114, 121/120, 153/152, 161/160, 196/195
Mapping: [⟨1 -1 8 9 5 13 17 12 4], ⟨0 5 -11 -12 -3 -18 -25 -15 1]]
Optimal tunings:
- WE: ~2 = 1201.4074 ¢, ~10/7 = 620.7185 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.9548 ¢
Optimal ET sequence: 29g, 31, 60e
Badness (Sintel): 1.04
Tritoni
Subgroup: 2.3.5.7.11
Comma list: 225/224, 385/384, 27783/27500
Mapping: [⟨1 -1 8 9 -11], ⟨0 5 -11 -12 28]]
Optimal tunings:
- WE: ~2 = 1201.0888 ¢, ~10/7 = 620.1733 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/7 = 619.6146 ¢
Optimal ET sequence: 31, 91, 122, 153d
Badness (Sintel): 1.50
Septimin
- For the 5-limit version, see Miscellaneous 5-limit temperaments #Septimin.
Septimin may be described as the 41 & 50 temperament. It is generated by a septimal minor third (7/6), which gives rise to the name, but the generator can be taken to be the octave complement, 12/7, such that eleven of them octave reduced give the perfect fifth; its ploidacot is thus eta-hendecacot. 91edo may be recommended as a tuning.
Subgroup: 2.3.5.7
Comma list: 225/224, 84035/82944
Mapping: [⟨1 -7 7 -5], ⟨0 11 -6 10]]
- mapping generators: ~2, ~12/7
- WE: ~2 = 1201.2452 ¢, ~12/7 = 937.3394 ¢
- error map: ⟨+1.245 +0.062 -1.633 -1.658]
- CWE: ~2 = 1200.0000 ¢, ~12/7 = 936.4036 ¢
- error map: ⟨0.000 -1.516 -4.735 -4.790]
Optimal ET sequence: 41, 91, 132d
Badness (Sintel): 1.38
11-limit
Subgroup: 2.3.5.7.11
Comma list: 225/224, 245/242, 385/384
Mapping: [⟨1 -7 7 -5 -2], ⟨0 11 -6 10 7]]
Optimal tunings:
- WE: ~2 = 1200.8059 ¢, ~12/7 = 936.9952 ¢
- CWE: ~2 = 1200.0000 ¢, ~12/7 = 936.3906 ¢
Optimal ET sequence: 41, 91, 223cdef
Badness (Sintel): 1.04
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 105/104, 144/143, 196/195, 245/242
Mapping: [⟨1 -7 7 -5 -2 -8], ⟨0 11 -6 10 7 15]]
Optimal tunings:
- WE: ~2 = 1200.5990 ¢, ~12/7 = 936.7670 ¢
- CWE: ~2 = 1200.0000 ¢, ~12/7 = 936.3196 ¢
Badness (Sintel): 0.955
Merman
- For the 5-limit version, see Miscellaneous 5-limit temperaments #Merman.
Merman may be described as the 41 & 43 temperament. Like tritonic, it is generated by a ~10/7 tritone, but here, seven generator steps give the interval class of 3. The ploidacot for this temperament is gamma-heptacot.
The name was likely derived from Triton, which was in turn derived from tritonic.
