15edt - Revision history

Fredg999: Text replacement - "[[The Riemann zeta function and tuning #" to "[[Riemann zeta function #"

2026-07-16T19:11:14Z

Text replacement - "[[The Riemann zeta function and tuning #" to "[[Riemann zeta function #"

← Older revision		Revision as of 19:11, 16 July 2026
Line 127:		Line 127:
	== JI approximation ==		== JI approximation ==
	=== Z function ===		=== Z function ===
	Below is a plot of the [[~~The~~ Riemann zeta function ~~and tuning~~ #Removing primes\|no-twos Z function]] in the vicinity of 15edt:		Below is a plot of the [[Riemann zeta function #Removing primes\|no-twos Z function]] in the vicinity of 15edt:

	[[File:15edt.png\|alt=15edt.png\|15edt.png]]		[[File:15edt.png\|alt=15edt.png\|15edt.png]]

FloraC: Text replacement - "octave" to "octave"

2026-07-14T14:15:51Z

Text replacement - "octave" to "octave"

← Older revision		Revision as of 14:15, 14 July 2026
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	== Theory ==		== Theory ==
	15edt corresponds to 9.4639…[[edo]]. It has [[harmonic]]s [[5/1\|5]] and [[13/1\|13]] closely in tune, but does not do so well for [[11/1\|11]], which is quite sharp. The main appeal of 15edt is that it allows for strong tritave equivalency, while supporting more conventional harmony. It achieves this with fantastic approximation of the [[4/1\|4th harmonic]], and terrible approximation of the [[~~2/1\|~~octave]]. In other words; 3:4:5 is available, but 4:5:6 is not. Like the octave, the [[7/1\|7th harmonic]] is about halfway between steps, so 6:7:8 is well approximated, but not 4:5:7. It also tempers out the syntonic comma, [[81/80]], in the 3.4.5 subgroup, as the major third is three perfect fourths below a tritave. As a 3.5.13-[[subgroup]] system, it tempers out [[2197/2187]] and [[3159/3125]], and if these commas are added, 15edt is related to the 2.3.5.13-subgroup temperament 19 & 123, which has a mapping {{mapping\| 1 0 0 0 \| 0 15 22 35 }}, where the generator, an approximate 27/25, has a [[POTE tuning]] of 126.773, very close to 15edt.		15edt corresponds to 9.4639…[[edo]]. It has [[harmonic]]s [[5/1\|5]] and [[13/1\|13]] closely in tune, but does not do so well for [[11/1\|11]], which is quite sharp. The main appeal of 15edt is that it allows for strong tritave equivalency, while supporting more conventional harmony. It achieves this with fantastic approximation of the [[4/1\|4th harmonic]], and terrible approximation of the [[octave]]. In other words; 3:4:5 is available, but 4:5:6 is not. Like the octave, the [[7/1\|7th harmonic]] is about halfway between steps, so 6:7:8 is well approximated, but not 4:5:7. It also tempers out the syntonic comma, [[81/80]], in the 3.4.5 subgroup, as the major third is three perfect fourths below a tritave. As a 3.5.13-[[subgroup]] system, it tempers out [[2197/2187]] and [[3159/3125]], and if these commas are added, 15edt is related to the 2.3.5.13-subgroup temperament 19 & 123, which has a mapping {{mapping\| 1 0 0 0 \| 0 15 22 35 }}, where the generator, an approximate 27/25, has a [[POTE tuning]] of 126.773, very close to 15edt.

