8539edo - Revision history

Eufalesio: Added Sagittal notation

2026-01-20T20:27:53Z

Added Sagittal notation

FloraC: /* Theory */ consolidation

2026-01-05T10:19:39Z

Theory: consolidation

← Older revision		Revision as of 10:19, 5 January 2026
Line 5:		Line 5:
	While it may strike many people as too large to be practical, 8539edo has seen actual use as a bookkeeping device to keep track of higher-limit intervals which have been allowed to freely modulate, and has been proposed as a [[interval size measure\|unit of interval measure]], the '''tina'''. This is because it is a very strong higher-limit system, [[consistency\|distinctly consistent]] through the [[27-odd-limit]]. It is a [[the Riemann zeta function and tuning #Zeta edo lists\|strict zeta]] tuning, and is also the first [[trivial temperament\|non-trivial]] edo to be consistent in the [[odd prime sum limit\|27-odd-prime-sum-limit]]. In the [[13-limit]], the only smaller systems with a lower logflat badness are {{EDOs\| 72, 270, 494, 5585 and 6079 }}; in the [[17-limit]], that becomes {{EDOs\| 72, 494, 1506, 3395 and 7033 }}. In the [[19-limit]], where it really shines, nothing beats it in terms of logflat badness until [[20203edo\|20203]].		While it may strike many people as too large to be practical, 8539edo has seen actual use as a bookkeeping device to keep track of higher-limit intervals which have been allowed to freely modulate, and has been proposed as a [[interval size measure\|unit of interval measure]], the '''tina'''. This is because it is a very strong higher-limit system, [[consistency\|distinctly consistent]] through the [[27-odd-limit]]. It is a [[the Riemann zeta function and tuning #Zeta edo lists\|strict zeta]] tuning, and is also the first [[trivial temperament\|non-trivial]] edo to be consistent in the [[odd prime sum limit\|27-odd-prime-sum-limit]]. In the [[13-limit]], the only smaller systems with a lower logflat badness are {{EDOs\| 72, 270, 494, 5585 and 6079 }}; in the [[17-limit]], that becomes {{EDOs\| 72, 494, 1506, 3395 and 7033 }}. In the [[19-limit]], where it really shines, nothing beats it in terms of logflat badness until [[20203edo\|20203]].

	It tempers out the [[~~Senior~~\|~~Senior~~ comma]] in the [[5-limit]], [[~~Very~~ high accuracy temperaments#Sepbizo-asegu (171 & 3566 & 4973)\|[-3 2 -17 ~~14⟩~~]] in the [[7-limit]], [[3294225/3294172]] in the [[11-limit]].		Some of the simpler commas it [[tempering out\|tempers out]] include the [[senior\|senior comma]] in the [[5-limit]]; [[very high accuracy temperaments #Sepbizo-asegu (171 & 3566 & 4973)\|{{monzo\| -3 2 -17 14 }}]] in the [[7-limit]]; [[3294225/3294172]] in the [[11-limit]]; [[123201/123200]] in the [[13-limit]]; [[28561/28560]], [[31213/31212]], [[37180/37179]] in the [[17-limit]]; 27456/27455, 43681/43680, 89376/89375 in the [[19-limit]]; 12168/12167, 16929/16928, 19551/19550, 21736/21735, [[25025/25024]], 43264/43263 among others in the [[23-limit]].

	~~Some of the simpler commas it [[tempering out\|tempers out]] include~~ [[123201/123200]] in the [[13-limit]]; [[28561/28560]], [[31213/31212]], [[37180/37179]] in the [[17-limit]]; 27456/27455, 43681/43680, 89376/89375 in the [[19-limit]]; 12168/12167, 16929/16928, 19551/19550, 21736/21735, [[25025/25024]], 43264/43263 among others in the [[23-limit]].

	Since it tempers out 12168/12167, it allows [[vicetertismic chords]] in the [[23-odd-limit]].		Since it tempers out 12168/12167, it allows [[vicetertismic chords]] in the [[23-odd-limit]].