Subgroup: 2.3.5.7
Comma list: 225/224, 2500000/2470629
Mapping: [⟨1 -2 10 11], ⟨0 7 -15 -16]]
- mapping generators: ~2, ~10/7
- WE: ~2 = 1200.3898 ¢, ~10/7 = 614.6413 ¢
- error map: ⟨+0.390 -0.435 -1.630 +1.634]
- CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.4073 ¢
- error map: ⟨0.000 -1.104 -2.423 +0.657]
Optimal ET sequence: 41, 84, 125
Badness (Sintel): 1.39
11-limit
Subgroup: 2.3.5.7.11
Comma list: 225/224, 441/440, 1344/1331
Mapping: [⟨1 -2 10 11 5], ⟨0 7 -15 -16 -3]]
Optimal tunings:
- WE: ~2 = 1199.9578 ¢, ~10/7 = 614.3720 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.3943 ¢
Optimal ET sequence: 41, 84, 125e
Badness (Sintel): 1.20
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 144/143, 225/224, 364/363, 441/440
Mapping: [⟨1 -2 10 11 5 -5], ⟨0 7 -15 -16 -3 17]]
Optimal tunings:
- WE: ~2 = 1199.7422 ¢, ~10/7 = 614.2110 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.3442 ¢
Optimal ET sequence: 41, 84, 125e, 209ef, 293ef
Badness (Sintel): 1.14
Mermaid
Subgroup: 2.3.5.7.11
Comma list: 225/224, 385/384, 532400/531441
Mapping: [⟨1 -2 10 11 -16], ⟨0 7 -15 -16 38]]
Optimal tunings:
- WE: ~2 = 1199.4973 ¢, ~10/7 = 614.7004 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.4470 ¢
Optimal ET sequence: 41, 84e, 125, 166
Badness (Sintel): 1.46
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 325/324, 385/384, 10648/10647
Mapping: [⟨1 -2 10 11 22 32], ⟨0 7 -15 -16 38 58]]
Optimal tunings:
- WE: ~2 = 1200.5126 ¢, ~10/7 = 614.7152 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/7 = 614.4562 ¢
Optimal ET sequence: 41, 84ef, 125f, 166
Badness (Sintel): 1.47
Slender
Slender tempers out the hewuermera comma in addition to the marvel comma, and may be described as the 31 & 32 temperament. This temperament has a generator of 49/48, three of which equal marvel's 16/15~15/14, and ten generators give 5/4. Its ploidacot is omega-13-cot.
The name was likely derived from slendro diesis, one of the names for the interval 49/48.
Subgroup: 2.3.5.7
Comma list: 225/224, 589824/588245
Mapping: [⟨1 2 2 3], ⟨0 -13 10 -6]]
- mapping generators: ~2, ~49/48
- WE: ~2 = 1200.3816 ¢, ~49/48 = 38.4256 ¢
- error map: ⟨+0.382 -0.725 -1.295 +1.765]
- CWE: ~2 = 1200.0000 ¢, ~49/48 = 38.4079 ¢
- error map: ⟨0.000 -1.257 -2.235 +0.727]
Optimal ET sequence: 31, 94, 125, 406c
Badness (Sintel): 1.44
11-limit
Subgroup: 2.3.5.7.11
Comma list: 225/224, 385/384, 1331/1323
Mapping: [⟨1 2 2 3 4], ⟨0 -13 10 -6 -17]]
Optimal tunings:
- WE: ~2 = 1199.4983 ¢, ~49/48 = 38.4030 ¢
- CWE: ~2 = 1200.0000 ¢, ~49/48 = 38.3775 ¢
Optimal ET sequence: 31, 63, 94, 125
Badness (Sintel): 0.838
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 275/273, 385/384, 1331/1323
Mapping: [⟨1 2 2 3 4 3], ⟨0 -13 10 -6 -17 22]]
Optimal tunings:
- WE: ~2 = 1200.1728 ¢, ~49/48 = 38.3192 ¢
- CWE: ~2 = 1200.0000 ¢, ~49/48 = 38.3129 ¢
Optimal ET sequence: 31, 63, 94
Badness (Sintel): 1.07
Triton
- For the 5-limit version, see Syntonic–kleismic equivalence continuum #Stump.
Triton may be described as the 19 & 21 temperament. Like tritonic, it is generated by a ~10/7 tritone, but here, three generator steps give the interval class of 3. The ploidacot for this temperament is alpha-tricot.
