Didacus - Revision history

FloraC at 10:20, 9 February 2026

2026-02-09T10:20:42Z

← Older revision		Revision as of 10:20, 9 February 2026
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	{{Infobox ~~Regtemp~~		{{Infobox regtemp
	\| Title = Didacus		\| Title = Didacus
	\| Subgroups = 2.5.7, 2.5.7.11		\| Subgroups = 2.5.7, 2.5.7.11
	\| Comma basis = [[3136/3125]] (2.5.7); <br> [[176/175]], [[1375/1372]] (2.5.7.11)		\| Comma basis = [[3136/3125]] (2.5.7); <br>[[176/175]], [[1375/1372]] (2.5.7.11)
	\| 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

FloraC: Cleanup on infobox

2026-02-09T10:01:08Z

Cleanup on infobox

← Older revision		Revision as of 10:01, 9 February 2026
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	\| Comma basis = [[3136/3125]] (2.5.7); <br> [[176/175]], [[1375/1372]] (2.5.7.11)		\| Comma basis = [[3136/3125]] (2.5.7); <br> [[176/175]], [[1375/1372]] (2.5.7.11)
	\| Edo join 1 = 6 \| Edo join 2 = 25		\| Edo join 1 = 6 \| Edo join 2 = 25
	\| ~~Generator~~ = 28/25 \| ~~Generator~~ tuning = ~~193~~.~~772~~ \| Optimization method = ~~POTE~~		\| Mapping = 1; 2 5 9
			\| Generators = 28/25 \| Generators tuning = 194.4 \| Optimization method = CWE
	\| MOS scales = [[1L 5s]], [[6L 1s]], [[6L 7s]], [[6L 13s]], [[6L 19s]]		\| MOS scales = [[1L 5s]], [[6L 1s]], [[6L 7s]], [[6L 13s]], [[6L 19s]]
	~~\| Mapping = 1; 2 5 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		\| Odd limit 2 = 2.5.7.11 11 \| Mistuning 2 = 4.13 \| Complexity 2 = 19
	\| Odd limit 2 = (2.5.7.11) 11 \| Mistuning 2 = 4.13 \| Complexity 2 = 19
	}}		}}
	'''Didacus''' is a temperament of the [[2.5.7 subgroup]], tempering out [[3136/3125]], the hemimean comma, 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. It presents one of the most efficient traversals of the no-threes subgroup, especially considering that some tunings of didacus extend neatly to 11 and 13 (as explained below).		'''Didacus''' is a temperament of the [[2.5.7 subgroup]], tempering out [[3136/3125]], the hemimean comma, 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. It presents one of the most efficient traversals of the no-threes subgroup, especially considering that some tunings of didacus extend neatly to 11 and 13 (as explained below).

Overthink: Added subroups to odd limits of infobox

2025-09-21T19:10:49Z

Added subroups to odd limits of infobox

← Older revision		Revision as of 19:10, 21 September 2025
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	\| MOS scales = [[1L 5s]], [[6L 1s]], [[6L 7s]], [[6L 13s]], [[6L 19s]]		\| MOS scales = [[1L 5s]], [[6L 1s]], [[6L 7s]], [[6L 13s]], [[6L 19s]]
	\| Mapping = 1; 2 5 9		\| Mapping = 1; 2 5 9
	\| Odd limit 1 = 7 \| Mistuning 1 = 1.22 \| Complexity 1 = 13		\| Odd limit 1 = (2.5.7) 7 \| Mistuning 1 = 1.22 \| Complexity 1 = 13
	\| Odd limit 2 = 11 \| Mistuning 2 = 4.13 \| Complexity 2 = 19		\| Odd limit 2 = (2.5.7.11) 11 \| Mistuning 2 = 4.13 \| Complexity 2 = 19
	}}		}}
	'''Didacus''' is a temperament of the [[2.5.7 subgroup]], tempering out [[3136/3125]], the hemimean comma, 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. It presents one of the most efficient traversals of the no-threes subgroup, especially considering that some tunings of didacus extend neatly to 11 and 13 (as explained below).		'''Didacus''' is a temperament of the [[2.5.7 subgroup]], tempering out [[3136/3125]], the hemimean comma, 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. It presents one of the most efficient traversals of the no-threes subgroup, especially considering that some tunings of didacus extend neatly to 11 and 13 (as explained below).