Eufalesio: Added lower limit comma temperings

2026-01-04T23:58:25Z

Added lower limit comma temperings

← Older revision		Revision as of 23:58, 4 January 2026
Line 5:		Line 5:
	While it may strike many people as too large to be practical, 8539edo has seen actual use as a bookkeeping device to keep track of higher-limit intervals which have been allowed to freely modulate, and has been proposed as a [[interval size measure\|unit of interval measure]], the '''tina'''. This is because it is a very strong higher-limit system, [[consistency\|distinctly consistent]] through the [[27-odd-limit]]. It is a [[the Riemann zeta function and tuning #Zeta edo lists\|strict zeta]] tuning, and is also the first [[trivial temperament\|non-trivial]] edo to be consistent in the [[odd prime sum limit\|27-odd-prime-sum-limit]]. In the [[13-limit]], the only smaller systems with a lower logflat badness are {{EDOs\| 72, 270, 494, 5585 and 6079 }}; in the [[17-limit]], that becomes {{EDOs\| 72, 494, 1506, 3395 and 7033 }}. In the [[19-limit]], where it really shines, nothing beats it in terms of logflat badness until [[20203edo\|20203]].		While it may strike many people as too large to be practical, 8539edo has seen actual use as a bookkeeping device to keep track of higher-limit intervals which have been allowed to freely modulate, and has been proposed as a [[interval size measure\|unit of interval measure]], the '''tina'''. This is because it is a very strong higher-limit system, [[consistency\|distinctly consistent]] through the [[27-odd-limit]]. It is a [[the Riemann zeta function and tuning #Zeta edo lists\|strict zeta]] tuning, and is also the first [[trivial temperament\|non-trivial]] edo to be consistent in the [[odd prime sum limit\|27-odd-prime-sum-limit]]. In the [[13-limit]], the only smaller systems with a lower logflat badness are {{EDOs\| 72, 270, 494, 5585 and 6079 }}; in the [[17-limit]], that becomes {{EDOs\| 72, 494, 1506, 3395 and 7033 }}. In the [[19-limit]], where it really shines, nothing beats it in terms of logflat badness until [[20203edo\|20203]].

	Some of the simpler commas it [[tempering out\|tempers out]] include [[123201/123200]] in the 13-limit; [[28561/28560]], [[31213/31212]], [[37180/37179]] in the 17-limit; 27456/27455, 43681/43680, 89376/89375 in the 19-limit; 12168/12167, 16929/16928, 19551/19550, 21736/21735, [[25025/25024]], 43264/43263 among others in the 23-limit.		It tempers out the [[Senior\|Senior comma]] in the [[5-limit]], [[Very high accuracy temperaments#Sepbizo-asegu (171 & 3566 & 4973)\|[-3 2 -17 14⟩]] in the [[7-limit]], [[3294225/3294172]] in the [[11-limit]].

			Some of the simpler commas it [[tempering out\|tempers out]] include [[123201/123200]] in the [[13-limit]]; [[28561/28560]], [[31213/31212]], [[37180/37179]] in the [[17-limit]]; 27456/27455, 43681/43680, 89376/89375 in the [[19-limit]]; 12168/12167, 16929/16928, 19551/19550, 21736/21735, [[25025/25024]], 43264/43263 among others in the [[23-limit]].

	Since it tempers out 12168/12167, it allows [[vicetertismic chords]] in the [[23-odd-limit]].		Since it tempers out 12168/12167, it allows [[vicetertismic chords]] in the [[23-odd-limit]].