Subgroup: 2.3.5.7
Comma list: 225/224, 1029/1000
Mapping: [⟨1 0 6 7], ⟨0 3 -7 -8]]
- mapping generators: ~2, ~10/7
- WE: ~2 = 1203.3828 ¢, ~10/7 = 632.9137 ¢
- error map: ⟨+3.383 -3.214 +3.587 -8.457]
- CWE: ~2 = 1200.0000 ¢, ~10/7 = 630.9827 ¢
- error map: ⟨0.000 -9.007 -3.192 -16.687]
Optimal ET sequence: 2, 17d, 19, 78bd, 97bd
Badness (Sintel): 1.50
11-limit
Subgroup: 2.3.5.7.11
Comma list: 45/44, 56/55, 1029/1000
Mapping: [⟨1 0 6 7 4], ⟨0 3 -7 -8 -1]]
Optimal tunings:
- WE: ~2 = 1201.3875 ¢, ~10/7 = 631.5852 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/7 = 630.8007 ¢
Optimal ET sequence: 2, 17d, 19
Badness (Sintel): 1.51
Marvolo
Subgroup: 2.3.5.7
Comma list: 225/224, 156250000/155649627
Mapping: [⟨1 2 1 1], ⟨0 -6 19 26]]
- mapping generators: ~2, ~21/20
- WE: ~2 = 1200.7714 ¢, ~21/20 = 83.4014 ¢
- error map: ⟨+0.772 -0.820 -0.916 +0.381]
- CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.3640 ¢
- error map: ⟨0.000 -2.139 -2.398 -1.362]
Optimal ET sequence: 29, 43, 72, 619bbccd, 691bbccd
Badness (Sintel): 2.11
11-limit
Subgroup: 2.3.5.7.11
Comma list: 225/224, 441/440, 4000/3993
Mapping: [⟨1 2 1 1 2], ⟨0 -6 19 26 21]]
Optimal tunings:
- WE: ~2 = 1200.7075 ¢, ~21/20 = 83.3888 ¢
- CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.3564 ¢
Optimal ET sequence: 29, 43, 72
Badness (Sintel): 0.958
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 225/224, 364/363, 441/440
Mapping: [⟨1 2 1 1 2 3], ⟨0 -6 19 26 21 10]]
Optimal tunings:
- WE: ~2 = 1200.9467 ¢, ~21/20 = 83.3956 ¢
- CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.3516 ¢
Optimal ET sequence: 29, 43, 72
Badness (Sintel): 0.887
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 169/168, 221/220, 225/224, 364/363, 441/440
Mapping: [⟨1 2 1 1 2 3 2], ⟨0 -6 19 26 21 10 30]]
Optimal tunings:
- WE: ~2 = 1200.9606 ¢, ~21/20 = 83.4030 ¢
- CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.3594 ¢
Optimal ET sequence: 29g, 43, 72
Badness (Sintel): 0.760
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 169/168, 210/209, 221/220, 225/224, 364/363, 441/440
Mapping: [⟨1 2 1 1 2 3 2 3], ⟨0 -6 19 26 21 10 30 18]]
Optimal tunings:
- WE: ~2 = 1200.7625 ¢, ~21/20 = 83.3895 ¢
- CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.3551 ¢
Optimal ET sequence: 29g, 43, 72
Badness (Sintel): 0.895
Enneaportent
Subgroup: 2.3.5.7
Comma list: 225/224, 40353607/40310784
Mapping: [⟨9 0 28 11], ⟨0 2 -1 2]]
- mapping generators: ~2592/2401, ~12005/6912
- WE: ~2592/2401 = 133.4174 ¢, ~12005/6912 = 950.7667 ¢ (~1728/1715 = 16.8452 ¢)
- error map: ⟨+0.756 -0.422 -1.395 +0.298]
- CWE: ~2592/2401 = 133.3333 ¢, ~12005/6912 = 950.2969 ¢ (~1728/1715 = 16.9636 ¢)
- error map: ⟨0.000 -1.361 -3.277 -1.565]
Optimal ET sequence: 9, 54, 63, 72, 495bccd, 567bcccd
Badness (Sintel): 2.37
11-limit
