78edo - Revision history

Overthink at 04:24, 26 October 2025

2025-10-26T04:24:55Z

← Older revision		Revision as of 04:24, 26 October 2025
Line 7:		Line 7:
	This tuning [[tempering out\|tempers out]] [[2048/2025]] in the [[5-limit]]; [[875/864]] and [[2401/2400]] in the [[7-limit]]; and [[100/99]], [[385/384]] and [[1375/1372]] in the [[11-limit]]. It provides the [[optimal patent val]] for 11-limit [[keen]] temperament.		This tuning [[tempering out\|tempers out]] [[2048/2025]] in the [[5-limit]]; [[875/864]] and [[2401/2400]] in the [[7-limit]]; and [[100/99]], [[385/384]] and [[1375/1372]] in the [[11-limit]]. It provides the [[optimal patent val]] for 11-limit [[keen]] temperament.

	Much like [[100edo\|100bddd]], the 78dd val can be used to construct an alternative to [[22edo]] for [[pajara]]. The large and small step sizes in this case have ratio 4:3. The width of the tempered perfect fifth is 707.7 [[cent]]s. The major third is 384.6 cents; less than two cents flat of just. The harmonic seventh is 984.6 cents, or about 15.8 cents sharp; hence this tuning prioritizes the 3- and 5-limits over the 7-limit, while still ensuring that no basic 7-limit intervals other than the tritones are more than 16 cents} off. The 22-note [[2mos]] generated in this way could be used to build straight-fretted guitars that would be {{w\|Augmented-fourths tuning\|tuned in tritones}}. The appeal of this scale is that it is less xenharmonic than [[22edo]] is, for listeners accustomed to 12edo. In particular, the 163.6 cents "flat minor whole tone" of 22edo is now 169.2 cents, making it more clearly a ''whole'' tone (albeit noticeably flat), rather than a neutral second.		Much like [[100edo\|100bddd]], the 78dd val can be used to construct an alternative to [[22edo]] for [[pajara]]. The large and small step sizes in this case have ratio 4:3. The width of the tempered perfect fifth is 707.7 [[cent]]s. The major third is 384.6 cents; less than two cents flat of just. The harmonic seventh is 984.6 cents, or about 15.8 cents sharp; hence this tuning prioritizes the 3- and 5-limits over the 7-limit, while still ensuring that no basic 7-limit intervals other than the tritones are more than 16 cents off. The 22-note [[2mos]] generated in this way could be used to build straight-fretted guitars that would be {{w\|Augmented-fourths tuning\|tuned in tritones}}. The appeal of this scale is that it is less xenharmonic than [[22edo]] is, for listeners accustomed to 12edo. In particular, the 163.6 cents "flat minor whole tone" of 22edo is now 169.2 cents, making it more clearly a ''whole'' tone (albeit noticeably flat), rather than a neutral second.

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

BudjarnLambeth at 08:56, 27 September 2025

2025-09-27T08:56:01Z

← Older revision		Revision as of 08:56, 27 September 2025
Line 17:		Line 17:
	== Intervals ==		== Intervals ==
	{{Interval table}}		{{Interval table}}

			== Scales ==
			* [[Maeve Gutierrez\|Gutierrez Moonglade scale]]

	== Instruments ==		== Instruments ==

Lucius Chiaraviglio: /* Intervals */ Insert Instruments section after this with link to Lumatone mapping for 78edo.

2025-07-02T09:41:07Z

Intervals: Insert Instruments section after this with link to Lumatone mapping for 78edo.

← Older revision		Revision as of 09:41, 2 July 2025
Line 18:		Line 18:
	{{Interval table}}		{{Interval table}}

			== Instruments ==

			[[Lumatone mapping for 78edo\|Lumatone mappings for 78edo]] are available.
	[[Category:Keen]]		[[Category:Keen]]

Lucius Chiaraviglio: /* Intervals */ Insert Music section after this, starting with Bryan Deister's ''microtonal improvisation in 78edo'' (2025)

2025-06-28T20:55:59Z

Intervals: Insert Music section after this, starting with Bryan Deister's ''microtonal improvisation in 78edo'' (2025)

← Older revision		Revision as of 20:55, 28 June 2025
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	[[Category:Keen]]		[[Category:Keen]]

			== Music ==
			; [[Bryan Deister]]
			* [https://www.youtube.com/shorts/jxYRAo6jHaE ''microtonal improvisation in 78edo''] (2025)

ArrowHead294 at 16:16, 20 February 2025

2025-02-20T16:16:43Z

← Older revision		Revision as of 16:16, 20 February 2025
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|78~~}}		{{ED intro}}

	== Theory ==		== Theory ==

FloraC: /* Theory */ cleanup

2024-06-27T08:12:57Z

Theory: cleanup

← Older revision		Revision as of 08:12, 27 June 2024
Line 7:		Line 7:
	This tuning [[tempering out\|tempers out]] [[2048/2025]] in the [[5-limit]]; [[875/864]] and [[2401/2400]] in the [[7-limit]]; and [[100/99]], [[385/384]] and [[1375/1372]] in the [[11-limit]]. It provides the [[optimal patent val]] for 11-limit [[keen]] temperament.		This tuning [[tempering out\|tempers out]] [[2048/2025]] in the [[5-limit]]; [[875/864]] and [[2401/2400]] in the [[7-limit]]; and [[100/99]], [[385/384]] and [[1375/1372]] in the [[11-limit]]. It provides the [[optimal patent val]] for 11-limit [[keen]] temperament.

