104edo - Revision history

Overthink: /* Intervals */ proper subgroup ordering

2026-05-11T23:01:30Z

Intervals: proper subgroup ordering

← Older revision		Revision as of 23:01, 11 May 2026
Line 165:		Line 165:
	! colspan="3" \| Approximate ratios		! colspan="3" \| Approximate ratios
	\|-		\|-
	! Of 2.3.7.11.13.17.19~~.25~~<br>subgroup		! Of 2.3.25.7.11.13.17.19<br>subgroup
	! Additional ratios of 5<br>tending sharp (104c val)		! Additional ratios of 5<br>tending sharp (104c val)
	! Additional ratios of 5<br>tending flat (patent val)		! Additional ratios of 5<br>tending flat (patent val)

Overthink: /* Theory */ dual-5

2026-05-11T23:01:12Z

Theory: dual-5

← Older revision		Revision as of 23:01, 11 May 2026
Line 7:		Line 7:
	Notably, 104edo inherits [[26edo]]'s accurate representation of the [[2.7.11 subgroup\|2.7.11-subgroup]], and thus supports [[orgone]] temperament in that subgroup.		Notably, 104edo inherits [[26edo]]'s accurate representation of the [[2.7.11 subgroup\|2.7.11-subgroup]], and thus supports [[orgone]] temperament in that subgroup.

	If prime 5 is desired, 104edo has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val\| 104 165 241 }} ([[patent val]]), tempers out [[3125/3072]], and [[support]]s [[magic]] temperament. The sharp major third val, {{val\| 104 165 242 }} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.		If prime 5 is desired, 104edo has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val\| 104 165 241 }} ([[patent val]]), tempers out [[3125/3072]], and [[support]]s [[magic]] temperament. The sharp major third val, {{val\| 104 165 242 }} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament. Additionally, it is viable to treat 104edo as dual-5, or as a 2.3.25.7.11.13.17.19 subgroup temperament.

	104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], 896/891, [[385/384]] and [[540/539]]. It also provides an excellent tuning for the rank-3 temperament pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, and the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val.		104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], 896/891, [[385/384]] and [[540/539]]. It also provides an excellent tuning for the rank-3 temperament pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, and the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val.

Overthink: /* Theory */ no-fives first; note 26edo's 2.7.11 and orgone

2026-05-11T22:52:51Z

Theory: no-fives first; note 26edo's 2.7.11 and orgone

← Older revision		Revision as of 22:52, 11 May 2026
Line 3:		Line 3:

	== Theory ==		== Theory ==
	104edo ~~has two different equally viable~~ 5-limit [[~~val~~]]s, ~~and both are useful. The flat major third val~~, ~~{{val\| 104 165 241 }} (~~[[~~patent val~~]]), ~~tempers out~~ [[~~3125~~/~~3072~~]], and [[~~support~~]]s [[~~magic~~]] temperament~~. The sharp~~ major third ~~val~~, {{val~~\| 104 165 242 }} (104c val)~~, ~~tempers out~~ [[~~2048/2025~~]] ~~and supports~~ [[~~diaschismic~~]] temperament.		104edo is a strong no-fives system, with good approximations up to the no-5 19-limit. In the [[2.3.7.11.13 subgroup\|2.3.7.11.13-subgroup]], it tempers out [[352/351]], [[364/363]], [[896/891]], [[2197/2187]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364.<!-- Add commas in 2.3.7.11.13.17.19 as well --> It is an excellent tuning for the 2.3.7.11.13-subgroup [[rank]]-3 [[parapyth]] temperament tempering out 352/351, 364/363, and 896/891, which maps [[14/11]] to the diatonic major third and [[13/11]] to the diatonic minor third, in fact providing the [[optimal patent val]]. Additionally, it supports the extension to prime 17 known as [[etypyth]], which maps 17/14 to the augmented second, though [[121edo]] is a more optimal tuning of it. It also provides the optimal patent val for the 2.3.7.11.13-subgroup {{nowrap\| 17 & 87 }} temperament tempering out 352/351, 364/363 and 2197/2187, which splits 3/1 into three ~13/9's, and can be considered a rank-2 reduction of parapyth.

