Didacus - Revision history

Vector at 01:51, 2 June 2026

2026-06-02T01:51:32Z

← Older revision		Revision as of 01:51, 2 June 2026
Line 9:		Line 9:
	\| Odd limit 1 = 2.5.7 7 \| Mistuning 1 = 1.22 \| Complexity 1 = 13		\| Odd limit 1 = 2.5.7 7 \| Mistuning 1 = 1.22 \| Complexity 1 = 13
	}}		}}
			[[File:Didacus13.png\|thumb\|Didacus 13-note MOS]]
	'''Didacus''' is a highly efficient temperament of the [[2.5.7 subgroup]], tempering out 3136/3125, such that two intervals of [[7/5]] reach the same point as three intervals of [[5/4]]; the generator is therefore a slightly narrowed (7/5)/(5/4) = [[28/25]], two of which stack to 5/4 and three of which stack to 7/5, meaning that the [[4:5:7]] chord is "locked" to (0 2 5) in terms of logarithmic size and generator steps. In the full [[7-limit]], Didacus tempering has the consequence of making 28/27 - 25/24 - 21/20 equidistant and 16/15 - 15/14 - 27/25 equidistant.		'''Didacus''' is a highly efficient temperament of the [[2.5.7 subgroup]], tempering out 3136/3125, such that two intervals of [[7/5]] reach the same point as three intervals of [[5/4]]; the generator is therefore a slightly narrowed (7/5)/(5/4) = [[28/25]], two of which stack to 5/4 and three of which stack to 7/5, meaning that the [[4:5:7]] chord is "locked" to (0 2 5) in terms of logarithmic size and generator steps. In the full [[7-limit]], Didacus tempering has the consequence of making 28/27 - 25/24 - 21/20 equidistant and 16/15 - 15/14 - 27/25 equidistant.

Lériendil: there's no 13 in the listed hemiwurschmidt intervals

2026-05-18T14:40:37Z

there's no 13 in the listed hemiwurschmidt intervals

← Older revision		Revision as of 14:40, 18 May 2026
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	! rowspan="2" \| 2.5.7 intervals !! colspan="3" \| Intervals of extensions		! rowspan="2" \| 2.5.7 intervals !! colspan="3" \| Intervals of extensions
	\|-		\|-
	! Tridecimal Didacus !! [[Luna and hemithirds#Intervals\|Hemithirds]] !! Hemiwürschmidt (~~2.3.5.7.11.13~~.23)		! Tridecimal Didacus !! [[Luna and hemithirds#Intervals\|Hemithirds]] !! Hemiwürschmidt (L11.23)

	\|-		\|-

Inthar at 12:29, 18 May 2026

2026-05-18T12:29:22Z

← Older revision		Revision as of 12:29, 18 May 2026
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	\| Title = Didacus		\| Title = Didacus
	\| Subgroups = 2.5.7		\| Subgroups = 2.5.7
	\| Comma basis = [[3136/3125]] (2.5.7)		\| Comma basis = 3136/3125 (2.5.7)
	\| Edo join 1 = 6 \| Edo join 2 = 25		\| Edo join 1 = 6 \| Edo join 2 = 25
	\| Mapping = 1; 2 5 9		\| Mapping = 1; 2 5 9
Line 9:		Line 9:
	\| Odd limit 1 = 2.5.7 7 \| Mistuning 1 = 1.22 \| Complexity 1 = 13		\| Odd limit 1 = 2.5.7 7 \| Mistuning 1 = 1.22 \| Complexity 1 = 13
	}}		}}
	'''Didacus''' is a highly efficient temperament of the [[2.5.7 subgroup]], tempering out [[3136/3125]], such that two intervals of [[7/5]] reach the same point as three intervals of [[5/4]]; the generator is therefore a slightly narrowed (7/5)/(5/4) = [[28/25]], two of which stack to 5/4 and three of which stack to 7/5, meaning that the [[4:5:7]] chord is "locked" to (0 2 5) in terms of logarithmic size and generator steps. In the full [[7-limit]], Didacus tempering has the consequence of making 28/27 - 25/24 - 21/20 equidistant and 16/15 - 15/14 - 27/25 equidistant.		'''Didacus''' is a highly efficient temperament of the [[2.5.7 subgroup]], tempering out 3136/3125, such that two intervals of [[7/5]] reach the same point as three intervals of [[5/4]]; the generator is therefore a slightly narrowed (7/5)/(5/4) = [[28/25]], two of which stack to 5/4 and three of which stack to 7/5, meaning that the [[4:5:7]] chord is "locked" to (0 2 5) in terms of logarithmic size and generator steps. In the full [[7-limit]], Didacus tempering has the consequence of making 28/27 - 25/24 - 21/20 equidistant and 16/15 - 15/14 - 27/25 equidistant.