	Using the patent val, it tempers out [[375/343]] and [[6561/6125]] in the 7-limit; [[81/77]], [[125/121]], and [[363/343]] in the 11-limit; [[65/63]], [[169/165]], [[585/539]], and [[1287/1225]] in the 13-limit; [[51/49]], [[121/119]], [[125/119]], [[189/187]], and [[195/187]] in the 17-limit (no-twos subgroup). With the patent [[4/1\|4]], it tempers out [[36/35]], [[64/63]], and 375/343 in the 3.4.5.7 subgroup; [[45/44]], [[80/77]], 81/77, and 363/343 in the 3.4.5.7.11 subgroup; [[52/49]], 65/63, [[65/64]], [[143/140]], and 169/165 in the 3.4.5.7.11.13 subgroup; 51/49, [[52/51]], [[85/84]], and 121/119 in the 3.4.5.7.11.13.17 subgroup ( that 15edt treated this way is essentially a retuning of [[19ed4]]). The [[kN subgroups\|215 subgroup]] of 15edt is 3.4.5.14.22.13.34, on which b15 tempers out the same commas as the patent val for [[30edt]].		Using the patent val, it tempers out [[375/343]] and [[6561/6125]] in the 7-limit; [[81/77]], [[125/121]], and [[363/343]] in the 11-limit; [[65/63]], [[169/165]], [[585/539]], and [[1287/1225]] in the 13-limit; [[51/49]], [[121/119]], [[125/119]], [[189/187]], and [[195/187]] in the 17-limit (no-twos subgroup). With the patent [[4/1\|4]], it tempers out [[36/35]], [[64/63]], and 375/343 in the 3.4.5.7 subgroup; [[45/44]], [[80/77]], 81/77, and 363/343 in the 3.4.5.7.11 subgroup; [[52/49]], 65/63, [[65/64]], [[143/140]], and 169/165 in the 3.4.5.7.11.13 subgroup; 51/49, [[52/51]], [[85/84]], and 121/119 in the 3.4.5.7.11.13.17 subgroup ( that 15edt treated this way is essentially a retuning of [[19ed4]]). The [[kN subgroups\|215 subgroup]] of 15edt is 3.4.5.14.22.13.34, on which b15 tempers out the same commas as the patent val for [[30edt]].

Lucius Chiaraviglio: /* Music */ Insert Instruments section before this, now that a Lumatone mapping for 15edt is available

2026-03-07T09:12:06Z

Music: Insert Instruments section before this, now that a Lumatone mapping for 15edt is available

← Older revision		Revision as of 09:12, 7 March 2026
Line 135:		Line 135:

	A short composition by [[User:Unque\|Unque]].		A short composition by [[User:Unque\|Unque]].

			== Instruments ==
			A [[Lumatone mapping for 15edt]] is now available.

	== Music ==		== Music ==

Lucius Chiaraviglio: /* Music */ Add Bryan Deister's ''15ed3 improv'' (2026)

2026-03-05T06:45:25Z

Music: Add Bryan Deister's ''15ed3 improv'' (2026)

← Older revision		Revision as of 06:45, 5 March 2026
Line 137:		Line 137:

	== Music ==		== Music ==
			; [[Bryan Deister]]
			* [https://www.youtube.com/shorts/THjsSiEjdvs ''15ed3 improv''] (2026)

	; [[nationalsolipsism]]		; [[nationalsolipsism]]
	* [https://www.youtube.com/watch?v=bC_Pc4jKm2k ''ox-idation''] (2012)		* [https://www.youtube.com/watch?v=bC_Pc4jKm2k ''ox-idation''] (2012)

	[[Category:Macrotonal]]		[[Category:Macrotonal]]

