26edt - Revision history

BudjarnLambeth: /* Theory */ mention more temperaments

2026-03-23T04:31:53Z

Theory: mention more temperaments

← Older revision		Revision as of 04:31, 23 March 2026
Line 8:		Line 8:

	While retaining 13edt's mapping of primes 3, 5, and 7, 26edt adds an accurate prime 17 to the mix, tempering out [[2025/2023]] to split the [[BPS]] generator of [[9/7]] into two intervals of [[17/15]]. This 17/15 generates [[Dubhe]] temperament and mos scales of {{mos scalesig\|8L 1s<3/1>\|link=1}} and {{mos scalesig\|9L 8s<3/1>\|link=1}} that can be used as a simple traversal of 26edt. Among the 3.5.7.17-[[subgroup]] intervals, the accuracy of [[21/17]] should be highlighted, forming a 21-strong [[consistent circle]] that traverses the edt.		While retaining 13edt's mapping of primes 3, 5, and 7, 26edt adds an accurate prime 17 to the mix, tempering out [[2025/2023]] to split the [[BPS]] generator of [[9/7]] into two intervals of [[17/15]]. This 17/15 generates [[Dubhe]] temperament and mos scales of {{mos scalesig\|8L 1s<3/1>\|link=1}} and {{mos scalesig\|9L 8s<3/1>\|link=1}} that can be used as a simple traversal of 26edt. Among the 3.5.7.17-[[subgroup]] intervals, the accuracy of [[21/17]] should be highlighted, forming a 21-strong [[consistent circle]] that traverses the edt.

			26 also supports the temperaments: [[mizar]] (generators ~1097.8c, ~49.7c) and [[bohlenic]] (1\13edt, ~11/1).

	Additionally, while still far from perfect, 26edt does slightly improve upon 13edt's approximation of harmonics 11 and 13, which turns out to be sufficient to allow 26edt to be [[consistent]] to the no-twos [[21-odd-limit]], and is in fact the first edt to achieve this.		Additionally, while still far from perfect, 26edt does slightly improve upon 13edt's approximation of harmonics 11 and 13, which turns out to be sufficient to allow 26edt to be [[consistent]] to the no-twos [[21-odd-limit]], and is in fact the first edt to achieve this.

ArrowHead294: Replace {{scale link}} with {{mos scalesig}}

2025-03-08T00:27:39Z

Replace {{scale link}} with {{mos scalesig}}

← Older revision		Revision as of 00:27, 8 March 2025
Line 7:		Line 7:
	A reason to double 13edt to 26edt is to approximate the [[8/1\|8th]], [[13/1\|13th]], [[17/1\|17th]], [[20/1\|20th]], and [[22/1\|22nd]] [[harmonic]]s particularly well{{dubious}}. Moreover, it has an exaggerated [[5L 2s (3/1-equivalent)\|triatonic]] scale with 11:16:21 supermajor triads, though only the 16:11 is particularly just due to its best 16 still being 28.04 cents sharp, or just about as bad as the 25 of 12edo (which is 27.373 cents sharp, an essentially just 100:63).		A reason to double 13edt to 26edt is to approximate the [[8/1\|8th]], [[13/1\|13th]], [[17/1\|17th]], [[20/1\|20th]], and [[22/1\|22nd]] [[harmonic]]s particularly well{{dubious}}. Moreover, it has an exaggerated [[5L 2s (3/1-equivalent)\|triatonic]] scale with 11:16:21 supermajor triads, though only the 16:11 is particularly just due to its best 16 still being 28.04 cents sharp, or just about as bad as the 25 of 12edo (which is 27.373 cents sharp, an essentially just 100:63).

