46edo - Revision history

Inthar: /* Regular temperaments */

2026-04-04T12:19:28Z

Regular temperaments

← Older revision		Revision as of 12:19, 4 April 2026
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	* [[Rodan]] with [[41edo]] (46edo is at the harder end)		* [[Rodan]] with [[41edo]] (46edo is at the harder end)
	* [[Diaschismic]] with [[34edo]]		* [[Diaschismic]] with [[34edo]]
			* [[Valentine]] with [[31edo]]

	== Notation ==		== Notation ==

Inthar: /* Scales */

2026-04-04T12:13:31Z

Scales

← Older revision		Revision as of 12:13, 4 April 2026
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	=== Scales ===		=== Scales ===
	46edo's 5-limit intervals are not found particularly early on in the chain of fifths, with 6/5 being a triple-diminished fifth and 5/4 a triple-augmented unison. 46edo has a 17-note chromatic scale generated by the perfect fifth, 3-3-3-2-3-3-2-3-3-3-2-3-3-2-3-3-2, however it is close to 17edo in that it lacks nearmajor and nearminor thirds, with supraminor and submajor thirds instead.		46edo's 5-limit intervals are not found particularly early on in the chain of fifths, with 6/5 being a triple-diminished fifth and 5/4 a triple-augmented unison. 46edo has a 17-note chromatic scale generated by the perfect fifth, 3-3-3-2-3-3-2-3-3-3-2-3-3-2-3-3-2, however it is close to 17edo in that it lacks nearmajor and nearminor thirds, with supraminor and submajor thirds instead.

			11-limit [[penslen]] is available in 46edo (because of Slendric and 385/384 tempering): 2-1-6-1-2-6-1-2-6-1-2-1-6-2-1-6.

	=== Regular temperaments ===		=== Regular temperaments ===

Inthar: /* JI approximation */

2026-04-04T12:10:08Z

JI approximation

← Older revision		Revision as of 12:10, 4 April 2026
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	{{Harmonics in ED\|46\|31\|0}}		{{Harmonics in ED\|46\|31\|0}}
	{\| class="wikitable"		{\| class="wikitable"
	\|+Thirds in ~~41edo~~		\|+Thirds in 46edo
	!Quality		!Quality
	\|Subminor		\|Subminor

Inthar: /* JI approximation */

2026-04-04T12:07:49Z

JI approximation

← Older revision		Revision as of 12:07, 4 April 2026
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	46edo is most accurately a 2.3.5.7.11.13.17.23 tuning. It has somewhat opposite tendencies to [[41edo]] in the 2.3.5.7.11.13 subgroup, though it has 41edo's [[Rodan]] characteristic of sharp prime 3 and flat prime 7 (more extremely). Hence their sum [[87edo]] is a near-optimal 13-limit Rodan tuning, with very accurate primes 5, 11, and 13. 46edo has extremely accurate approximations of 14/11 and 10/9.		46edo is most accurately a 2.3.5.7.11.13.17.23 tuning. It has somewhat opposite tendencies to [[41edo]] in the 2.3.5.7.11.13 subgroup, though it has 41edo's [[Rodan]] characteristic of sharp prime 3 and flat prime 7 (more extremely). Hence their sum [[87edo]] is a near-optimal 13-limit Rodan tuning, with very accurate primes 5, 11, and 13. 46edo has extremely accurate approximations of 14/11 and 10/9.

	Because 46edo is not a meantone system, the best diatonic to use for 5-limit harmony is the Zarlino diatonic scale (LMsLMLs), tuned in 41edo as 8-7-4-8-7-8-4. However, it also features a MOS diatonic of 8-8-3-8-8-8-3.		Because 46edo is not a meantone system, the best diatonic to use for 5-limit harmony is the Zarlino diatonic scale (LMsLMLs), tuned in 41edo as 8-7-4-8-7-8-4. However, it also features a [[gentle tuning\|neogothic]] MOS diatonic of 8-8-3-8-8-8-3.

	{{Harmonics in ED\|46\|31\|0}}		{{Harmonics in ED\|46\|31\|0}}

Inthar: /* Regular temperaments */

2026-04-04T12:06:29Z

Regular temperaments

← Older revision		Revision as of 12:06, 4 April 2026
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	46edo shares		46edo shares
	* [[Rodan]] with [[41edo]] (46edo is at the harder end)		* [[Rodan]] with [[41edo]] (46edo is at the harder end)
	* [[Diaschismic]] with 34edo		* [[Diaschismic]] with [[34edo]]

	== Notation ==		== Notation ==

Inthar: /* Regular temperaments */

2026-04-04T12:06:14Z

Regular temperaments

← Older revision		Revision as of 12:06, 4 April 2026
Line 61:		Line 61:

	=== Regular temperaments ===		=== Regular temperaments ===
	46edo shares [[Rodan]] with 41edo ~~and~~ [[Diaschismic]] with 34edo.		46edo shares
			* [[Rodan]] with [[41edo]] (46edo is at the harder end)
			* [[Diaschismic]] with 34edo

	== Notation ==		== Notation ==

Inthar: /* JI approximation */

2026-04-04T12:04:53Z

JI approximation

← Older revision		Revision as of 12:04, 4 April 2026
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	== Theory ==		== Theory ==
	=== JI approximation ===		=== JI approximation ===
	~~41edo~~ is most accurately a 2.3.5.7.11.13.17.23 tuning. It has somewhat opposite tendencies to [[41edo]] in the 2.3.5.7.11.13 subgroup, though it has 41edo's [[Rodan]] characteristic of sharp prime 3 and flat prime 7 (more extremely). Hence their sum [[87edo]] is a near-optimal 13-limit Rodan tuning, with very accurate primes 5, 11, and 13.		46edo is most accurately a 2.3.5.7.11.13.17.23 tuning. It has somewhat opposite tendencies to [[41edo]] in the 2.3.5.7.11.13 subgroup, though it has 41edo's [[Rodan]] characteristic of sharp prime 3 and flat prime 7 (more extremely). Hence their sum [[87edo]] is a near-optimal 13-limit Rodan tuning, with very accurate primes 5, 11, and 13. 46edo has extremely accurate approximations of 14/11 and 10/9.

