Sensipent family - Revision history

FloraC: Move browser here

2026-10-11T09:06:52Z

Move browser here

@@ Line 29: / Line 29: @@
 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.
-Bison adds [[6144/6125]] with a semioctave period. Subpental adds [[3136/3125]] or [[19683/19600]] with a generator of ~56/45; two generator steps make the original. Trisensory adds [[1728/1715]] with a 1/3-octave period. Heinz adds [[1029/1024]] with a generator of ~48/35; three make the original. Catafourth adds [[2401/2400]] with a generator of ~189/125; four make the original. Finally, browser adds [[16875/16807]] with a generator of ~90/49; five make the original.
+Bison adds [[6144/6125]] with a semioctave period. Subpental adds [[3136/3125]] or [[19683/19600]] with a generator of ~56/45; two generator steps make the original. Trisensory adds [[1728/1715]] with a 1/3-octave period. Heinz adds [[1029/1024]] with a generator of ~48/35; three make the original. Catafourth adds [[2401/2400]] with a generator of ~189/125; four make the original. Finally, browser adds [[16875/16807]] with a generator of ~90/49; five make the original. All are considered below.
 ==== Subgroup extensions ====
@@ Line 719: / Line 714: @@
 Badness (Sintel): 0.896
 == Subgroup extensions ==

FloraC: Move catafourth here

2026-10-11T08:59:23Z

Move catafourth here

@@ Line 29: / Line 29: @@
 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.
-Bison adds [[6144/6125]] with a semioctave period. Subpental adds [[3136/3125]] or [[19683/19600]] with a generator of ~56/45; two generator steps make the original. Trisensory adds [[1728/1715]] with a 1/3-octave period. Heinz adds [[1029/1024]] with a generator of ~48/35; three make the original. Catafourth adds [[2401/2400]] with a generator of ~250/189; four make the original. Finally, browser adds [[16875/16807]] with a generator of ~49/45; five make the original.
+Bison adds [[6144/6125]] with a semioctave period. Subpental adds [[3136/3125]] or [[19683/19600]] with a generator of ~56/45; two generator steps make the original. Trisensory adds [[1728/1715]] with a 1/3-octave period. Heinz adds [[1029/1024]] with a generator of ~48/35; three make the original. Catafourth adds [[2401/2400]] with a generator of ~189/125; four make the original. Finally, browser adds [[16875/16807]] with a generator of ~90/49; five make the original.
 Temperaments discussed elsewhere include:
 * ''[[Browser]]'' → [[Canopic clan #Browser|Canopic clan]] (+16875/16807)
-Considered below are sensi, sensei, warrior, bison, subpental, trisensory and heinz.
+The rest are considered below.
 ==== Subgroup extensions ====
@@ Line 670: / Line 669: @@
 Badness (Sintel): 1.12
 == Subgroup extensions ==

Godtone: /* Subgroup extensions */ smaller ratios are first in tempered-together tildes right? also grammar

2026-10-01T19:24:19Z

Subgroup extensions: smaller ratios are first in tempered-together tildes right? also grammar

← Older revision		Revision as of 19:24, 1 October 2026
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	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, which is important as it is the difference between 5/4 and 4/3 (whose product is 5/3 at 2 gens). This gives the 2.3.5.31-subgroup version of sensipent discussed in [[#Subgroup extensions_2\|#Subgroup extensions]]. It is essentially the only simple and accurate extension that preserves sensipent's fine-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, which is important as it is the difference between 5/4 and 4/3 (whose product is 5/3 at 2 gens). This gives the 2.3.5.31-subgroup version of sensipent discussed in [[#Subgroup extensions_2\|#Subgroup extensions]]. It is essentially the only simple and accurate extension that preserves sensipent's fine-tempered 5-limit structure.