FloraC: Expand the harmonics table a little

2025-10-02T16:02:11Z

Expand the harmonics table a little

← Older revision		Revision as of 16:02, 2 October 2025
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	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|8539\|columns=9}}		{{Harmonics in equal\|8539\|columns=11}}
	{{Harmonics in equal\|8539\|columns=9\|start=10\|collapsed=true\|title=Approximation of prime harmonics in 8539edo (continued)}}		{{Harmonics in equal\|8539\|columns=11\|start=12\|collapsed=true\|title=Approximation of prime harmonics in 8539edo (continued)}}

	=== Subsets and supersets ===		=== Subsets and supersets ===

MisterShafXen: /* Theory */

2025-05-04T18:32:31Z

Theory

← Older revision		Revision as of 18:32, 4 May 2025
Line 5:		Line 5:
	While it may strike many people as too large to be practical, 8539edo has seen actual use as a bookkeeping device to keep track of higher-limit intervals which have been allowed to freely modulate, and has been proposed as a [[interval size measure\|unit of interval measure]], the '''tina'''. This is because it is a very strong higher-limit system, [[consistency\|distinctly consistent]] through the [[27-odd-limit]]. It is a [[the Riemann zeta function and tuning #Zeta edo lists\|strict zeta]] tuning, and is also the first [[trivial temperament\|non-trivial]] edo to be consistent in the [[odd prime sum limit\|27-odd-prime-sum-limit]]. In the [[13-limit]], the only smaller systems with a lower logflat badness are {{EDOs\| 72, 270, 494, 5585 and 6079 }}; in the [[17-limit]], that becomes {{EDOs\| 72, 494, 1506, 3395 and 7033 }}. In the [[19-limit]], where it really shines, nothing beats it in terms of logflat badness until [[20203edo\|20203]].		While it may strike many people as too large to be practical, 8539edo has seen actual use as a bookkeeping device to keep track of higher-limit intervals which have been allowed to freely modulate, and has been proposed as a [[interval size measure\|unit of interval measure]], the '''tina'''. This is because it is a very strong higher-limit system, [[consistency\|distinctly consistent]] through the [[27-odd-limit]]. It is a [[the Riemann zeta function and tuning #Zeta edo lists\|strict zeta]] tuning, and is also the first [[trivial temperament\|non-trivial]] edo to be consistent in the [[odd prime sum limit\|27-odd-prime-sum-limit]]. In the [[13-limit]], the only smaller systems with a lower logflat badness are {{EDOs\| 72, 270, 494, 5585 and 6079 }}; in the [[17-limit]], that becomes {{EDOs\| 72, 494, 1506, 3395 and 7033 }}. In the [[19-limit]], where it really shines, nothing beats it in terms of logflat badness until [[20203edo\|20203]].

	Some of the simpler commas it [[tempering out\|tempers out]] include [[123201/123200]] in the 13-limit; [[28561/28560]], [[31213/31212]], [[37180/37179]] in the 17-limit; 27456/27455, 43681/43680, 89376/89375 in the 19-limit; 12168/12167, 16929/16928, 19551/19550, 21736/21735, 25025/25024, 43264/43263 among others in the 23-limit.		Some of the simpler commas it [[tempering out\|tempers out]] include [[123201/123200]] in the 13-limit; [[28561/28560]], [[31213/31212]], [[37180/37179]] in the 17-limit; 27456/27455, 43681/43680, 89376/89375 in the 19-limit; 12168/12167, 16929/16928, 19551/19550, 21736/21735, [[25025/25024]], 43264/43263 among others in the 23-limit.

	Since it tempers out 12168/12167, it allows [[vicetertismic chords]] in the [[23-odd-limit]].		Since it tempers out 12168/12167, it allows [[vicetertismic chords]] in the [[23-odd-limit]].

FloraC: /* Theory */ extend the harmonics table to the 61-limit

2025-03-11T14:21:03Z

Theory: extend the harmonics table to the 61-limit

← Older revision		Revision as of 14:21, 11 March 2025
Line 3:		Line 3:

