39edt - Revision history

BudjarnLambeth: /* Theory */ add line with more temperaments

2026-03-23T04:24:10Z

Theory: add line with more temperaments

← Older revision		Revision as of 04:24, 23 March 2026
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	== Theory ==		== Theory ==
	It is a strong no-twos 13-limit system, a fact first noted by [[Paul Erlich]]; in fact it has a better no-twos 13-[[throdd limit]] relative error than any other edt up to [[914edt]]. Like [[26edt]] and [[52edt]], it is a multiple of 13edt and so contains the Bohlen-Pierce scale, being [[contorted]] in the no-twos 7-limit, tempering out the same BP commas, [[245/243]] and [[3125/3087]], as 13edt. In the [[11-limit]] it tempers out [[1331/1323]] and in the [[13-limit]] [[275/273]], [[1575/1573]], and [[847/845]]. An efficient traversal is therefore given by [[Mintra]] temperament, which in the 13-limit tempers out 275/273 and 1331/1323 alongside 245/243, and is generated by the interval of [[11/7]], which serves as a [[macrodiatonic]] "superpyth" fourth and splits the [[BPS]] generator of [[9/7]], up a tritave, in three.		It is a strong no-twos 13-limit system, a fact first noted by [[Paul Erlich]]; in fact it has a better no-twos 13-[[throdd limit]] relative error than any other edt up to [[914edt]]. Like [[26edt]] and [[52edt]], it is a multiple of 13edt and so contains the Bohlen-Pierce scale, being [[contorted]] in the no-twos 7-limit, tempering out the same BP commas, [[245/243]] and [[3125/3087]], as 13edt. In the [[11-limit]] it tempers out [[1331/1323]] and in the [[13-limit]] [[275/273]], [[1575/1573]], and [[847/845]]. An efficient traversal is therefore given by [[Mintra]] temperament, which in the 13-limit tempers out 275/273 and 1331/1323 alongside 245/243, and is generated by the interval of [[11/7]], which serves as a [[macrodiatonic]] "superpyth" fourth and splits the [[BPS]] generator of [[9/7]], up a tritave, in three.

			39edt also supports the temperaments: [[suhail]] (generators ~634.1c, ~49.7c), [[erigone]] (3/1, ~682.4c), [[electra]] (3/1, ~536.1c), [[bohlenic]] (1\13edt, ~11/1) and [[deneb]] (3/1, ~892.6c).

	If octaves are inserted, 39edt is related to the {{nowrap\|49f & 172f}} temperament in the full 13-limit, known as [[Sensamagic clan#Triboh\|triboh]], tempering out 245/243, 275/273, 847/845 and 1575/1573, which has mapping [{{val\|1 0 0 0 0 0}}, {{val\|0 39 57 69 85 91}}]. This has a POTE generator which is an approximate 77/75 of 48.822 cents. 39edt is the ninth [[The Riemann zeta function and tuning#Removing primes\|no-twos zeta peak edt]].		If octaves are inserted, 39edt is related to the {{nowrap\|49f & 172f}} temperament in the full 13-limit, known as [[Sensamagic clan#Triboh\|triboh]], tempering out 245/243, 275/273, 847/845 and 1575/1573, which has mapping [{{val\|1 0 0 0 0 0}}, {{val\|0 39 57 69 85 91}}]. This has a POTE generator which is an approximate 77/75 of 48.822 cents. 39edt is the ninth [[The Riemann zeta function and tuning#Removing primes\|no-twos zeta peak edt]].

