26ed5 - Revision history

BudjarnLambeth: Collapse harmonics tables, correct error (octave to pentave)

2025-01-05T23:46:13Z

Collapse harmonics tables, correct error (octave to pentave)

← Older revision		Revision as of 23:46, 5 January 2025
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	=== Prime subgroups ===		=== Prime subgroups ===
	Pure-~~octaves~~ 26ed5 is incompatible with [[prime limit]] tuning. Of all primes up to 41, 5 and 41 are the only two it approximates well. Many of 26ed5’s 'near-miss' [[prime]]s are tuned sharp, so 26ed5 can be made to work more normally by [[Octave shrinking\|compressing]] 26ed5’s [[equave]], making [[5/1]] slightly flat but still okay and the other primes more in-tune.		Pure-[[pentave]]s 26ed5 is incompatible with [[prime limit]] tuning. Of all primes up to 41, 5 and 41 are the only two it approximates well. Many of 26ed5’s 'near-miss' [[prime]]s are tuned sharp, so 26ed5 can be made to work more normally by [[Octave shrinking\|compressing]] 26ed5’s [[equave]], making [[5/1]] slightly flat but still okay and the other primes more in-tune.

	A good compressed tuning of 26ed5 is [[46ed17]], which transforms 26ed5 from a 5.41 tuning to a 3.5.11.17.23.43 tuning. The 3/1 in 46ed17 isn’t that good, with similar error to [[5edo]], but it’s a huge improvement on 26ed5. And the 5, 11, 17, 23 and 43 are genuinely solid approximations. Other tunings which are almost identical to 46ed17, and so provide those same benefits, are [[8ed18/11]] and [[20ed24/7]].		A good compressed tuning of 26ed5 is [[46ed17]], which transforms 26ed5 from a 5.41 tuning to a 3.5.11.17.23.43 tuning. The 3/1 in 46ed17 isn’t that good, with similar error to [[5edo]], but it’s a huge improvement on 26ed5. And the 5, 11, 17, 23 and 43 are genuinely solid approximations. Other tunings which are almost identical to 46ed17, and so provide those same benefits, are [[8ed18/11]] and [[20ed24/7]].

	If one attempts to [[Octave stretch\|stretch]] 26ed5 instead of compress, one will not find any tunings that approximate primes well until reaching [[11edo]], so only compression is viable, not stretching.		If one attempts to [[Octave stretch\|stretch]] 26ed5 instead of compress, one will not find any tunings that approximate primes well until reaching [[11edo]], so only compression is viable, not stretching.
			{{Harmonics in equal
			\| steps = 26
			\| num = 5
			\| denom = 1
			\| intervals = prime
			\| collapsed = 1
			\| start = 1
			\| title = Prime harmonics 2 to 31 (26ed5)
			}}
			{{Harmonics in equal
			\| steps = 26
			\| num = 5
			\| denom = 1
			\| intervals = prime
			\| collapsed = 1
			\| start = 12
			\| title = Prime harmonics 37 to 79 (26ed5)
			}}

	=== Composite subgroups ===		=== Composite subgroups ===
	If one ~~ignores~~ primes and ~~focuses on~~ integers ~~in general~~, 26ed5 can instead be used as a strong tuning for the giant [[subgroup]]:		If one does not restrict to primes and allows all integers, pure-pentaves 26ed5 can instead be used as a strong tuning for the giant [[subgroup]]:

	'''5.6.12.22.32.44.49.52.56'''		'''5.6.12.22.32.44.49.52.56'''
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	Or it can be a strong tuning for any smaller subgroup that is contained within that group.		Or it can be a strong tuning for any smaller subgroup that is contained within that group.

	~~=== Tables of harmonics ===~~
	{{Harmonics in equal		{{Harmonics in equal
	\| steps = 26		\| steps = 26
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	\| denom = 1		\| denom = 1
	\| intervals = integer		\| intervals = integer
			\| collapsed = 1
	\| start = 1		\| start = 1
	\| title = ~~Harmonics~~ 2 to 12 (26ed5)		\| title = Integer harmonics 2 to 12 (26ed5)
	}}		}}
	{{Harmonics in equal		{{Harmonics in equal
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	\| collapsed = 1		\| collapsed = 1
	\| start = 12		\| start = 12
	\| title = ~~Harmonics~~ 13 to 23 (26ed5)		\| title = Integer harmonics 13 to 23 (26ed5)
	}}		}}
	{{Harmonics in equal		{{Harmonics in equal
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	\| collapsed = 1		\| collapsed = 1
	\| start = 23		\| start = 23
	\| title = ~~Harmonics~~ 24 to 34 (26ed5)		\| title = Integer harmonics 24 to 34 (26ed5)
	}}		}}
	{{Harmonics in equal		{{Harmonics in equal
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	\| collapsed = 1		\| collapsed = 1
	\| start = 34		\| start = 34
	\| title = ~~Harmonics~~ 35 to 45 (26ed5)		\| title = Integer harmonics 35 to 45 (26ed5)
	}}		}}
	{{Harmonics in equal		{{Harmonics in equal
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	\| collapsed = 1		\| collapsed = 1
	\| start = 45		\| start = 45
	\| title = ~~Harmonics~~ 46 to 56 (26ed5)		\| title = Integer harmonics 46 to 56 (26ed5)
	}}		}}
	{{Harmonics in equal		{{Harmonics in equal
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	\| collapsed = 1		\| collapsed = 1
	\| start = 56		\| start = 56
	\| title = ~~Harmonics~~ 57 to 68 (26ed5)		\| title = Integer harmonics 57 to 68 (26ed5)
	\| columns = 12		\| columns = 12
	}}		}}

BudjarnLambeth: /* Table */

2025-01-05T05:56:38Z

Table

← Older revision		Revision as of 05:56, 5 January 2025
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	As a [[MOS scale]], it is an example of the scale [[13L 1s (5/1-equivalent)]]. The 2/1-equivalent version would be [[13L 1s]].		As a [[MOS scale]], it is an example of the scale [[13L 1s (5/1-equivalent)]]. The 2/1-equivalent version would be [[13L 1s]].