Subgroup: 2.3.5.7.11
Comma list: 225/224, 385/384, 12005/11979
Mapping: [⟨9 0 28 11 24], ⟨0 2 -1 2 1]]
Optimal tunings:
- WE: ~121/112 = 133.4071 ¢, ~210/121 = 950.7131 ¢ (~99/98 = 16.8633 ¢)
- CWE: ~121/112 = 133.3333 ¢, ~210/121 = 950.2994 ¢ (~99/98 = 16.9661 ¢)
Optimal ET sequence: 9, 54, 63, 72
Badness (Sintel): 1.01
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 225/224, 364/363, 1716/1715
Mapping: [⟨9 0 28 11 24 19], ⟨0 2 -1 2 1 2]]
Optimal tunings:
- WE: ~14/13 = 133.4245 ¢, ~26/15 = 950.9362 ¢ (~105/104 = 16.9650 ¢)
- CWE: ~14/13 = 133.3333 ¢, ~26/15 = 950.4364 ¢ (~99/98 = 17.1031 ¢)
Optimal ET sequence: 9, 54, 63, 72
Badness (Sintel): 0.922
Gracecordial
- For the 5-limit version, see Schismic–Pythagorean equivalence continuum #Gracecordial (5-limit).
Subgroup: 2.3.5.7
Comma list: 225/224, 781250000/771895089
Mapping: [⟨1 0 34 63], ⟨0 1 -20 -38]]
- mapping generators: ~2, ~3
- WE: ~2 = 1200.4904 ¢, ~3/2 = 701.1103 ¢
- error map: ⟨+0.490 -0.354 -1.655 +1.241]
- CWE: ~2 = 1200.3333 ¢, ~3/2 = 700.8112 ¢
- error map: ⟨0.000 -1.144 -2.537 +0.349]
Optimal ET sequence: 12, …, 113, 125, 238c, 363c
Badness (Sintel): 2.44
11-limit
Subgroup: 2.3.5.7.11
Comma list: 225/224, 385/384, 236328125/234365481
Mapping: [⟨1 0 34 63 -90], ⟨0 1 -20 -38 59]]
Optimal tunings:
- WE: ~2 = 1200.5571 ¢, ~3/2 = 701.1589 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8328 ¢
Optimal ET sequence: 12e, 113, 125, 238c
Badness (Sintel): 2.96
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 325/324, 385/384, 831875/830466
Mapping: [⟨1 0 34 63 -90 -66], ⟨0 1 -20 -38 59 44]]
Optimal tunings:
- WE: ~2 = 1200.6282 ¢, ~3/2 = 701.2080 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8421 ¢
Optimal ET sequence: 12e, 113, 125f, 238cf
Badness (Sintel): 2.16
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 225/224, 273/272, 325/324, 385/384, 4928/4913
Mapping: [⟨1 0 34 63 -90 -66 -7], ⟨0 1 -20 -38 59 44 7]]
Optimal tunings:
- WE: ~2 = 1200.5058 ¢, ~3/2 = 701.1360 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8414 ¢
Optimal ET sequence: 12e, 113, 125f, 238cf
Badness (Sintel): 1.96
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 1445/1444
Mapping: [⟨1 0 34 63 -90 -66 -7 9], ⟨0 1 -20 -38 59 44 7 -3]]
Optimal tunings:
- WE: ~2 = 1200.4418 ¢, ~3/2 = 701.0999 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8425 ¢
Optimal ET sequence: 12e, 113, 125f, 238cf
Badness (Sintel): 1.71
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 225/224, 273/272, 324/323, 325/324, 385/384, 460/459, 529/528
Mapping: [⟨1 0 34 63 -90 -66 -7 9 -43], ⟨0 1 -20 -38 59 44 7 -3 30]]
Optimal tunings:
- WE: ~2 = 1200.4641 ¢, ~3/2 = 701.1145 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8444 ¢
Optimal ET sequence: 12e, 113, 238cfi
Badness (Sintel): 1.57
29-limit
