FloraC: /* Twothirdtonic */ restore the octave-complement generator tuning

2026-08-08T13:44:31Z

Twothirdtonic: restore the octave-complement generator tuning

← Older revision		Revision as of 13:44, 8 August 2026
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	== Twothirdtonic ==		== Twothirdtonic ==
	Twothirdtonic tempers out 6144/6125, the [[porwell comma]], in addition to the senga, and may be described as the {{nowrap\| 37 & 46 }} temperament, generated by one third of a [[5/4\|classical major third]] that represents [[15/14]], [[14/13]], and [[13/12]] in the [[13-limit]] interpretation. Note that in the data below, the generator is taken to be its [[octave complement]], thirteen of which [[octave reduction\|octave reduced]] make the [[3/2\|perfect fifth]]; it follows that the [[ploidacot]] for this temperament is 11-sheared 13-cot. [[46edo]] may be recommended as a tuning.		Twothirdtonic tempers out 6144/6125, the [[porwell comma]], in addition to the senga, and may be described as the {{nowrap\| 37 & 46 }} temperament, generated by one third of a [[5/4\|classical major third]] that represents [[15/14]], [[14/13]], and [[13/12]] in the [[13-limit]] interpretation. Note that in the data below, the generator is taken to be its [[octave complement]], thirteen of which [[octave reduction\|octave reduced]] make the [[3/2\|perfect fifth]]; it follows that the [[ploidacot]] for this temperament is 11-sheared 13-cot (or triseph). [[46edo]] may be recommended as a tuning.

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7
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	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[WE]]: ~2 = 1199.3074{{c}}, ~28/15 = 1068.9820{{c}}		* [[WE]]: ~2 = 1199.3074{{c}}, ~28/15 = 1068.9820{{c}} (~15/14 = 130.3255{{c}})
	: [[error map]]: {{val\| -0.693 +1.736 +3.278 -5.176 }}		: [[error map]]: {{val\| -0.693 +1.736 +3.278 -5.176 }}
	* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/15 = 1069.5746{{c}}		* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/15 = 1069.5746{{c}} (~15/14 = 130.4254{{c}})
	: error map: {{val\| 0.000 +2.515 +4.962 -3.505 }}		: error map: {{val\| 0.000 +2.515 +4.962 -3.505 }}

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	Optimal tunings:		Optimal tunings:
	* WE: ~2 = 1199.7068{{c}}, ~28/15 = 1069.3084{{c}}		* WE: ~2 = 1199.7068{{c}}, ~28/15 = 1069.3084{{c}} (~15/14 = 130.3983{{c}})
	* CWE: ~2 = 1200.0000{{c}}, ~28/15 = 1069.5600{{c}}		* CWE: ~2 = 1200.0000{{c}}, ~28/15 = 1069.5600{{c}} (~15/14 = 130.4400{{c}})

	{{Optimal ET sequence\|legend=0\| 9, 28b, 37, 46 }}		{{Optimal ET sequence\|legend=0\| 9, 28b, 37, 46 }}
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	Optimal tunings:		Optimal tunings:
	* WE: ~2 = 1199.9531{{c}}, ~13/7 = 1069.5492{{c}}		* WE: ~2 = 1199.9531{{c}}, ~13/7 = 1069.5492{{c}} (~14/13 = 130.4039{{c}})
	* CWE: ~2 = 1200.0000{{c}}, ~13/7 = 1069.5893{{c}}		* CWE: ~2 = 1200.0000{{c}}, ~13/7 = 1069.5893{{c}} (~14/13 = 130.4107{{c}})

	{{Optimal ET sequence\|legend=0\| 9, 28b, 37, 46 }}		{{Optimal ET sequence\|legend=0\| 9, 28b, 37, 46 }}

FloraC: Birth with leapday and twothirdtonic moved here

2026-08-08T09:58:03Z

Birth with leapday and twothirdtonic moved here

New page

{{Technical data page}}
This is a collection of [[rank-2 temperament|rank-2]] '''sengic temperaments''', which [[tempering out|temper out]] the [[senga]] ({{monzo|legend=1| 1 -3 -2 3 }}, [[ratio]]: 686/675), with [[S-expression]] S13⋅S14<sup>2</sup>.

Temperaments discussed elsewhere are:
* ''[[Fluorine]]'' (+21/20) → [[Septisemi temperaments #Fluorine|Septisemi temperaments]]
* ''[[Octokaidecal]]'' (+28/27) → [[Trienstonic clan #Octokaidecal|Trienstonic clan]]
* ''[[Beatles]]'' (+64/63) → [[Archytas clan #Beatles|Archytas clan]]
* ''[[Liese]]'' (+81/80) → [[Meantone family #Liese|Meantone family]]
* ''[[Squirrel]]'' (+32805/32768) → [[Schismatic family #Squirrel|Schismatic family]]
* [[Negri]] (+49/48) → [[Semaphoresmic clan #Beatles|Semaphoresmic clan]]
* ''[[Progression]]'' (+36/35) → [[Mint temperaments #Progression|Mint temperaments]]
* ''[[Echidnic]]'' (+1029/1024) → [[Diaschismic family #Echidnic|Diaschismic family]]
* [[Sensi]] (+126/125) → [[Sensipent family #Sensi|Sensipent family]]
* ''[[Niner]]'' (+128/125) → [[Augmented family #Niner|Augmented family]]
* ''[[Ammonite]]'' (+250/243) → [[Porcupine family #Ammonite|Porcupine family]]
* ''[[Sycamore]]'' (+875/864) → [[Sycamore family #Septimal sycamore|Sycamore family]]

Considered below are leapday and twothirdtonic, in the order of increasing [[badness]].

