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{{Main| Garibaldi }}
{{Main| Garibaldi }}


Garibaldi tempers out the [[garischisma]], equating the [[64/63|septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double diminished octave (C-Cbb), or down-minor seventh (C-vBb) with the down-arrow representing the comma step. It necessitates a sharper fifth than pure. Its [[S-expression]]-based comma list is {[[5120/5103|S8/S9]], [[225/224|S15]]}.  
Garibaldi tempers out the [[garischisma]], equating the [[64/63|septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double-diminished octave (C–C𝄫), or down-minor seventh (C-vB♭) with the down-arrow representing the comma step. It necessitates a sharper fifth than pure. Its [[S-expression]]-based comma list is {[[5120/5103|S8/S9]], [[225/224|S15]]}.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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=== Cassandra ===
=== Cassandra ===
Cassandra is one of the best extensions of garibaldi to the 11- and 13-limit as well as the 2.3.5.7.11.13.19 subgroup.  
Cassandra is one of the best extensions of garibaldi to the 11- and 13-limit as well as the 2.3.5.7.11.13.19 subgroup, though it comes with a higher complexity.  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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{{Main| Pontiac }}
{{Main| Pontiac }}


Pontiac tempers out the [[ragisma]], rendering a very accurate 7-limit microtemperament. The 7/4 is found at +39 fifths, represented by the quintuple augmented third (C-Exx#), or triple-up major sixth (C-^<sup>3</sup>A).  
Pontiac tempers out the [[ragisma]], rendering a very accurate 7-limit microtemperament. The 7/4 is found at +39 fifths, represented by the quintuple-augmented third (C-E𝄪𝄪♯), or triple-up major sixth (C-^<sup>3</sup>A).  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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=== Helenoid ===
=== Helenoid ===
The helenoid temperament ({{nowrap| 53 & 118 }}) is closely related to the helenus temperament, but with the ragisma rather than the [[225/224|marvel comma]] tempered out.
Helenoid may be described as {{nowrap| 53 & 118 }}, and is closely related to the helenus temperament, differing only by the mapping of 7.  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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=== Ponta ===
=== Ponta ===
The ponta temperament ({{nowrap| 53 & 171 }}) tempers out the [[540/539|swetisma]] and the ragisma.
Ponta tempers out [[540/539]] and may be described as {{nowrap| 171 & 224 }}. [[224edo]] itself makes for an excellent tuning.  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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=== Pontic ===
=== Pontic ===
The pontic temperament ({{nowrap| 118 & 171 }}) tempers out the [[441/440|werckisma]] and the ragisma.
Pontic temperament tempers out [[441/440]] and may be described as {{nowrap| 118 & 171 }}. [[289edo]] may be recommended as a tuning.  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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=== Bipont ===
=== Bipont ===
The bipont temperament ({{nowrap| 118 & 224 }}) has a period of half octave and tempers out the [[3025/3024|lehmerisma (3025/3024)]] and the [[9801/9800|kalisma (9801/9800)]].
Bipont tempers out the [[3025/3024|lehmerisma (3025/3024)]] and the [[9801/9800|kalisma (9801/9800)]]. It may be described as {{nowrap| 118 & 224 }}. It has a period of half octave and a ploidacot signature of diploid monocot. [[342edo]] may be recommended as a tuning.  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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== Grackle ==
== Grackle ==
Grackle tempers out {{monzo| -44 26 0 1 }}. The 7/4 is found at -26 fifths, represented by the triple diminished ninth (C-Dbbbb), or double-down minor seventh (C-vvBb), which is to say, two comma steps are required to bend the Pythagorean minor seventh to the septimal one.  
Grackle tempers out {{monzo| -44 26 0 1 }} so 7/4 is found at -26 fifths, represented by the triple-diminished ninth (C–D𝄫𝄫) or double-down minor seventh (C–vvB♭). Two comma steps are required to bend the Pythagorean minor seventh to the septimal one.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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See [[Archytas clan #Schism]].  
See [[Archytas clan #Schism]].  


Schism is a relatively low-accuracy extension as it tempers out the septimal comma. The 7/4 is found at -2 fifths, represented by the minor seventh (C-Bb). 12edo is recommendable tuning, though 29edo (29d val), 41edo (41d val), and 53edo (53d val) can be used.
Schism is a relatively low-accuracy extension as it tempers out the septimal comma. The 7/4 is found at -2 fifths, represented by the minor seventh (C–B♭). 12edo is recommendable tuning, though 29edo (29d val), 41edo (41d val), and 53edo (53d val) can be used.