Squib at 13:34, 20 June 2025

2025-06-20T13:34:34Z

← Older revision		Revision as of 13:34, 20 June 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	15edt corresponds to 9.4639…[[edo]]. It has [[harmonic]]s [[5/1\|5]] and [[13/1\|13]] closely in tune, but does not do so well for [[11/1\|11]], which is quite sharp. The main appeal of 15edt is that it allows for strong tritave equivalency, while supporting more conventional harmony. It achieves this with fantastic approximation of the [[4/1\|4th harmonic]], and terrible approximation of the [[2/1\|octave]]. In other words; 3:4:5 is available, but 4:5:6 is not. Like the octave, the [[7/1\|7th harmonic]] is about halfway between steps, so 6:7:8 is well approximated, but not 4:6:7. It also tempers out the syntonic comma, [[81/80]], in the 3.4.5 subgroup, as the major third is three perfect fourths below a tritave. As a 3.5.13-[[subgroup]] system, it tempers out [[2197/2187]] and [[3159/3125]], and if these commas are added, 15edt is related to the 2.3.5.13-subgroup temperament 19 & 123, which has a mapping {{mapping\| 1 0 0 0 \| 0 15 22 35 }}, where the generator, an approximate 27/25, has a [[POTE tuning]] of 126.773, very close to 15edt.		15edt corresponds to 9.4639…[[edo]]. It has [[harmonic]]s [[5/1\|5]] and [[13/1\|13]] closely in tune, but does not do so well for [[11/1\|11]], which is quite sharp. The main appeal of 15edt is that it allows for strong tritave equivalency, while supporting more conventional harmony. It achieves this with fantastic approximation of the [[4/1\|4th harmonic]], and terrible approximation of the [[2/1\|octave]]. In other words; 3:4:5 is available, but 4:5:6 is not. Like the octave, the [[7/1\|7th harmonic]] is about halfway between steps, so 6:7:8 is well approximated, but not 4:5:7. It also tempers out the syntonic comma, [[81/80]], in the 3.4.5 subgroup, as the major third is three perfect fourths below a tritave. As a 3.5.13-[[subgroup]] system, it tempers out [[2197/2187]] and [[3159/3125]], and if these commas are added, 15edt is related to the 2.3.5.13-subgroup temperament 19 & 123, which has a mapping {{mapping\| 1 0 0 0 \| 0 15 22 35 }}, where the generator, an approximate 27/25, has a [[POTE tuning]] of 126.773, very close to 15edt.

	Using the patent val, it tempers out [[375/343]] and [[6561/6125]] in the 7-limit; [[81/77]], [[125/121]], and [[363/343]] in the 11-limit; [[65/63]], [[169/165]], [[585/539]], and [[1287/1225]] in the 13-limit; [[51/49]], [[121/119]], [[125/119]], [[189/187]], and [[195/187]] in the 17-limit (no-twos subgroup). With the patent [[4/1\|4]], it tempers out [[36/35]], [[64/63]], and 375/343 in the 3.4.5.7 subgroup; [[45/44]], [[80/77]], 81/77, and 363/343 in the 3.4.5.7.11 subgroup; [[52/49]], 65/63, [[65/64]], [[143/140]], and 169/165 in the 3.4.5.7.11.13 subgroup; 51/49, [[52/51]], [[85/84]], and 121/119 in the 3.4.5.7.11.13.17 subgroup ( that 15edt treated this way is essentially a retuning of [[19ed4]]). The [[kN subgroups\|215 subgroup]] of 15edt is 3.4.5.14.22.13.34, on which b15 tempers out the same commas as the patent val for [[30edt]].		Using the patent val, it tempers out [[375/343]] and [[6561/6125]] in the 7-limit; [[81/77]], [[125/121]], and [[363/343]] in the 11-limit; [[65/63]], [[169/165]], [[585/539]], and [[1287/1225]] in the 13-limit; [[51/49]], [[121/119]], [[125/119]], [[189/187]], and [[195/187]] in the 17-limit (no-twos subgroup). With the patent [[4/1\|4]], it tempers out [[36/35]], [[64/63]], and 375/343 in the 3.4.5.7 subgroup; [[45/44]], [[80/77]], 81/77, and 363/343 in the 3.4.5.7.11 subgroup; [[52/49]], 65/63, [[65/64]], [[143/140]], and 169/165 in the 3.4.5.7.11.13 subgroup; 51/49, [[52/51]], [[85/84]], and 121/119 in the 3.4.5.7.11.13.17 subgroup ( that 15edt treated this way is essentially a retuning of [[19ed4]]). The [[kN subgroups\|215 subgroup]] of 15edt is 3.4.5.14.22.13.34, on which b15 tempers out the same commas as the patent val for [[30edt]].