	== Theory ==		== Theory ==
	While it may strike many people as too large to be practical, 8539edo has seen actual use as a bookkeeping device to keep track of higher-limit intervals which have been allowed to freely modulate, and has been proposed as a [[interval size measure\|unit of interval measure]], the '''tina'''. This is because it is a very strong higher-limit system, ~~distinctly~~ [[consistent]] through the 27-odd-limit. It is a [[the Riemann zeta function and tuning #Zeta ~~EDO~~ lists\|strict zeta]] tuning, and is also the first [[trivial temperament\|non-trivial]] edo to be consistent in the ~~27-~~[[odd prime sum limit\|odd-prime-sum-limit]]. In the 13-limit, the only smaller systems with a lower logflat badness are {{EDOs\| 72, 270, 494, 5585 and 6079 }}; in the 17-limit, that becomes {{EDOs\| 72, 494, 1506, 3395 and 7033 }}. In the 19-limit, where it really shines, nothing beats it in terms of logflat badness until [[20203edo\|20203]].		While it may strike many people as too large to be practical, 8539edo has seen actual use as a bookkeeping device to keep track of higher-limit intervals which have been allowed to freely modulate, and has been proposed as a [[interval size measure\|unit of interval measure]], the '''tina'''. This is because it is a very strong higher-limit system, [[consistency\|distinctly consistent]] through the [[27-odd-limit]]. It is a [[the Riemann zeta function and tuning #Zeta edo lists\|strict zeta]] tuning, and is also the first [[trivial temperament\|non-trivial]] edo to be consistent in the [[odd prime sum limit\|27-odd-prime-sum-limit]]. In the [[13-limit]], the only smaller systems with a lower logflat badness are {{EDOs\| 72, 270, 494, 5585 and 6079 }}; in the [[17-limit]], that becomes {{EDOs\| 72, 494, 1506, 3395 and 7033 }}. In the [[19-limit]], where it really shines, nothing beats it in terms of logflat badness until [[20203edo\|20203]].

	Some of the simpler commas it [[tempering out\|tempers out]] include [[123201/123200]] in the 13-limit; [[28561/28560]], [[31213/31212]], [[37180/37179]] in the 17-limit; 27456/27455, 43681/43680, 89376/89375 in the 19-limit; 12168/12167, 16929/16928, 19551/19550, 21736/21735, 25025/25024, 43264/43263 among others in the 23-limit.		Some of the simpler commas it [[tempering out\|tempers out]] include [[123201/123200]] in the 13-limit; [[28561/28560]], [[31213/31212]], [[37180/37179]] in the 17-limit; 27456/27455, 43681/43680, 89376/89375 in the 19-limit; 12168/12167, 16929/16928, 19551/19550, 21736/21735, 25025/25024, 43264/43263 among others in the 23-limit.
Line 10:		Line 10:

	=== Prime harmonics ===		=== Prime harmonics ===
	{{Harmonics in equal\|8539\|columns=12}}		{{Harmonics in equal\|8539\|columns=9}}
			{{Harmonics in equal\|8539\|columns=9\|start=10\|collapsed=true\|title=Approximation of prime harmonics in 8539edo (continued)}}

	=== Subsets and supersets ===		=== Subsets and supersets ===

ArrowHead294 at 16:31, 20 February 2025

2025-02-20T16:31:40Z

← Older revision		Revision as of 16:31, 20 February 2025
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|8539~~}}		{{ED intro}}

	== Theory ==		== Theory ==

FloraC: Expand

2025-01-12T10:12:56Z

Expand

← Older revision		Revision as of 10:12, 12 January 2025
Line 2:		Line 2:
	{{EDO intro\|8539}}		{{EDO intro\|8539}}

