Hemipyth - Revision history

Overthink: /* MOS patterns */ + idiosyncratic terms template; style

2026-03-11T22:07:33Z

MOS patterns: + idiosyncratic terms template; style

← Older revision		Revision as of 22:07, 11 March 2026
Line 273:		Line 273:

	== MOS patterns ==		== MOS patterns ==
			{{Idiosyncratic terms\|The mos names for hemipyth[14], hemipyth[24], and hemipyth[34] are proposals described on [[TAMNAMS_Extension #Naming mos descendants]].}}
	By default hemipyth[n] refers to a [[MOS]] pattern of size n inside the octave i.e. there are always two √2 periods per octave. The generator is √3 unless otherwise stated.		By default hemipyth[n] refers to a [[MOS]] pattern of size n inside the octave i.e. there are always two √2 periods per octave. The generator is √3 unless otherwise stated.

Line 279:		Line 280:
	\|-		\|-
	! hemipyth[n]		! hemipyth[n]
	!TAMNAMS name!! MOS pattern !! hardness (untempered)		! TAMNAMS name !! MOS pattern !! hardness (untempered)
	\|-		\|-
	\| hemipyth[4]		\| hemipyth[4]
	\|biwood\|\| [[2L 2s]] \|\| 1.4094		\| biwood \|\| [[2L 2s]] \|\| 1.4094
	\|-		\|-
	\| hemipyth[6]		\| hemipyth[6]
	\|citric\|\| [[4L 2s]] \|\| 2.4424		\| citric \|\| [[4L 2s]] \|\| 2.4424
	\|-		\|-
	\| hemipyth[10]		\| hemipyth[10]
	\|lime\|\| [[4L 6s]] \|\| 1.4424		\| lime \|\| [[4L 6s]] \|\| 1.4424
	\|-		\|-
	\| hemipyth[14]		\| hemipyth[14]
	\|m-chro lime\|\| [[10L 4s]] \|\| 2.260		\| m-chro lime \|\| [[10L 4s]] \|\| 2.260
	\|-		\|-
	\| hemipyth[24]		\| hemipyth[24]
	\|f-enhar lime\|\| [[10L 14s]] \|\| 1.260		\| f-enhar lime \|\| [[10L 14s]] \|\| 1.260
	\|-		\|-
	\|hemipyth[34]		\| hemipyth[34]
	\|paso-lime		\| paso-lime
	\|[[24L 10s]]		\| [[24L 10s]]
	\|3.8459		\| 3.8459
	\|-		\|-
	\|hemipyth[58]		\| hemipyth[58]
	\|		\|
	\|[[24L 34s]]		\| [[24L 34s]]
	\|2.8459		\| 2.8459
	\|-		\|-
	\|hemipyth[82]		\| hemipyth[82]
	\|		\|
	\|[[24L 58s]]		\| [[24L 58s]]
	\|1.8459		\| 1.8459
	\|-		\|-
	\|hemipyth[106]		\| hemipyth[106]
	\|		\|
	\|[[82L 24s]]		\| [[82L 24s]]
	\|1.1822		\| 1.1822
	\|}		\|}

Overthink: linking, + "primes"

2026-03-11T22:02:00Z

linking, + "primes"

← Older revision		Revision as of 22:02, 11 March 2026
Line 1:		Line 1:
	{{texchars}}		{{texchars}}
	A '''hemipyth''' (or '''"hemipythagorean"''') interval is an [[interval]] in the <math>\sqrt{2}\,.\sqrt{3}</math> [[subgroup]] i.e. intervals that can be constructed by multiplying half-integer powers of 2 and 3. Hemipythagorean is the pure tuning of [[Ploidacot/Diploid dicot\|diploid dicot]].		A '''hemipyth''' (or '''"hemipythagorean"''') interval is an [[interval]] in the <math>\sqrt{2}\,.\sqrt{3}-</math>[[subgroup]]; i.e. intervals that can be constructed by multiplying half-integer powers of primes [[2/1\|2]] and [[3/1\|3]]. Hemipythagorean is the pure tuning of [[Ploidacot/Diploid dicot\|diploid dicot]].