== Bischismic ==
== Bischismic ==
Bischismic tempers out 3136/3125, the [[hemimean comma]], as well as 321489/320000, the [[varunisma]], and may be described as the {{nowrap| 118 & 130 }} temperament. The octave is split in halves, so the [[ploidacot]] of this temperament is diploid monocot. In schismic, -10 fifths make the interval class of 10/9. Bischismic then finds [[7/4]] by a stack of two [[10/9]]'s plus a semi-octave period, and in the [[11-limit]], it simply finds [[11/8]] by a stack of three [[10/9]]'s. [[248edo]] and [[378edo]] make for excellent tunings in both cases.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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== Kleischismic ==
== Kleischismic ==
Kleischismic tempers out 1500625/1492992, the [[uniwiz comma]], and may be described as the {{nowrap| 94 & 118 }} temperament. The generator is a infrafifth, two of which plus a semi-octave period make the [[3/1|3rd]] [[harmonic]]; its [[ploidacot]] is thus diploid alpha-dicot. In schismic, 10 fifths make the interval class of [[9/5]]. Kleischismic then finds [[7/4]] by that minus a [[36/35]] quartertone, which is the aforementioned generator minus a semi-octave period. The generator stands in for [[16/11]] and the quartertone stands in for [[33/32]] in the [[11-limit]]. [[212edo]] and [[330edo]] in the 330e val may be recommended as tunings.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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== Salsa ==
== Salsa ==
Salsa tempers out 245/243, the [[sensamagic comma]], and may be described as the {{nowrap| 41 & 65 }} temperament. It has a neutral third as a generator; its [[ploidacot]] is dicot. In fact it is related to [[hemififths]], from which this less accurate temperament only differs by the mapping of [[5/1|5]].
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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== Hemischis ==
== Hemischis ==
Hemischis tempers out 6144/6125, the [[porwell comma]], as well as 19683/19600, the [[cataharry comma]], and may be described as the {{nowrap| 53 & 130 }} temperament. Its [[ploidacot]] is alpha-dicot.
The [[S-expression]]-based comma list for 13-limit hemischis is {[[540/539|S12/S14]], [[676/675|S13/S15 = S26]], [[729/728|S27]], [[4096/4095|S64]], ([[4225/4224|S65]])}. Tempering out [[169/168]] ({{S|13}}), [[225/224]] ({{S|15}}) or [[625/624]] ({{S|25}}) leads to [[53edo]] while tempering out [[24192/24167]] ([[S-expression|S12/S13]]), [[10985/10976]] ([[S-expression|S13/S14]]), [[43904/43875]] ([[S-expression|S14/S15]]) or [[2401/2400]] ([[S-expression|S49]]) leads to [[130edo]] and implies S12, S13, S14, and S15 are tempered together.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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=== 13-limit ===
=== 13-limit ===
Its [[S-expression]]-based comma list is {[[540/539|S12/S14]], [[676/675|S13/S15 = S26]], [[729/728|S27]], [[4096/4095|S64]](, [[4225/4224|S65]])}. Tempering out [[169/168|S13]], [[225/224|S15]] or [[625/624|S25]] leads to [[53edo]] (through [[Catakleismic]]) while tempering out [[24192/24167|S12/S13]], [[10985/10976|S13/S14]], [[43904/43875|S14/S15]] or [[2401/2400|S49]] (implying S12 = S13 = S14 = S15) leads to [[130edo]].
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


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== Term ==
== Term ==
Term tempers out the [[landscape comma]], mapping ~63/50 to the 1/3-octave period. It can be described as {{nowrap| 12 & 171 }}, and is the unique temperament that equates a syntonic~Pythagorean comma with a stack of three [[marvel comma]]s. A [[septimal comma]] is then found as a stack of four marvel commas. In some 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma]], with three kleismas making a comma, so this temperament may be useful for modeling that. [[171edo]] makes for an excellent tuning.  
Term tempers out the [[landscape comma]], mapping [[63/50]] to the 1/3-octave period. It can be described as {{nowrap| 12 & 171 }}, and is the unique temperament that equates a syntonic~Pythagorean comma with a stack of three [[marvel comma]]s. A [[septimal comma]] is then found as a stack of four marvel commas. In some 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma]], with three kleismas making a comma, so this temperament may be useful for modeling that. [[171edo]] makes for an excellent tuning.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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=== Terminal ===
=== Terminal ===
The terminal temperament ({{nowrap| 12 & 159 }}) tempers out 441/440 and 4375/4356. In this temperament, 44/35 and 63/50 are represented as one period of 1/3 octave.  
Terminal tempers out 441/440 and 4375/4356, and may be described as {{nowrap| 159 & 171 }}. In this temperament, 44/35 and 63/50 are represented as one period of 1/3 octave.  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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=== Terminator ===
=== Terminator ===
Terminator tempers out 540/539, and may be described as {{nowrap| 171 & 183 }}.
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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=== Semiterm ===
=== Semiterm ===
The semiterm temperament ({{nowrap| 12 & 342 }}) has a period of 1/6 octave and tempers out [[9801/9800]] (kalisma) and 151263/151250 (odiheim comma).
The semiterm temperament tempers out [[9801/9800]] (kalisma) as well as [[151263/151250]] (odiheim comma), and may be described as {{nowrap| 12 & 342 }}. It has a period of 1/6 octave and its ploidacot is hexaploid monocot.  