Subgroup: 2.3.5.7.11.13.17.19.23.29
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 494/493
Mapping: [⟨1 0 34 63 -90 -66 -7 9 -43 -49], ⟨0 1 -20 -38 59 44 7 -3 30 34]]
Optimal tunings:
- WE: ~2 = 1200.4400 ¢, ~3/2 = 701.0986 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8428 ¢
Optimal ET sequence: 12e, 113, 125f, 238cfi
Badness (Sintel): 1.50
31-limit
Subgroup: 2.3.5.7.11.13.17.19.23.29.31
Comma list: 225/224, 273/272, 290/289, 324/323, 325/324, 385/384, 460/459, 465/464, 494/493
Mapping: [⟨1 0 34 63 -90 -66 -7 9 -43 -49 -79], ⟨0 1 -20 -38 59 44 7 -3 30 34 53]]
Optimal tunings:
- WE: ~2 = 1200.4178 ¢, ~3/2 = 701.0822 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8396 ¢
Optimal ET sequence: 12e, 113, 125f, 238cfi
Badness (Sintel): 1.53
Gracecord
Subgroup: 2.3.5.7.11
Comma list: 225/224, 441/440, 109375/107811
Mapping: [⟨1 0 34 63 89], ⟨0 1 -20 -38 -54]]
Optimal tunings:
- WE: ~2 = 1200.6064 ¢, ~3/2 = 701.2398 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8718 ¢
Optimal ET sequence: 12, …, 101cd, 113
Badness (Sintel): 2.21
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 364/363, 441/440, 6125/6084
Mapping: [⟨1 0 34 63 89 113], ⟨0 1 -20 -38 -54 -69]]
Optimal tunings:
- WE: ~2 = 1200.6225 ¢, ~3/2 = 701.2539 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8781 ¢
Optimal ET sequence: 12f, …, 101cdf, 113
Badness (Sintel): 1.83
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 225/224, 364/363, 441/440, 595/594, 2000/1989
Mapping: [⟨1 0 34 63 89 113 -7], ⟨0 1 -20 -38 -54 -69 7]]
Optimal tunings:
- WE: ~2 = 1200.3308 ¢, ~3/2 = 701.0632 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8654 ¢
Optimal ET sequence: 12f, 101cdf, 113
Badness (Sintel): 1.87
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 210/209, 225/224, 324/323, 364/363, 400/399, 665/663
Mapping: [⟨1 0 34 63 89 113 -7 9], ⟨0 1 -20 -38 -54 -69 7 -3]]
Optimal tunings:
- WE: ~2 = 1200.2658 ¢, ~3/2 = 701.0213 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 700.8629 ¢
Optimal ET sequence: 12f, 101cdf, 113
Badness (Sintel): 1.68
Alphorn
Subgroup: 2.3.5.7
Comma list: 225/224, 5764801/5668704
Mapping: [⟨1 -7 5 -9], ⟨0 16 -5 22]]
- mapping generators: ~2, ~35/24
- WE: ~2 = 1201.3004 ¢, ~35/24 = 644.4767 ¢
- error map: ⟨+1.300 +0.569 -2.195 -2.043]
- CWE: ~2 = 1200.3333 ¢, ~35/24 = 643.8137 ¢
- error map: ⟨0.000 -0.936 -5.382 -4.924]
Optimal ET sequence: 13d, 28d, 41, 151cd, 192cdd, 233ccdd
Badness (Sintel): 3.27
11-limit
Subgroup: 2.3.5.7.11
Comma list: 225/224, 385/384, 12250/11979
Mapping: [⟨1 -7 5 -9 4], ⟨0 16 -5 22 -1]]
Optimal tunings:
- WE: ~2 = 1200.5123 ¢, ~16/11 = 644.1307 ¢
- CWE: ~2 = 1200.0000 ¢, ~16/11 = 643.8662 ¢
Optimal ET sequence: 13d, 28d, 41
Badness (Sintel): 2.43
Misneb
- For the 5-limit version, see Miscellaneous 5-limit temperaments #Misneb.
