Sensipent family - Revision history

FloraC: Move the opening note to a reasonable place. Don't shock the readers with stuff like 31/24 or 40/31 without proper context

2026-08-24T04:24:08Z

Move the opening note to a reasonable place. Don't shock the readers with stuff like 31/24 or 40/31 without proper context

← Older revision		Revision as of 04:24, 24 August 2026
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	[[Regular temperament\|Temperaments]] of the '''sensipent family''' [[tempering out\|temper out]] the [[sensipent comma]], 78732/78125, also known as medium semicomma.		[[Regular temperament\|Temperaments]] of the '''sensipent family''' [[tempering out\|temper out]] the [[sensipent comma]], 78732/78125, also known as medium semicomma.

	Note that optimal ET sequences here often omit some [[val]] of [[84edo]], which is a good tuning for interpreting the gen of sqrt(5/3) as [[40/31]][[~]][[31/24]] s.t 24:31:40 is a chord of [0, 1, 2] gens, by preferring a flatter fifth as in [[#Sensipent (2.3.5.31 subgroup)\|the 2.3.5.31 subgroup extension]], as the 5-limit generator is both complex and relatively high in damage for its complexity.

	== Sensipent ==		== Sensipent ==
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	==== Subgroup extensions ====		==== Subgroup extensions ====
	The generator of sensipent can be accurately interpreted as [[31/24]]~[[40/31]], tempering out [[961/960]] ({{S\|31}}), so that the [[31-limit]] quartertones [[32/31]] and [[31/30]] are equated, as sensipent splits [[16/15]] into two equal parts. This gives the 2.3.5.31-subgroup version of sensipent discussed in [[#Other subgroup extensions]]. It is essentially the only simple and accurate extension that preserves sensipent's tempered 5-limit structure.		The generator of sensipent can be accurately interpreted as [[31/24]][[~]][[40/31]], tempering out [[961/960]] ({{S\|31}}), so that the [[31-limit]] quartertones [[32/31]] and [[31/30]] are equated, as sensipent splits [[16/15]] into two equal parts. This gives the 2.3.5.31-subgroup version of sensipent discussed in [[#Other subgroup extensions]]. It is essentially the only simple and accurate extension that preserves sensipent's tempered 5-limit structure.

			This extension can be laid onto many other extensions. Note that optimal ET sequences here often omit some [[val]] of [[84edo]], which is a good tuning for interpreting the generator as 31/24~40/31 such that 24:31:40 is a chord of [0, 1, 2] generator steps, by preferring a flatter fifth as in the 2.3.5.31-subgroup extension, as the 5-limit generator is both complex and relatively high in damage for its complexity.

	[[#Sensible\|Sensible]] is a less sparse subgroup extension present in smaller edo tunings like [[111edo]] at the cost of a little accuracy, considered immediately below.		[[#Sensible\|Sensible]] is a less sparse subgroup extension present in smaller edo tunings like [[111edo]] at the cost of a little accuracy, considered immediately below.

FloraC: + intro to subgroup extensions

2026-08-24T04:10:26Z

+ intro to subgroup extensions

← Older revision		Revision as of 04:10, 24 August 2026
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	=== Overview to extensions ===		=== Overview to extensions ===
			==== Full 7-limit extensions ====
	The second comma of the comma list determines which 7-limit family member we are looking at. Sensi adds [[126/125]]. Sensei adds [[225/224]]. Warrior adds [[5120/5103]]. These are all strong extensions that use the same period and generator as sensipent.		The second comma of the comma list determines which 7-limit family member we are looking at. Sensi adds [[126/125]]. Sensei adds [[225/224]]. Warrior adds [[5120/5103]]. These are all strong extensions that use the same period and generator as sensipent.

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	Temperaments discussed elsewhere include:		Temperaments discussed elsewhere include:
	* ''[[Catafourth]]'' → [[Breedsmic temperaments #Catafourth\|Breedsmic temperaments]] (+2401/2400)		* ''[[Catafourth]]'' → [[Breedsmic temperaments #Catafourth\|Breedsmic temperaments]] (+2401/2400)
	* ''[[Browser]]'' → [[~~Mirkwai~~ clan #Browser\|~~Mirkwai~~ clan]] (+16875/16807)		* ''[[Browser]]'' → [[Canopic clan #Browser\|Canopic clan]] (+16875/16807)

	Considered below are sensi, sensei, warrior, bison, subpental, trisensory and heinz.		Considered below are sensi, sensei, warrior, bison, subpental, trisensory and heinz.