FloraC: Text replacement - "Prime-optimized" to "Norm-based"

2025-09-12T14:08:23Z

Text replacement - "Prime-optimized" to "Norm-based"

← Older revision		Revision as of 14:08, 12 September 2025
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	=== Optimized tunings ===		=== Optimized tunings ===
	{\| class="wikitable mw-collapsible mw-collapsed"		{\| class="wikitable mw-collapsible mw-collapsed"
	\|+ style="font-size: 105%; white-space: nowrap;" \| ~~Prime~~-~~optimized~~ tunings		\|+ style="font-size: 105%; white-space: nowrap;" \| Norm-based tunings
	\|-		\|-
	! rowspan="2" \| Weight-skew\Order !! colspan="2" \| Euclidean		! rowspan="2" \| Weight-skew\Order !! colspan="2" \| Euclidean

Lériendil: added extensia

2025-07-15T20:19:23Z

added extensia

← Older revision		Revision as of 20:19, 15 July 2025
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	\| 195.623		\| 195.623
	\|		\|
	\| '''~~upper~~ bound of (2.5.7.11) 11-odd-limit diamond tradeoff'''		\| '''Upper bound of (2.5.7.11) 11-odd-limit diamond tradeoff'''
	\|-		\|-
	\| [[49edo\|8\49]]		\| [[49edo\|8\49]]

Lériendil at 19:33, 15 July 2025

2025-07-15T19:33:33Z

Show changes

Lériendil at 12:17, 9 July 2025

2025-07-09T12:17:01Z

← Older revision		Revision as of 12:17, 9 July 2025
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	It also has a simple extension to prime 11 - undecimal didacus, by tempering out [[176/175]], the valinorsma, so that (5/4)<sup>2</sup> is equated to [[11/7]] and 9 generators stack to [[11/4]]; prime 13 can be found by tempering out [[640/637]], equating [[16/13]] to [[49/40]], and thereby putting the 13th harmonic 8 generators down. Beyond tridecimal didacus, further extensions to primes 17 and 19, known as roulette and mediantone, are also possible, sharing in common the interpretation of the generator as [[19/17]].		It also has a simple extension to prime 11 - undecimal didacus, by tempering out [[176/175]], the valinorsma, so that (5/4)<sup>2</sup> is equated to [[11/7]] and 9 generators stack to [[11/4]]; prime 13 can be found by tempering out [[640/637]], equating [[16/13]] to [[49/40]], and thereby putting the 13th harmonic 8 generators down. Beyond tridecimal didacus, further extensions to primes 17 and 19, known as roulette and mediantone, are also possible, sharing in common the interpretation of the generator as [[19/17]].

	As for prime 3, while didacus has as a weak extension, among others, [[septimal meantone]] (didacus is every other step of septimal meantone, and has an interpretation such that the generator represents [[9/8]]~[[10/9]], known as ''isra''), strong extensions that include 3 are rather complex. Hemithirds (25 & 31) tempers out [[1029/1024]] to find the fifth at [[3/2]] [[~]] ([[8/7]])<sup>3</sup>, and therefore the 3rd harmonic 15 generators down; and hemiwürschmidt (31 & 37) tempers out [[2401/2400]] so that ([[5/4]])<sup>8</sup> is equated to [[6/1]], finding the 3rd harmonic 16 generators up (and as described for the page for 5-limit [[würschmidt]], there is also a free extension to find [[23/1]] at 28 generators). These two mappings intersect in 31edo, though the latter hews closer to the optimal range for undecimal didacus specifically.		As for prime 3, while didacus has as a weak extension, among others, [[septimal meantone]] (didacus is every other step of septimal meantone, and has an interpretation such that the generator represents [[9/8]]~[[10/9]], known as ''isra''), strong extensions that include 3 are rather complex. [[Hemithirds]] (25 & 31) tempers out [[1029/1024]] to find the fifth at [[3/2]] [[~]] ([[8/7]])<sup>3</sup>, and therefore the 3rd harmonic 15 generators down; and [[hemiwürschmidt]] (31 & 37) tempers out [[2401/2400]] so that ([[5/4]])<sup>8</sup> is equated to [[6/1]], finding the 3rd harmonic 16 generators up (and as described for the page for 5-limit [[würschmidt]], there is also a free extension to find [[23/1]] at 28 generators). These two mappings intersect in 31edo, though the latter hews closer to the optimal range for undecimal didacus specifically.

	For technical data, see [[Hemimean clan #Didacus]].		For technical data, see [[Hemimean clan #Didacus]].