Subgroup: 2.3.5.7
Comma list: 225/224, 4194304/4117715
Mapping: [⟨1 -12 15 1], ⟨0 15 -14 2]]
- mapping generators: ~2, ~15/8
- WE: ~2 = 1199.7642 ¢, ~15/8 = 1086.5513 ¢
- error map: ⟨-0.236 -0.856 -1.569 +4.041]
- CWE: ~2 = 1200.0000 ¢, ~15/8 = 1086.7633 ¢
- error map: ⟨0.000 -0.506 -0.999 +4.701]
Optimal ET sequence: 21, 32, 53
Badness (Sintel): 3.57
11-limit
Subgroup: 2.3.5.7.11
Comma list: 99/98, 176/175, 1310720/1294139
Mapping: [⟨1 -12 15 1 27], ⟨0 15 -14 2 -26]]
Optimal tunings:
- WE: ~2 = 1200.1654 ¢, ~15/8 = 1086.8269 ¢
- CWE: ~2 = 1200.0000 ¢, ~15/8 = 1086.6766 ¢
Optimal ET sequence: 21, 32e, 53, 127
Badness (Sintel): 2.82
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 99/98, 176/175, 640/637, 847/845
Mapping: [⟨1 -12 15 1 27 20], ⟨0 15 -14 2 -26 -18]]
Optimal tunings:
- WE: ~2 = 1200.1687 ¢, ~15/8 = 1086.8295 ¢
- CWE: ~2 = 1200.0000 ¢, ~15/8 = 1086.6757 ¢
Optimal ET sequence: 21, 32e, 53, 127
Badness (Sintel): 1.88
Musneb
Subgroup: 2.3.5.7.11
Comma list: 225/224, 385/384, 66550/64827
Mapping: [⟨1 3 1 3 6], ⟨0 -15 14 -2 -27]]
Optimal tunings:
- WE: ~2 = 1200.0839 ¢, ~15/8 = 1086.9343 ¢
- CWE: ~2 = 1200.0000 ¢, ~15/8 = 1086.8593 ¢
Optimal ET sequence: 21e, 32, 53
Badness (Sintel): 2.89
Untriton
- For the 5-limit version, see Miscellaneous 5-limit temperaments #Untriton.
Named by Petr Pařízek in 2011[1], untriton may be described as the 51 & 53 temperament. Like tritonic, it is generated by a ~10/7 tritone, but here, nine generator steps give the interval class of 3. The ploidacot for this temperament is delta-enneacot.
Subgroup: 2.3.5.7
Comma list: 225/224, 125000000/121060821
Mapping: [⟨1 -3 12 13], ⟨0 9 -19 -20]]
- mapping generators: ~2, ~10/7
- WE: ~2 = 1199.8275 ¢, ~10/7 = 611.2710 ¢
- error map: ⟨-0.172 +0.002 -2.533 +3.511]
- CWE: ~2 = 1200.0000 ¢, ~10/7 = 611.3614 ¢
- error map: ⟨0.000 +0.298 -2.181 +3.946]
Badness (Sintel): 3.64
11-limit
Subgroup: 2.3.5.7.11
Comma list: 121/120, 225/224, 22000/21609
Mapping: [⟨1 -3 12 13 6], ⟨0 9 -19 -20 -5]]
Optimal tunings:
- WE: ~2 = 1200.3591 ¢, ~10/7 = 611.5569 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/7 = 611.3690 ¢
Badness (Sintel): 2.46
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 121/120, 225/224, 275/273, 1040/1029
Mapping: [⟨1 -3 12 13 6 20], ⟨0 9 -19 -20 -5 -32]]
Optimal tunings:
- WE: ~2 = 1200.4078 ¢, ~10/7 = 611.5536 ¢
- CWE: ~2 = 1200.0000 ¢, ~10/7 = 611.3392 ¢
Badness (Sintel): 1.96
Naiadical
Named by Xenllium in 2026, naiadical may be described as the 21 & 29 temperament.