			==== Subgroup extensions ====
			The generator of sensipent can be accurately interpreted as [[31/24]]~[[40/31]], tempering out [[961/960]] ({{S\|31}}), so that the [[31-limit]] quartertones [[32/31]] and [[31/30]] are equated, as sensipent splits [[16/15]] into two equal parts. This gives the 2.3.5.31-subgroup version of sensipent discussed in [[#Other subgroup extensions]]. It is essentially the only simple and accurate extension that preserves sensipent's tempered 5-limit structure.

			[[#Sensible\|Sensible]] is a less sparse subgroup extension present in smaller edo tunings like [[111edo]] at the cost of a little accuracy, considered immediately below.

	=== Sensible ===		=== Sensible ===
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	=== Bisensi ===		=== Bisensi ===
	Bisensi has a 1/2-octave period and the generator can be taken as ~9/7 or its semi-octave complement, ~11/10. Its ploidacot is diploid delta-heptacot (pergen (P8/2, ccP5/7)).		Bisensi has a 1/2-octave period and the generator can be taken as ~9/7 or its semi-octave complement, ~11/10. Its ploidacot is diploid delta-heptacot (pergen (P8/2, ccP5/7)). (As 84edo is even, one might be interested to know its [[17-limit]] val is 84def, corresponding to a strong flat tendency.)

	(As 84edo is even, one might be interested to know its [[17-limit]] val is 84def, corresponding to a strong flat tendency.)

	Subgroup: 2.3.5.7.11		Subgroup: 2.3.5.7.11
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	== Other subgroup extensions ==		== Other subgroup extensions ==
	=== Sensipent (2.3.5.31 subgroup) ===		=== Sensipent (2.3.5.31 subgroup) ===
	The generator of sensipent can be accurately interpreted as [[31/24]]~[[40/31]], tempering out [[961/960]] ({{s\|31}}), so that the [[31-limit]] quarter-tones [[32/31]] and [[31/30]] are equated, as sensipent splits [[16/15]] into two equal parts. This is essentially the only simple and accurate extension that preserves sensipent's tempered [[5-limit]] structure.

	~~For a less sparse subgroup present in smaller edo tunings like [[111edo]] at the cost of a little accuracy, see the extension to the 2.3.5.11.17.31 subgroup [[#Sensible]].~~

	Subgroup: 2.3.5.31		Subgroup: 2.3.5.31

Godtone: note significance of this observation

2026-08-23T17:27:16Z

note significance of this observation

← Older revision		Revision as of 17:27, 23 August 2026
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	[[Regular temperament\|Temperaments]] of the '''sensipent family''' [[tempering out\|temper out]] the [[sensipent comma]], 78732/78125, also known as medium semicomma.		[[Regular temperament\|Temperaments]] of the '''sensipent family''' [[tempering out\|temper out]] the [[sensipent comma]], 78732/78125, also known as medium semicomma.

	Note that optimal ET sequences here often omit some [[val]] of [[84edo]].		Note that optimal ET sequences here often omit some [[val]] of [[84edo]], which is a good tuning for interpreting the gen of sqrt(5/3) as [[40/31]][[~]][[31/24]] s.t 24:31:40 is a chord of [0, 1, 2] gens, by preferring a flatter fifth as in [[#Sensipent (2.3.5.31 subgroup)\|the 2.3.5.31 subgroup extension]], as the 5-limit generator is both complex and relatively high in damage for its complexity.

	== Sensipent ==		== Sensipent ==

Godtone: /* Bisensi */ note val of 84 used

2026-08-23T17:16:58Z

Bisensi: note val of 84 used

← Older revision		Revision as of 17:16, 23 August 2026
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	=== Bisensi ===		=== Bisensi ===
	Bisensi has a 1/2-octave period and the generator can be taken as ~9/7 or its semi-octave complement, ~11/10. Its ploidacot is diploid delta-heptacot (pergen (P8/2, ccP5/7)).		Bisensi has a 1/2-octave period and the generator can be taken as ~9/7 or its semi-octave complement, ~11/10. Its ploidacot is diploid delta-heptacot (pergen (P8/2, ccP5/7)).