	Because 46edo is not a meantone system, the best diatonic to use for 5-limit harmony is the Zarlino diatonic scale (LMsLMLs), tuned in 41edo as 8-7-4-8-7-8-4. However, it also features a MOS diatonic of 8-8-3-8-8-8-3.		Because 46edo is not a meantone system, the best diatonic to use for 5-limit harmony is the Zarlino diatonic scale (LMsLMLs), tuned in 41edo as 8-7-4-8-7-8-4. However, it also features a MOS diatonic of 8-8-3-8-8-8-3.

Inthar: Created page with "'''46edo''', or 46 equal divisions of the octave, is an equal tuning with a step size of approximately 26.1 cents. It is known for its relatively good approximation of 13-limit just intonation. == Theory == === JI approximation === 41edo is most accurately a 2.3.5.7.11.13.17.23 tuning. It has somewhat opposite tendencies to 41edo in the 2.3.5.7.11.13 subgroup, though it has 41edo's Rodan characteristic of sharp prime 3 and flat prime 7 (more extremely). Hence th..."

2026-04-04T12:03:35Z

Created page with "'''46edo''', or 46 equal divisions of the octave, is an equal tuning with a step size of approximately 26.1 cents. It is known for its relatively good approximation of 13-limit just intonation. == Theory == === JI approximation === 41edo is most accurately a 2.3.5.7.11.13.17.23 tuning. It has somewhat opposite tendencies to 41edo in the 2.3.5.7.11.13 subgroup, though it has 41edo's Rodan characteristic of sharp prime 3 and flat prime 7 (more extremely). Hence th..."

New page

'''46edo''', or 46 equal divisions of the octave, is an equal tuning with a step size of approximately 26.1 cents. It is known for its relatively good approximation of 13-limit just intonation.

== Theory ==
=== JI approximation ===
41edo is most accurately a 2.3.5.7.11.13.17.23 tuning. It has somewhat opposite tendencies to [[41edo]] in the 2.3.5.7.11.13 subgroup, though it has 41edo's [[Rodan]] characteristic of sharp prime 3 and flat prime 7 (more extremely). Hence their sum [[87edo]] is a near-optimal 13-limit Rodan tuning, with very accurate primes 5, 11, and 13.

Because 46edo is not a meantone system, the best diatonic to use for 5-limit harmony is the Zarlino diatonic scale (LMsLMLs), tuned in 41edo as 8-7-4-8-7-8-4. However, it also features a MOS diatonic of 8-8-3-8-8-8-3.

{{Harmonics in ED|46|31|0}}
{| class="wikitable"
|+Thirds in 41edo
!Quality
|Subminor
|'''Farminor'''
|Nearminor
|Supraminor
|Submajor
|Nearmajor
|'''Farmajor'''
|Supermajor
|-
!Cents
|261
|'''287'''
|313
|339
|365
|391
|'''417'''
|443
|-
!Just interpretation
|7/6
|'''13/11'''
|6/5
|11/9, 17/14
|16/13, 21/17
|5/4
|'''14/11'''
|9/7, 13/10
|-
!Steps
|10
|'''11'''
|12
|13
|14
|15
|'''16'''
|17
|}
Thirds available in the diatonic scale generated by stacking the perfect fifth are bolded.

=== Chords ===
46edo has four different flavors of minor and major intervals but lacks true neutrals and interordinals. Its subminor and supermajor intervals approximate simpler septimal ratios such as 7/4 and 9/7, while its nearminor and nearmajor intervals approximate classical 5-limit harmony which includes ratios like 5/4 and 9/5, and its farmajor and farminor intervals approximate more complex [[neogothic]] triads 22:28:33 and 22:26:33. Its supraminor triad approximates 14:17:21.

As a result, 46edo has eight qualities of tertian, fifth-bounded triads: supermajor, farmajor, nearmajor, submajor, supraminor, nearminor, farminor, subminor. 46edo lacks true interordinal intervals, so as for latal fourth-bounded triads, there are only four qualities.

=== Scales ===
46edo's 5-limit intervals are not found particularly early on in the chain of fifths, with 6/5 being a triple-diminished fifth and 5/4 a triple-augmented unison. 46edo has a 17-note chromatic scale generated by the perfect fifth, 3-3-3-2-3-3-2-3-3-3-2-3-3-2-3-3-2, however it is close to 17edo in that it lacks nearmajor and nearminor thirds, with supraminor and submajor thirds instead.

=== Regular temperaments ===
46edo shares [[Rodan]] with 41edo and [[Diaschismic]] with 34edo.

== Notation ==
46edo can be notated with diatonic notation plus ups and downs, which naturally reflect 46edo's structure, as 5/4 is downmajor, 81/64 is major, and 9/7 is upmajor. (In fact, "up" can be declared equivalent to "super"/"supra" and "down" equivalent to "sub".)

== Practice ==
Rodan isomorphic layout for 46edo: +1\46 up-left, +10\46 up, +9\46 up-right

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