	Notable hemipyth intervals include the neutral third <math>\sqrt{\frac{3}{2}} = \frac{\sqrt{3}}{\sqrt{2}}</math>, semioctave <math>\sqrt{2}</math>, and the semifourth <math>\sqrt{\frac{4}{3}} = \frac{2}{\sqrt{3}}</math>.		Notable hemipyth intervals include the [[neutral third]] <math>\sqrt{\frac{3}{2}} = \frac{\sqrt{3}}{\sqrt{2}}</math>, [[semioctave]] <math>\sqrt{2}</math>, and the [[semifourth]] <math>\sqrt{\frac{4}{3}} = \frac{2}{\sqrt{3}}</math>.

	Many temperaments naturally produce intervals that split ~{{sfrac\|3\|2}}, ~2, or ~{{sfrac\|4\|3}} exactly in half and can thus be interpreted as neutral thirds, semioctaves, or semifourths within the temperament.		Many temperaments naturally produce intervals that split ~{{sfrac\|3\|2}}, ~2, or ~{{sfrac\|4\|3}} exactly in half and can thus be interpreted as neutral thirds, semioctaves, or semifourths within the temperament.

Overthink: /* Music */ remove todo with nonexistent categories

2026-03-11T21:57:46Z

Music: remove todo with nonexistent categories

← Older revision		Revision as of 21:57, 11 March 2026
Line 318:		Line 318:

	== Music ==		== Music ==
	~~{{todo\|inline-1\| Make more music }}~~

	[[File:The_Hymn_of_Pergele.mp3]]		[[File:The_Hymn_of_Pergele.mp3]]