	104edo ~~with the flat third is especially notable as an excellent tuning for magic temperament, providing the~~ [[~~optimal patent val~~]] ~~for 11-limit magic and~~ the ~~13-limit magic extension~~ [[~~necromancy]]~~. ~~In the 5-limit it tempers out the magic comma, 3125/3072; in the~~ 7~~-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the~~ 11-~~limit, [[100/99]], [[896/891~~]], ~~[[385/384]]~~ and [[~~540/539]]. It provides an excellent tuning also for the rank-3 temperaments pairing 100/99 with 225/224 ([[apollo~~]] temperament~~), 245/243 or 875/864, or the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val~~.		Notably, 104edo inherits [[26edo]]'s accurate representation of the [[2.7.11 subgroup\|2.7.11-subgroup]], and thus supports [[orgone]] temperament in that subgroup.

	~~104 with the sharp third~~ is ~~excellent for 11-, 13-~~, ~~or 17-limit diaschismic. It tempers out 2048/2025 in the~~ 5-limit, [[~~126/125~~]] and ~~[[5120/5103]] in the 7-limit~~, [[~~176/175~~]] ~~and 896/891 in the 11-limit~~, [[~~196~~/~~195~~]], [[~~352/351~~]] ~~and~~ [[~~364/363~~]] ~~in the 13-limit and~~ [[~~136~~/~~135~~]] and [[~~256/255~~]] ~~in the 17-limit~~.		If prime 5 is desired, 104edo has two different equally viable 5-limit [[val]]s, and both are useful. The flat major third val, {{val\| 104 165 241 }} ([[patent val]]), tempers out [[3125/3072]], and [[support]]s [[magic]] temperament. The sharp major third val, {{val\| 104 165 242 }} (104c val), tempers out [[2048/2025]] and supports [[diaschismic]] temperament.

	~~104~~ is ~~also~~ notable as ~~a no~~-~~fives system~~; ~~on 2.3.~~7~~.11.13~~, it tempers out ~~352~~/~~351~~, ~~364~~/~~363~~, 896/891, [[~~2197~~/~~2187~~]], [[~~10648~~/~~10647~~]]~~, 16807~~/~~16731~~, ~~20449~~/~~20412, 21632~~/~~21609~~, and ~~26411~~/~~26364. It is~~ the optimal patent val for ~~the {{nowrap\|~~ 17 ~~& 87 }} 2.3~~.7.11~~.13 subgroup temperament tempering out~~ 352/351, 364/363 and ~~2197~~/~~2187, which has a 13~~/~~9 generator, three of which give a 3~~.		104edo with the flat third is especially notable as an excellent tuning for magic temperament, providing the [[optimal patent val]] for 11-limit magic and the 13-limit magic extension [[necromancy]]. In the 5-limit it tempers out the magic comma, 3125/3072; in the 7-limit, it tempers out [[225/224]], [[245/243]] and [[875/864]]; and in the 11-limit, [[100/99]], 896/891, [[385/384]] and [[540/539]]. It also provides an excellent tuning for the rank-3 temperament pairing 100/99 with 225/224 ([[apollo]] temperament), 245/243 or 875/864, and the rank-4 temperament tempering out 100/99, for which it gives the optimal patent val.

			104edo with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]], 352/351 and 364/363 in the 13-limit, and [[136/135]] and [[256/255]] in the 17-limit.

	=== Prime harmonics ===		=== Prime harmonics ===

FloraC: Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

2026-03-13T13:33:14Z

Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

← Older revision		Revision as of 13:33, 13 March 2026
Line 154:		Line 154:
	\| [[Bosonic]]		\| [[Bosonic]]
	\|}		\|}
	<nowiki />* [[Normal ~~lists~~\|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal lists~~\|minimal form]] in parentheses if distinct		<nowiki />* [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form\|minimal form]] in parentheses if distinct

	== Intervals ==		== Intervals ==

Overthink: /* Intervals */ mark inconsistent 6/5

2025-12-18T00:56:23Z

Intervals: mark inconsistent 6/5

← Older revision		Revision as of 00:56, 18 December 2025
Line 339:		Line 339:
	\|		\|
	\|		\|
	\| [[6/5]], ''[[40/33]]''		\| ''[[6/5]]'', ''[[40/33]]''
	\|-		\|-
	\| 29		\| 29