	====~~Table~~====		====Intervals====
	{\| class="wikitable mw-collapsible"		{\| class="wikitable mw-collapsible"
	\|+		\|+

BudjarnLambeth: /* Intervals */

2025-01-05T05:54:54Z

Intervals

← Older revision		Revision as of 05:54, 5 January 2025
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	!16		!16
	!1714.7		!1714.7
	\|81/30		\|32/12, 81/30
	\|27/10		\|8/3, 27/10
	\|-		\|-
	!17		!17

BudjarnLambeth: /* Intervals */

2025-01-05T05:51:25Z

Intervals

← Older revision		Revision as of 05:51, 5 January 2025
Line 174:		Line 174:
	!18		!18
	!1929.0		!1929.0
	\|		\|110/36, 104/34
	\|		\|55/18, 52/17
	\|-		\|-
	!19		!19
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	!22		!22
	!2357.7		!2357.7
	\|		\|117/30
	\|		\|39/10
	\|-		\|-
	!23		!23

BudjarnLambeth: /* Intervals */

2025-01-05T05:46:07Z

Intervals

← Older revision		Revision as of 05:46, 5 January 2025
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	!20		!20
	!2143.3		!2143.3
	\|		\|(32/12)x(63/49)
	\|		\|24/7
	\|-		\|-
	!21		!21

BudjarnLambeth: /* Intervals */

2025-01-05T05:38:32Z

Intervals

← Older revision		Revision as of 05:38, 5 January 2025
Line 209:		Line 209:
	!25		!25
	!2679.1		!2679.1
	\|		\|56/12
	\|		\|14/3
	\|-		\|-
	!26		!26
Line 291:		Line 291:
	\|25		\|25
	\|2679.1		\|2679.1
	\|		\|14/3
	\|-		\|-
	\|26		\|26

BudjarnLambeth: /* Intervals */

2025-01-05T05:31:35Z

Intervals

← Older revision		Revision as of 05:31, 5 January 2025
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	! colspan="2" \|Just intonation approximation		! colspan="2" \|Just intonation approximation
	\|-		\|-
	!<small>5.6.12.22… subgroup (described above)</small>		!<small>5.6.12.22… subgroup <br>described above</small>
	!<small>5.6.12.22… subgroup~~;</small>~~ <br>~~<small>~~with ratios simplified</small>		!<small>5.6.12.22… subgroup <br>(with ratios simplified)</small>
	\|-		\|-
	!1		!1

BudjarnLambeth: /* Prime subgroups */

2025-01-05T05:29:54Z

Prime subgroups

← Older revision		Revision as of 05:29, 5 January 2025
Line 7:		Line 7:
	Pure-octaves 26ed5 is incompatible with [[prime limit]] tuning. Of all primes up to 41, 5 and 41 are the only two it approximates well. Many of 26ed5’s 'near-miss' [[prime]]s are tuned sharp, so 26ed5 can be made to work more normally by [[Octave shrinking\|compressing]] 26ed5’s [[equave]], making [[5/1]] slightly flat but still okay and the other primes more in-tune.		Pure-octaves 26ed5 is incompatible with [[prime limit]] tuning. Of all primes up to 41, 5 and 41 are the only two it approximates well. Many of 26ed5’s 'near-miss' [[prime]]s are tuned sharp, so 26ed5 can be made to work more normally by [[Octave shrinking\|compressing]] 26ed5’s [[equave]], making [[5/1]] slightly flat but still okay and the other primes more in-tune.

	A good compressed tuning of 26ed5 is [[46ed17]], which transforms 26ed5 from a 5.41 tuning to a 3.5.11.17.23.43 tuning. The 3/1 in 46ed17 isn’t that good, ~~comparable~~ to [[5edo]], but it’s a huge improvement on 26ed5. And the 5, 11, 17, 23 and 43 are genuinely solid approximations. Other tunings which are almost identical to 46ed17, and so provide those same benefits, are [[8ed18/11]] and [[20ed24/7]].		A good compressed tuning of 26ed5 is [[46ed17]], which transforms 26ed5 from a 5.41 tuning to a 3.5.11.17.23.43 tuning. The 3/1 in 46ed17 isn’t that good, with similar error to [[5edo]], but it’s a huge improvement on 26ed5. And the 5, 11, 17, 23 and 43 are genuinely solid approximations. Other tunings which are almost identical to 46ed17, and so provide those same benefits, are [[8ed18/11]] and [[20ed24/7]].

	If one attempts to [[Octave stretch\|stretch]] 26ed5 instead of compress, one will not find any tunings that approximate primes well until reaching [[11edo]], so only compression is viable, not stretching.		If one attempts to [[Octave stretch\|stretch]] 26ed5 instead of compress, one will not find any tunings that approximate primes well until reaching [[11edo]], so only compression is viable, not stretching.

BudjarnLambeth: Remove all the fractional subgroup stuff because it turned out to be redundant and needlessly complicated: integers work better for this specific scale

2025-01-05T05:29:18Z

Remove all the fractional subgroup stuff because it turned out to be redundant and needlessly complicated: integers work better for this specific scale

Show changes

BudjarnLambeth at 05:10, 5 January 2025

2025-01-05T05:10:33Z

Show changes