Overthink: /* Equal temperaments */ links, improve writing

2026-03-11T21:49:48Z

Equal temperaments: links, improve writing

← Older revision		Revision as of 21:49, 11 March 2026
Line 16:		Line 16:
	\|+ style="font-size: 105%;" \| List of edo mappings with full or partial hemipyth support		\|+ style="font-size: 105%;" \| List of edo mappings with full or partial hemipyth support
	\|-		\|-
	! Edo (warts) !! Has <math>\sqrt{2}</math> !! Has <math>\sqrt{\frac{3}{2}}</math> !! Has <math>\sqrt{\frac{4}{3}}</math>		! Edo ([[Wart notation\|warts]]) !! Has <math>\sqrt{2}</math> !! Has <math>\sqrt{\frac{3}{2}}</math> !! Has <math>\sqrt{\frac{4}{3}}</math>
	\|-		\|-
	\| 2 \|\| yes \|\| no \|\| no		\| [[2edo\|2]] \|\| yes \|\| no \|\| no
	\|-		\|-
	\| 3 \|\| no \|\| yes \|\| no		\| [[3edo\|3]] \|\| no \|\| yes \|\| no
	\|-		\|-
	\| 4 \|\| yes \|\| yes \|\| yes		\| [[4edo\|4]] \|\| yes \|\| yes \|\| yes
	\|-		\|-
	\| 5 \|\| no \|\| no \|\| yes		\| [[5edo\|5]] \|\| no \|\| no \|\| yes
	\|-		\|-
	\| 6 \|\| yes \|\| yes \|\| yes		\| [[6edo\|6]] \|\| yes \|\| yes \|\| yes
	\|-		\|-
	\| 7 \|\| no \|\| yes \|\| no		\| [[7edo\|7]] \|\| no \|\| yes \|\| no
	\|-		\|-
	\| 8 \|\| yes \|\| no \|\| no		\| [[8edo\|8]] \|\| yes \|\| no \|\| no
	\|-		\|-
	\| 9 \|\| no \|\| no \|\| yes		\| [[9edo\|9]] \|\| no \|\| no \|\| yes
	\|-		\|-
	\| 10 \|\| yes \|\| yes \|\| yes		\| [[10edo\|10]] \|\| yes \|\| yes \|\| yes
	\|-		\|-
	\| 11 \|\| no \|\| yes \|\| no		\| [[11edo\|11]] \|\| no \|\| yes \|\| no
	\|-		\|-
	\| 12 \|\| yes \|\| no \|\| no		\| [[12edo\|12]] \|\| yes \|\| no \|\| no
	\|-		\|-
	\| 13 \|\| no \|\| yes \|\| no		\| [[13edo\|13]] \|\| no \|\| yes \|\| no
	\|-		\|-
	\| 13b \|\| no \|\| no \|\| yes		\| 13b \|\| no \|\| no \|\| yes
	\|-		\|-
	\| 14 \|\| yes \|\| yes \|\| yes		\| [[14edo\|14]] \|\| yes \|\| yes \|\| yes
	\|-		\|-
	\| 15 \|\| no \|\| no \|\| yes		\| [[15edo\|15]] \|\| no \|\| no \|\| yes
	\|-		\|-
	\| 16 \|\| yes \|\| no \|\| no		\| [[16edo\|16]] \|\| yes \|\| no \|\| no
	\|-		\|-
	\| 17 \|\| no \|\| yes \|\| no		\| [[17edo\|17]] \|\| no \|\| yes \|\| no
	\|-		\|-
	\| 18 \|\| yes \|\| no \|\| no		\| [[18edo\|18]] \|\| yes \|\| no \|\| no
	\|-		\|-
	\| 18b \|\| yes \|\| yes \|\| yes		\| 18b \|\| yes \|\| yes \|\| yes
	\|-		\|-
	\| 19 \|\| no \|\| no \|\| yes		\| [[19edo\|19]] \|\| no \|\| no \|\| yes
	\|-		\|-
	\| 20* \|\| yes \|\| yes \|\| yes		\| [[20edo\|20]]* \|\| yes \|\| yes \|\| yes
	\|-		\|-
	\| 20b \|\| yes \|\| no \|\| no		\| 20b \|\| yes \|\| no \|\| no
	\|-		\|-
	\| 21 \|\| no \|\| yes \|\| no		\| [[21edo\|21]] \|\| no \|\| yes \|\| no
	\|-		\|-
	\| 22 \|\| yes \|\| no \|\| no		\| [[22edo\|22]] \|\| yes \|\| no \|\| no
	\|-		\|-
	\| 23 \|\| no \|\| no \|\| yes		\| [[23edo\|23]] \|\| no \|\| no \|\| yes
	\|-		\|-
	\| 24 \|\| yes \|\| yes \|\| yes		\| [[24edo\|24]] \|\| yes \|\| yes \|\| yes
	\|}		\|}

	<nowiki>*</nowiki> Above the patent val of 20edo results in the same tuning as the patent val of 10edo, so it adds nothing new.		<nowiki>*</nowiki> Above the patent val of 20edo results in the same tuning as the patent val of 10edo, so it adds nothing new.

	Note how in hemipyth the patent val of 24edo is not tuned the same as 12edo's patent val. In fact 24edo is arguably the smallest edo where all of the important hemipyth intervals are tuned reasonably accurately.		Note how in hemipyth the patent val of 24edo is not tuned the same as 12edo's patent val. In fact, 24edo is arguably the smallest edo where all of the important hemipyth intervals are tuned reasonably accurately.

	Other edos with hemipyth-supporting patent vals are 28, 30, 34, 38, 44, 48, 52, 54, 58, etc. 58edo is the first one to reduce the absolute error of the neutral third generator compared to 24edo~~. You need~~ to go all the way to 82edo in order to get an improvement in terms of relative error.		Other edos with hemipyth-supporting patent vals are {{edos\|28, 30, 34, 38, 44, 48, 52, 54, 58}}, etc. 58edo is the first one to reduce the absolute error of the neutral third generator compared to 24edo, though one needs to go all the way to [[82edo]] in order to get an improvement in terms of relative error.