	[[31edo]] is a very good tuning of Didacus, with its generator 5\31 (which is the "mean tone" of 31edo); but [[25edo]], [[37edo]], and [[68edo]] among others are good tunings as well. As this generator tends to be slightly less than 1/6 of the octave, [[MOS]] scales of Didacus tend to consist of 6 long intervals interspersed by sequences of diesis-sized steps (representing [[50/49]]~[[128/125]]), therefore bearing similar properties to those of [[Slendric]].		[[31edo]] is a very good tuning of Didacus, with its generator 5\31 (which is the "mean tone" of 31edo); but [[25edo]], [[37edo]], and [[68edo]] among others are good tunings as well. As this generator tends to be slightly less than 1/6 of the octave, [[MOS]] scales of Didacus tend to consist of 6 long intervals interspersed by sequences of diesis-sized steps (representing [[50/49]]~[[128/125]]), therefore bearing similar properties to those of [[Slendric]].

Inthar at 21:52, 24 April 2026

2026-04-24T21:52:37Z

← Older revision		Revision as of 21:52, 24 April 2026
Line 13:		Line 13:
	[[31edo]] is a very good tuning of Didacus, with its generator 5\31 (which is the "mean tone" of 31edo); but [[25edo]], [[37edo]], and [[68edo]] among others are good tunings as well. As this generator tends to be slightly less than 1/6 of the octave, [[MOS]] scales of Didacus tend to consist of 6 long intervals interspersed by sequences of diesis-sized steps (representing [[50/49]]~[[128/125]]), therefore bearing similar properties to those of [[Slendric]].		[[31edo]] is a very good tuning of Didacus, with its generator 5\31 (which is the "mean tone" of 31edo); but [[25edo]], [[37edo]], and [[68edo]] among others are good tunings as well. As this generator tends to be slightly less than 1/6 of the octave, [[MOS]] scales of Didacus tend to consist of 6 long intervals interspersed by sequences of diesis-sized steps (representing [[50/49]]~[[128/125]]), therefore bearing similar properties to those of [[Slendric]].

	Septimal [[Meantone]] is a relatively inaccurate weak extension of Didacus to prime 3. It may be used in the 2.9.5.7 subgroup as a strong extension. A more accurate but complex full-7-limit extension is Hemiwurschmidt ({{e\|31}} & {{e\|68}}) which adds the [[Wurschmidt]] relation (5/4)<sup>8</sup> ~= 3/2.		Septimal [[Meantone]] is a relatively inaccurate weak extension of Didacus to prime 3. It may be used in the 2.9.5.7 subgroup as a strong extension. A more accurate but complex full-7-limit extension is Hemiwurschmidt ({{e\|31}} & {{e\|37}}) which adds the [[Wurschmidt]] relation (5/4)<sup>8</sup> ~= 3/2.
	== Interval chain ==		== Interval chain ==
	In the following table, odd harmonics and subharmonics 1–35 are labeled in '''bold'''.		In the following table, odd harmonics and subharmonics 1–35 are labeled in '''bold'''.

Inthar at 21:51, 24 April 2026

2026-04-24T21:51:48Z

← Older revision		Revision as of 21:51, 24 April 2026
Line 13:		Line 13:
	[[31edo]] is a very good tuning of Didacus, with its generator 5\31 (which is the "mean tone" of 31edo); but [[25edo]], [[37edo]], and [[68edo]] among others are good tunings as well. As this generator tends to be slightly less than 1/6 of the octave, [[MOS]] scales of Didacus tend to consist of 6 long intervals interspersed by sequences of diesis-sized steps (representing [[50/49]]~[[128/125]]), therefore bearing similar properties to those of [[Slendric]].		[[31edo]] is a very good tuning of Didacus, with its generator 5\31 (which is the "mean tone" of 31edo); but [[25edo]], [[37edo]], and [[68edo]] among others are good tunings as well. As this generator tends to be slightly less than 1/6 of the octave, [[MOS]] scales of Didacus tend to consist of 6 long intervals interspersed by sequences of diesis-sized steps (representing [[50/49]]~[[128/125]]), therefore bearing similar properties to those of [[Slendric]].