Lériendil: /* Theory */

2025-06-19T04:18:27Z

Theory

← Older revision		Revision as of 04:18, 19 June 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	15edt corresponds to 9.4639…[[edo]]. It has [[harmonic]]s [[5/1\|5]] and [[13/1\|13]] closely in tune, but does not do so well for [[11/1\|11]], which is quite sharp. The main appeal of 15edt is that it allows for strong tritave equivalency, while supporting more conventional harmony. It achieves this with fantastic approximation of the [[4/1\|4th harmonic]], and terrible approximation of the [[2/1\|octave]]. In other words; 3:4:5 is available, but 4:5:6 is not. Like the octave, the [[7/1\|7th harmonic]] is about halfway between steps, so 6:7:8 is well approximated, but not 4:6:7.		15edt corresponds to 9.4639…[[edo]]. It has [[harmonic]]s [[5/1\|5]] and [[13/1\|13]] closely in tune, but does not do so well for [[11/1\|11]], which is quite sharp. The main appeal of 15edt is that it allows for strong tritave equivalency, while supporting more conventional harmony. It achieves this with fantastic approximation of the [[4/1\|4th harmonic]], and terrible approximation of the [[2/1\|octave]]. In other words; 3:4:5 is available, but 4:5:6 is not. Like the octave, the [[7/1\|7th harmonic]] is about halfway between steps, so 6:7:8 is well approximated, but not 4:6:7. It also tempers out the syntonic comma, [[81/80]], in the 3.4.5 subgroup, as the major third is three perfect fourths below a tritave. As a 3.5.13-[[subgroup]] system, it tempers out [[2197/2187]] and [[3159/3125]], and if these commas are added, 15edt is related to the 2.3.5.13-subgroup temperament 19 & 123, which has a mapping {{mapping\| 1 0 0 0 \| 0 15 22 35 }}, where the generator, an approximate 27/25, has a [[POTE tuning]] of 126.773, very close to 15edt.

	~~15edt is associated with~~ [[~~tempering out~~]] the ~~mowgli comma~~, ~~{{monzo\| 0 22~~ -~~15 }}~~ in the 5-limit, ~~which fixes~~ [[5/3]] ~~to 7\15edt; in an octave context~~, ~~this temperament is supported by~~ [[~~19edo~~]] ~~but has an~~ [[~~optimal patent val~~]] of [[~~303edo~~]]. ~~It also~~ tempers out the ~~syntonic comma~~, ~~{{monzo\| -4 4 -1 }}~~, as the ~~major third is three perfect fourths below a tritave~~. ~~As a 3~~.5.~~13-~~[[~~subgroup~~]] ~~system~~, ~~it tempers out~~ [[~~2197~~/~~2187~~]] ~~and~~ [[~~3159~~/~~3125~~]], and ~~if these commas are added, 15edt is related to~~ the 2.3.5.13-subgroup ~~temperament 19 & 123~~, ~~which has a mapping {{mapping\| 1 0 0 0 \| 0 15 22 35 }}~~, ~~where the generator~~, ~~an approximate 27~~/~~25, has~~ a [[~~POTE tuning~~]] of ~~126~~.~~773~~, ~~very close to 15edt~~.		Using the patent val, it tempers out [[375/343]] and [[6561/6125]] in the 7-limit; [[81/77]], [[125/121]], and [[363/343]] in the 11-limit; [[65/63]], [[169/165]], [[585/539]], and [[1287/1225]] in the 13-limit; [[51/49]], [[121/119]], [[125/119]], [[189/187]], and [[195/187]] in the 17-limit (no-twos subgroup). With the patent [[4/1\|4]], it tempers out [[36/35]], [[64/63]], and 375/343 in the 3.4.5.7 subgroup; [[45/44]], [[80/77]], 81/77, and 363/343 in the 3.4.5.7.11 subgroup; [[52/49]], 65/63, [[65/64]], [[143/140]], and 169/165 in the 3.4.5.7.11.13 subgroup; 51/49, [[52/51]], [[85/84]], and 121/119 in the 3.4.5.7.11.13.17 subgroup ( that 15edt treated this way is essentially a retuning of [[19ed4]]). The [[kN subgroups\|215 subgroup]] of 15edt is 3.4.5.14.22.13.34, on which b15 tempers out the same commas as the patent val for [[30edt]].