Overthink: /* Theory */ fix

2025-12-25T13:35:13Z

Theory: fix

← Older revision		Revision as of 13:35, 25 December 2025
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	== Theory ==		== Theory ==
	It is a strong no-twos 13-limit system, a fact first noted by [[Paul Erlich]]; in fact it has a better no-twos 13-[[~~odd~~ limit]] relative error than any other edt up to [[914edt]]. Like [[26edt]] and [[52edt]], it is a multiple of 13edt and so contains the Bohlen-Pierce scale, being [[contorted]] in the no-twos 7-limit, tempering out the same BP commas, [[245/243]] and [[3125/3087]], as 13edt. In the [[11-limit]] it tempers out [[1331/1323]] and in the [[13-limit]] [[275/273]], [[1575/1573]], and [[847/845]]. An efficient traversal is therefore given by [[Mintra]] temperament, which in the 13-limit tempers out 275/273 and 1331/1323 alongside 245/243, and is generated by the interval of [[11/7]], which serves as a [[macrodiatonic]] "superpyth" fourth and splits the [[BPS]] generator of [[9/7]], up a tritave, in three.		It is a strong no-twos 13-limit system, a fact first noted by [[Paul Erlich]]; in fact it has a better no-twos 13-[[throdd limit]] relative error than any other edt up to [[914edt]]. Like [[26edt]] and [[52edt]], it is a multiple of 13edt and so contains the Bohlen-Pierce scale, being [[contorted]] in the no-twos 7-limit, tempering out the same BP commas, [[245/243]] and [[3125/3087]], as 13edt. In the [[11-limit]] it tempers out [[1331/1323]] and in the [[13-limit]] [[275/273]], [[1575/1573]], and [[847/845]]. An efficient traversal is therefore given by [[Mintra]] temperament, which in the 13-limit tempers out 275/273 and 1331/1323 alongside 245/243, and is generated by the interval of [[11/7]], which serves as a [[macrodiatonic]] "superpyth" fourth and splits the [[BPS]] generator of [[9/7]], up a tritave, in three.

	If octaves are inserted, 39edt is related to the {{nowrap\|49f & 172f}} temperament in the full 13-limit, known as [[Sensamagic clan#Triboh\|triboh]], tempering out 245/243, 275/273, 847/845 and 1575/1573, which has mapping [{{val\|1 0 0 0 0 0}}, {{val\|0 39 57 69 85 91}}]. This has a POTE generator which is an approximate 77/75 of 48.822 cents. 39edt is the ninth [[The Riemann zeta function and tuning#Removing primes\|no-twos zeta peak edt]].		If octaves are inserted, 39edt is related to the {{nowrap\|49f & 172f}} temperament in the full 13-limit, known as [[Sensamagic clan#Triboh\|triboh]], tempering out 245/243, 275/273, 847/845 and 1575/1573, which has mapping [{{val\|1 0 0 0 0 0}}, {{val\|0 39 57 69 85 91}}]. This has a POTE generator which is an approximate 77/75 of 48.822 cents. 39edt is the ninth [[The Riemann zeta function and tuning#Removing primes\|no-twos zeta peak edt]].

Contribution: /* Approximation to JI */

2025-10-27T18:13:14Z

Approximation to JI

← Older revision		Revision as of 18:13, 27 October 2025
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	The Tenney–Euclidean regular temperement in the 3.5.7.11.13 subgroup mapped with [⟨39 57 69 85 91]] gives 48.82201 ¢.		The Tenney–Euclidean regular temperement in the 3.5.7.11.13 subgroup mapped with [⟨39 57 69 85 91]] gives 48.82201 ¢.

	69ed7, with a step size of 48.82356 ¢, is an equal division that ~~approximate~~ this area better than 39edt.		[[69ed7]], with a step size of 48.82356 ¢, is an equal division that approximates this area better than 39edt.

	== Music ==		== Music ==

Contribution: /* Approximation to JI */

2025-10-27T18:12:04Z

Approximation to JI

← Older revision		Revision as of 18:12, 27 October 2025
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	== Approximation to JI ==		== Approximation to JI ==

	According to the finite Euler product with sigma = 1, the 3.5.7.11.13 subgroup gets its maxima at 48.82085 ¢. With sigma = 1/2, ~~it gives~~ 48.82100 ¢.		According to the finite Euler product with sigma = 1, the 3.5.7.11.13 subgroup gets its maxima at 48.82085 ¢. With sigma = 1/2, the maxima is 48.82100 ¢.