Overthink: /* Intervals */ 28/27 isn't inconsistent

2025-12-07T19:59:20Z

Intervals: 28/27 isn't inconsistent

← Older revision		Revision as of 19:59, 7 December 2025
Line 199:		Line 199:
	\| 5		\| 5
	\| 57.7		\| 57.7
	\| ''[[28/27]]'', [[33/32]]		\| [[28/27]], [[33/32]]
	\| ''[[26/25]]''		\| ''[[26/25]]''
	\| ''[[25/24]]'', ''[[36/35]]''		\| ''[[25/24]]'', ''[[36/35]]''

FloraC: /* Intervals */ +some ratios

2025-04-09T14:05:54Z

Intervals: +some ratios

← Older revision		Revision as of 14:05, 9 April 2025
Line 175:		Line 175:
	\| 1		\| 1
	\| 11.5		\| 11.5
	\| [[~~100~~/99]], [[~~225~~/~~224~~]]		\| [[144/143]], [[169/168]]
	\|		\| ''[[91/90]]'', [[121/120]]
	\|		\| [[105/104]], [[196/195]]
	\|-		\|-
	\| 2		\| 2
	\| 23.1		\| 23.1
	\| [[64/63]]		\| [[64/63]], [[99/98]]
	\| [[81/80]], ''[[~~225~~/~~224~~]]''		\| [[81/80]], [[100/99]], ''[[105/104]]''
	\| ''[[50/49]]''		\| ''[[50/49]]'', ''[[55/54]]'', [[91/90]], ''[[121/120]]''
	\|-		\|-
	\| 3		\| 3
	\| 34.6		\| 34.6
	\| [[49/48]], [[50/49]]		\| [[49/48]], [[50/49]]
	\|		\| [[55/54]]
	\| ''[[81/80]]'', ''[[126/125]]''		\| ''[[40/39]]'', [[45/44]], ''[[81/80]]'', ''[[126/125]]''
	\|-		\|-
	\| 4		\| 4
	\| 46.2		\| 46.2
	\|		\|
	\| [[36/35]], ''[[50/49]]''		\| [[36/35]], [[40/39]], ''[[45/44]]'', ''[[50/49]]''
	\|		\|
	\|-		\|-
Line 200:		Line 200:
	\| 57.7		\| 57.7
	\| ''[[28/27]]'', [[33/32]]		\| ''[[28/27]]'', [[33/32]]
	\|		\| ''[[26/25]]''
	\| ''[[25/24]]'', ''[[36/35]]''		\| ''[[25/24]]'', ''[[36/35]]''
	\|-		\|-
	\| 6		\| 6
	\| 69.2		\| 69.2
	\| [[25/24]]		\| [[25/24]], [[26/25]], [[27/26]]
	\|		\|
	\|		\|
Line 213:		Line 213:
	\| [[22/21]]		\| [[22/21]]
	\| [[21/20]], ''[[25/24]]''		\| [[21/20]], ''[[25/24]]''
	\| ''[[20/19]]''		\| ''[[20/19]]'', ''[[26/25]]''
	\|-		\|-
	\| 8		\| 8
Line 295:		Line 295:
	\| 21		\| 21
	\| 242.3		\| 242.3
	\|		\| [[38/33]]
	\|		\|
	\| [[15/13]]		\| [[15/13]]
	\|-		\|-
Line 319:		Line 319:
	\| 25		\| 25
	\| 288.5		\| 288.5
	\| [[32/27]], [[13/11]]		\| [[13/11]], [[32/27]]
	\| ''[[20/17]]''		\| ''[[20/17]]''
	\|		\|
Line 325:		Line 325:
	\| 26		\| 26
	\| 300.0		\| 300.0
	\| [[25/21]], [[19/16]]		\| [[19/16]], [[25/21]]
	\|		\|
	\|		\|

FloraC: /* Intervals */ reduce cent values to one decimal place, per discussion on Discord; sort ratios; -some excessively complex ratios

2025-04-09T13:36:29Z

Intervals: reduce cent values to one decimal place, per discussion on Discord; sort ratios; -some excessively complex ratios