	While retaining 13edt's mapping of primes 3, 5, and 7, 26edt adds an accurate prime 17 to the mix, tempering out [[2025/2023]] to split the [[BPS]] generator of [[9/7]] into two intervals of [[17/15]]. This 17/15 generates [[Dubhe]] temperament and mos scales of {{sl\|8L 1s}} and {{sl\|9L 8s}} that can be used as a simple traversal of 26edt. Among the 3.5.7.17-[[subgroup]] intervals, the accuracy of [[21/17]] should be highlighted, forming a 21-strong [[consistent circle]] that traverses the edt.		While retaining 13edt's mapping of primes 3, 5, and 7, 26edt adds an accurate prime 17 to the mix, tempering out [[2025/2023]] to split the [[BPS]] generator of [[9/7]] into two intervals of [[17/15]]. This 17/15 generates [[Dubhe]] temperament and mos scales of {{mos scalesig\|8L 1s<3/1>\|link=1}} and {{mos scalesig\|9L 8s<3/1>\|link=1}} that can be used as a simple traversal of 26edt. Among the 3.5.7.17-[[subgroup]] intervals, the accuracy of [[21/17]] should be highlighted, forming a 21-strong [[consistent circle]] that traverses the edt.

	Additionally, while still far from perfect, 26edt does slightly improve upon 13edt's approximation of harmonics 11 and 13, which turns out to be sufficient to allow 26edt to be [[consistent]] to the no-twos [[21-odd-limit]], and is in fact the first edt to achieve this.		Additionally, while still far from perfect, 26edt does slightly improve upon 13edt's approximation of harmonics 11 and 13, which turns out to be sufficient to allow 26edt to be [[consistent]] to the no-twos [[21-odd-limit]], and is in fact the first edt to achieve this.

FloraC: Fix dead links, -redundant categories, and misc. cleanup

2025-02-10T17:13:40Z

Fix dead links, -redundant categories, and misc. cleanup

← Older revision		Revision as of 17:13, 10 February 2025
Line 5:		Line 5:
	26edt corresponds to 16.404…[[edo]]. It is [[contorted]] in the 7-limit, tempering out the same commas, [[245/243]] and [[3125/3087]], as [[13edt]]. In the 11-limit it tempers out 125/121 and 3087/3025, in the 13-limit 175/169, 147/143, and 847/845, and in the 17-limit 119/117. It is the seventh [[The Riemann zeta function and tuning#Removing primes\|zeta peak tritave division]].		26edt corresponds to 16.404…[[edo]]. It is [[contorted]] in the 7-limit, tempering out the same commas, [[245/243]] and [[3125/3087]], as [[13edt]]. In the 11-limit it tempers out 125/121 and 3087/3025, in the 13-limit 175/169, 147/143, and 847/845, and in the 17-limit 119/117. It is the seventh [[The Riemann zeta function and tuning#Removing primes\|zeta peak tritave division]].

	A reason to double 13edt to 26edt is to approximate the 8th, 13th, 17th, 20th, and 22nd ~~harmonics~~ particularly well{{~~Dubious~~}}. Moreover, it has an exaggerated ~~diatonic~~ scale with 11:16:21 supermajor triads, though only the 16:11 is particularly just due to its best 16 still being 28.04 cents sharp, or just about as bad as the 25 of 12edo (which is 27.373 cents sharp, an essentially just 100:63).		A reason to double 13edt to 26edt is to approximate the [[8/1\|8th]], [[13/1\|13th]], [[17/1\|17th]], [[20/1\|20th]], and [[22/1\|22nd]] [[harmonic]]s particularly well{{dubious}}. Moreover, it has an exaggerated [[5L 2s (3/1-equivalent)\|triatonic]] scale with 11:16:21 supermajor triads, though only the 16:11 is particularly just due to its best 16 still being 28.04 cents sharp, or just about as bad as the 25 of 12edo (which is 27.373 cents sharp, an essentially just 100:63).