	== Notation ==		== Notation ==

Overthink: /* Temperament interpretations */ fix expression

2026-03-11T21:43:32Z

Temperament interpretations: fix expression

← Older revision		Revision as of 21:43, 11 March 2026
Line 253:		Line 253:
	\|}		\|}

	Some possible interpretations for <math>\sim\frac{3}{2}</math> are:		Some possible interpretations for <math>\sim\sqrt{\frac{3}{2}}</math> are:

	{\| class="wikitable"		{\| class="wikitable"

Eufalesio: /* Temperament interpretations */

2026-03-11T10:51:53Z

Temperament interpretations

← Older revision		Revision as of 10:51, 11 March 2026
Line 265:		Line 265:
	\|-		\|-
	\| [[Breed family#Breed\|breed]]\|\| ~{{sfrac\|49\|40}} \|\| no \|\| no (2.3.5.7)		\| [[Breed family#Breed\|breed]]\|\| ~{{sfrac\|49\|40}} \|\| no \|\| no (2.3.5.7)
			\|-
			\|[[Very high accuracy temperaments#Euzenius\|euzenius]]
			\|~{{sfrac\|6250\|5103}}
			\|no
			\|no (2.3.5.7)
	\|}		\|}

Eufalesio: Edited some things

2026-02-14T00:07:16Z

Edited some things

← Older revision		Revision as of 00:07, 14 February 2026
Line 209:		Line 209:
	\| [[hemidim]] \|\| ~{{sfrac\|36\|25}} \|\| ~{{sfrac\|25\|21}} \|\| ~{{sfrac\|7\|6}} \|\| yes \|\| yes		\| [[hemidim]] \|\| ~{{sfrac\|36\|25}} \|\| ~{{sfrac\|25\|21}} \|\| ~{{sfrac\|7\|6}} \|\| yes \|\| yes
	\|-		\|-
	\| [[~~greenland~~]] \|\| ~{{sfrac\|99\|70}} \|\| ~{{sfrac\|49\|40}} \|\| ~~~{{sfrac\|15\|13}}~~~{{sfrac\|231\|200}} \|\| no \|\| no		\| [[baldur]]\|\| ~{{sfrac\|99\|70}} \|\| ~{{sfrac\|49\|40}} \|\| ~{{sfrac\|231\|200}} \|\| no \|\| no
	\|-		\|-
	\| [[semisema]] \|\| ~{{sfrac\|108\|77}} \|\| ~{{sfrac\|11\|9}} \|\| ~{{sfrac\|7\|6}} \|\| no \|\| yes		\| [[semisema]] \|\| ~{{sfrac\|108\|77}} \|\| ~{{sfrac\|11\|9}} \|\| ~{{sfrac\|7\|6}} \|\| no \|\| yes
Line 233:		Line 233:
	\| [[semitonic]] \|\| ~{{sfrac\|17\|12}} \|\| no \|\| yes (2.3.17)		\| [[semitonic]] \|\| ~{{sfrac\|17\|12}} \|\| no \|\| yes (2.3.17)
	\|-		\|-
	\| [[kalismic temperaments\|kalismic]] \|\| ~{{sfrac\|99\|70}} \|\| no \|\| no		\| [[kalismic temperaments\|kalismic]] \|\| ~{{sfrac\|99\|70}} \|\| no \|\| no (2.3.5.7.11)
	\|}		\|}

Line 246:		Line 246:
	\|-		\|-
	\| [[barbados]] \|\| ~{{sfrac\|26\|15}} \|\| no \|\| yes (2.3.{{sfrac\|13\|5}})		\| [[barbados]] \|\| ~{{sfrac\|26\|15}} \|\| no \|\| yes (2.3.{{sfrac\|13\|5}})
			\|-
			\|[[Catalog of rank-4 temperaments#Seascape (160083/160000)\|seascape]]
			\|~{{sfrac\|400\|231}}
			\|no
			\|no (2.3.5.7.11)
	\|}		\|}

Line 259:		Line 264:
	\| [[Rastmic clan#Neutral\|neutral]] \|\| ~{{sfrac\|11\|9}} \|\| no \|\| yes (2.3.11)		\| [[Rastmic clan#Neutral\|neutral]] \|\| ~{{sfrac\|11\|9}} \|\| no \|\| yes (2.3.11)
	\|-		\|-
	\| [[~~jove~~]] \|\| ~~~{{sfrac\|11\|9}}~~~{{sfrac\|49\|40}} \|\| no \|\| no		\| [[Breed family#Breed\|breed]]\|\| ~{{sfrac\|49\|40}} \|\| no \|\| no (2.3.5.7)
	\|}		\|}

	== MOS patterns ==		== MOS patterns ==
	By default hemipyth[n] refers to a [[MOS]] pattern of size n inside the octave i.e. there are always two periods per octave. The generator is √3 unless otherwise stated.		By default hemipyth[n] refers to a [[MOS]] pattern of size n inside the octave i.e. there are always two √2 periods per octave. The generator is √3 unless otherwise stated.