	Much like [[100edo\|100bddd]], the 78dd val can be used to construct an alternative to [[22edo]] for [[pajara]]. The large and small step sizes in this case have ratio 4:3. The width of the tempered perfect fifth is 707.7{{cent}}. The major third is 384.6~~{{cent}}~~; less than two cents flat of just. The harmonic seventh is 984.6~~{{cent}}~~, or about 15.8~~{{cent}}~~ sharp; hence this tuning prioritizes the 3- and 5-limits over the 7-limit, while still ensuring that no basic 7-limit intervals other than the tritones are more than 16~~{{cent}~~} off. The 22-note ~~2MOS~~ generated in this way could be used to build straight-fretted guitars that would be {{w\|Augmented-fourths tuning\|tuned in tritones}}. The appeal of this scale is that it is less xenharmonic than [[22edo]] is, for listeners accustomed to 12edo. In particular, the 163.6~~{{cent}}~~ "flat minor whole tone" of 22edo is now 169.2~~{{cent}}~~, making it more clearly a ''whole'' tone (albeit noticeably flat), rather than a neutral second.		Much like [[100edo\|100bddd]], the 78dd val can be used to construct an alternative to [[22edo]] for [[pajara]]. The large and small step sizes in this case have ratio 4:3. The width of the tempered perfect fifth is 707.7 [[cent]]s. The major third is 384.6 cents; less than two cents flat of just. The harmonic seventh is 984.6 cents, or about 15.8 cents sharp; hence this tuning prioritizes the 3- and 5-limits over the 7-limit, while still ensuring that no basic 7-limit intervals other than the tritones are more than 16 cents} off. The 22-note [[2mos]] generated in this way could be used to build straight-fretted guitars that would be {{w\|Augmented-fourths tuning\|tuned in tritones}}. The appeal of this scale is that it is less xenharmonic than [[22edo]] is, for listeners accustomed to 12edo. In particular, the 163.6 cents "flat minor whole tone" of 22edo is now 169.2 cents, making it more clearly a ''whole'' tone (albeit noticeably flat), rather than a neutral second.

	=== Odd harmonics ===		=== Odd harmonics ===
Line 13:		Line 13:

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 78 factors into {{factorization\|78}}, 78edo has subset edos {{EDOs\| 2, 3, 6, 13, 26, and 39 }}. [[156edo]], which doubles it, is a notable tuning.		Since 78 factors into {{factorization\|78}}, 78edo has subset edos {{EDOs\| 2, 3, 6, 13, 26, and 39 }}. [[156edo]], which doubles it, is a notable tuning.

	== Intervals ==		== Intervals ==

FloraC: +subsets and supersets

2024-06-27T08:10:57Z

+subsets and supersets

← Older revision		Revision as of 08:10, 27 June 2024
Line 11:		Line 11:
	=== Odd harmonics ===		=== Odd harmonics ===
	{{Harmonics in equal\|78}}		{{Harmonics in equal\|78}}

			=== Subsets and supersets ===
			Since 78 factors into {{factorization\|78}}, 78edo has subset edos {{EDOs\| 2, 3, 6, 13, 26, and 39 }}. [[156edo]], which doubles it, is a notable tuning.

	== Intervals ==		== Intervals ==

FloraC: +one-liner overview

2024-06-26T07:36:32Z

+one-liner overview

← Older revision		Revision as of 07:36, 26 June 2024
Line 3:		Line 3:

	== Theory ==		== Theory ==
			78edo is [[consistent]] to the [[7-odd-limit]], but the error of [[harmonic]] [[3/1\|3]], inherited from [[39edo]], is quite large for the size of the system.

	This tuning [[tempering out\|tempers out]] [[2048/2025]] in the [[5-limit]]; [[875/864]] and [[2401/2400]] in the [[7-limit]]; and [[100/99]], [[385/384]] and [[1375/1372]] in the [[11-limit]]. It provides the [[optimal patent val]] for 11-limit [[keen]] temperament.		This tuning [[tempering out\|tempers out]] [[2048/2025]] in the [[5-limit]]; [[875/864]] and [[2401/2400]] in the [[7-limit]]; and [[100/99]], [[385/384]] and [[1375/1372]] in the [[11-limit]]. It provides the [[optimal patent val]] for 11-limit [[keen]] temperament.