Subgroup: 2.3.5.7
Comma list: 225/224, 823543/800000
Mapping: [⟨1 -4 11 9], ⟨0 9 -14 -10]]
- mapping generators: ~2, ~32/21
- WE: ~2 = 1202.1198 ¢, ~32/21 = 745.4675 ¢
- error map: ⟨+2.120 -1.227 +0.459 -4.423]
- CWE: ~2 = 1200.0000 ¢, ~32/21 = 744.1318 ¢
- error map: ⟨0.000 -4.769 -4.159 -10.144]
Optimal ET sequence: 21, 29, 50, 79d, 129cdd, 179bcddd
Badness (Sintel): 3.67
11-limit
Subgroup: 2.3.5.7.11
Comma list: 225/224, 245/242, 1617/1600
Mapping: [⟨1 -4 11 9 14], ⟨0 9 -14 -10 -17]]
Optimal tunings:
- WE: ~2 = 1201.9008 ¢, ~21/16 = 745.3867 ¢
- CWE: ~2 = 1200.0000 ¢, ~32/21 = 744.1777 ¢
Optimal ET sequence: 21, 29, 50, 79d
Badness (Sintel): 2.00
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 105/104, 196/195, 245/242, 1001/1000
Mapping: [⟨1 -4 11 9 14 13], ⟨0 9 -14 -10 -17 -15]]
Optimal tunings:
- WE: ~2 = 1201.7863 ¢, ~20/13 = 745.3344 ¢
- CWE: ~2 = 1200.0000 ¢, ~20/13 = 744.1931 ¢
Optimal ET sequence: 21, 29, 50, 79d
Badness (Sintel): 1.43
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 105/104, 170/169, 196/195, 221/220, 245/242
Mapping: [⟨1 -4 11 9 14 13 14], ⟨0 9 -14 -10 -17 -15 -16]]
Optimal tunings:
- WE: ~2 = 1201.9208 ¢, ~20/13 = 745.3976 ¢
- CWE: ~2 = 1200.0000 ¢, ~20/13 = 744.1669 ¢
Optimal ET sequence: 21, 29g, 50, 79dg
Badness (Sintel): 1.26
Quintannic
Named by Scott Dakota, quintannic may be described as the 43 & 60 temperament.