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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=== Hemiterm ===
=== Hemiterm ===
The hemiterm temperament tempers out [[3025/3024]] (lehmerisma), and may be described as {{nowrap| 159 & 183 }}. Its ploidacot is triploid beta-dicot.
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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== Altinex ==
== Altinex ==
Altinex is an alternative to [[#Hemiterm|hemiterm]] and may be described as {{nowrap| 24 & 159 }}. [[159edo]] itself makes for a recommendable tuning.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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== Squirrel ==
== Squirrel ==
The squirrel temperament ({{nowrap| 29 & 36 }}) has a ~11/10 generator, three of which give the fourth (~4/3), and thirteen of which give 7/4 with octave reduction.
Squirrel tempers out 686/675, the [[sengic comma]], and may be described as {{nowrap| 29 & 36 }}. It has a [[~]][[11/10]] generator, three of which give the fourth ([[4/3]]), and thirteen of which give [[7/4]] with octave reduction. Its [[ploidacot]] is omega-tricot.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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== Tertiaschis ==
== Tertiaschis ==
The tertiaschis temperament ({{nowrap| 94 & 159 }}) has a ~11/10 generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel]], but tempers out 1071875/1062882 for prime 7.  
Tertiaschis may be described as {{nowrap| 94 & 159 }}. It has a [[~]][[11/10]] generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel|squirrel]], but tempers out 1071875/1062882 for prime 7.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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== Countertertiaschis ==
== Countertertiaschis ==
The countertertiaschis temperament ({{nowrap| 159 & 224 }}) has a ~11/10 generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel]], but tempers out 244140625/243045684 for prime 7.  
Countertertiaschis may be described as {{nowrap| 159 & 224 }}. It has a [[~]][[11/10]] generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel|squirrel]], but tempers out 244140625/243045684 for prime 7.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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== Quadrant ==
== Quadrant ==
The ''quadrant'' temperament ({{nowrap| 12 & 224 }}) has a period of quarter octave and tempers out the [[dimcomp comma]], 390625/388962. In this temperament, 25/21 is mapped into quarter octave.
Quadrant tempers out 390625/388962, the [[dimcomp comma]], and maps [[25/21]] to the 1/4-octave period. It may be decribed as the {{nowrap| 12 & 212 }} temperament; its ploidacot is tetraploid monocot. Just as [[#Term|term]] equates the syntonic~Pythagorean comma with three [[marvel comma]]s, quadrant equates the syntonic~Pythagorean comma with four. A [[septimal comma]] is then found as a stack of five marvel commas.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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== Sesquiquartififths ==
== Sesquiquartififths ==
Sesquiquartififths tempers out 2401/2400, the [[breedsma]], and may be described as the {{nowrap| 41 & 171 }} temperament. It splits the fifth into four; its [[ploidacot]] is thus tetracot.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


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=== Sesquart ===
=== Sesquart ===
Sesquart is the main [[11-limit|11-]] and [[13-limit]] extension of sesquiquartififths of practical interest, as it identifies the neutral third with [[11/9]], which is realized in [[41edo]], [[89edo]], [[130edo]], and [[171edo]] also makes for a possible tuning.
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


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Badness (Sintel): 0.925
Badness (Sintel): 0.925


===== Sesquartia =====
===== Heartia =====
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17


Comma list: 243/242, 364/363, 441/440, 595/594, 3584/3575
Comma list: 243/242, 256/255, 273/272, 364/363, 441/440


