Xenharmonic Wiki

Didacus: Difference between revisions

← Older edit
VisualWikitext
added 6edo detemper info
m added extensia
 
(14 intermediate revisions by the same user not shown)
Line 3: Line 3:
| 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
| Odd limit 2 = 11 | Mistuning 2 = 4.13 | Complexity 2 = 19
| Odd limit 2 = 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 ratios. 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).  


[[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 scale]]s 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 scale]]s 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]].
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]], 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 174: Line 174:


=== The hexatonic framework ===
=== The hexatonic framework ===
The 2.5.7 subgroup can be crudely approximated by [[6edo]], which is itself technically a didacus tuning as 5/4 spans 2 steps and 7/5 spans 3. Every other didacus tuning is essentially a dietic inflection of this basic hexatonic structure. Therefore, the intervals of didacus can be organized according to how many steps of [[6edo]], or equivalently the 6-note MOS, they correspond to. They can be labeled "wholetone", "ditone", "tritone", etc., and inflected so that "minor" intervals are those just below a step of 6edo, and "major" intervals are just above.  Below are the intervals of the symmetric mode of Didacus[25] ([[6L&nbsp;19s]]) in undecimal CEE tuning.
The 2.5.7 subgroup can be crudely approximated by [[6edo]], which is itself technically a didacus tuning as 5/4 spans 2 steps and 7/5 spans 3. Every other didacus tuning is essentially a dietic inflection of this basic hexatonic structure. Therefore, the intervals of didacus can be organized according to how many steps of 6edo, or equivalently the 6-note MOS, they correspond to. They can be labeled "wholetone", "ditone", "tritone", etc., and inflected so that "minor" intervals are those just below a step of 6edo, and "major" intervals are just above, whereas the unison, octave, and generators can be labeled "perfect" instead.  Below are the intervals within 10 generators of the unison in undecimal CEE tuning.