	Septimal [[Meantone]] is a relatively inaccurate weak extension of Didacus to prime 3. It may be used in the 2.9.5.7 subgroup as a strong extension. A more accurate but complex full-7-limit extension is Hemiwurschmidt ({{e\|31}} & {{e\|99}}) which adds the [[Wurschmidt]] relation (5/4)<sup>8</sup> ~= 3/2.		Septimal [[Meantone]] is a relatively inaccurate weak extension of Didacus to prime 3. It may be used in the 2.9.5.7 subgroup as a strong extension. A more accurate but complex full-7-limit extension is Hemiwurschmidt ({{e\|31}} & {{e\|68}}) which adds the [[Wurschmidt]] relation (5/4)<sup>8</sup> ~= 3/2.
	== Interval chain ==		== Interval chain ==
	In the following table, odd harmonics and subharmonics 1–35 are labeled in '''bold'''.		In the following table, odd harmonics and subharmonics 1–35 are labeled in '''bold'''.

Inthar at 20:13, 24 April 2026

2026-04-24T20:13:45Z

← Older revision		Revision as of 20:13, 24 April 2026
Line 13:		Line 13:
	[[31edo]] is a very good tuning of Didacus, with its generator 5\31 (which is the "mean tone" of 31edo); but [[25edo]], [[37edo]], and [[68edo]] among others are good tunings as well. As this generator tends to be slightly less than 1/6 of the octave, [[MOS]] scales of Didacus tend to consist of 6 long intervals interspersed by sequences of diesis-sized steps (representing [[50/49]]~[[128/125]]), therefore bearing similar properties to those of [[Slendric]].		[[31edo]] is a very good tuning of Didacus, with its generator 5\31 (which is the "mean tone" of 31edo); but [[25edo]], [[37edo]], and [[68edo]] among others are good tunings as well. As this generator tends to be slightly less than 1/6 of the octave, [[MOS]] scales of Didacus tend to consist of 6 long intervals interspersed by sequences of diesis-sized steps (representing [[50/49]]~[[128/125]]), therefore bearing similar properties to those of [[Slendric]].

	Septimal [[Meantone]] is a relatively inaccurate weak extension of Didacus to prime 3. It may be used in the 2.9.5.7 subgroup as a strong extension.		Septimal [[Meantone]] is a relatively inaccurate weak extension of Didacus to prime 3. It may be used in the 2.9.5.7 subgroup as a strong extension. A more accurate but complex full-7-limit extension is Hemiwurschmidt ({{e\|31}} & {{e\|99}}) which adds the [[Wurschmidt]] relation (5/4)<sup>8</sup> ~= 3/2.

	== Interval chain ==		== Interval chain ==
	In the following table, odd harmonics and subharmonics 1–35 are labeled in '''bold'''.		In the following table, odd harmonics and subharmonics 1–35 are labeled in '''bold'''.

Inthar at 06:06, 22 April 2026

2026-04-22T06:06:08Z

← Older revision		Revision as of 06:06, 22 April 2026
Line 9:		Line 9:
	\| Odd limit 1 = 2.5.7 7 \| Mistuning 1 = 1.22 \| Complexity 1 = 13		\| Odd limit 1 = 2.5.7 7 \| Mistuning 1 = 1.22 \| Complexity 1 = 13
	}}		}}
	'''Didacus''' is a highly efficient temperament of the [[2.5.7 subgroup]], tempering out [[3136/3125]], such that two intervals of [[7/5]] reach the same point as three intervals of [[5/4]]; the generator is therefore (7/5)/(5/4) = [[28/25]] ~~(slightly narrowed)~~, two of which stack to 5/4 and three of which stack to 7/5, meaning that the [[4:5:7]] chord is "locked" to (0 2 5) in terms of logarithmic size and generator steps. In the full [[7-limit]], Didacus tempering has the consequence of making 28/27 - 25/24 - 21/20 equidistant and 16/15 - 15/14 - 27/25 equidistant.		'''Didacus''' is a highly efficient temperament of the [[2.5.7 subgroup]], tempering out [[3136/3125]], such that two intervals of [[7/5]] reach the same point as three intervals of [[5/4]]; the generator is therefore a slightly narrowed (7/5)/(5/4) = [[28/25]], two of which stack to 5/4 and three of which stack to 7/5, meaning that the [[4:5:7]] chord is "locked" to (0 2 5) in terms of logarithmic size and generator steps. In the full [[7-limit]], Didacus tempering has the consequence of making 28/27 - 25/24 - 21/20 equidistant and 16/15 - 15/14 - 27/25 equidistant.