	{\| class="wikitable"		{\| class="wikitable"
	\|+ style="font-size: 105%;" \| MOS patterns of hemipyth		\|+ style="font-size: 105%;" \| MOS patterns of hemipyth
	\|-		\|-
	! hemipyth[n] !! MOS pattern !! hardness (untempered)		! hemipyth[n]
			!TAMNAMS name!! MOS pattern !! hardness (untempered)
			\|-
			\| hemipyth[4]
			\|biwood\|\| [[2L 2s]] \|\| 1.4094
			\|-
			\| hemipyth[6]
			\|citric\|\| [[4L 2s]] \|\| 2.4424
			\|-
			\| hemipyth[10]
			\|lime\|\| [[4L 6s]] \|\| 1.4424
			\|-
			\| hemipyth[14]
			\|m-chro lime\|\| [[10L 4s]] \|\| 2.260
	\|-		\|-
	\| hemipyth[4] \|\| [[~~2L 2s~~]] \|\| 1.41		\| hemipyth[24]
			\|f-enhar lime\|\| [[10L 14s]] \|\| 1.260
	\|-		\|-
	\| hemipyth[6] \|\| [[~~4L 2s~~]] \|~~\| 2~~.44		\|hemipyth[34]
			\|paso-lime
			\|[[24L 10s]]
			\|3.8459
	\|-		\|-
	\| hemipyth[10] \|\| [[~~4L 6s~~]] \|~~\| 1~~.44		\|hemipyth[58]
			\|
			\|[[24L 34s]]
			\|2.8459
	\|-		\|-
	\| hemipyth[14] \|\| [[~~10L 4s~~]] \|~~\| 2~~.26		\|hemipyth[82]
			\|
			\|[[24L 58s]]
			\|1.8459
	\|-		\|-
	\| hemipyth[24] \|\| [[~~10L 14s~~]] \|\| 1.26		\|hemipyth[106]
			\|
			\|[[82L 24s]]
			\|1.1822
	\|}		\|}

Xenllium: /* Music */

2026-02-05T11:40:31Z

Music

← Older revision		Revision as of 11:40, 5 February 2026
Line 287:		Line 287:

	The Hymn of Pergele, a short piece in {{nowrap\|[[Hemipyth]][10] 4{{!}}4(2)}} (Pacific mode of [[4L 6s]]), written by [[User:2^67-1\|Cole]].		The Hymn of Pergele, a short piece in {{nowrap\|[[Hemipyth]][10] 4{{!}}4(2)}} (Pacific mode of [[4L 6s]]), written by [[User:2^67-1\|Cole]].

			== See also ==
			* [[Hemipent]] – √2.√3.√5 subgroup

	[[Category:Rank 2]]		[[Category:Rank 2]]
	[[Category:Subgroup]]		[[Category:Subgroup]]
	[[Category:Listen]]		[[Category:Listen]]

Xenllium at 03:31, 5 February 2026

2026-02-05T03:31:13Z

← Older revision		Revision as of 03:31, 5 February 2026
Line 127:		Line 127:
	Luckily we don't need to introduce any more generalizations to the notation to indicate <math>\sqrt{\frac{4}{3}}</math>. It's a neutral {{sfrac\|2\|1\|2}} or a α{{demiflat2}} (alp semiflat) w.r.t middle C.		Luckily we don't need to introduce any more generalizations to the notation to indicate <math>\sqrt{\frac{4}{3}}</math>. It's a neutral {{sfrac\|2\|1\|2}} or a α{{demiflat2}} (alp semiflat) w.r.t middle C.