	The Tenney–Euclidean regular temperement in the 3.5.7.11.13 subgroup mapped with [⟨39 57 69 85 91]] gives 48.82201 ¢.		The Tenney–Euclidean regular temperement in the 3.5.7.11.13 subgroup mapped with [⟨39 57 69 85 91]] gives 48.82201 ¢.

Contribution: /* Approximation to JI */

2025-10-27T18:11:26Z

Approximation to JI

@@ Line 350: / Line 350: @@
 == Approximation to JI ==
-=== No-2 zeta peak ===
+According to the finite Euler product with sigma = 1, the 3.5.7.11.13 subgroup gets its maxima at 48.82085 ¢. With sigma = 1/2, it gives 48.82100 ¢.
-Every 7 steps of the [[172edo|172f]] val is an excellent approximation of the ninth no-2 zeta peak in the 15-limit.
+The Tenney–Euclidean regular temperement in the 3.5.7.11.13 subgroup mapped with [⟨39 57 69 85 91]] gives 48.82201 ¢.
 == Music ==

Phanomium: /* Music */ Added my song Polygonal.

2025-06-30T13:30:41Z

Music: Added my song Polygonal.

← Older revision		Revision as of 13:30, 30 June 2025
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	; [[Francium]]		; [[Francium]]
	* [https://www.youtube.com/watch?v=jstg4_B0jfY ''Strange Juice''] (2025)		* [https://www.youtube.com/watch?v=jstg4_B0jfY ''Strange Juice''] (2025)
			;[https://www.youtube.com/@PhanomiumMusic Phanomium]
			* ''[https://www.youtube.com/watch?v=GX79ZX1Z8C8 Polygonal]'' (2025)

Francium: +music

2025-06-12T11:29:56Z

+music

← Older revision		Revision as of 11:29, 12 June 2025
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	Every 7 steps of the [[172edo\|172f]] val is an excellent approximation of the ninth no-2 zeta peak in the 15-limit.		Every 7 steps of the [[172edo\|172f]] val is an excellent approximation of the ninth no-2 zeta peak in the 15-limit.

			== Music ==
			; [[Francium]]
			* [https://www.youtube.com/watch?v=jstg4_B0jfY ''Strange Juice''] (2025)

ArrowHead294: /* Intervals */

2025-06-09T15:56:31Z

Intervals

← Older revision		Revision as of 15:56, 9 June 2025
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	! [[Cent]]s		! [[Cent]]s
	! [[Hekt]]s		! [[Hekt]]s
	! [[4L 5s (3/1-equivalent)\|Enneatonic]] degree		! [[4L 5s (3/1-equivalent)\|Enneatonic]]<br />degree
	! Corresponding~~<br />~~3.5.7.11.13 subgroup<br />intervals		! Corresponding 3.5.7.11.13 subgroup<br />intervals
	! [[Lambda ups and downs notation\|Lambda]]<br />(sLsLsLsLs,<br />{{nowrap\|J {{=}} 1/1}})		! [[Lambda ups and downs notation\|Lambda]]<br />(sLsLsLsLs,<br />{{nowrap\|J {{=}} 1/1}})
	! Mintaka[7]<br />(E macro-Phrygian)		! Mintaka[7]<br />(E macro-Phrygian)

Contribution: /* Approximation to JI */

2025-06-07T21:14:18Z

Approximation to JI

← Older revision		Revision as of 21:14, 7 June 2025
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	\|}		\|}

	Every 7 steps of the 172f val is an excellent approximation of the ninth no-2 zeta peak in the 15-limit.		Every 7 steps of the [[172edo\|172f]] val is an excellent approximation of the ninth no-2 zeta peak in the 15-limit.

Contribution: /* Approximation to JI */

2025-06-07T21:13:53Z

Approximation to JI

← Older revision		Revision as of 21:13, 7 June 2025
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	\|2.509907328		\|2.509907328
	\|}		\|}

			Every 7 steps of the 172f val is an excellent approximation of the ninth no-2 zeta peak in the 15-limit.