Subgroup: 2.3.5.7
Comma list: 225/224, 9805926501/9765625000
Mapping: [⟨1 1 5 7], ⟨0 5 -23 -36]]
- mapping generators: ~2, ~10000/9261
- WE: ~2 = 1200.9803 ¢, ~10000/9261 = 139.9522 ¢
- error map: ⟨+0.980 -1.214 -0.313 -0.243]
- CWE: ~2 = 1200.0000 ¢, ~10000/9261 = 139.8184 ¢
- error map: ⟨0.000 -2.863 -2.136 -2.287]
Optimal ET sequence: 43, 60, 103, 266bcd, 369bcd
Badness (Sintel): 3.81
11-limit
Subgroup: 2.3.5.7.11
Comma list: 225/224, 441/440, 43923/43750
Mapping: [⟨1 1 5 7 8], ⟨0 5 -23 -36 -39]]
Optimal tunings:
- WE: ~2 = 1201.0031 ¢, ~320/297 = 139.9435 ¢
- CWE: ~2 = 1200.0000 ¢, ~320/297 = 139.8053 ¢
Optimal ET sequence: 43, 60e, 103, 369bcdeee, 472bbcddeee
Badness (Sintel): 1.74
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 441/440, 1001/1000, 1188/1183
Mapping: [⟨1 1 5 7 8 3], ⟨0 5 -23 -36 -39 6]]
Optimal tunings:
- WE: ~2 = 1200.8354 ¢, ~13/12 = 139.9095 ¢
- CWE: ~2 = 1200.0000 ¢, ~13/12 = 139.7997 ¢
Optimal ET sequence: 43, 60e, 103
Badness (Sintel): 1.35
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 225/224, 273/272, 375/374, 441/440, 891/884
Mapping: [⟨1 1 5 7 8 3 7], ⟨0 5 -23 -36 -39 6 -25]]
Optimal tunings:
- WE: ~2 = 1200.7402 ¢, ~13/12 = 139.9015 ¢
- CWE: ~2 = 1200.0000 ¢, ~13/12 = 139.8038 ¢
Optimal ET sequence: 43, 60e, 103
Badness (Sintel): 1.17
Gwazy
- For the 5-limit version, see Very high accuracy temperaments #Kwazy.
Named by Petr Pařízek in 2011[1], gwazy may be described as the 22 & 74 temperament.
Subgroup: 2.3.5.7
Comma list: 225/224, 5971968/5764801
Mapping: [⟨2 1 6 4], ⟨0 8 -5 6]]
- mapping generators: ~2401/1728, ~35/32
- WE: ~2401/1728 = 599.7132 ¢, ~35/32 = 162.5806 ¢
- error map: ⟨-0.574 -1.597 -0.937 +5.510]
- CWE: ~2401/1728 = 600.0000 ¢, ~35/32 = 162.6388 ¢
- error map: ⟨0.000 -0.844 +0.492 +7.007]
Optimal ET sequence: 22, 74, 96, 118d
Badness (Sintel): 4.53
11-limit
Subgroup: 2.3.5.7.11
Comma list: 99/98, 176/175, 65536/65219
Mapping: [⟨2 1 6 4 8], ⟨0 8 -5 6 -4]]
Optimal tunings:
- WE: ~363/256 = 599.8517 ¢, ~11/10 = 162.5518 ¢
- CWE: ~363/256 = 600.0000 ¢, ~11/10 = 162.5863 ¢
Optimal ET sequence: 22, 74, 96
Badness (Sintel): 2.26
Tertiosec
- For the 5-limit version, see Miscellaneous 5-limit temperaments #Tertiosec.
Tertiosec may be described as the 21 & 75 temperament. It was initially named tertiomar by Petr Pařízek in 2011[1], but was changed to tertiosec in 2012[2].
Subgroup: 2.3.5.7
Comma list: 225/224, 14495514624/13841287201
Mapping: [⟨3 -1 12 7], ⟨0 8 -7 2]]
- mapping generators: ~3072/2401, ~2048/1715
- WE: ~3072/2401 = 399.8257 ¢, ~2048/1715 = 287.5920 ¢
- error map: ⟨-0.523 -1.044 -1.549 +5.138]
- CWE: ~3072/2401 = 400.0000 ¢, ~2048/1715 = 287.7088 ¢
- error map: ⟨0.000 -0.284 -0.276 +6.592]
Optimal ET sequence: 21, 54, 75, 96, 171d
Badness (Sintel): 10.9
11-limit
Subgroup: 2.3.5.7.11
Comma list: 225/224, 3840/3773, 12005/11979
Mapping: [⟨3 -1 12 7 14], ⟨0 8 -7 2 -5]]
Optimal tunings:
- WE: ~44/35 = 399.6550 ¢, ~33/28 = 287.5803 ¢
- CWE: ~44/35 = 400.0000 ¢, ~33/28 = 287.8224 ¢
Optimal ET sequence: 21, 54, 75e
Badness (Sintel): 5.74