== Leapday ==
{{Main| Leapday }}
: ''For the 5-limit version, see [[Miscellaneous 5-limit temperaments #Leapday]].''

Leapday tempers out 5120/5103, the [[aberschisma]], in addition to the senga, and may be described as the {{nowrap| 29 & 46 }} temperament. It extends [[leapfrog]], such that [[7/4]] is found by 15 generators up, as a double-augmented fifth (a major sixth and a diesis). 5/4 is found by a tritone above that, as a triple-augmented unison (a minor third and two dieses). [[46edo]] itself is an excellent tuning for this.

Leapday is more notable in the higher limits than the lower, as it nails the 13-limit pretty well from identifying [[14/11]] by a major third and [[13/11]] by a minor third, tempering out not only [[352/351]] and [[364/363]] but [[91/90]], [[121/120]], [[169/168]] and [[196/195]]. It can be further extended to include the [[17/1|17th]] and [[23/1|23rd]] [[harmonic]]s. Adding 17 would fix the valid diamond monotone tuning to 46edo, however.

Leapday has an alternative extension called [[porwell temperaments #Polypyth|polypyth]], which tempers out the same 5-limit comma as leapday, but with the porwell comma ([[6144/6125]]) rather than the aberschisma tempered out.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 686/675, 5120/5103

{{Mapping|legend=1| 1 0 -31 -21 | 0 1 21 15 }}
: mapping generators: ~2, ~3

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.7167{{c}}, ~3/2 = 704.0971{{c}}
: [[error map]]: {{val| -0.283 +1.859 +2.559 -5.669 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 704.2504{{c}}
: error map: {{val| 0.000 +2.295 +2.945 -5.070 }}

{{Optimal ET sequence|legend=1| 17c, 29, 46 }}

[[Badness]] (Sintel): 2.43

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 121/120, 441/440, 686/675

Mapping: {{mapping| 1 0 -31 -21 -14 | 0 1 21 15 11 }}

Optimal tunings:
* WE: ~2 = 1200.0731{{c}}, ~3/2 = 704.2933{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2538{{c}}

{{Optimal ET sequence|legend=0| 17c, 29, 46 }}

Badness (Sintel): 1.28

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 121/120, 169/168, 352/351

Mapping: {{mapping| 1 0 -31 -21 -14 -9 | 0 1 21 15 11 8 }}

Optimal tunings:
* WE: ~2 = 1200.4758{{c}}, ~3/2 = 704.4930{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2346{{c}}

{{Optimal ET sequence|legend=0| 17c, 29, 46, 121def }}

Badness (Sintel): 1.02

=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17

Comma list: 91/90, 121/120, 136/135, 154/153, 169/168

Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 | 0 1 21 15 11 8 24 }}

Optimal tunings:
* WE: ~2 = 1200.4818{{c}}, ~3/2 = 704.5121{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2507{{c}}

{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 121defg }}

Badness (Sintel): 0.910

=== 2.3.5.7.11.13.17.23 subgroup ===
Subgroup: 2.3.5.7.11.13.17.23

Comma list: 91/90, 121/120, 136/135, 154/153, 161/160, 169/168

Mapping: {{mapping| 1 0 -31 -21 -14 -9 -34 -5 | 0 1 21 15 11 8 24 6 }}

Optimal tunings:
* WE: ~2 = 1200.5169{{c}}, ~3/2 = 704.5279{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 704.2450{{c}}

{{Optimal ET sequence|legend=0| 17cg, 29g, 46, 121defg }}

Badness (Sintel): 0.872

== Twothirdtonic ==
Twothirdtonic tempers out 6144/6125, the [[porwell comma]], in addition to the senga, and may be described as the {{nowrap| 37 & 46 }} temperament, generated by one third of a [[5/4|classical major third]] that represents [[15/14]], [[14/13]], and [[13/12]] in the [[13-limit]] interpretation. Note that in the data below, the generator is taken to be its [[octave complement]], thirteen of which [[octave reduction|octave reduced]] make the [[3/2|perfect fifth]]; it follows that the [[ploidacot]] for this temperament is 11-sheared 13-cot. [[46edo]] may be recommended as a tuning.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 686/675, 6144/6125

{{Mapping|legend=1| 1 -10 5 -7 | 0 13 -3 11 }}
: mapping generators: ~2, ~28/15

[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.3074{{c}}, ~28/15 = 1068.9820{{c}}
: [[error map]]: {{val| -0.693 +1.736 +3.278 -5.176 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~28/15 = 1069.5746{{c}}
: error map: {{val| 0.000 +2.515 +4.962 -3.505 }}

{{Optimal ET sequence|legend=1| 9, 28b, 37, 46 }}

[[Badness]] (Sintel): 2.52

=== 11-limit ===
Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175, 686/675

Mapping: {{mapping| 1 -10 5 -7 -1 | 0 13 -3 11 5 }}

Optimal tunings:
* WE: ~2 = 1199.7068{{c}}, ~28/15 = 1069.3084{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~28/15 = 1069.5600{{c}}

{{Optimal ET sequence|legend=0| 9, 28b, 37, 46 }}

Badness (Sintel): 1.35

=== 13-limit ===
Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 121/120, 169/168, 176/175

Mapping: {{mapping| 1 -10 5 -7 -1 -7 | 0 13 -3 11 5 12 }}

Optimal tunings:
* WE: ~2 = 1199.9531{{c}}, ~13/7 = 1069.5492{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~13/7 = 1069.5893{{c}}

{{Optimal ET sequence|legend=0| 9, 28b, 37, 46 }}

Badness (Sintel): 1.07

[[Category:Sengic temperaments| ]]

Sengic temperaments - Revision history

FloraC: /* Twothirdtonic */ restore the octave-complement generator tuning

FloraC: Birth with leapday and twothirdtonic moved here