			(As 84edo is even, one might be interested to know its [[17-limit]] val is 84def, corresponding to a strong flat tendency.)

	Subgroup: 2.3.5.7.11		Subgroup: 2.3.5.7.11

Godtone at 17:12, 23 August 2026

2026-08-23T17:12:44Z

← Older revision		Revision as of 17:12, 23 August 2026
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	{{See also\| Sensipent #Sendai interval table }}		{{See also\| Sensipent #Sendai interval table }}

	Sendai is an accurate extension of sensipent with primes [[23/16\|23]] and [[29/16\|29]] found by [[User:VIxen\|VIxen]]. It is named after the body of acquis designed to prevent disaster risk and improve civil protection through international cooperation and after the city in Japan of the same name where it was signed (and where an international music competition is held).		Sendai is an accurate extension of sensipent with primes [[23/16\|23]] and [[29/16\|29]] found by [[User:VIxen\|VIxen]]. It is named after the body of acquis designed to prevent disaster risk and improve civil protection through international cooperation and after the city in Japan of the same name where it was signed (and where an international music competition is held). Note that [[84edo]] is an alternative possible [[patent val]] tuning not appearing in the optimal ET sequence.

	Subgroup: 2.3.5.23.29.31		Subgroup: 2.3.5.23.29.31

Godtone at 17:10, 23 August 2026

2026-08-23T17:10:49Z

← Older revision		Revision as of 17:10, 23 August 2026
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	The aforementioned extension with prime 17 through tempering out 1089/1088 implies tempering out [[256/255]] ({{s\|16}}), as {{nowrap\| 256/255 {{=}} (22/17)/(165/128) }}.		The aforementioned extension with prime 17 through tempering out 1089/1088 implies tempering out [[256/255]] ({{s\|16}}), as {{nowrap\| 256/255 {{=}} (22/17)/(165/128) }}.

	Sensible uses the accurate mapping of prime 31 in sensipent, so that the sensible generator serves many roles in subgroup harmony, but it is not ~[[9/7]] or ~[[13/10]] which would incur more damage. Its [[S-expression]]-based comma list {{nowrap\| is {([[8019/8000\|S9/S10]], [[256/255\|S16]],) [[529/528\|S23]], [[576/575\|S24]], [[961/960\|S31]], [[1024/1023\|S32]], [[1089/1088\|S33]]} }} implying also tempering out [[496/495]] (S31⋅S32) and [[528/527]] (S32⋅S33) as well as [[16337/16335]] (S31/S33) = ([[17/15]])/([[33/31]])<sup>2</sup>. A notable [[patent val]] tuning not appearing in the optimal ET sequence is [[157edo]].		Sensible uses the accurate mapping of prime 31 in sensipent, so that the sensible generator serves many roles in subgroup harmony, but it is not ~[[9/7]] or ~[[13/10]] which would incur more damage. Its [[S-expression]]-based comma list {{nowrap\| is {([[8019/8000\|S9/S10]], [[256/255\|S16]],) [[529/528\|S23]], [[576/575\|S24]], [[961/960\|S31]], [[1024/1023\|S32]], [[1089/1088\|S33]]} }} implying also tempering out [[496/495]] (S31⋅S32) and [[528/527]] (S32⋅S33) as well as [[16337/16335]] (S31/S33) = ([[17/15]])/([[33/31]])<sup>2</sup>. A notable [[patent val]] tuning not appearing in the optimal ET sequence is [[157edo]]. A notable non-patent val is [[84edo\|84g]].

	Subgroup: 2.3.5.11		Subgroup: 2.3.5.11

Godtone: note missingness of 84edo

2026-08-23T17:05:14Z

note missingness of 84edo

← Older revision		Revision as of 17:05, 23 August 2026
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	[[Regular temperament\|Temperaments]] of the '''sensipent family''' [[tempering out\|temper out]] the [[sensipent comma]], 78732/78125, also known as medium semicomma.		[[Regular temperament\|Temperaments]] of the '''sensipent family''' [[tempering out\|temper out]] the [[sensipent comma]], 78732/78125, also known as medium semicomma.