Lériendil at 19:12, 28 June 2025

2025-06-28T19:12:21Z

← Older revision		Revision as of 19:12, 28 June 2025
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	\| Subgroups = 2.5.7, 2.5.7.11		\| Subgroups = 2.5.7, 2.5.7.11
	\| Comma basis = [[3136/3125]] (2.5.7); <br> [[176/175]], [[1375/1372]] (2.5.7.11)		\| Comma basis = [[3136/3125]] (2.5.7); <br> [[176/175]], [[1375/1372]] (2.5.7.11)
	\| Edo join 1 = 6 \| Edo join 2 = 19		\| Edo join 1 = 6 \| Edo join 2 = 25
	\| Generator = 28/25 \| Generator tuning = 193.772 \| Optimization method = POTE		\| Generator = 28/25 \| Generator tuning = 193.772 \| Optimization method = POTE
	\| MOS scales = [[1L 5s]], [[6L 1s]], [[6L 7s]]		\| MOS scales = [[1L 5s]], [[6L 1s]], [[6L 7s]], [[6L 13s]], [[6L 19s]]
	\| Mapping = 1; 2 5 9		\| Mapping = 1; 2 5 9
	\| Odd limit 1 = 7 \| Mistuning 1 = 1.22 \| Complexity 1 = 13		\| Odd limit 1 = 7 \| Mistuning 1 = 1.22 \| Complexity 1 = 13

Lériendil at 11:12, 20 June 2025

2025-06-20T11:12:43Z

← Older revision		Revision as of 11:12, 20 June 2025
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	\| 20/13, 77/50		\| 20/13, 77/50
	\| '''32/21'''		\| '''32/21'''
	\| 75/49~~, 23/15~~		\| 23/15, 75/49
	\|-		\|-
	\| 11		\| 11
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	\| 25/13, 77/40		\| 25/13, 77/40
	\| 40/21		\| 40/21
	\| 48/25~~, 23/12~~		\| 23/12, 48/25
	\|-		\|-
	\| 13		\| 13
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	\| 77/64, 110/91		\| 77/64, 110/91
	\| 25/21		\| 25/21
	\| 6/5~~, 115/96~~		\| 6/5
	\|-		\|-
	\| 15		\| 15

Lériendil at 23:41, 4 May 2025

2025-05-04T23:41:20Z

← Older revision		Revision as of 23:41, 4 May 2025
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	=== Isomorphism with Sirius ===		=== Isomorphism with Sirius ===
	One of the more peculiar properties of the Didacus temperament is its relationship with the [[3.5.7 subgroup]] temperament [[Sirius]]. Sirius tempers out [[3125/3087]] which is the ~~Don Page comma~~ between ~~7/5 and~~ [[5/3]]~~: it is the difference between 5/3~~ stacked twice and 7/5 stacked thrice, so that 5/3 spans three generators (representing (5/3)/(7/5) = [[25/21]]) and 7/5 spans two. Therefore [[7/3]], the [[tritave]]-reduced harmonic 7, is split into 5 equal parts, 3 of which represent 5/3, the tritave-reduced harmonic 5, and we can see that the 3:5:7 chord in Sirius and 4:5:7 chord in Didacus are isomorphic to each other's complement.		One of the more peculiar properties of the Didacus temperament is its relationship with the [[3.5.7 subgroup]] temperament [[Sirius]]. Sirius tempers out [[3125/3087]] which is the difference between [[5/3]] stacked twice and 7/5 stacked thrice, so that 5/3 spans three generators (representing (5/3)/(7/5) = [[25/21]]) and 7/5 spans two. Therefore [[7/3]], the [[tritave]]-reduced harmonic 7, is split into 5 equal parts, 3 of which represent 5/3, the tritave-reduced harmonic 5, and we can see that the 3:5:7 chord in Sirius and 4:5:7 chord in Didacus are isomorphic to each other's complement.

	Even more interestingly, Sirius also has [[6L 1s (3/1-equivalent)\|6L 1s]] and [[6L 7s (3/1-equivalent)\|6L 7s]] MOS scales, which have the same shape as the 7- and 13-note MOSes of Didacus, and a descendant 19-note MOS scale. While the cardinalities of scales diverge between the temperaments after 19 notes, a quite analogous hexatonic picture to the previous section can be constructed for Sirius, and in sharing this structure despite the massive stretch between octaves and tritaves, Didacus and Sirius provide a unique avenue for transferring consonant octave-repeating no-threes harmony into consonant tritave-repeating no-twos harmony.		Even more interestingly, Sirius also has [[6L 1s (3/1-equivalent)\|6L 1s]] and [[6L 7s (3/1-equivalent)\|6L 7s]] MOS scales, which have the same shape as the 7- and 13-note MOSes of Didacus, and a descendant 19-note MOS scale. While the cardinalities of scales diverge between the temperaments after 19 notes, a quite analogous hexatonic picture to the previous section can be constructed for Sirius, and in sharing this structure despite the massive stretch between octaves and tritaves, Didacus and Sirius provide a unique avenue for transferring consonant octave-repeating no-threes harmony into consonant tritave-repeating no-twos harmony.