	~~Using the patent val, it tempers out~~ [[~~375/343]] and [[6561/6125~~]] in the ~~7-limit; [[81/77]]~~, ~~[[125/121]], and [[363/343]] in the 11~~-~~limit; [[65/63]], [[169/165]], [[585/539]], and [[1287/1225]]~~ in the ~~13-limit;~~ [[~~51/49]], [[121/119]], [[125/119]], [[189/187]], and [[195/187]] in the 17~~-limit ~~(no-twos subgroup). With the patent [[4/1\|4~~]], ~~it tempers out~~ [[~~36/35]], [[64/63]], and 375~~/~~343 in the~~ 3~~.4.5.7 subgroup; [[45/44~~]]~~, [[80/77]], 81/77, and 363/343 in the 3.4.5.~~7~~.11 subgroup~~; ~~[[52/49]], 65/63, [[65/64]], [[143/140]], and 169/165~~ in ~~the 3.4.5.7.11.13 subgroup; 51/49~~, ~~[[52/51]], [[85/84]], and 121/119 in the 3.4.5.7.11.13.17 subgroup ( that 15edt treated~~ this ~~way~~ is ~~essentially a retuning of~~ [[~~19ed4~~]]~~). The~~ [[~~kN subgroups\|215 subgroup~~]] of ~~15edt is 3.4.5.14.22.13.34, on which b15 tempers out the same commas as the patent val for~~ [[~~30edt~~]].		15edt is also associated with [[tempering out]] the mowgli comma, {{monzo\| 0 22 -15 }} in the [[5-limit]], which fixes [[5/3]] to 7\15edt; in an octave context, this temperament is supported by [[19edo]] but has an [[optimal patent val]] of [[303edo]].

	=== Harmonics ===		=== Harmonics ===

Squib at 02:48, 19 June 2025

2025-06-19T02:48:44Z

← Older revision		Revision as of 02:48, 19 June 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	15edt corresponds to 9.4639…[[edo]]. It has [[harmonic]]s [[5/1\|5]] and [[13/1\|13]] closely in tune, but does not do so well for ~~[[7/1\|7]] and~~ [[11/1\|11]], which ~~are~~ quite sharp. The main appeal of 15edt is that it allows for strong tritave equivalency, while supporting more conventional harmony. It achieves this with fantastic approximation of the [[4/1\|4th harmonic]], and terrible approximation of the [[2/1\|octave]]. In other words; 3:4:5 is available, but 4:5:6 is not.		15edt corresponds to 9.4639…[[edo]]. It has [[harmonic]]s [[5/1\|5]] and [[13/1\|13]] closely in tune, but does not do so well for [[11/1\|11]], which is quite sharp. The main appeal of 15edt is that it allows for strong tritave equivalency, while supporting more conventional harmony. It achieves this with fantastic approximation of the [[4/1\|4th harmonic]], and terrible approximation of the [[2/1\|octave]]. In other words; 3:4:5 is available, but 4:5:6 is not. Like the octave, the [[7/1\|7th harmonic]] is about halfway between steps, so 6:7:8 is well approximated, but not 4:6:7.

	15edt is associated with [[tempering out]] the mowgli comma, {{monzo\| 0 22 -15 }} in the 5-limit, which fixes [[5/3]] to 7\15edt; in an octave context, this temperament is supported by [[19edo]] but has an [[optimal patent val]] of [[303edo]]. As a 3.5.13-[[subgroup]] system, it tempers out [[2197/2187]] and [[3159/3125]], and if these commas are added, 15edt is related to the 2.3.5.13-subgroup temperament 19 & 123, which has a mapping {{mapping\| 1 0 0 0 \| 0 15 22 35 }}, where the generator, an approximate 27/25, has a [[POTE tuning]] of 126.773, very close to 15edt.		15edt is associated with [[tempering out]] the mowgli comma, {{monzo\| 0 22 -15 }} in the 5-limit, which fixes [[5/3]] to 7\15edt; in an octave context, this temperament is supported by [[19edo]] but has an [[optimal patent val]] of [[303edo]]. It also tempers out the syntonic comma, {{monzo\| -4 4 -1 }}, as the major third is three perfect fourths below a tritave. As a 3.5.13-[[subgroup]] system, it tempers out [[2197/2187]] and [[3159/3125]], and if these commas are added, 15edt is related to the 2.3.5.13-subgroup temperament 19 & 123, which has a mapping {{mapping\| 1 0 0 0 \| 0 15 22 35 }}, where the generator, an approximate 27/25, has a [[POTE tuning]] of 126.773, very close to 15edt.