	Much like [[100edo\|100bddd]], the 78dd val can be used to construct an alternative to 22edo for pajara. The large and small step sizes in this case have ratio 4:3. The width of the tempered perfect fifth is 707.7{{cent}}. The major third is 384.6{{cent}}; less than two cents flat of just. The harmonic seventh is 984.6{{cent}}, or about 15.8{{cent}} sharp; hence this tuning prioritizes the 3- and 5-limits over the 7-limit, while still ensuring that no basic 7-limit intervals other than the tritones are more than 16{{cent}} off. The 22-note 2MOS generated in this way could be used to build straight-fretted guitars that would be {{w\|Augmented-fourths tuning\|tuned in tritones}}. The appeal of this scale is that it is less xenharmonic than [[22edo]] is, for listeners accustomed to 12edo. In particular, the 163.6{{cent}} "flat minor whole tone" of 22edo is now 169.2{{cent}}, making it more clearly a ''whole'' tone (albeit noticeably flat), rather than a neutral second.		Much like [[100edo\|100bddd]], the 78dd val can be used to construct an alternative to [[22edo]] for [[pajara]]. The large and small step sizes in this case have ratio 4:3. The width of the tempered perfect fifth is 707.7{{cent}}. The major third is 384.6{{cent}}; less than two cents flat of just. The harmonic seventh is 984.6{{cent}}, or about 15.8{{cent}} sharp; hence this tuning prioritizes the 3- and 5-limits over the 7-limit, while still ensuring that no basic 7-limit intervals other than the tritones are more than 16{{cent}} off. The 22-note 2MOS generated in this way could be used to build straight-fretted guitars that would be {{w\|Augmented-fourths tuning\|tuned in tritones}}. The appeal of this scale is that it is less xenharmonic than [[22edo]] is, for listeners accustomed to 12edo. In particular, the 163.6{{cent}} "flat minor whole tone" of 22edo is now 169.2{{cent}}, making it more clearly a ''whole'' tone (albeit noticeably flat), rather than a neutral second.

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

FloraC: Cleanup

2024-06-26T07:35:23Z

Cleanup

← Older revision		Revision as of 07:35, 26 June 2024
Line 3:		Line 3:

	== Theory ==		== Theory ==
			This tuning [[tempering out\|tempers out]] [[2048/2025]] in the [[5-limit]]; [[875/864]] and [[2401/2400]] in the [[7-limit]]; and [[100/99]], [[385/384]] and [[1375/1372]] in the [[11-limit]]. It provides the [[optimal patent val]] for 11-limit [[keen]] temperament.

			Much like [[100edo\|100bddd]], the 78dd val can be used to construct an alternative to 22edo for pajara. The large and small step sizes in this case have ratio 4:3. The width of the tempered perfect fifth is 707.7{{cent}}. The major third is 384.6{{cent}}; less than two cents flat of just. The harmonic seventh is 984.6{{cent}}, or about 15.8{{cent}} sharp; hence this tuning prioritizes the 3- and 5-limits over the 7-limit, while still ensuring that no basic 7-limit intervals other than the tritones are more than 16{{cent}} off. The 22-note 2MOS generated in this way could be used to build straight-fretted guitars that would be {{w\|Augmented-fourths tuning\|tuned in tritones}}. The appeal of this scale is that it is less xenharmonic than [[22edo]] is, for listeners accustomed to 12edo. In particular, the 163.6{{cent}} "flat minor whole tone" of 22edo is now 169.2{{cent}}, making it more clearly a ''whole'' tone (albeit noticeably flat), rather than a neutral second.

			=== Odd harmonics ===
	{{Harmonics in equal\|78}}		{{Harmonics in equal\|78}}
	This tuning tempers out 2048/2025 in the [[5-limit]]; 875/864 and 2401/2400 in the [[7-limit]]; and 100/99, 385/384 and 1375/1372 in the [[11-limit]]. It provides the [[optimal patent val]] for 11-limit [[Diaschismic_family\|keen temperament]].

	Much like [[100edo\|100bddd]], the 78dd val can be used to construct an alternative to 22edo for pajara. The large and small step sizes in this case have ratio 4:3. The width of the tempered perfect fifth is 707.7{{cent}}. The major third is 384.6{{cent}}; less than two cents flat of just. The harmonic seventh is 984.6{{cent}}, or about 15.8{{cent}} sharp; hence this tuning prioritizes the 3- and 5-limits over the 7-limit, while still ensuring that no basic 7-limit intervals other than the tritones are more than 16{{cent}} off. The 22-note 2MOS generated in this way could be used to build straight-fretted guitars that would be [https://en.wikipedia.org/wiki/Augmented-fourths_tuning tuned in tritones]. The appeal of this scale is that it is less xenharmonic than [[22edo]] is, for listeners accustomed to 12edo. In particular, the 163.6{{cent}} "flat minor whole tone" of 22edo is now 169.2{{cent}}, making it more clearly a ''whole'' tone (albeit noticeably flat), rather than a neutral second.

	== Intervals ==		== Intervals ==

FloraC: -redundant categories

2023-11-27T14:43:07Z

-redundant categories

← Older revision		Revision as of 14:43, 27 November 2023
Line 11:		Line 11:
	{{Interval table}}		{{Interval table}}

	~~[[Category:Equal divisions of the octave\|##]] <!-- 2-digit number -->~~
	[[Category:Keen]]		[[Category:Keen]]