Show changes

FloraC: /* Theory */ +octave stretch

2025-04-09T12:18:45Z

Theory: +octave stretch

← Older revision		Revision as of 12:18, 9 April 2025
Line 9:		Line 9:
	104 with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]], [[352/351]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.		104 with the sharp third is excellent for 11-, 13-, or 17-limit diaschismic. It tempers out 2048/2025 in the 5-limit, [[126/125]] and [[5120/5103]] in the 7-limit, [[176/175]] and 896/891 in the 11-limit, [[196/195]], [[352/351]] and [[364/363]] in the 13-limit and [[136/135]] and [[256/255]] in the 17-limit.

	104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, [[2197/2187]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364. It is the optimal patent val for the {{nowrap\|17 &~~amp;~~ 87}} 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.		104 is also notable as a no-fives system; on 2.3.7.11.13, it tempers out 352/351, 364/363, 896/891, [[2197/2187]], [[10648/10647]], 16807/16731, 20449/20412, 21632/21609, and 26411/26364. It is the optimal patent val for the {{nowrap\| 17 & 87 }} 2.3.7.11.13 subgroup temperament tempering out 352/351, 364/363 and 2197/2187, which has a 13/9 generator, three of which give a 3.

	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|104}}		{{Harmonics in equal\|104}}

			=== Octave stretch ===
			104edo's approximations of harmonics 3, 7, 11, and 13 can all be improved if slightly compressing the octave is acceptable, using tunings such as [[269ed6]], which is also suitable for the full 13-limit and beyond, using the 104c val. A greater focus on prime 5 could lead to more heavily compressed tunings such as [[165edt]].

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 104 factors into 2<sup>3</sup> × 13, it has subset edos {{EDOs\| 2, 4, 8, 13, 26, and 52 }}.		Since 104 factors into primes as {{nowrap\| 2<sup>3</sup> × 13 }}, 104edo has subset edos {{EDOs\| 2, 4, 8, 13, 26, and 52 }}.

	== Regular temperament properties ==		== Regular temperament properties ==

ArrowHead294 at 16:31, 27 February 2025

2025-02-27T16:31:24Z

← Older revision		Revision as of 16:31, 27 February 2025
Line 32:		Line 32:
	\| {{monzo\| 165 -104 }}		\| {{monzo\| 165 -104 }}
	\| {{mapping\| 104 165 }}		\| {{mapping\| 104 165 }}
	\| ~~−0~~.597		\| −0.597
	\| 0.596		\| 0.596
	\| 5.17		\| 5.17
Line 39:		Line 39:
	\| 2048/2025, {{monzo\| 0 22 -15 }}		\| 2048/2025, {{monzo\| 0 22 -15 }}
	\| {{mapping\| 104 165 242 }} (104c)		\| {{mapping\| 104 165 242 }} (104c)
	\| ~~−1~~.258		\| −1.258
	\| 1.054		\| 1.054
	\| 9.14		\| 9.14
Line 46:		Line 46:
	\| 126/125, 2048/2025, 117649/116640		\| 126/125, 2048/2025, 117649/116640
	\| {{mapping\| 104 165 242 292 }} (104c)		\| {{mapping\| 104 165 242 292 }} (104c)
	\| ~~−0~~.980		\| −0.980
	\| 1.032		\| 1.032
	\| 8.95		\| 8.95
Line 53:		Line 53:
	\| 126/125, 176/175, 896/891, 14641/14580		\| 126/125, 176/175, 896/891, 14641/14580
	\| {{mapping\| 104 165 242 292 360 }} (104c)		\| {{mapping\| 104 165 242 292 360 }} (104c)
	\| ~~−0~~.930		\| −0.930
	\| 0.929		\| 0.929
	\| 8.05		\| 8.05
Line 60:		Line 60:
	\| 126/125, 176/175, 196/195, 364/363, 2197/2187		\| 126/125, 176/175, 196/195, 364/363, 2197/2187
	\| {{mapping\| 104 165 242 292 360 385 }} (104c)		\| {{mapping\| 104 165 242 292 360 385 }} (104c)
	\| ~~−0~~.855		\| −0.855
	\| 0.864		\| 0.864
	\| 7.49		\| 7.49