			== Theory ==
	While it may strike many people as too large to be practical, 8539edo has seen actual use as a bookkeeping device to keep track of higher-limit intervals which have been allowed to freely modulate, and has been proposed as a [[interval size measure\|unit of interval measure]], the '''tina'''. This is because it is a very strong higher-limit system, distinctly [[consistent]] through the 27-odd-limit. It is a [[the Riemann zeta function and tuning #Zeta EDO lists\|strict zeta]] tuning, and is also the first [[trivial temperament\|non-trivial]] edo to be consistent in the 27-[[odd prime sum limit\|odd-prime-sum-limit]]. In the 13-limit, the only smaller systems with a lower logflat badness are {{EDOs\| 72, 270, 494, 5585 and 6079 }}; in the 17-limit, that becomes {{EDOs\| 72, 494, 1506, 3395 and 7033 }}. In the 19-limit, where it really shines, nothing beats it in terms of logflat badness until [[20203edo\|20203]].		While it may strike many people as too large to be practical, 8539edo has seen actual use as a bookkeeping device to keep track of higher-limit intervals which have been allowed to freely modulate, and has been proposed as a [[interval size measure\|unit of interval measure]], the '''tina'''. This is because it is a very strong higher-limit system, distinctly [[consistent]] through the 27-odd-limit. It is a [[the Riemann zeta function and tuning #Zeta EDO lists\|strict zeta]] tuning, and is also the first [[trivial temperament\|non-trivial]] edo to be consistent in the 27-[[odd prime sum limit\|odd-prime-sum-limit]]. In the 13-limit, the only smaller systems with a lower logflat badness are {{EDOs\| 72, 270, 494, 5585 and 6079 }}; in the 17-limit, that becomes {{EDOs\| 72, 494, 1506, 3395 and 7033 }}. In the 19-limit, where it really shines, nothing beats it in terms of logflat badness until [[20203edo\|20203]].

	Some of the simpler commas it [[tempering out\|tempers out]] include [[123201/123200]] in the 13-limit; [[28561/28560]], [[31213/31212]], [[37180/37179]] in the 17-limit; 27456/27455, 43681/43680, 89376/89375 in the 19-limit; 12168/12167, 16929/16928, 19551/19550, 21736/21735, 25025/25024, 43264/43263 among others in the 23-limit.		Some of the simpler commas it [[tempering out\|tempers out]] include [[123201/123200]] in the 13-limit; [[28561/28560]], [[31213/31212]], [[37180/37179]] in the 17-limit; 27456/27455, 43681/43680, 89376/89375 in the 19-limit; 12168/12167, 16929/16928, 19551/19550, 21736/21735, 25025/25024, 43264/43263 among others in the 23-limit.

			Since it tempers out 12168/12167, it allows [[vicetertismic chords]] in the [[23-odd-limit]].

	=== Prime harmonics ===		=== Prime harmonics ===

FloraC: Oops, wrong addition

2024-11-05T17:04:33Z

Oops, wrong addition

← Older revision		Revision as of 17:04, 5 November 2024
Line 4:		Line 4:
	While it may strike many people as too large to be practical, 8539edo has seen actual use as a bookkeeping device to keep track of higher-limit intervals which have been allowed to freely modulate, and has been proposed as a [[interval size measure\|unit of interval measure]], the '''tina'''. This is because it is a very strong higher-limit system, distinctly [[consistent]] through the 27-odd-limit. It is a [[the Riemann zeta function and tuning #Zeta EDO lists\|strict zeta]] tuning, and is also the first [[trivial temperament\|non-trivial]] edo to be consistent in the 27-[[odd prime sum limit\|odd-prime-sum-limit]]. In the 13-limit, the only smaller systems with a lower logflat badness are {{EDOs\| 72, 270, 494, 5585 and 6079 }}; in the 17-limit, that becomes {{EDOs\| 72, 494, 1506, 3395 and 7033 }}. In the 19-limit, where it really shines, nothing beats it in terms of logflat badness until [[20203edo\|20203]].		While it may strike many people as too large to be practical, 8539edo has seen actual use as a bookkeeping device to keep track of higher-limit intervals which have been allowed to freely modulate, and has been proposed as a [[interval size measure\|unit of interval measure]], the '''tina'''. This is because it is a very strong higher-limit system, distinctly [[consistent]] through the 27-odd-limit. It is a [[the Riemann zeta function and tuning #Zeta EDO lists\|strict zeta]] tuning, and is also the first [[trivial temperament\|non-trivial]] edo to be consistent in the 27-[[odd prime sum limit\|odd-prime-sum-limit]]. In the 13-limit, the only smaller systems with a lower logflat badness are {{EDOs\| 72, 270, 494, 5585 and 6079 }}; in the 17-limit, that becomes {{EDOs\| 72, 494, 1506, 3395 and 7033 }}. In the 19-limit, where it really shines, nothing beats it in terms of logflat badness until [[20203edo\|20203]].

