72edo - Revision history

Inthar: /* Temperaments */

2026-04-13T04:32:50Z

Temperaments

← Older revision		Revision as of 04:32, 13 April 2026
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	Due to this, you can build scales in 72edo by starting with 12edo or 24edo as a base and altering certain notes.		Due to this, you can build scales in 72edo by starting with 12edo or 24edo as a base and altering certain notes.

	72edo is the definitive tuning for 11-limit [[Marvel]] and [[Miracle]].		72edo is the definitive edo tuning for 11-limit [[Marvel]] and [[Miracle]].

	== Notation ==		== Notation ==

Inthar: /* Temperaments */

2026-04-13T04:32:20Z

Temperaments

← Older revision		Revision as of 04:32, 13 April 2026
Line 18:		Line 18:
	Due to this, you can build scales in 72edo by starting with 12edo or 24edo as a base and altering certain notes.		Due to this, you can build scales in 72edo by starting with 12edo or 24edo as a base and altering certain notes.

	72edo is the definitive 11-limit [[Marvel]] ~~tuning~~.		72edo is the definitive tuning for 11-limit [[Marvel]] and [[Miracle]].

	== Notation ==		== Notation ==

Inthar: /* Temperaments */

2026-04-13T04:31:12Z

Temperaments

← Older revision		Revision as of 04:31, 13 April 2026
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	Due to this, you can build scales in 72edo by starting with 12edo or 24edo as a base and altering certain notes.		Due to this, you can build scales in 72edo by starting with 12edo or 24edo as a base and altering certain notes.

			72edo is the definitive 11-limit [[Marvel]] tuning.

	== Notation ==		== Notation ==

Inthar: /* Section header */

2026-04-10T22:45:13Z

Section header

← Older revision		Revision as of 22:45, 10 April 2026
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	72edo's best approximations of the odd prime harmonics up to 17 are all flat, especially 13. As such, slightly stretching the tuning so that the octave is about ⅚ of a cent sharp of just can be considered as optimizing it.		72edo's best approximations of the odd prime harmonics up to 17 are all flat, especially 13. As such, slightly stretching the tuning so that the octave is about ⅚ of a cent sharp of just can be considered as optimizing it.

	=== ~~Section header~~ ===		=== Temperaments ===
	72edo may be conceptualized as the intersection of two distinct extensions of [[~~compton~~]]: septimal ~~compton~~ itself (setting the 64/63 Archytas comma to twice the 81/80 syntonic comma) and "hours" (24 & 72), which divides each step of 12edo into two and reaches septimal intervals by altering interordinals by a comma step. In this way, 72edo provides a useful way to conceptualize no-13s just intonation (and to a lesser extent prime 13 as well) with respect to the conventional 12-form.		72edo may be conceptualized as the intersection of two distinct extensions of [[Compton]]: septimal Compton itself (setting the 64/63 Archytas comma to twice the 81/80 syntonic comma) and "hours" (24 & 72), which divides each step of 12edo into two and reaches septimal intervals by altering interordinals by a comma step. In this way, 72edo provides a useful way to conceptualize no-13s just intonation (and to a lesser extent prime 13 as well) with respect to the conventional 12-form.

	Due to this, you can build scales in 72edo by starting with 12edo or 24edo as a base and altering certain notes.		Due to this, you can build scales in 72edo by starting with 12edo or 24edo as a base and altering certain notes.

	== Notation ==		== Notation ==

Vector at 19:39, 10 April 2026

2026-04-10T19:39:39Z

← Older revision		Revision as of 19:39, 10 April 2026
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	=== Octave stretch ===		=== Octave stretch ===
	72edo's best approximations of the odd prime harmonics up to 17 are all flat, especially 13. As such, slightly stretching the tuning so that the octave is about ⅚ of a cent sharp of just can be considered as optimizing it.		72edo's best approximations of the odd prime harmonics up to 17 are all flat, especially 13. As such, slightly stretching the tuning so that the octave is about ⅚ of a cent sharp of just can be considered as optimizing it.

			=== Section header ===
			72edo may be conceptualized as the intersection of two distinct extensions of [[compton]]: septimal compton itself (setting the 64/63 Archytas comma to twice the 81/80 syntonic comma) and "hours" (24 & 72), which divides each step of 12edo into two and reaches septimal intervals by altering interordinals by a comma step. In this way, 72edo provides a useful way to conceptualize no-13s just intonation (and to a lesser extent prime 13 as well) with respect to the conventional 12-form.

			Due to this, you can build scales in 72edo by starting with 12edo or 24edo as a base and altering certain notes.

	== Notation ==		== Notation ==

Vector at 21:18, 7 April 2026

2026-04-07T21:18:37Z

← Older revision		Revision as of 21:18, 7 April 2026
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	=== Just notation systems ===		=== Just notation systems ===
	72edo is large enough that using notation systems designed for just intonation becomes useful. All of the primary just intonation systems, including Sagittal Evo and prime-factor Sagittal, agree on the accidentals for the 11-limit in 72edo, except for Neutral FJS and Ben Johnston.		72edo is large enough that using notation systems designed for just intonation becomes useful. All of the primary just intonation systems, including Sagittal Evo and prime-factor Sagittal, agree on the accidentals for the 11-limit in 72edo, except for Neutral FJS (which simply relabels the 11-sharp as a semisharp) and Ben Johnston.
	{\| class="wikitable"		{\| class="wikitable"
	\|+		\|+