Mapping: {{mapping| 1 1 7 5 2 -2 -6 | 0 4 -32 -15 10 39 69 }}
Mapping: {{mapping| 1 1 7 5 2 -2 0 | 0 4 -32 -15 10 39 28 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.8902{{c}}, ~72/65 = 175.4077{{c}}
* WE: ~2 = 1199.6422{{c}}, ~72/65 = 175.3338{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.4234{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3857{{c}}


{{Optimal ET sequence|legend=0| 41, 130, 171 }}
{{Optimal ET sequence|legend=0| 41, 89, 130g }}


Badness (Sintel): 1.18
Badness (Sintel): 1.45


====== 19-limit ======
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 243/242, 361/360, 364/363, 441/440, 456/455, 595/594
Comma list: 171/170, 243/242, 256/255, 273/272, 324/323, 441/440


Mapping: {{mapping| 1 1 7 5 2 -2 -6 6 | 0 4 -32 -15 10 39 69 -12 }}
Mapping: {{mapping| 1 1 7 5 2 -2 0 6 | 0 4 -32 -15 10 39 28 -12 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9864{{c}}, ~21/19 = 175.4169{{c}}
* WE: ~2 = 1199.7499{{c}}, ~21/19 = 175.3432{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.4189{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.3797{{c}}


{{Optimal ET sequence|legend=0| 41, 130, 171 }}
{{Optimal ET sequence|legend=0| 41, 89, 130g }}


Badness (Sintel): 1.24
Badness (Sintel): 1.40


====== 23-limit ======
===== Sesquartia =====
Subgroup: 2.3.5.7.11.13.17.19.23
Subgroup: 2.3.5.7.11.13.17


Comma list: 243/242, 323/322, 361/360, 364/363, 441/440, 456/455, 595/594
Comma list: 243/242, 364/363, 441/440, 595/594, 3584/3575


Mapping: {{mapping| 1 1 7 5 2 -2 -6 6 -6 | 0 4 -32 -15 10 39 69 -12 72 }}
Mapping: {{mapping| 1 1 7 5 2 -2 -6 | 0 4 -32 -15 10 39 69 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.9606{{c}}, ~21/19 = 175.4067{{c}}
* WE: ~2 = 1199.8902{{c}}, ~72/65 = 175.4077{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.4123{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.4234{{c}}


{{Optimal ET sequence|legend=0| 41i, 130, 171 }}
{{Optimal ET sequence|legend=0| 41, 130, 171 }}


Badness (Sintel): 1.36
Badness (Sintel): 1.18


===== Heartia =====
====== 19-limit ======
Subgroup: 2.3.5.7.11.13.17
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 243/242, 256/255, 273/272, 364/363, 441/440
Comma list: 243/242, 361/360, 364/363, 441/440, 456/455, 595/594


Mapping: {{mapping| 1 1 7 5 2 -2 0 | 0 4 -32 -15 10 39 28 }}
Mapping: {{mapping| 1 1 7 5 2 -2 -6 6 | 0 4 -32 -15 10 39 69 -12 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.6422{{c}}, ~72/65 = 175.3338{{c}}
* WE: ~2 = 1199.9864{{c}}, ~21/19 = 175.4169{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3857{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.4189{{c}}


{{Optimal ET sequence|legend=0| 41, 89, 130g }}
{{Optimal ET sequence|legend=0| 41, 130, 171 }}


Badness (Sintel): 1.45
Badness (Sintel): 1.24


====== 19-limit ======
====== 23-limit ======
Subgroup: 2.3.5.7.11.13.17.19
Subgroup: 2.3.5.7.11.13.17.19.23


Comma list: 171/170, 243/242, 256/255, 273/272, 324/323, 441/440
Comma list: 243/242, 323/322, 361/360, 364/363, 441/440, 456/455, 595/594


Mapping: {{mapping| 1 1 7 5 2 -2 0 6 | 0 4 -32 -15 10 39 28 -12 }}
Mapping: {{mapping| 1 1 7 5 2 -2 -6 6 -6 | 0 4 -32 -15 10 39 69 -12 72 }}


Optimal tunings:  
Optimal tunings:  
* WE: ~2 = 1199.7499{{c}}, ~21/19 = 175.3432{{c}}
* WE: ~2 = 1199.9606{{c}}, ~21/19 = 175.4067{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.3797{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.4123{{c}}


{{Optimal ET sequence|legend=0| 41, 89, 130g }}
{{Optimal ET sequence|legend=0| 41i, 130, 171 }}


Badness (Sintel): 1.40
Badness (Sintel): 1.36


===== Hearty =====
===== Hearty =====
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