	While retaining 13edt's mapping of primes 3, 5, and 7, 26edt adds an accurate prime 17 to the mix, tempering out [[2025/2023]] to split the [[~~Bohlen−Pierce−Stearns~~]] generator of [[9/7]] into two intervals of [[17/15]]. This 17/15 generates [[Dubhe]] temperament and ~~MOS~~ scales of {{sl\|8L 1s}} and {{sl\|9L 8s}} that can be used as a simple traversal of 26edt. Among the 3.5.7.17 [[subgroup]] intervals, the accuracy of [[21/17]] should be highlighted, forming a 21-strong [[consistent circle]] that traverses the edt.		While retaining 13edt's mapping of primes 3, 5, and 7, 26edt adds an accurate prime 17 to the mix, tempering out [[2025/2023]] to split the [[BPS]] generator of [[9/7]] into two intervals of [[17/15]]. This 17/15 generates [[Dubhe]] temperament and mos scales of {{sl\|8L 1s}} and {{sl\|9L 8s}} that can be used as a simple traversal of 26edt. Among the 3.5.7.17-[[subgroup]] intervals, the accuracy of [[21/17]] should be highlighted, forming a 21-strong [[consistent circle]] that traverses the edt.

	Additionally, while still far from perfect, 26edt does slightly improve upon 13edt's approximation of harmonics 11 and 13, which turns out to be sufficient to allow 26edt to be [[consistent]] to the no-twos 21-[[odd limit]], and is in fact the first edt to achieve this.		Additionally, while still far from perfect, 26edt does slightly improve upon 13edt's approximation of harmonics 11 and 13, which turns out to be sufficient to allow 26edt to be [[consistent]] to the no-twos [[21-odd-limit]], and is in fact the first edt to achieve this.
	{{Harmonics in equal\|26\|3\|1\|~~intervals~~=~~prime~~}}
			=== Harmonics ===
			{{Harmonics in equal\|26\|3\|1}}
			{{Harmonics in equal\|26\|3\|1\|columns=12\|start=12\|collapsed=true\|title=Approximation of harmonics in 26edt (continued)}}

	== Intervals ==		== Intervals ==
Line 19:		Line 22:
	! [[Hekt]]s		! [[Hekt]]s
	! [[4L 5s (3/1-equivalent)\|Enneatonic]] degree		! [[4L 5s (3/1-equivalent)\|Enneatonic]] degree
	! Corresponding<br />3.5.7.17 subgroup intervals		! Corresponding<br>3.5.7.17 subgroup intervals
	! Dubhe<br />(LLLLLLLLs,<br />J = 1/1)		! Dubhe<br>(LLLLLLLLs,<br />J = 1/1)
	! [[Lambda ups and downs notation\|Lambda]]<br />(sLsLsLsLs,<br />E = 1/1)		! [[Lambda ups and downs notation\|Lambda]]<br>(sLsLsLsLs,<br />E = 1/1)
	\|-		\|-
	\| 0		\| 0
Line 241:		Line 244:

	=== Connection to 26edo ===		=== Connection to 26edo ===
	It is a weird coincidence{{~~Dubious~~}} how 26edt intones many [[26edo]] intervals within ~~±6~~.5{{c}} when it is supposed to have nothing to do with this other tuning:		It is a weird coincidence{{dubious}} how 26edt intones many [[26edo]] intervals within ±6.5{{c}} when it is supposed to have nothing to do with this other tuning:

	{\| class="wikitable right-all"		{\| class="wikitable right-all"
Line 284:		Line 287:

	== Music ==		== Music ==
	* ''The Eel And Loach To Attack In Lasciviousness Are Insane'': [https://www.youtube.com/watch?v=AhWJ2yJsODs video] \| [http://micro.soonlabel.com/gene_ward_smith/Others/Omega9/Omega9%20-%20The%20Eel%20And%20Loach%20To%20Attack%20In%20Lasciviousness%20Are%20Insane.mp3 ~~MP3] by Omega9~~		; [[Omega9]]
			* ''The Eel And Loach To Attack In Lasciviousness Are Insane'' – [https://www.youtube.com/watch?v=AhWJ2yJsODs video] \| [https://web.archive.org/web/20201127012842/http://micro.soonlabel.com/gene_ward_smith/Others/Omega9/Omega9%20-%20The%20Eel%20And%20Loach%20To%20Attack%20In%20Lasciviousness%20Are%20Insane.mp3 play]
	~~[[Category:edt]]~~
	~~[[Category:tritave]]~~
	~~[[category:nonoctave]~~]

