3159811edo - Revision history

BudjarnLambeth: Mathematical interest

2025-08-10T22:09:49Z

Mathematical interest

← Older revision		Revision as of 22:09, 10 August 2025
Line 1:		Line 1:
	{{~~Niche~~}}{{clear}}		{{Mathematical interest}}{{clear}}
	{{Infobox ET\|Consistency=65\|Distinct consistency=65}}		{{Infobox ET\|Consistency=65\|Distinct consistency=65}}
	{{ED intro}}		{{ED intro}}

ArrowHead294 at 15:56, 7 July 2025

2025-07-07T15:56:28Z

← Older revision		Revision as of 15:56, 7 July 2025
Line 14:		Line 14:
	== Scales ==		== Scales ==
	=== Harmonic scales ===		=== Harmonic scales ===
	3159811edo accurately approximates [[32afdo\|mode 32]] of the [[harmonic series]]. Additionally, unlike in [[10edo]]'s approximation of mode 4, [[87edo]]'s approximation of mode 8, or [[311edo]]'s approximation of mode 16, all interval pairs are distinguished.		As mentioned, 3159811edo accurately approximates [[32afdo\|mode 32]] of the [[harmonic series]]. Additionally, unlike in [[10edo]]'s approximation of [[4afdo\|mode 4]], [[87edo]]'s approximation of [[8afdo\|mode 8]], or [[311edo]]'s approximation of [[16afdo\|mode 16]], all interval pairs are distinguished.

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

ArrowHead294: /* Harmonic scales */

2025-07-07T15:54:37Z

Harmonic scales

← Older revision		Revision as of 15:54, 7 July 2025
Line 14:		Line 14:
	== Scales ==		== Scales ==
	=== Harmonic scales ===		=== Harmonic scales ===
	3159811edo accurately approximates [[32afdo\|mode 32]] of the [[harmonic series]]. Additionally, unlike in [[87edo]]'s approximation of mode 8 or [[311edo]]'s approximation of mode 16, all interval pairs are distinguished.		3159811edo accurately approximates [[32afdo\|mode 32]] of the [[harmonic series]]. Additionally, unlike in [[10edo]]'s approximation of mode 4, [[87edo]]'s approximation of mode 8, or [[311edo]]'s approximation of mode 16, all interval pairs are distinguished.

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

BudjarnLambeth: /* Harmonic scales */ add link

2025-06-05T03:31:46Z

Harmonic scales: add link

← Older revision		Revision as of 03:31, 5 June 2025
Line 14:		Line 14:
	== Scales ==		== Scales ==
	=== Harmonic scales ===		=== Harmonic scales ===
	3159811edo accurately approximates mode 32 of the [[harmonic series]]. Additionally, unlike in [[87edo]]'s approximation of mode 8 or [[311edo]]'s approximation of mode 16, all interval pairs are distinguished.		3159811edo accurately approximates [[32afdo\|mode 32]] of the [[harmonic series]]. Additionally, unlike in [[87edo]]'s approximation of mode 8 or [[311edo]]'s approximation of mode 16, all interval pairs are distinguished.

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

ArrowHead294: /* Harmonic scales */

2025-06-04T20:02:26Z

Harmonic scales

← Older revision		Revision as of 20:02, 4 June 2025
Line 40:		Line 40:
	\| 5/4		\| 5/4
	\|-		\|-
	! ~~… in~~ cents		! …in cents
	\| 0		\| 0
	\| 53.273		\| 53.273
Line 85:		Line 85:
	\| 3/2		\| 3/2
	\|-		\|-
	! ~~… in~~ cents		! …in cents
	\| 429.062		\| 429.062
	\| 470.781		\| 470.781