Vector at 21:18, 7 April 2026

2026-04-07T21:18:02Z

← Older revision		Revision as of 21:18, 7 April 2026
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	=== Just notation systems ===		=== Just notation systems ===
	72edo is large enough that using notation systems designed for just intonation becomes useful. All of the primary just intonation systems, including Sagittal Evo, agree on the accidentals for the 11-limit in 72edo, except for Neutral FJS and Ben Johnston.		72edo is large enough that using notation systems designed for just intonation becomes useful. All of the primary just intonation systems, including Sagittal Evo and prime-factor Sagittal, agree on the accidentals for the 11-limit in 72edo, except for Neutral FJS and Ben Johnston.
	{\| class="wikitable"		{\| class="wikitable"
	\|+		\|+

Vector at 21:17, 7 April 2026

2026-04-07T21:17:27Z

← Older revision		Revision as of 21:17, 7 April 2026
Line 19:		Line 19:

	=== Just notation systems ===		=== Just notation systems ===
	72edo is large enough that using notation systems designed for just intonation becomes useful. All of the primary just intonation systems, including Sagittal Evo, agree on the accidentals for the 11-limit in 72edo, except for Neutral FJS.		72edo is large enough that using notation systems designed for just intonation becomes useful. All of the primary just intonation systems, including Sagittal Evo, agree on the accidentals for the 11-limit in 72edo, except for Neutral FJS and Ben Johnston.
	{\| class="wikitable"		{\| class="wikitable"
	\|+		\|+
	!Steps		!Steps
	~~!FJS/HEJI~~
	!Neutral FJS		!Neutral FJS
	!Sagittal		!Sagittal/Color/FJS/HEJI
	\|-		\|-
	\|0		\|0
	\| colspan="3" \|natural		\| colspan="2" \|natural
	\|-		\|-
	\|1		\|1
	\| colspan="2" ~~\|5-sharp~~		\| colspan="2" \|5-comma up
	\|5-comma up
	\|-		\|-
	\|2		\|2
	\| colspan="2" ~~\|7-sharp~~		\| colspan="2" \|7-comma up
	\|7-comma up
	\|-		\|-
	\|3		\|3
	~~\|11-sharp~~
	\|semisharp		\|semisharp
	\|11 M-diesis up		\|11 M-diesis up
	\|-		\|-
	\|4		\|4
	\| colspan="2" ~~\|7-flat sharp~~		\| colspan="2" \|7-comma down sharp
	\|7-comma down sharp
	\|-		\|-
	\|5		\|5
	\| colspan="2" ~~\|5-flat sharp~~		\| colspan="2" \|5-comma down sharp
	\|5-comma down sharp
	\|-		\|-
	\|6		\|6
	\| colspan="3" \|sharp		\| colspan="2" \|sharp
	\|}		\|}

Vector at 21:14, 7 April 2026

2026-04-07T21:14:47Z

@@ Line 12: / Line 12: @@
 === Octave stretch ===
 edo's best approximations of the odd prime harmonics up to 17 are all flat, especially 13. As such, slightly stretching the tuning so that the octave is about ⅚ of a cent sharp of just can be considered as optimizing it.
 == Use in software ==

Overthink: /* Theory */ a few more links

2026-03-21T03:42:39Z

Theory: a few more links

← Older revision		Revision as of 03:42, 21 March 2026
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	{{Harmonics in ED\|72\|31}}		{{Harmonics in ED\|72\|31}}

	Its highly divisible fourth and fifth lead to a wide range of notable tonal structures. It divides the fifth and the fourth in half, leading to 24edo's neutral/mosh scales and semiquartal scale, respectively, and it also divides the fifth into three ~[[8/7]]<nowiki/>s, leading to 36edo's Slendric scales. Combining these divisions yields Miracle scales, dividing the fifth into six ~[[16/15]]<nowiki/>s.		Its highly divisible fourth and fifth lead to a wide range of notable tonal structures. It divides the fifth and the fourth in half, leading to 24edo's [[mosh\|neutral/mosh]] scales and [[semiquartal]] scale, respectively, and it also divides the fifth into three ~[[8/7]]<nowiki/>s, leading to 36edo's Slendric scales. Combining these divisions yields [[Miracle]] scales, dividing the fifth into six ~[[16/15]]<nowiki/>s.

	72edo can be treated as six rings of 12edo, where differences between notes in two different rings can be seen as combining a higher prime with the 3-limit. For example, the 12edo rings can be referred to as ring 0 for the root, ring 1 for scale degrees 6n+1, ring 2 for scale degrees 6n+2, and so on. The root note combined with a note on ring 5 can be interpreted as a 5-limit interval, and combining the root note with a note on ring 4 gives a [[2.3.7 subgroup\|2.3.7-subgroup]] (septal) interval. This makes extending [[Diatonic notation\|standard diatonic notation]] straightforward. Existing notations for 72edo include ups and downs notation (with and without quarter-tone accidentals), Maneri-Sims notation, and Ivan Wyschnegradsky's notation.		72edo can be treated as six rings of 12edo, where differences between notes in two different rings can be seen as combining a higher prime with the 3-limit. For example, the 12edo rings can be referred to as ring 0 for the root, ring 1 for scale degrees 6n+1, ring 2 for scale degrees 6n+2, and so on. The root note combined with a note on ring 5 can be interpreted as a 5-limit interval, and combining the root note with a note on ring 4 gives a [[2.3.7 subgroup\|2.3.7-subgroup]] (septal) interval. This makes extending [[Diatonic notation\|standard diatonic notation]] straightforward. Existing notations for 72edo include ups and downs notation (with and without quarter-tone accidentals), Maneri-Sims notation, and Ivan Wyschnegradsky's notation.