	Using the patent val, it tempers out [[375/343]] and [[6561/6125]] in the 7-limit; [[81/77]], [[125/121]], and [[363/343]] in the 11-limit; [[65/63]], [[169/165]], [[585/539]], and [[1287/1225]] in the 13-limit; [[51/49]], [[121/119]], [[125/119]], [[189/187]], and [[195/187]] in the 17-limit (no-twos subgroup). With the patent [[4/1\|4]], it tempers out [[36/35]], [[64/63]], and 375/343 in the 3.4.5.7 subgroup; [[45/44]], [[80/77]], 81/77, and 363/343 in the 3.4.5.7.11 subgroup; [[52/49]], 65/63, [[65/64]], [[143/140]], and 169/165 in the 3.4.5.7.11.13 subgroup; 51/49, [[52/51]], [[85/84]], and 121/119 in the 3.4.5.7.11.13.17 subgroup ( that 15edt treated this way is essentially a retuning of [[19ed4]]). The [[kN subgroups\|215 subgroup]] of 15edt is 3.4.5.14.22.13.34, on which b15 tempers out the same commas as the patent val for [[30edt]].		Using the patent val, it tempers out [[375/343]] and [[6561/6125]] in the 7-limit; [[81/77]], [[125/121]], and [[363/343]] in the 11-limit; [[65/63]], [[169/165]], [[585/539]], and [[1287/1225]] in the 13-limit; [[51/49]], [[121/119]], [[125/119]], [[189/187]], and [[195/187]] in the 17-limit (no-twos subgroup). With the patent [[4/1\|4]], it tempers out [[36/35]], [[64/63]], and 375/343 in the 3.4.5.7 subgroup; [[45/44]], [[80/77]], 81/77, and 363/343 in the 3.4.5.7.11 subgroup; [[52/49]], 65/63, [[65/64]], [[143/140]], and 169/165 in the 3.4.5.7.11.13 subgroup; 51/49, [[52/51]], [[85/84]], and 121/119 in the 3.4.5.7.11.13.17 subgroup ( that 15edt treated this way is essentially a retuning of [[19ed4]]). The [[kN subgroups\|215 subgroup]] of 15edt is 3.4.5.14.22.13.34, on which b15 tempers out the same commas as the patent val for [[30edt]].

Wendy gunk: Rearranged some stuff, and added a simpler explanation as to what 15edt's quirks are before the comma infodump

2025-04-08T17:46:48Z

Rearranged some stuff, and added a simpler explanation as to what 15edt's quirks are before the comma infodump

← Older revision		Revision as of 17:46, 8 April 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	15edt corresponds to 9.4639…[[edo]]. It has [[harmonic]]s [[5/1\|5]] and [[13/1\|13]] closely in tune, but does not do so well for [[7/1\|7]] and [[11/1\|11]], which are quite sharp. It ~~is associated~~ with [[~~tempering out~~]] ~~the mowgli comma~~, ~~{{monzo\| 0 22 -15 }} in~~ the ~~5-limit, which fixes~~ [[5/~~3]] to 7\15edt; in an~~ octave ~~context, this temperament is supported by [[19edo]] but has an [[optimal patent val]] of [[303edo~~]]. ~~As a~~ 3.5~~.13-[[subgroup]] system~~, ~~it tempers out [[2197/2187]] and [[3159/3125]], and if these commas are added, 15edt~~ is related to the 2.3.5.13-subgroup temperament 19 & 123, which has a mapping {{mapping\| 1 0 0 0 \| 0 15 22 35 }}, where the generator, an approximate 27/25, has a [[POTE tuning]] of 126.773, very close to 15edt.		15edt corresponds to 9.4639…[[edo]]. It has [[harmonic]]s [[5/1\|5]] and [[13/1\|13]] closely in tune, but does not do so well for [[7/1\|7]] and [[11/1\|11]], which are quite sharp. The main appeal of 15edt is that it allows for strong tritave equivalency, while supporting more conventional harmony. It achieves this with fantastic approximation of the [[4/1\|4th harmonic]], and terrible approximation of the [[2/1\|octave]]. In other words; 3:4:5 is available, but 4:5:6 is not.