	This extension can be applied to 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 ~~the~~ 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.		This extension can be applied to 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 40/31~31/24 such that 24:31:40 is tuned as 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]] is a less sparse subgroup extension present in edo tunings like [[111edo]] at the cost of a little accuracy. [[Sendai]] is perhaps more accurate (esp. w.r.t. prime 31) but the subgroup is very sparse, only having one prime that sensible doesn't (29). In general, one should consider combining multiple interpretations of the sensi genchain structure, but the mappings may conflict/be too high in damage, but generally, up to warts, [[65edo]] and 111edo are good general tunings, while [[46edo]] is good for lower-accuracy interpretations. All support sensible by patent val, and in that regard {{nowrap\| 65 + 111 = 176g }} is almost patent val.		[[Sensible]] is a less sparse subgroup extension present in edo tunings like [[111edo]] at the cost of a little accuracy. [[Sendai]] is perhaps more accurate (esp. w.r.t. prime 31) but the subgroup is very sparse, only having one prime that sensible doesn't (29). In general, one should consider combining multiple interpretations of the sensi genchain structure, but the mappings may conflict/be too high in damage, but generally, up to warts, [[65edo]] and 111edo are good general tunings, while [[46edo]] is good for lower-accuracy interpretations. All support sensible by patent val, and in that regard {{nowrap\| 65 + 111 = 176g }} is almost patent val.

Godtone: /* Subgroup extensions */ fix link

2026-10-01T16:00:03Z

Subgroup extensions: fix link

← Older revision		Revision as of 16:00, 1 October 2026
Line 42:		Line 42:
	This extension can be applied to 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 the 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.		This extension can be applied to 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 the 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]] is a less sparse subgroup extension present in edo tunings like [[111edo]] at the cost of a little accuracy. [[~~#Sendai\|~~Sendai]] is perhaps more accurate (esp. w.r.t. prime 31) but the subgroup is very sparse, only having one prime that sensible doesn't (29). In general, one should consider combining multiple interpretations of the sensi genchain structure, but the mappings may conflict/be too high in damage, but generally, up to warts, [[65edo]] and 111edo are good general tunings, while [[46edo]] is good for lower-accuracy interpretations. All support sensible by patent val, and in that regard {{nowrap\| 65 + 111 = 176g }} is almost patent val.		[[Sensible]] is a less sparse subgroup extension present in edo tunings like [[111edo]] at the cost of a little accuracy. [[Sendai]] is perhaps more accurate (esp. w.r.t. prime 31) but the subgroup is very sparse, only having one prime that sensible doesn't (29). In general, one should consider combining multiple interpretations of the sensi genchain structure, but the mappings may conflict/be too high in damage, but generally, up to warts, [[65edo]] and 111edo are good general tunings, while [[46edo]] is good for lower-accuracy interpretations. All support sensible by patent val, and in that regard {{nowrap\| 65 + 111 = 176g }} is almost patent val.

	== Sensi ==		== Sensi ==

Godtone: /* Subgroup extensions */ fix link

2026-10-01T15:58:24Z

Subgroup extensions: fix link

← Older revision		Revision as of 15:58, 1 October 2026
Line 42:		Line 42:
	This extension can be applied to 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 the 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.		This extension can be applied to 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 the 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 edo tunings like [[111edo]] at the cost of a little accuracy. [[#Sendai\|Sendai]] is perhaps more accurate (esp. w.r.t. prime 31) but the subgroup is very sparse, only having one prime that sensible doesn't (29). In general, one should consider combining multiple interpretations of the sensi genchain structure, but the mappings may conflict/be too high in damage, but generally, up to warts, [[65edo]] and 111edo are good general tunings, while [[46edo]] is good for lower-accuracy interpretations. All support sensible by patent val, and in that regard {{nowrap\| 65 + 111 = 176g }} is almost patent val.		[[Sensible]] is a less sparse subgroup extension present in edo tunings like [[111edo]] at the cost of a little accuracy. [[#Sendai\|Sendai]] is perhaps more accurate (esp. w.r.t. prime 31) but the subgroup is very sparse, only having one prime that sensible doesn't (29). In general, one should consider combining multiple interpretations of the sensi genchain structure, but the mappings may conflict/be too high in damage, but generally, up to warts, [[65edo]] and 111edo are good general tunings, while [[46edo]] is good for lower-accuracy interpretations. All support sensible by patent val, and in that regard {{nowrap\| 65 + 111 = 176g }} is almost patent val.

	== Sensi ==		== Sensi ==

Godtone: /* Sensible (2.3.5.11) */ 241 is notable and explain its placement in terms of tuning

2026-10-01T15:46:05Z

Sensible (2.3.5.11): 241 is notable and explain its placement in terms of tuning

← Older revision		Revision as of 15:46, 1 October 2026
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	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]].		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]].

