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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
Line 16: Line 16:
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]].  
Line 106: Line 106:
| 20/13, 77/50
| 20/13, 77/50
| '''32/21'''
| '''32/21'''
| 75/49, 23/15
| 23/15, 75/49
|-
|-
| 11
| 11
Line 120: Line 120:
| 25/13, 77/40
| 25/13, 77/40
| 40/21
| 40/21
| 48/25, 23/12
| 23/12, 48/25
|-
|-
| 13
| 13
Line 134: Line 134:
| 77/64, 110/91
| 77/64, 110/91
| 25/21
| 25/21
| 6/5, 115/96
| 6/5
|-
|-
| 15
| 15
Line 279: Line 279:


Similarly to tertian harmony in diatonic and [[chain of fifths]]-based systems, a system of harmony for didacus can be constructed based on these hexatonic categories. The fundamental chord, 4:5:7:8, splits the hexatone into intervals of 2, 3, and 1 tones respectively, so that 4:5:7 is a tritone stacked atop a ditone, 5:7:8 is a wholetone stacked atop a tritone, and 7:8:10 is a ditone stacked atop a wholetone. We can then take these chords' complements to achieve the other permutations of 1, 2, and 3. Notably, other chords one may want to use, such as 8:11:14, also fit into this format; 8:11:14 is a ditone stacked atop a tritone, and in that fashion much can be obtained from creating different harmonies from inflections of hexatonic interval categories.
Similarly to tertian harmony in diatonic and [[chain of fifths]]-based systems, a system of harmony for didacus can be constructed based on these hexatonic categories. The fundamental chord, 4:5:7:8, splits the hexatone into intervals of 2, 3, and 1 tones respectively, so that 4:5:7 is a tritone stacked atop a ditone, 5:7:8 is a wholetone stacked atop a tritone, and 7:8:10 is a ditone stacked atop a wholetone. We can then take these chords' complements to achieve the other permutations of 1, 2, and 3. Notably, other chords one may want to use, such as 8:11:14, also fit into this format; 8:11:14 is a ditone stacked atop a tritone, and in that fashion much can be obtained from creating different harmonies from inflections of hexatonic interval categories.
=== 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 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.


== Tunings ==
== Tunings ==
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The below tuning spectrum assumes undecimal didacus.
The below tuning spectrum assumes undecimal didacus.