			Note that optimal ET sequences here often omit some [[val]] of [[84edo]].

	== Sensipent ==		== Sensipent ==

FloraC: /* Bison */ update

2026-08-02T08:34:41Z

Bison: update

← Older revision		Revision as of 08:34, 2 August 2026
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	== Bison ==		== Bison ==
	Bison has a 1/2-octave period and the generator can be taken as ~162/125 or its semi-octave complement, ~35/32. Its ploidacot is diploid delta-heptacot (pergen (P8/2, ccP5/7))~~. Related page: [[Bison/Eliora's Approach]]~~.		Bison has a 1/2-octave period and the generator can be taken as [[~]][[162/125]] or its semi-octave complement, ~[[35/32]]. Its [[ploidacot]] is diploid delta-heptacot ([[pergen]] (P8/2, ccP5/7)).

	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7

Hkm: Undo revision 232169 by Hkm (talk)

2026-06-12T05:12:38Z

Undo revision 232169 by Hkm (talk)

← Older revision		Revision as of 05:12, 12 June 2026
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	Sensi tempers out [[245/243]], [[686/675]] and [[4375/4374]] in addition to [[126/125]], and can be described as the {{nowrap\| 19 & 27 }} temperament. It has as a generator half the size of a slightly wide major sixth, which gives an interval sharp of 9/7 and flat of 13/10, both of which can be used to identify it, as 2.3.5.7.13 sensi (sensation) tempers out 91/90. 22/17, in the middle, is even closer to the generator. [[46edo]] is an excellent sensi tuning, and [[mos scale]]s of size 8, 11, 19 and 27 are available.		Sensi tempers out [[245/243]], [[686/675]] and [[4375/4374]] in addition to [[126/125]], and can be described as the {{nowrap\| 19 & 27 }} temperament. It has as a generator half the size of a slightly wide major sixth, which gives an interval sharp of 9/7 and flat of 13/10, both of which can be used to identify it, as 2.3.5.7.13 sensi (sensation) tempers out 91/90. 22/17, in the middle, is even closer to the generator. [[46edo]] is an excellent sensi tuning, and [[mos scale]]s of size 8, 11, 19 and 27 are available.

	=== ~~Sensi~~ ===		=== Septimal sensi ===
	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7

Hkm: Undo revision 232170 by Hkm (talk)

2026-06-12T05:11:52Z

Undo revision 232170 by Hkm (talk)

← Older revision		Revision as of 05:11, 12 June 2026
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	Sensi tempers out [[245/243]], [[686/675]] and [[4375/4374]] in addition to [[126/125]], and can be described as the {{nowrap\| 19 & 27 }} temperament. It has as a generator half the size of a slightly wide major sixth, which gives an interval sharp of 9/7 and flat of 13/10, both of which can be used to identify it, as 2.3.5.7.13 sensi (sensation) tempers out 91/90. 22/17, in the middle, is even closer to the generator. [[46edo]] is an excellent sensi tuning, and [[mos scale]]s of size 8, 11, 19 and 27 are available.		Sensi tempers out [[245/243]], [[686/675]] and [[4375/4374]] in addition to [[126/125]], and can be described as the {{nowrap\| 19 & 27 }} temperament. It has as a generator half the size of a slightly wide major sixth, which gives an interval sharp of 9/7 and flat of 13/10, both of which can be used to identify it, as 2.3.5.7.13 sensi (sensation) tempers out 91/90. 22/17, in the middle, is even closer to the generator. [[46edo]] is an excellent sensi tuning, and [[mos scale]]s of size 8, 11, 19 and 27 are available.

			=== Sensi ===
	[[Subgroup]]: 2.3.5.7		[[Subgroup]]: 2.3.5.7

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	[[Badness]] (Sintel): 0.648		[[Badness]] (Sintel): 0.648

	=== 2.3.5.7.13 subgroup (sensation) ===		==== 2.3.5.7.13 subgroup (sensation) ====
	Subgroup: 2.3.5.7.13		Subgroup: 2.3.5.7.13