	[[31edo]] is a very good tuning of Didacus, with its generator 5\31 (which is the "mean tone" of 31edo); but [[25edo]], [[37edo]], and [[68edo]] among others are good tunings as well. As this generator tends to be slightly less than 1/6 of the octave, [[MOS]] scales of Didacus tend to consist of 6 long intervals interspersed by sequences of diesis-sized steps (representing [[50/49]]~[[128/125]]), therefore bearing similar properties to those of [[Slendric]].		[[31edo]] is a very good tuning of Didacus, with its generator 5\31 (which is the "mean tone" of 31edo); but [[25edo]], [[37edo]], and [[68edo]] among others are good tunings as well. As this generator tends to be slightly less than 1/6 of the octave, [[MOS]] scales of Didacus tend to consist of 6 long intervals interspersed by sequences of diesis-sized steps (representing [[50/49]]~[[128/125]]), therefore bearing similar properties to those of [[Slendric]].

Inthar at 06:00, 22 April 2026

2026-04-22T06:00:48Z

← Older revision		Revision as of 06:00, 22 April 2026
Line 9:		Line 9:
	\| Odd limit 1 = 2.5.7 7 \| Mistuning 1 = 1.22 \| Complexity 1 = 13		\| Odd limit 1 = 2.5.7 7 \| Mistuning 1 = 1.22 \| Complexity 1 = 13
	}}		}}
	'''Didacus''' is a highly efficient temperament of the [[2.5.7 subgroup]], tempering out [[3136/3125]], such that two intervals of [[7/5]] reach the same point as three intervals of [[5/4]]; the generator is therefore (7/5)/(5/4) = [[28/25]], two of which stack to 5/4 and three of which stack to 7/5, meaning that the [[4:5:7]] chord is "locked" to (0 2 5) in terms of logarithmic size and generator steps. In the full [[7-limit]], Didacus tempering has the consequence of making 28/27 - 25/24 - 21/20 equidistant and 16/15 - 15/14 - 27/25 equidistant.		'''Didacus''' is a highly efficient temperament of the [[2.5.7 subgroup]], tempering out [[3136/3125]], such that two intervals of [[7/5]] reach the same point as three intervals of [[5/4]]; the generator is therefore (7/5)/(5/4) = [[28/25]] (slightly narrowed), two of which stack to 5/4 and three of which stack to 7/5, meaning that the [[4:5:7]] chord is "locked" to (0 2 5) in terms of logarithmic size and generator steps. In the full [[7-limit]], Didacus tempering has the consequence of making 28/27 - 25/24 - 21/20 equidistant and 16/15 - 15/14 - 27/25 equidistant.

	[[31edo]] is a very good tuning of Didacus, with its generator 5\31 (which is the "mean tone" of 31edo); but [[25edo]], [[37edo]], and [[68edo]] among others are good tunings as well. As this generator tends to be slightly less than 1/6 of the octave, [[MOS]] scales of Didacus tend to consist of 6 long intervals interspersed by sequences of diesis-sized steps (representing [[50/49]]~[[128/125]]), therefore bearing similar properties to those of [[Slendric]].		[[31edo]] is a very good tuning of Didacus, with its generator 5\31 (which is the "mean tone" of 31edo); but [[25edo]], [[37edo]], and [[68edo]] among others are good tunings as well. As this generator tends to be slightly less than 1/6 of the octave, [[MOS]] scales of Didacus tend to consist of 6 long intervals interspersed by sequences of diesis-sized steps (representing [[50/49]]~[[128/125]]), therefore bearing similar properties to those of [[Slendric]].

