124edo - Revision history

Overthink: /* Theory */ fixed subgroup

2025-11-20T23:19:17Z

Theory: fixed subgroup

← Older revision		Revision as of 23:19, 20 November 2025
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	== Theory ==		== Theory ==
	124edo is closely related to [[31edo]], but the [[patent val]]s differ on the [[mapping]] for [[3/1\|3]]. The equal temperament [[tempering out\|tempers out]] 2048/2025 ([[diaschisma]]) and {{monzo\| -6 -24 19 }} in the 5-limit. Using the patent val, it tempers out [[3136/3125]], [[4000/3969]], and 33614/32805 in the 7-limit; [[385/384]], 1232/1215, 1331/1323, and 3773/3750 in the 11-limit; [[196/195]], [[364/363]], 572/567, [[625/624]], and [[1001/1000]] in the 13-limit. Note that although its sharp fifth is slightly closer to just, both fifths are about equally off in both directions, and its [[9/1\|9th harmonic]] is especially accurate as a result, so it can be considered a [[dual-fifth system]], in which it performs very well in the 2.9.5.7.11.13.17.19.23.37 [[subgroup]] (the dual-fifth ~~no-31's~~ [[37-limit]]), which is arguably the right way to analyze its approximations of JI. Also interesting is that one may want to double the number of notes to add a fifth closer to just, but this causes the relative errors of other primes to double leading to [[consistency\|inconsistencies]], so its most reasonable and capable conceptualization seems to be that of a dual-fifth system.		124edo is closely related to [[31edo]], but the [[patent val]]s differ on the [[mapping]] for [[3/1\|3]]. The equal temperament [[tempering out\|tempers out]] 2048/2025 ([[diaschisma]]) and {{monzo\| -6 -24 19 }} in the 5-limit. Using the patent val, it tempers out [[3136/3125]], [[4000/3969]], and 33614/32805 in the 7-limit; [[385/384]], 1232/1215, 1331/1323, and 3773/3750 in the 11-limit; [[196/195]], [[364/363]], 572/567, [[625/624]], and [[1001/1000]] in the 13-limit. Note that although its sharp fifth is slightly closer to just, both fifths are about equally off in both directions, and its [[9/1\|9th harmonic]] is especially accurate as a result, so it can be considered a [[dual-fifth system]], in which it performs very well in the 2.9.5.7.11.13.17.19.23.37 [[subgroup]] (the dual-fifth [[23-limit]] add-37), which is arguably the right way to analyze its approximations of JI. Also interesting is that one may want to double the number of notes to add a fifth closer to just, but this causes the relative errors of other primes to double leading to [[consistency\|inconsistencies]], so its most reasonable and capable conceptualization seems to be that of a dual-fifth system.

	=== Odd harmonics ===		=== Odd harmonics ===

Contribution at 15:00, 2 July 2025

2025-07-02T15:00:28Z

← Older revision		Revision as of 15:00, 2 July 2025
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	{{Harmonics in equal\|124\|columns=12}}		{{Harmonics in equal\|124\|columns=12}}
	{{Harmonics in equal\|124\|columns=12\|start=13\|collapsed=1\|title=Approximation of odd harmonics in 124edo (continued)}}		{{Harmonics in equal\|124\|columns=12\|start=13\|collapsed=1\|title=Approximation of odd harmonics in 124edo (continued)}}

			=== No-3 approach ===
			If prime 3 is ignored, 124edo represents the no-3 28-integer-limit consistently. 124edo is distinctly consistent within the no-3 26-integer-limit.

	== Intervals ==		== Intervals ==

ArrowHead294 at 19:22, 20 February 2025

2025-02-20T19:22:56Z

Show changes

Contribution: /* Intervals */

2024-08-11T14:41:50Z

Intervals

← Older revision		Revision as of 14:41, 11 August 2024
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	\|2.0		\|2.0
	\|'''<big><u>2/1</u></big>'''		\|'''<big><u>2/1</u></big>'''
	\|P1 + 1 oct		\|P1 +1 oct
	\|-		\|-
	\|colspan="4"\|		\|colspan="4"\|