ArrowHead294: /* Scales */

2025-06-04T20:02:03Z

Scales

← Older revision		Revision as of 20:02, 4 June 2025
Line 41:		Line 41:
	\|-		\|-
	! … in cents		! … in cents
	\| 0.000		\| 0
	\| 53.273		\| 53.273
	\| 104.955		\| 104.955
Line 128:		Line 128:
	\| 7/4		\| 7/4
	\|-		\|-
	! ~~… in~~ cents		! …in cents
	\| 737.652		\| 737.652
	\| 772.627		\| 772.627
Line 171:		Line 171:
	\| 2/1		\| 2/1
	\|-		\|-
	! ~~… in~~ cents		! …in cents
	\| 999.468		\| 999.468
	\| 1029.577		\| 1029.577

ArrowHead294: /* Harmonic scales */

2025-06-04T19:32:31Z

Harmonic scales

← Older revision		Revision as of 19:32, 4 June 2025
Line 14:		Line 14:
	== Scales ==		== Scales ==
	=== Harmonic scales ===		=== Harmonic scales ===
	3159811edo accurately approximates mode 32 of the [[harmonic series]]. ~~All~~ interval pairs are distinguished.		3159811edo accurately approximates mode 32 of the [[harmonic series]]. Additionally, unlike in [[87edo]]'s approximation of mode 8 or [[311edo]]'s approximation of mode 16, all interval pairs are distinguished.

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

ArrowHead294 at 23:54, 3 June 2025

2025-06-03T23:54:37Z

← Older revision		Revision as of 23:54, 3 June 2025
Line 1:		Line 1:

	{{Niche}}{{clear}}		{{Niche}}{{clear}}
	{{Infobox ET\|Consistency=65\|Distinct consistency=65}}		{{Infobox ET\|Consistency=65\|Distinct consistency=65}}
Line 5:		Line 4:

	== Theory ==		== Theory ==
	Although its step size is far ~~beyond~~ the human melodic [[just-noticeable difference]], 3159811edo is [[consistent]] in the 65-odd-limit with a lower [[relative error]] than any previous equal temperaments in the 61-limit. It is the smallest edo which is purely consistent{{idio}} in the 63-odd-limit (i.e. does not exceed 25% relative error on the first 63 harmonics of the [[harmonic series]]).		Although its step size is far smaller than the human melodic [[just-noticeable difference]], 3159811edo is [[consistent]] in the 65-odd-limit with a lower [[relative error]] than any previous equal temperaments in the 61-limit. It is the smallest edo which is purely consistent{{idio}} in the 63-odd-limit (i.e. does not exceed 25% relative error on the first 63 harmonics of the [[harmonic series]]).

	While not practical to build an acoustic instrument for, one potential use of this system is in electronic music production, where free modulation between higher-limit JI intervals is desired. Instead of keeping track of the intervals directly, the number of steps to the octave for an interval could simply be added or subtracted from one note to get to the next. However, like all other equal temperaments, the consistency of this tuning is finite, and the sequence of intervals may eventually start to deviate from their true JI counterparts.		While not practical to build an acoustic instrument for, one potential use of this system is in electronic music production, where free modulation between higher-limit JI intervals is desired. Instead of keeping track of the intervals directly, the number of steps to the octave for an interval could simply be added or subtracted from one note to get to the next. However, like all other equal temperaments, the consistency of this tuning is finite, and the sequence of intervals may eventually start to deviate from their true JI counterparts.

ArrowHead294: /* Scales */

2025-06-03T22:41:09Z

Scales

← Older revision		Revision as of 22:41, 3 June 2025
Line 163:		Line 163:
	\|-		\|-
	! JI ratios		! JI ratios
	\| ~~999.468~~		\| 57/32
	\| ~~1029.577~~		\| 29/16
	\| ~~1059.172~~		\| 59/32
	\| ~~1088.269~~		\| 15/8
	\| ~~1116.885~~		\| 61/32
	\| ~~1145.036~~		\| 31/16
	\| ~~1172.736~~		\| 63/32
	\| ~~1200.000~~		\| 2/1
	\|-		\|-
	! … in cents		! … in cents