{| class="wikitable"
{| class="wikitable center-all left-1"
|-
|-
! Steps of 6edo
! Steps of 6edo
| Unison
| '''Unison'''
| Wholetone
| '''Wholetone'''
| Ditone
| '''Ditone'''
| Tritone
| '''Tritone'''
| Tetratone
| '''Tetratone'''
| Pentatone
| '''Pentatone'''
| Hexatone
| '''Hexatone'''
|-
|- style="background-color: #DFDFDF;"
! "Augmented" interval
! "Augmented" interval
| 67.79
| 33.89
| 262.14
| 228.24
| 456.49
| 456.49
| 650.84
| 650.84
Line 195: Line 195:
| 1039.54
| 1039.54
|  
|  
|-
|-  
! JI intervals represented
! JI intervals represented
| 26/25
| 50/49, 56/55, 65/64
| 64/55, 65/56
| 8/7, 25/22
| 13/10, 64/49
| 13/10, 64/49
| 16/11
| 16/11
Line 204: Line 204:
| 20/11, 64/35
| 20/11, 64/35
|  
|  
|-
|- style="background-color: #DFDFDF;"
! "Major" interval
! "Major" interval
| 33.89
|  
| 228.24
|  
| 422.60
| 422.60
| 616.95
| 616.95
| 811.30
| 811.30
| 1005.65
|  
| ''1200.00''
|  
|-
|-  
! JI intervals represented
! JI intervals represented
| 50/49, 56/55, 65/64
|  
| 8/7, 25/22
|  
| 14/11, 32/25
| 14/11, 32/25
| 10/7
| 10/7
| 8/5, 35/22
| 8/5, 35/22
|
|
|- style="background-color: #DFDFDF;"
! "Perfect" interval
| ''0.00''
| 194.35
|
|
|
| 1005.65
| ''1200.00''
|-
! JI intervals represented
| ''1/1''
| 28/25
|
|
|
| 25/14
| 25/14
| ''2/1''
| ''2/1''
|-
|- style="background-color: #DFDFDF;"
! "Minor" interval
! "Minor" interval
| ''0.00''
|  
| 194.35
|  
| 388.70
| 388.70
| 583.05
| 583.05
| 777.40
| 777.40
| 971.76
|  
| 1166.11
|  
|-
|-  
! JI intervals represented
! JI intervals represented
| 1/1
|  
| 28/25
|  
| 5/4, 44/35
| 5/4, 44/35
| 7/5
| 7/5
| 11/7, 25/16
| 11/7, 25/16
| 7/4, 44/25
|  
| 49/25, 55/28
|  
|-
|- style="background-color: #DFDFDF;"
! "Diminished" interval
! "Diminished" interval
|  
|  
Line 247: Line 265:
| 549.16
| 549.16
| 743.51
| 743.51
| 937.86
| 971.76
| 1132.21
| 1166.11
|-
|-  
! JI intervals represented
! JI intervals represented
|  
|  
Line 256: Line 274:
| 11/8
| 11/8
| 20/13, 49/32
| 20/13, 49/32
| 55/32
| 7/4, 44/25
| 25/13
| 49/25, 55/28
|}
|}
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 ==
Line 300: Line 325:
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
|-
|-
Line 309: Line 335:
|  
|  
| '''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 314: 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 319: Line 347:
|  
|  
| 189.474
| 189.474
| ↓ ''[[Spell]]'' (49/48)
| 19e val
| 19e val
|-
|-
Line 324: Line 353:
| [[125/112]]
| [[125/112]]
| 190.115
| 190.115
|
| Full-comma
| Full-comma
|-
|-
Line 329: Line 359:
|  
|  
| 190.909
| 190.909
|
| 44dee val
| 44dee val
|-
|-
Line 334: Line 365:
|  
|  
| 192.000
| 192.000
| ↑ Spell <br> ↓ [[Luna and hemithirds#Tuning spectrum|Hemithirds]] (1029/1024)
|  
|  
|-
|-
Line 339: Line 371:
|  
|  
| 192.593
| 192.593
|
| 81ee val
| 81ee val
|-
|-
Line 344: Line 377:
|  
|  
| 192.857
| 192.857
|
| 56e val
| 56e val
|-
|-
Line 349: Line 383:
|  
|  
| 193.103
| 193.103
|
| 87e val
| 87e val
|-
|-
Line 354: 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 359: Line 395:
|  
|  
| 193.220
| 193.220
|
| 118ee val
| 118ee val
|-
|-
Line 364: Line 401:
|  
|  
| 193.548
| 193.548
| ↑ Hemithirds <br> ↓ ''[[Hemiwurschmidt]]'' (2401/2400)
|  
|  
|-
|-
Line 369: Line 407:
| [[35/32]]  
| [[35/32]]  
| 193.591
| 193.591
|
| 3/7-comma
| 3/7-comma
|-
|-
Line 374: Line 413:
| [[7/4]]
| [[7/4]]
| 193.765
| 193.765
|
| 2/5-comma
| 2/5-comma
|-  
|-  
Line 379: Line 419:
|  
|  
| 193.789
| 193.789
|
| 161e val
| 161e val
|-
|-
Line 384: Line 425:
|  
|  
| 193.846
| 193.846
|
| 130e val
| 130e val
|-
|-
Line 389: Line 431:
| [[49/40]]  
| [[49/40]]  
| 193.917
| 193.917
|
| 3/8-comma
| 3/8-comma
|-
|-
Line 394: Line 437:
|  
|  
| 193.939
| 193.939
|
|  
|  
|-
|-
Line 399: Line 443:
|  
|  
| 194.012
| 194.012
|
| 167e val
| 167e val
|-
|-
Line 404: Line 449:
|  
|  
| 194.118
| 194.118
|
|  
|  
|-
|-
Line 409: 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 414: Line 461:
|  
|  
| 194.286
| 194.286
|
|  
|  
|-
|-
Line 419: Line 467:
|  
|  
| 194.366
| 194.366
|
|  
|  
|-
|-
Line 424: Line 473:
| [[11/8]]
| [[11/8]]
| 194.591
| 194.591
|
|  
|  
|-
|-
Line 429: Line 479:
|  
|  
| 194.595
| 194.595
| ↑ Hemiwurschmidt
|  
|  
|-
|-
Line 434: Line 485:
| [[196/125]]
| [[196/125]]
| 194.678
| 194.678
|
| 1/4-comma
| 1/4-comma
|-
|-
Line 439: Line 491:
|  
|  
| 194.872
| 194.872
|
| 117d val
| 117d val
|-
|-
Line 444: Line 497:
|  
|  
| 195.000
| 195.000
|
|  
|  
|-
|-
Line 449: Line 503:
| [[11/10]]
| [[11/10]]
| 195.001
| 195.001
|
|  
|  
|-
|-
Line 454: Line 509:
|  
|  
| 195.349
| 195.349
|
|  
|  
|-
|-
Line 459: 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 469: Line 527:
| [[28/25]]
| [[28/25]]
| 196.198
| 196.198
|
| Untempered tuning
| Untempered tuning
|-
|-
Line 474: Line 533:
|  
|  
| 196.364
| 196.364
|
| 55de val
| 55de val
|-
|-
Line 479: 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'''
|}
|}
Retrieved from "https://en.xen.wiki/w/Didacus"