Contribution: /* Intervals */

2024-08-11T14:41:24Z

Intervals

← Older revision		Revision as of 14:41, 11 August 2024
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	\|2.0		\|2.0
	\|'''<big><u>2/1</u></big>'''		\|'''<big><u>2/1</u></big>'''
	\|P1 + ~~1oct~~		\|P1 + 1 oct
	\|-		\|-
	\|colspan="4"\|		\|colspan="4"\|

Contribution: /* Intervals */

2024-08-11T14:41:01Z

Intervals

← Older revision		Revision as of 14:41, 11 August 2024
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	\|1.934		\|1.934
	\|<big>31/16</big>, 27/14, 33/17, <small>35/18</small>, <small>25/13</small>, <small>123/64</small>, <small>39/20</small>, <small>23/12</small>		\|<big>31/16</big>, 27/14, 33/17, <small>35/18</small>, <small>25/13</small>, <small>123/64</small>, <small>39/20</small>, <small>23/12</small>
	\|vvA7, ^^d1 +~~1oct~~		\|vvA7, ^^d1 +1 oct
	\|-		\|-
	\|119		\|119
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	\|1.9449		\|1.9449
	\|'''<big><u>35/18</u></big>''', <big>33/17</big>, 39/20, 31/16, 125/64, <small>45/23</small>, <small>27/14</small>		\|'''<big><u>35/18</u></big>''', <big>33/17</big>, 39/20, 31/16, 125/64, <small>45/23</small>, <small>27/14</small>
	\|^^^d1 +~~1oct~~		\|^^^d1 +1 oct
	\|-		\|-
	\|120		\|120
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	\|1.9558		\|1.9558
	\|'''<big><u>45/23</u></big>''', <big>125/64</big>, 39/20, <small>35/18</small>, <small>63/32</small>, <small>33/17</small>		\|'''<big><u>45/23</u></big>''', <big>125/64</big>, 39/20, <small>35/18</small>, <small>63/32</small>, <small>33/17</small>
	\|v<sup>4</sup>1 +~~1oct~~		\|v<sup>4</sup>1 +1 oct
	\|-		\|-
	\|121		\|121
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	\|1.9667		\|1.9667
	\|'''<big>63/32</big>''', <small>45/23</small>, <small>125/64</small>, <small>39/20</small>, <small>127/64</small>		\|'''<big>63/32</big>''', <small>45/23</small>, <small>125/64</small>, <small>39/20</small>, <small>127/64</small>
	\|vvv1 +~~1oct~~		\|vvv1 +1 oct
	\|-		\|-
	\|122		\|122
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	\|1.9778		\|1.9778
	\|127/64, 63/32		\|127/64, 63/32
	\|vv1 +~~1oct~~		\|vv1 +1 oct
	\|-		\|-
	\|123		\|123
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	\|1.9889		\|1.9889
	\|<big>127/64</big>		\|<big>127/64</big>
	\|v1 +~~1oct~~		\|v1 +1 oct
	\|-		\|-
	\|124		\|124
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	\|2.0		\|2.0
	\|'''<big><u>2/1</u></big>'''		\|'''<big><u>2/1</u></big>'''
	\|P1 +1oct		\|P1 + 1oct
	\|-		\|-
	\|colspan="4"\|		\|colspan="4"\|

Contribution: /* Intervals */

2024-08-10T15:38:01Z

Intervals

← Older revision		Revision as of 15:38, 10 August 2024
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	!JI Ratio Approximations		!JI Ratio Approximations
	!Ups and Downs Notation		!Ups and Downs Notation
			[[val]] <124 196] (124b)
	\|-		\|-
	\|0		\|0

Contribution: /* Intervals */

2024-08-10T15:35:35Z

Intervals

← Older revision		Revision as of 15:35, 10 August 2024
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	== Intervals ==		== Intervals ==
	{\| class="wikitable mw-collapsible mw-collapsed"		{\| class="wikitable mw-collapsible mw-collapsed center-5"
	\|+124 EDO Table of Intervals		\|+124 EDO Table of Intervals
	!Step		!Step
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	!Ratio		!Ratio
	!JI Ratio Approximations		!JI Ratio Approximations
	!~~JI Ratio Approximations~~		!Ups and Downs Notation
	\|-		\|-
	\|0		\|0

Contribution at 15:34, 10 August 2024

2024-08-10T15:34:38Z

Show changes

Contribution: /* Intervals */

2024-08-10T15:34:17Z

Intervals

Show changes