{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4 left-5"
! EDO<br>generator
! EDO<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged interval)]]*
! [[Eigenmonzo|Eigenmonzo<br>(unchanged interval)]]*
! Generator (¢)
! Generator (¢)
! Extension
! Comments
! Comments
|-
|-
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|  
|  
| '''171.429'''
| '''171.429'''
|
| 7dee val, '''lower bound of (2.5.7) 7-odd-limit diamond monotone'''
| 7dee val, '''lower bound of (2.5.7) 7-odd-limit diamond monotone'''
|-
|-
Line 334: Line 341:
|  
|  
| '''184.615'''
| '''184.615'''
|
| 13e val, '''lower bound of (2.5.7.11) 11-odd-limit diamond monotone'''
| 13e val, '''lower bound of (2.5.7.11) 11-odd-limit diamond monotone'''
|-
|-
Line 339: Line 347:
|  
|  
| 189.474
| 189.474
| ↓ ''[[Spell]]'' (49/48)
| 19e val
| 19e val
|-
|-
Line 344: Line 353:
| [[125/112]]
| [[125/112]]
| 190.115
| 190.115
|
| Full-comma
| Full-comma
|-
|-
Line 349: Line 359:
|  
|  
| 190.909
| 190.909
|
| 44dee val
| 44dee val
|-
|-
Line 354: Line 365:
|  
|  
| 192.000
| 192.000
| ↑ Spell <br> ↓ [[Luna and hemithirds#Tuning spectrum|Hemithirds]] (1029/1024)
|  
|  
|-
|-
Line 359: Line 371:
|  
|  
| 192.593
| 192.593
|
| 81ee val
| 81ee val
|-
|-
Line 364: Line 377:
|  
|  
| 192.857
| 192.857
|
| 56e val
| 56e val
|-
|-
Line 369: Line 383:
|  
|  
| 193.103
| 193.103
|
| 87e val
| 87e val
|-
|-
Line 374: Line 389:
| [[5/4]]
| [[5/4]]
| 193.157
| 193.157
|
| 1/2-comma, '''lower bound of (2.5.7.11) 7- and 11-odd-limit diamond tradeoff'''
| 1/2-comma, '''lower bound of (2.5.7.11) 7- and 11-odd-limit diamond tradeoff'''
|-  
|-  
Line 379: Line 395:
|  
|  
| 193.220
| 193.220
|
| 118ee val
| 118ee val
|-
|-
Line 384: Line 401:
|  
|  
| 193.548
| 193.548
| ↑ Hemithirds <br> ↓ ''[[Hemiwurschmidt]]'' (2401/2400)
|  
|  
|-
|-
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| [[35/32]]  
| [[35/32]]  
| 193.591
| 193.591
|
| 3/7-comma
| 3/7-comma
|-
|-
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| [[7/4]]
| [[7/4]]
| 193.765
| 193.765
|
| 2/5-comma
| 2/5-comma
|-  
|-  
Line 399: Line 419:
|  
|  
| 193.789
| 193.789
|
| 161e val
| 161e val
|-
|-
Line 404: Line 425:
|  
|  
| 193.846
| 193.846
|
| 130e val
| 130e val
|-
|-
Line 409: Line 431:
| [[49/40]]  
| [[49/40]]  
| 193.917
| 193.917
|
| 3/8-comma
| 3/8-comma
|-
|-
Line 414: Line 437:
|  
|  
| 193.939
| 193.939
|
|  
|  
|-
|-
Line 419: Line 443:
|  
|  
| 194.012
| 194.012
|
| 167e val
| 167e val
|-
|-
Line 424: Line 449:
|  
|  
| 194.118
| 194.118
|
|  
|  
|-
|-
Line 429: Line 455:
| [[7/5]]
| [[7/5]]
| 194.171
| 194.171
|
| 1/3-comma, '''upper bound of (2.5.7) 7-odd-limit diamond tradeoff'''
| 1/3-comma, '''upper bound of (2.5.7) 7-odd-limit diamond tradeoff'''
|-
|-
Line 434: Line 461:
|  
|  
| 194.286
| 194.286
|
|  
|  
|-
|-
Line 439: Line 467:
|  
|  
| 194.366
| 194.366
|
|  
|  
|-
|-
Line 444: Line 473:
| [[11/8]]
| [[11/8]]
| 194.591
| 194.591
|
|  
|  
|-
|-
Line 449: Line 479:
|  
|  
| 194.595
| 194.595
| ↑ Hemiwurschmidt
|  
|  
|-
|-
Line 454: Line 485:
| [[196/125]]
| [[196/125]]
| 194.678
| 194.678
|
| 1/4-comma
| 1/4-comma
|-
|-
Line 459: Line 491:
|  
|  
| 194.872
| 194.872
|
| 117d val
| 117d val
|-
|-
Line 464: Line 497:
|  
|  
| 195.000
| 195.000
|
|  
|  
|-
|-
Line 469: Line 503:
| [[11/10]]
| [[11/10]]
| 195.001
| 195.001
|
|  
|  
|-
|-
Line 474: Line 509:
|  
|  
| 195.349
| 195.349
|
|  
|  
|-
|-
Line 479: Line 515:
| [[11/7]]
| [[11/7]]
| 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]]
|  
|  
| 195.918
| 195.918
|
|  
|  
|-
|-
Line 489: Line 527:
| [[28/25]]
| [[28/25]]
| 196.198
| 196.198
|
| Untempered tuning
| Untempered tuning
|-
|-
Line 494: Line 533:
|  
|  
| 196.364
| 196.364
|
| 55de val
| 55de val
|-
|-
Line 499: Line 539:
|  
|  
| '''200.000'''
| '''200.000'''
|
| '''Upper bound of (no-threes) 7- and 11-odd-limit diamond monotone'''
| '''Upper bound of (no-threes) 7- and 11-odd-limit diamond monotone'''
|}
|}
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