	Nicknames are still assigned to make it easier to talk about the [[5L 4s]] scale generated by <math>\sim~~\vsp~~\sqrt{\frac{4}{3}}</math> against the octave.		Nicknames are still assigned to make it easier to talk about the [[5L 4s]] scale generated by <math>\sim\sqrt{\frac{4}{3}}</math> against the octave.

	{\| class="wikitable"		{\| class="wikitable"
Line 179:		Line 179:
	The neutral third receives only half of the tuning damage of the fifth so it has a strong character even if the fifth isn't tuned very pure. The irrational nature of <math>\sqrt{\frac{3}{2}}</math> also makes it more tolerant of imprecise tuning.		The neutral third receives only half of the tuning damage of the fifth so it has a strong character even if the fifth isn't tuned very pure. The irrational nature of <math>\sqrt{\frac{3}{2}}</math> also makes it more tolerant of imprecise tuning.

	The same goes for the semifourth. A poorly tuned ~{{sfrac\|4\|3}} still results in a decent <math>\sim~~\vsp~~\sqrt{\frac{4}{3}}</math> (assuming it's featured in the tuning in the first place).		The same goes for the semifourth. A poorly tuned ~{{sfrac\|4\|3}} still results in a decent <math>\sim\sqrt{\frac{4}{3}}</math> (assuming it's featured in the tuning in the first place).

	=== Signposts ===		=== Signposts ===
Line 187:		Line 187:

	== Temperament interpretations ==		== Temperament interpretations ==
	Under [[ploidacot]] classification diploid temperaments feature <math>\sim~~\vsp~~\sqrt{2}</math>, dicot temperaments have <math>\sim~~\vsp~~\sqrt{\frac{3}{2}}</math> and alpha-dicot temperaments feature <math>\sim~~\vsp~~\sqrt{\frac{4}{3}}</math> (by virtue of having a <math>\sim~~\vsp~~\sqrt{3}</math>).		Under [[ploidacot]] classification diploid temperaments feature <math>\sim\sqrt{2}</math>, dicot temperaments have <math>\sim\sqrt{\frac{3}{2}}</math> and alpha-dicot temperaments feature <math>\sim\sqrt{\frac{4}{3}}</math> (by virtue of having a <math>\sim\sqrt{3}</math>).

	Full hemipyth support is indicated by at least "diploid dicot". Examples include:		Full hemipyth support is indicated by at least "diploid dicot". Examples include:
Line 193:		Line 193:
	\|+ style="font-size: 105%;" \| Higher-prime interpretations of hemipyth intervals		\|+ style="font-size: 105%;" \| Higher-prime interpretations of hemipyth intervals
	\|-		\|-
	! Temperament !! <math>\sim~~\vsp~~\sqrt{2}</math> !! <math>\sim~~\vsp~~\sqrt{\frac{3}{2}}</math> !! <math>\sim~~\vsp~~\sqrt{\frac{4}{3}}</math> !! contorted !! rank-2		! Temperament !! <math>\sim\sqrt{2}</math> !! <math>\sim\sqrt{\frac{3}{2}}</math> !! <math>\sim\sqrt{\frac{4}{3}}</math> !! contorted !! rank-2
	\|-		\|-
	\| [[decimal]] \|\| ~{{sfrac\|7\|5}} \|\| ~{{sfrac\|5\|4}} \|\| ~{{sfrac\|7\|6}} \|\| no \|\| yes		\| [[decimal]] \|\| ~{{sfrac\|7\|5}} \|\| ~{{sfrac\|5\|4}} \|\| ~{{sfrac\|7\|6}} \|\| no \|\| yes
Line 218:		Line 218:
	\|}		\|}

	Above contorted tunings don't have a <math>\sim~~\vsp~~\sqrt{2}</math> period with a <math>\sim~~\vsp~~\sqrt{3}</math> generator, but introduce further splits. Higher than rank-2 temperaments introduce further structure that goes beyond basic hemipyth.		Above contorted tunings don't have a <math>\sim\sqrt{2}</math> period with a <math>\sim\sqrt{3}</math> generator, but introduce further splits. Higher than rank-2 temperaments introduce further structure that goes beyond basic hemipyth.