ArrowHead294: Testing

2025-06-03T11:49:00Z

Testing

← Older revision		Revision as of 11:49, 3 June 2025
Line 42:		Line 42:
	\|-		\|-
	! … in cents		! … in cents
	\| ~~{{cents\|1/1\|prec=3}}~~		\| 0.000
	\| ~~{{cents\|33/32\|prec=3}}~~		\| 53.273
	\| ~~{{cents\|17/16\|prec=3}}~~		\| 104.955
	\| ~~{{cents\|35/32\|prec=3}}~~		\| 155.140
	\| ~~{{cents\|9/8\|prec=3}}~~		\| 203.910
	\| ~~{{cents\|37/32\|prec=3}}~~		\| 251.344
	\| ~~{{cents\|19/16\|prec=3}}~~		\| 297.513
	\| ~~{{cents\|39/32\|prec=3}}~~		\| 342.483
	\| ~~{{cents\|5/4\|prec=3}}~~		\| 386.314
	\|-		\|-
	! Degrees in 3159811edo		! Degrees in 3159811edo
Line 87:		Line 87:
	\|-		\|-
	! … in cents		! … in cents
	\| ~~{{cents\|41/32\|prec=3}}~~		\| 429.062
	\| ~~{{cents\|21/16\|prec=3}}~~		\| 470.781
	\| ~~{{cents\|43/32\|prec=3}}~~		\| 511.518
	\| ~~{{cents\|11/8\|prec=3}}~~		\| 551.318
	\| ~~{{cents\|45/32\|prec=3}}~~		\| 590.224
	\| ~~{{cents\|23/16\|prec=3}}~~		\| 628.274
	\| ~~{{cents\|47/32\|prec=3}}~~		\| 665.507
	\| ~~{{cents\|3/2\|prec=3}}~~		\| 701.955
	\|-		\|-
	! Degrees in 3159811edo		! Degrees in 3159811edo
Line 130:		Line 130:
	\|-		\|-
	! … in cents		! … in cents
	\| ~~{{cents\|49/32\|prec=3}}~~		\| 737.652
	\| ~~{{cents\|25/16\|prec=3}}~~		\| 772.627
	\| ~~{{cents\|51/32\|prec=3}}~~		\| 806.910
	\| ~~{{cents\|13/8\|prec=3}}~~		\| 840.528
	\| ~~{{cents\|53/32\|prec=3}}~~		\| 873.505
	\| ~~{{cents\|27/16\|prec=3}}~~		\| 905.865
	\| ~~{{cents\|55/32\|prec=3}}~~		\| 937.632
	\| ~~{{cents\|7/4\|prec=3}}~~		\| 968.826
	\|-		\|-
	! Degrees in 3159811edo		! Degrees in 3159811edo
Line 163:		Line 163:
	\|-		\|-
	! JI ratios		! JI ratios
	\| ~~{{cents\|57/32\|prec=3}}~~		\| 999.468
	\| ~~{{cents\|29/16\|prec=3}}~~		\| 1029.577
	\| ~~{{cents\|59/32\|prec=3}}~~		\| 1059.172
	\| ~~{{cents\|15/8\|prec=3}}~~		\| 1088.269
	\| ~~{{cents\|61/32\|prec=3}}~~		\| 1116.885
	\| ~~{{cents\|31/16\|prec=3}}~~		\| 1145.036
	\| ~~{{cents\|63/32\|prec=3}}~~		\| 1172.736
	\| ~~{{cents\|2/1\|prec=3}}~~		\| 1200.000
	\|-		\|-
	! … in cents		! … in cents
Line 193:		Line 193:
	\|}		\|}

	* The scale in adjacent steps is 140277, 136089, 132144, 128421, 124902, 121571, 118413, 115415, 112565, 109852, 107267, 104801, 102446, 100194, 98039, 95975, 93996, 92097, 90273, 88520, 86834, 85211, 83647, 82140, 80686, 79283, 77927, 76618, 75351, 74126, 72940, 71791.		The scale in adjacent steps is 140277, 136089, 132144, 128421, 124902, 121571, 118413, 115415, 112565, 109852, 107267, 104801, 102446, 100194, 98039, 95975, 93996, 92097, 90273, 88520, 86834, 85211, 83647, 82140, 80686, 79283, 77927, 76618, 75351, 74126, 72940, 71791.