	~~Using the patent val, it tempers out~~ [[~~375/343]] and [[6561/6125~~]] in the 7-limit; [[81/77]], [[~~125/121~~]]~~, and~~ [[~~363/343~~]] ~~in the 11-limit;~~ [[~~65/63~~]], [[~~169/165~~]], [[~~585~~/~~539~~]], and [[~~1287~~/~~1225~~]] in the 13-~~limit; [[51/49]]~~, ~~[[121/119]]~~, ~~[[125~~/~~119]]~~, [[~~189/187~~]], ~~and [[195/187]] in the 17-limit (no-twos subgroup)~~.		15edt is associated with [[tempering out]] the mowgli comma, {{monzo\| 0 22 -15 }} in the 5-limit, which fixes [[5/3]] to 7\15edt; in an octave context, this temperament is supported by [[19edo]] but has an [[optimal patent val]] of [[303edo]]. As a 3.5.13-[[subgroup]] system, it tempers out [[2197/2187]] and [[3159/3125]], and if these commas are added, 15edt is related to the 2.3.5.13-subgroup temperament 19 & 123, which has a mapping {{mapping\| 1 0 0 0 \| 0 15 22 35 }}, where the generator, an approximate 27/25, has a [[POTE tuning]] of 126.773, very close to 15edt.

	~~15edt is also notable for having a good approximation of~~ the ~~4th harmonic or double octave~~. With the patent [[4/1\|4]], it tempers out [[36/35]], [[64/63]], and 375/343 in the 3.4.5.7 subgroup; [[45/44]], [[80/77]], 81/77, and 363/343 in the 3.4.5.7.11 subgroup; [[52/49]], 65/63, [[65/64]], [[143/140]], and 169/165 in the 3.4.5.7.11.13 subgroup; 51/49, [[52/51]], [[85/84]], and 121/119 in the 3.4.5.7.11.13.17 subgroup (~~note~~ that 15edt treated this way is essentially a retuning of [[19ed4]]). The [[kN subgroups\|215 subgroup]] of 15edt is 3.4.5.14.22.13.34, on which b15 tempers out the same commas as the patent val for [[30edt]].		Using the patent val, it tempers out [[375/343]] and [[6561/6125]] in the 7-limit; [[81/77]], [[125/121]], and [[363/343]] in the 11-limit; [[65/63]], [[169/165]], [[585/539]], and [[1287/1225]] in the 13-limit; [[51/49]], [[121/119]], [[125/119]], [[189/187]], and [[195/187]] in the 17-limit (no-twos subgroup). With the patent [[4/1\|4]], it tempers out [[36/35]], [[64/63]], and 375/343 in the 3.4.5.7 subgroup; [[45/44]], [[80/77]], 81/77, and 363/343 in the 3.4.5.7.11 subgroup; [[52/49]], 65/63, [[65/64]], [[143/140]], and 169/165 in the 3.4.5.7.11.13 subgroup; 51/49, [[52/51]], [[85/84]], and 121/119 in the 3.4.5.7.11.13.17 subgroup ( that 15edt treated this way is essentially a retuning of [[19ed4]]). The [[kN subgroups\|215 subgroup]] of 15edt is 3.4.5.14.22.13.34, on which b15 tempers out the same commas as the patent val for [[30edt]].