FloraC: Restore the corresponding edo

2025-02-10T17:00:16Z

Restore the corresponding edo

← Older revision		Revision as of 17:00, 10 February 2025
Line 2:		Line 2:
	{{ED intro}}		{{ED intro}}

	26edt is [[contorted]] in the 7-limit, tempering out the same commas, [[245/243]] and [[3125/3087]], as [[13edt]]. In the 11-limit it tempers out 125/121 and 3087/3025, in the 13-limit 175/169, 147/143, and 847/845, and in the 17-limit 119/117. It is the seventh [[The Riemann zeta function and tuning#Removing primes\|zeta peak tritave division]].		== Theory ==
			26edt corresponds to 16.404…[[edo]]. It is [[contorted]] in the 7-limit, tempering out the same commas, [[245/243]] and [[3125/3087]], as [[13edt]]. In the 11-limit it tempers out 125/121 and 3087/3025, in the 13-limit 175/169, 147/143, and 847/845, and in the 17-limit 119/117. It is the seventh [[The Riemann zeta function and tuning#Removing primes\|zeta peak tritave division]].

	A reason to double 13edt to 26edt is to approximate the 8th, 13th, 17th, 20th, and 22nd harmonics particularly well{{Dubious}}. Moreover, it has an exaggerated diatonic scale with 11:16:21 supermajor triads, though only the 16:11 is particularly just due to its best 16 still being 28.04 cents sharp, or just about as bad as the 25 of 12edo (which is 27.373 cents sharp, an essentially just 100:63).		A reason to double 13edt to 26edt is to approximate the 8th, 13th, 17th, 20th, and 22nd harmonics particularly well{{Dubious}}. Moreover, it has an exaggerated diatonic scale with 11:16:21 supermajor triads, though only the 16:11 is particularly just due to its best 16 still being 28.04 cents sharp, or just about as bad as the 25 of 12edo (which is 27.373 cents sharp, an essentially just 100:63).

	~~== Theory ==~~
	While retaining 13edt's mapping of primes 3, 5, and 7, 26edt adds an accurate prime 17 to the mix, tempering out [[2025/2023]] to split the [[Bohlen−Pierce−Stearns]] generator of [[9/7]] into two intervals of [[17/15]]. This 17/15 generates [[Dubhe]] temperament and MOS scales of {{sl\|8L 1s}} and {{sl\|9L 8s}} that can be used as a simple traversal of 26edt. Among the 3.5.7.17 [[subgroup]] intervals, the accuracy of [[21/17]] should be highlighted, forming a 21-strong [[consistent circle]] that traverses the edt.		While retaining 13edt's mapping of primes 3, 5, and 7, 26edt adds an accurate prime 17 to the mix, tempering out [[2025/2023]] to split the [[Bohlen−Pierce−Stearns]] generator of [[9/7]] into two intervals of [[17/15]]. This 17/15 generates [[Dubhe]] temperament and MOS scales of {{sl\|8L 1s}} and {{sl\|9L 8s}} that can be used as a simple traversal of 26edt. Among the 3.5.7.17 [[subgroup]] intervals, the accuracy of [[21/17]] should be highlighted, forming a 21-strong [[consistent circle]] that traverses the edt.