Inthar at 20:36, 18 April 2026

2026-04-18T20:36:24Z

← Older revision		Revision as of 20:36, 18 April 2026
Line 9:		Line 9:
	\| Odd limit 1 = 2.5.7 7 \| Mistuning 1 = 1.22 \| Complexity 1 = 13		\| Odd limit 1 = 2.5.7 7 \| Mistuning 1 = 1.22 \| Complexity 1 = 13
	}}		}}
	'''Didacus''' is a highly efficient temperament of the [[2.5.7 subgroup]], tempering out [[3136/3125]], such that two intervals of [[7/5]] reach the same point as three intervals of [[5/4]]; the generator is therefore (7/5)/(5/4) = [[28/25]], two of which stack to 5/4 and three of which stack to 7/5, meaning that the [[4:5:7]] chord is "locked" to (0 2 5) in terms of logarithmic size and generator steps. In a full [[7=limit]] ~~temperament~~, Didacus tempering has the consequence of making 28/27 - 25/24 - 21/20 equidistant and 16/15 - 15/14 - 27/25 equidistant.		'''Didacus''' is a highly efficient temperament of the [[2.5.7 subgroup]], tempering out [[3136/3125]], such that two intervals of [[7/5]] reach the same point as three intervals of [[5/4]]; the generator is therefore (7/5)/(5/4) = [[28/25]], two of which stack to 5/4 and three of which stack to 7/5, meaning that the [[4:5:7]] chord is "locked" to (0 2 5) in terms of logarithmic size and generator steps. In the full [[7-limit]], Didacus tempering has the consequence of making 28/27 - 25/24 - 21/20 equidistant and 16/15 - 15/14 - 27/25 equidistant.

	[[31edo]] is a very good tuning of Didacus, with its generator 5\31 (which is the "mean tone" of 31edo); but [[25edo]], [[37edo]], and [[68edo]] among others are good tunings as well. As this generator tends to be slightly less than 1/6 of the octave, [[MOS]] scales of Didacus tend to consist of 6 long intervals interspersed by sequences of diesis-sized steps (representing [[50/49]]~[[128/125]]), therefore bearing similar properties to those of [[Slendric]].		[[31edo]] is a very good tuning of Didacus, with its generator 5\31 (which is the "mean tone" of 31edo); but [[25edo]], [[37edo]], and [[68edo]] among others are good tunings as well. As this generator tends to be slightly less than 1/6 of the octave, [[MOS]] scales of Didacus tend to consist of 6 long intervals interspersed by sequences of diesis-sized steps (representing [[50/49]]~[[128/125]]), therefore bearing similar properties to those of [[Slendric]].

Inthar at 20:36, 18 April 2026

2026-04-18T20:36:06Z

← Older revision		Revision as of 20:36, 18 April 2026
Line 9:		Line 9:
	\| Odd limit 1 = 2.5.7 7 \| Mistuning 1 = 1.22 \| Complexity 1 = 13		\| Odd limit 1 = 2.5.7 7 \| Mistuning 1 = 1.22 \| Complexity 1 = 13
	}}		}}
	'''Didacus''' is a highly efficient temperament of the [[2.5.7 subgroup]], tempering out [[3136/3125]], such that two intervals of [[7/5]] reach the same point as three intervals of [[5/4]]; the generator is therefore (7/5)/(5/4) = [[28/25]], two of which stack to 5/4 and three of which stack to 7/5, meaning that the [[4:5:7]] chord is "locked" to (0 2 5) in terms of logarithmic size and generator steps. In a full ~~2.3.5.~~7 temperament, Didacus tempering has the consequence of making 28/27 - 25/24 - 21/20 equidistant and 16/15 - 15/14 - 27/25 equidistant.		'''Didacus''' is a highly efficient temperament of the [[2.5.7 subgroup]], tempering out [[3136/3125]], such that two intervals of [[7/5]] reach the same point as three intervals of [[5/4]]; the generator is therefore (7/5)/(5/4) = [[28/25]], two of which stack to 5/4 and three of which stack to 7/5, meaning that the [[4:5:7]] chord is "locked" to (0 2 5) in terms of logarithmic size and generator steps. In a full [[7=limit]] temperament, Didacus tempering has the consequence of making 28/27 - 25/24 - 21/20 equidistant and 16/15 - 15/14 - 27/25 equidistant.

	[[31edo]] is a very good tuning of Didacus, with its generator 5\31 (which is the "mean tone" of 31edo); but [[25edo]], [[37edo]], and [[68edo]] among others are good tunings as well. As this generator tends to be slightly less than 1/6 of the octave, [[MOS]] scales of Didacus tend to consist of 6 long intervals interspersed by sequences of diesis-sized steps (representing [[50/49]]~[[128/125]]), therefore bearing similar properties to those of [[Slendric]].		[[31edo]] is a very good tuning of Didacus, with its generator 5\31 (which is the "mean tone" of 31edo); but [[25edo]], [[37edo]], and [[68edo]] among others are good tunings as well. As this generator tends to be slightly less than 1/6 of the octave, [[MOS]] scales of Didacus tend to consist of 6 long intervals interspersed by sequences of diesis-sized steps (representing [[50/49]]~[[128/125]]), therefore bearing similar properties to those of [[Slendric]].