	=== Harmonics ===		=== Harmonics ===

Lériendil: cleanup 2

2025-02-10T15:04:56Z

cleanup 2

← Older revision		Revision as of 15:04, 10 February 2025
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	== Theory ==		== Theory ==
	15edt corresponds to 9.4639…[[edo]]. It has [[harmonic]]s [[5/1\|5]] and [[13/1\|13]] closely in tune, but does not do so well for [[7/1\|7]] and [[11/1\|11]], which are quite sharp. It [[tempering ~~out\|tempers~~ out]] the mowgli comma, {{monzo\| 0 22 -15 }} in the 5-limit, which is ~~tempered out~~ by [[19edo]] but has an [[optimal patent val]] of [[303edo]]. As a 3.5.13-[[subgroup]] system, it tempers out [[2197/2187]] and [[3159/3125]]~~. Using the patent val~~, ~~it tempers out [[375/343]]~~ and ~~[[6561/6125]] in the 7-limit; [[81/77]], [[125/121]]~~, and [[363/343]] in the 11-limit; [[65/63]], [[169/165]], [[585/539]], and [[1287/1225]] in the 13-limit; [[51/49]], [[121/119]], [[125/119]], [[189/187]], and [[195/187]] in the 17-limit (no-twos subgroup). 15edt is related to the 2.3.5.13-subgroup temperament 19 & 123, which has a mapping {{mapping\| 1 0 0 0 \| 0 15 22 35 }}, where the generator, an approximate 27/25, has a [[POTE tuning]] of 126.773, very close to 15edt.		15edt corresponds to 9.4639…[[edo]]. It has [[harmonic]]s [[5/1\|5]] and [[13/1\|13]] closely in tune, but does not do so well for [[7/1\|7]] and [[11/1\|11]], which are quite sharp. It is associated with [[tempering out]] the mowgli comma, {{monzo\| 0 22 -15 }} in the 5-limit, which fixes [[5/3]] to 7\15edt; in an octave context, this temperament is supported by [[19edo]] but has an [[optimal patent val]] of [[303edo]]. As a 3.5.13-[[subgroup]] system, it tempers out [[2197/2187]] and [[3159/3125]], and if these commas are added, 15edt is related to the 2.3.5.13-subgroup temperament 19 & 123, which has a mapping {{mapping\| 1 0 0 0 \| 0 15 22 35 }}, where the generator, an approximate 27/25, has a [[POTE tuning]] of 126.773, very close to 15edt.

	With the patent [[4/1\|4]], it tempers out [[36/35]], [[64/63]], and 375/343 in the 4.3.5.7 subgroup; [[45/44]], [[80/77]], 81/77, and 363/343 in the 4.3.5.7.11 subgroup; [[52/49]], 65/63, [[65/64]], [[143/140]], and 169/165 in the 4.3.5.7.11.13 subgroup; 51/49, [[52/51]], [[85/84]], and 121/119 in the 4.3.5.7.11.13.17 subgroup (~~as well as~~ [[19ed4]]). The [[kN subgroups\|215 subgroup]] of 15edt is 4.3.5.14.22.13.34, on which b15 tempers out the same commas as the patent val for [[30edt]].		Using the patent val, it tempers out [[375/343]] and [[6561/6125]] in the 7-limit; [[81/77]], [[125/121]], and [[363/343]] in the 11-limit; [[65/63]], [[169/165]], [[585/539]], and [[1287/1225]] in the 13-limit; [[51/49]], [[121/119]], [[125/119]], [[189/187]], and [[195/187]] in the 17-limit (no-twos subgroup).

			15edt is also notable for having a good approximation of the 4th harmonic or double octave. With the patent [[4/1\|4]], it tempers out [[36/35]], [[64/63]], and 375/343 in the 3.4.5.7 subgroup; [[45/44]], [[80/77]], 81/77, and 363/343 in the 3.4.5.7.11 subgroup; [[52/49]], 65/63, [[65/64]], [[143/140]], and 169/165 in the 3.4.5.7.11.13 subgroup; 51/49, [[52/51]], [[85/84]], and 121/119 in the 3.4.5.7.11.13.17 subgroup (note that 15edt treated this way is essentially a retuning of [[19ed4]]). The [[kN subgroups\|215 subgroup]] of 15edt is 3.4.5.14.22.13.34, on which b15 tempers out the same commas as the patent val for [[30edt]].

	=== Harmonics ===		=== Harmonics ===

FloraC: Cleanup

2025-02-10T10:12:00Z

Cleanup

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