ArrowHead294 at 15:40, 10 February 2025

2025-02-10T15:40:13Z

← Older revision		Revision as of 15:40, 10 February 2025
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	26edt ~~divides the tritave ([[3/1]]) into 26 equal parts of 73.152 cents each, corresponding to 16.404[[edo]]. It~~ is [[contorted]] in the 7-limit, tempering out the same commas, [[245/243]] and [[3125/3087]], as [[13edt]]. In the 11-limit it tempers out 125/121 and 3087/3025, in the 13-limit 175/169, 147/143, and 847/845, and in the 17-limit 119/117. It is the seventh [[The Riemann zeta function and tuning#Removing primes\|zeta peak tritave division]].		{{ED intro}}

			26edt is [[contorted]] in the 7-limit, tempering out the same commas, [[245/243]] and [[3125/3087]], as [[13edt]]. In the 11-limit it tempers out 125/121 and 3087/3025, in the 13-limit 175/169, 147/143, and 847/845, and in the 17-limit 119/117. It is the seventh [[The Riemann zeta function and tuning#Removing primes\|zeta peak tritave division]].

	A reason to double 13edt to 26edt is to approximate the 8th, 13th, 17th, 20th, and 22nd harmonics particularly well{{Dubious}}. Moreover, it has an exaggerated diatonic scale with 11:16:21 supermajor triads, though only the 16:11 is particularly just due to its best 16 still being 28.04 cents sharp, or just about as bad as the 25 of 12edo (which is 27.373 cents sharp, an essentially just 100:63).		A reason to double 13edt to 26edt is to approximate the 8th, 13th, 17th, 20th, and 22nd harmonics particularly well{{Dubious}}. Moreover, it has an exaggerated diatonic scale with 11:16:21 supermajor triads, though only the 16:11 is particularly just due to its best 16 still being 28.04 cents sharp, or just about as bad as the 25 of 12edo (which is 27.373 cents sharp, an essentially just 100:63).

	== Theory ==		== Theory ==
	While retaining 13edt's mapping of primes 3, 5, and 7, 26edt adds an accurate prime 17 to the mix, tempering out [[2025/2023]] to split the [[Bohlen−Pierce−Stearns]] generator of [[9/7]] into two intervals of [[17/15]]. This 17/15 generates [[Dubhe]] temperament and MOS scales of {{sl\|8L 1s~~\|3/1~~}} and {{sl\|9L 8s~~\|3/1~~}} that can be used as a simple traversal of 26edt. Among the 3.5.7.17 [[subgroup]] intervals, the accuracy of [[21/17]] should be highlighted, forming a 21-strong [[consistent circle]] that traverses the edt.		While retaining 13edt's mapping of primes 3, 5, and 7, 26edt adds an accurate prime 17 to the mix, tempering out [[2025/2023]] to split the [[Bohlen−Pierce−Stearns]] generator of [[9/7]] into two intervals of [[17/15]]. This 17/15 generates [[Dubhe]] temperament and MOS scales of {{sl\|8L 1s}} and {{sl\|9L 8s}} that can be used as a simple traversal of 26edt. Among the 3.5.7.17 [[subgroup]] intervals, the accuracy of [[21/17]] should be highlighted, forming a 21-strong [[consistent circle]] that traverses the edt.

	Additionally, while still far from perfect, 26edt does slightly improve upon 13edt's approximation of harmonics 11 and 13, which turns out to be sufficient to allow 26edt to be [[consistent]] to the no-twos 21-[[odd limit]], and is in fact the first edt to achieve this.		Additionally, while still far from perfect, 26edt does slightly improve upon 13edt's approximation of harmonics 11 and 13, which turns out to be sufficient to allow 26edt to be [[consistent]] to the no-twos 21-[[odd limit]], and is in fact the first edt to achieve this.

ArrowHead294 at 16:19, 19 January 2025

2025-01-19T16:19:54Z

← Older revision		Revision as of 16:19, 19 January 2025
Line 239:		Line 239:

	=== Connection to 26edo ===		=== Connection to 26edo ===
			It is a weird coincidence{{Dubious}} how 26edt intones many [[26edo]] intervals within ±6.5{{c}} when it is supposed to have nothing to do with this other tuning:
	It is a weird coincidence {{Dubious}} how 26edt intones many [[26edo]] intervals within ~~plus or minus~~ 6.5 ~~cents~~ when it is supposed to have nothing to do with this other tuning:

	{\| class="wikitable right-all"		{\| class="wikitable right-all"

ArrowHead294 at 14:02, 9 January 2025

2025-01-09T14:02:16Z

Show changes

Lériendil: /* Intervals */

2024-10-11T22:03:03Z

Intervals

← Older revision		Revision as of 22:03, 11 October 2024
Line 19:		Line 19:
	! [[4L 5s (3/1-equivalent)\|Enneatonic]] degree		! [[4L 5s (3/1-equivalent)\|Enneatonic]] degree
	! Corresponding		! Corresponding
	3.5.7.17 subgroup ~~<br>~~		3.5.7.17 subgroup intervals
	intervals
	! Dubhe		! Dubhe
	(LLLLLLLLs, <br>		(LLLLLLLLs, <br>

Lériendil: /* Intervals */

2024-10-11T16:53:42Z

Intervals

← Older revision		Revision as of 16:53, 11 October 2024
Line 56:		Line 56:
	\| 150		\| 150
	\| N2		\| N2
	\| [[17/15]] (+2.8c)		\| [[135/119]] (+1.1c); [[17/15]] (+2.8c)
	\| K		\| K
	\| ^F, vF#, vGb		\| ^F, vF#, vGb
Line 104:		Line 104:
	\| 450		\| 450
	\| N4/sd5		\| N4/sd5
	\| [[51/35]] (+6.6c); [[25/17]] (-9.3c)		\| [[51/35]] (+6.6c); [[119/81]] (-7.6c); [[25/17]] (-9.3c)
	\| M		\| M
	\| ^H, vH#, vJb		\| ^H, vH#, vJb
Line 120:		Line 120:
	\| 550		\| 550
	\| Sa4/N5		\| Sa4/N5
	\| [[27/17]] (+3.8c)		\| [[119/75]] (+5.5c); [[27/17]] (+3.8c)
	\| Nb		\| Nb
	\| vJ, ^H#, ^Jb		\| vJ, ^H#, ^Jb
Line 152:		Line 152:
	\| 750		\| 750
	\| N6/sd7		\| N6/sd7
	\| [[17/9]] (-3.8c)		\| [[225/119]] (-5.5c); [[17/9]] (-3.8c)
	\| O		\| O
	\| ^A, vA#, vBb		\| ^A, vA#, vBb
Line 168:		Line 168:
	\| 850		\| 850
	\| Sa6/N7		\| Sa6/N7
	\| [[35/17]] (-6.6c); [[51/25]] (+9.3c)		\| [[35/17]] (-6.6c); [[243/119]] (+7.6c); [[51/25]] (+9.3c)
	\| Pb		\| Pb
	\| vB, ^A#, ^Bb		\| vB, ^A#, ^Bb
Line 216:		Line 216:
	\| 1150		\| 1150
	\| N9		\| N9
	\| [[45/17]] (-2.8c)		\| [[119/45]] (-1.1c); [[45/17]] (-2.8c)
	\| Rb		\| Rb
	\| vD, ^C#, ^Db		\| vD, ^C#, ^Db

Lériendil: /* Theory */

2024-10-11T16:23:01Z

Theory

← Older revision		Revision as of 16:23, 11 October 2024
Line 8:		Line 8:

	Additionally, while still far from perfect, 26edt does slightly improve upon 13edt's approximation of harmonics 11 and 13, which turns out to be sufficient to allow 26edt to be [[consistent]] to the no-twos 21-[[odd limit]], and is in fact the first edt to achieve this.		Additionally, while still far from perfect, 26edt does slightly improve upon 13edt's approximation of harmonics 11 and 13, which turns out to be sufficient to allow 26edt to be [[consistent]] to the no-twos 21-[[odd limit]], and is in fact the first edt to achieve this.
	{{Harmonics in equal\|26\|3\|1\|intervals=~~odd\|start=59~~}}		{{Harmonics in equal\|26\|3\|1\|intervals=prime}}

	== Intervals ==		== Intervals ==