	Finally, despite not appearing in the final optimal ET sequence, [[241edo]] is a good tuning for it via the val {{nowrap\| 65 + 176g {{=}} 241g. }}		Finally, despite not appearing in the final optimal ET sequence, [[241edo]] is a good tuning for it via the val {{nowrap\| 65 + 176g {{=}} 241g. }} (Specifically, in the sequence of edos 111, 176, 241, 65 we have increasingly good 5-limit, an essentially-just 11/8 in 111 that gets slightly sharper, a 23/16 that gets slightly flatter and becomes essentially-just in 65, and a 31 that is essentially-just in 241, leaving only the sharp 17 as encouraging a tuning closer to 111 than 65, and with more unusual complex subgroup harmony extensions (or simpler but less accurate ones) available in most of these tunings.)

	Subgroup: 2.3.5.11		Subgroup: 2.3.5.11

Godtone: /* Sensible (2.3.5.11) */ note 65 + 176g = 241g tuning

2026-10-01T15:32:15Z

Sensible (2.3.5.11): note 65 + 176g = 241g tuning

← Older revision		Revision as of 15:32, 1 October 2026
Line 695:		Line 695:

	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]].		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]].

			Finally, despite not appearing in the final optimal ET sequence, [[241edo]] is a good tuning for it via the val {{nowrap\| 65 + 176g {{=}} 241g. }}

	Subgroup: 2.3.5.11		Subgroup: 2.3.5.11

Godtone: /* Subgroup extensions */ readd info in hopefully more readable way

2026-10-01T15:11:04Z

Subgroup extensions: readd info in hopefully more readable way

← Older revision		Revision as of 15:11, 1 October 2026
Line 38:		Line 38:

	==== 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 [[#Subgroup extensions_2\|#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, which is important as it is the difference between 5/4 and 4/3 (whose product is 5/3 at 2 gens). This gives the 2.3.5.31-subgroup version of sensipent discussed in [[#Subgroup extensions_2\|#Subgroup extensions]]. It is essentially the only simple and accurate extension that preserves sensipent's fine-tempered 5-limit structure.

	This extension can be applied to 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.		This extension can be applied to 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 the 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. [[#Sendai\|Sendai]] is perhaps more accurate but the subgroup is very sparse, only having one prime that sensible doesn't (29). In general, one should consider combining multiple interpretations of the sensi genchain structure, but the mappings may conflict/be too high in damage, but generally, up to warts, [[65edo]] and 111edo are good general tunings, while [[46edo]] is good for lower-accuracy interpretations. All support sensible by patent val, and in that regard {{nowrap\| 65 + 111 = 176g }} is almost patent val.		[[#Sensible\|Sensible]] is a less sparse subgroup extension present in edo tunings like [[111edo]] at the cost of a little accuracy. [[#Sendai\|Sendai]] is perhaps more accurate (esp. w.r.t. prime 31) but the subgroup is very sparse, only having one prime that sensible doesn't (29). In general, one should consider combining multiple interpretations of the sensi genchain structure, but the mappings may conflict/be too high in damage, but generally, up to warts, [[65edo]] and 111edo are good general tunings, while [[46edo]] is good for lower-accuracy interpretations. All support sensible by patent val, and in that regard {{nowrap\| 65 + 111 = 176g }} is almost patent val.

	== Sensi ==		== Sensi ==

FloraC: Fix a link

2026-10-01T14:00:17Z

Fix a link

← Older revision		Revision as of 14:00, 1 October 2026
Line 38:		Line 38:

	==== 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 [[#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 [[#Subgroup extensions_2\|#Subgroup extensions]]. It is essentially the only simple and accurate extension that preserves sensipent's tempered 5-limit structure.

	This extension can be applied to 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.		This extension can be applied to 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.

FloraC: Fix heading level

2026-10-01T13:59:24Z

Fix heading level

← Older revision		Revision as of 13:59, 1 October 2026
Line 25:		Line 25:
	[[Badness]] (Sintel): 0.826		[[Badness]] (Sintel): 0.826

	==== Overview to extensions ====		=== Overview to extensions ===
	=== Full 7-limit 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.