	Some possible interpretations for <math>\sim~~\vsp~~\sqrt{2}</math> are:		Some possible interpretations for <math>\sim\sqrt{2}</math> are:

	{\| class="wikitable"		{\| class="wikitable"
	\|+ style="font-size: 105%;" \| Higher-prime interpretations of <math>\sqrt{2}</math>		\|+ style="font-size: 105%;" \| Higher-prime interpretations of <math>\sqrt{2}</math>
	\|-		\|-
	! Temperament !! <math>\sim~~\vsp~~\sqrt{2}</math> !! contorted !! rank-2		! Temperament !! <math>\sim\sqrt{2}</math> !! contorted !! rank-2
	\|-		\|-
	\| [[jubilic]] \|\| ~{{sfrac\|7\|5}} \|\| no \|\| yes (2.5.7)		\| [[jubilic]] \|\| ~{{sfrac\|7\|5}} \|\| no \|\| yes (2.5.7)
Line 236:		Line 236:
	\|}		\|}

	Some possible interpretations for <math>\sim~~\vsp~~\sqrt{3}</math> are:		Some possible interpretations for <math>\sim\sqrt{3}</math> are:

	{\| class="wikitable"		{\| class="wikitable"
	\|+ style="font-size: 105%;" \| Higher-prime interpretations of <math>\sqrt{3}</math>		\|+ style="font-size: 105%;" \| Higher-prime interpretations of <math>\sqrt{3}</math>
	\|-		\|-
	! Temperament !! <math>\sim~~\vsp~~\sqrt{3}</math> !! contorted !! rank-2		! Temperament !! <math>\sim\sqrt{3}</math> !! contorted !! rank-2
	\|-		\|-
	\| [[semaphore]] \|\| ~{{sfrac\|7\|4}} \|\| no \|\| yes (2.3.7)		\| [[semaphore]] \|\| ~{{sfrac\|7\|4}} \|\| no \|\| yes (2.3.7)
Line 248:		Line 248:
	\|}		\|}

	Some possible interpretations for <math>\sim~~\vsp~~\frac{3}{2}</math> are:		Some possible interpretations for <math>\sim\frac{3}{2}</math> are:

	{\| class="wikitable"		{\| class="wikitable"
	\|+ style="font-size: 105%;" \| Higher-prime interpretations of <math>\sqrt{\frac{3}{2}}</math>		\|+ style="font-size: 105%;" \| Higher-prime interpretations of <math>\sqrt{\frac{3}{2}}</math>
	\|-		\|-
	! Temperament !! <math>\sqrt{~\frac{3}{2}}</math> !! contorted !! rank-2		! Temperament !! <math>\sim\sqrt{\frac{3}{2}}</math> !! contorted !! rank-2
	\|-		\|-
	\| [[dicot]] \|\| ~{{sfrac\|5\|4}} \|\| no \|\| yes (2.3.5)		\| [[dicot]] \|\| ~{{sfrac\|5\|4}} \|\| no \|\| yes (2.3.5)

Overthink: /* Equal temperaments */ Replaced asterisk template with nowiki

2025-12-02T22:53:44Z

Equal temperaments: Replaced asterisk template with nowiki

← Older revision		Revision as of 22:53, 2 December 2025
Line 71:		Line 71:
	\|}		\|}

	~~{{asterisk}}~~ Above the patent val of 20edo results in the same tuning as the patent val of 10edo, so it adds nothing new.		<nowiki>*</nowiki> Above the patent val of 20edo results in the same tuning as the patent val of 10edo, so it adds nothing new.

	Note how in hemipyth the patent val of 24edo is not tuned the same as 12edo's patent val. In fact 24edo is arguably the smallest edo where all of the important hemipyth intervals are tuned reasonably accurately.		Note how in hemipyth the patent val of 24edo is not tuned the same as 12edo's patent val. In fact 24edo is arguably the smallest edo where all of the important hemipyth intervals are tuned reasonably accurately.