	Some of the simpler commas it [[tempering out\|tempers out]] include [[123201/123200]] in the 13-limit; [[28561/28560]], [[31213/31212]], [[37180/37179]] in the 17-limit; ~~10830/10829,~~ 27456/27455, 43681/43680, 89376/89375 in the 19-limit; 12168/12167, 16929/16928, 19551/19550, 21736/21735, 25025/25024, 43264/43263 among others in the 23-limit.		Some of the simpler commas it [[tempering out\|tempers out]] include [[123201/123200]] in the 13-limit; [[28561/28560]], [[31213/31212]], [[37180/37179]] in the 17-limit; 27456/27455, 43681/43680, 89376/89375 in the 19-limit; 12168/12167, 16929/16928, 19551/19550, 21736/21735, 25025/25024, 43264/43263 among others in the 23-limit.

	=== Prime harmonics ===		=== Prime harmonics ===

FloraC: +more commas

2024-11-05T17:00:57Z

+more commas

← Older revision		Revision as of 17:00, 5 November 2024
Line 4:		Line 4:
	While it may strike many people as too large to be practical, 8539edo has seen actual use as a bookkeeping device to keep track of higher-limit intervals which have been allowed to freely modulate, and has been proposed as a [[interval size measure\|unit of interval measure]], the '''tina'''. This is because it is a very strong higher-limit system, distinctly [[consistent]] through the 27-odd-limit. It is a [[the Riemann zeta function and tuning #Zeta EDO lists\|strict zeta]] tuning, and is also the first [[trivial temperament\|non-trivial]] edo to be consistent in the 27-[[odd prime sum limit\|odd-prime-sum-limit]]. In the 13-limit, the only smaller systems with a lower logflat badness are {{EDOs\| 72, 270, 494, 5585 and 6079 }}; in the 17-limit, that becomes {{EDOs\| 72, 494, 1506, 3395 and 7033 }}. In the 19-limit, where it really shines, nothing beats it in terms of logflat badness until [[20203edo\|20203]].		While it may strike many people as too large to be practical, 8539edo has seen actual use as a bookkeeping device to keep track of higher-limit intervals which have been allowed to freely modulate, and has been proposed as a [[interval size measure\|unit of interval measure]], the '''tina'''. This is because it is a very strong higher-limit system, distinctly [[consistent]] through the 27-odd-limit. It is a [[the Riemann zeta function and tuning #Zeta EDO lists\|strict zeta]] tuning, and is also the first [[trivial temperament\|non-trivial]] edo to be consistent in the 27-[[odd prime sum limit\|odd-prime-sum-limit]]. In the 13-limit, the only smaller systems with a lower logflat badness are {{EDOs\| 72, 270, 494, 5585 and 6079 }}; in the 17-limit, that becomes {{EDOs\| 72, 494, 1506, 3395 and 7033 }}. In the 19-limit, where it really shines, nothing beats it in terms of logflat badness until [[20203edo\|20203]].

	Some of the simpler commas it [[tempering out\|tempers out]] include [[123201/123200]] in the 13-limit; [[28561/28560]], [[31213/31212]], [[37180/37179]] in the 17-limit; 27456/27455, 43681/43680, 89376/89375 in the 19-limit; 12168/12167, 19551/19550, 21736/21735, 25025/25024, 43264/43263 among others in the 23-limit.		Some of the simpler commas it [[tempering out\|tempers out]] include [[123201/123200]] in the 13-limit; [[28561/28560]], [[31213/31212]], [[37180/37179]] in the 17-limit; 10830/10829, 27456/27455, 43681/43680, 89376/89375 in the 19-limit; 12168/12167, 16929/16928, 19551/19550, 21736/21735, 25025/25024, 43264/43263 among others in the 23-limit.

	=== Prime harmonics ===		=== Prime harmonics ===