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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 = ??? | Complexity 2 = 19
| Odd limit 2 = 11 | Mistuning 2 = 4.13 | Complexity 2 = 19
}}
}}
'''Didacus''' is an temperament of the [[2.5.7 subgroup]], tempering out [[3136/3125]], the hemimean comma, such that two intervals of [[7/5]] reach 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, notable for being one of the most efficient traversals of the no-threes subgroup. [[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]].
'''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).  


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]]. Further extensions to primes 13, 17, and 19, known as roulette and mediantone, are also possible.
[[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]].


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 spans the optimal range for undecimal didacus specifically.
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.


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


== Interval chain ==
== Theory ==
=== 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'''.  


{| class="wikitable sortable center-all right-2"
{| class="wikitable sortable center-all right-2"
|-
|-
! rowspan="3" | &#35; !! rowspan="3" | Cents* !! colspan="3" | Approximate ratios
! rowspan="3" | &#35; !! rowspan="3" | Cents* !! colspan="4" | Approximate ratios
|-
|-
! rowspan="2" | 2.5.7.11 intervals !! colspan="3" | Intervals of extensions
! rowspan="2" | 2.5.7 intervals !! colspan="3" | Intervals of extensions
|-
|-
! Hemithirds !! Hemiwürschmidt
! Tridecimal didacus !! [[Luna and hemithirds#Intervals|Hemithirds]] !! Hemiwürschmidt
|-
|-
| 0
| 0
| 0.0
| 0.0
| '''1/1'''
| '''1/1'''
|
|
|
|
|
Line 37: Line 41:
| 1
| 1
| 194.4
| 194.4
| 28/25, 49/44, 55/49
| 28/25, 125/112
| 49/44, 55/49
|
|
|  
|  
Line 43: Line 48:
| 2
| 2
| 388.9
| 388.9
| '''5/4''', 44/35
| '''5/4'''
| 44/35
|
|
| 144/115
| 144/115
Line 50: Line 56:
| 583.3
| 583.3
| 7/5
| 7/5
|
| 128/91
|  
|  
|-
|-
| 4
| 4
| 777.7
| 777.7
| 11/7, '''25/16'''
| '''25/16'''
| 11/7
|
|
| 36/23
| 36/23
Line 61: Line 68:
| 5
| 5
| 972.1
| 972.1
| '''7/4''', 44/25
| '''7/4'''
| 44/25, 160/91
|
|
| 184/105
| 184/105
Line 67: Line 75:
| 6
| 6
| 1166.6
| 1166.6
| 49/25, 55/28
| 49/25, 125/64
| 55/28, 128/65
|
|
| 96/49, 45/23
| 96/49, 45/23
Line 73: Line 82:
| 7
| 7
| 161.0
| 161.0
| 11/10, '''35/32'''
| '''35/32'''
| 11/10, 100/91
|  
|  
| 23/21, 126/115
| 23/21, 126/115
Line 79: Line 89:
| 8
| 8
| 355.4
| 355.4
| 49/40, 121/98
| 49/40
| '''16/13'''
| 128/105
| 128/105
| 60/49, 92/75
| 60/49, 92/75
Line 85: Line 96:
| 9
| 9
| 549.9
| 549.9
| 175/128
| '''11/8'''
| '''11/8'''
|  
|  
Line 91: Line 103:
| 10
| 10
| 744.3
| 744.3
| 49/32, 77/50
| 49/32
| 20/13, 77/50
| '''32/21'''
| '''32/21'''
| 75/49, 23/15
| 23/15, 75/49
|-
|-
| 11
| 11
| 938.7
| 938.7
| 55/32, 121/70
|
| 55/32, 112/65
| 128/75
| 128/75
| 12/7
| 12/7
Line 103: Line 117:
| 12
| 12
| 1133.1
| 1133.1
| 77/40
|  
| 25/13, 77/40
| 40/21
| 40/21
| 48/25, 23/12
| 23/12, 48/25
|-
|-
| 13
| 13
| 127.6
| 127.6
| 121/112
|  
| 14/13
| '''16/15'''
| '''16/15'''
| 15/14
| 15/14
Line 115: Line 131:
| 14
| 14
| 322.0
| 322.0
| 77/64, 121/100
|
| 77/64, 110/91
| 25/21
| 25/21
| 6/5, 115/96
| 6/5
|-
|-
| 15
| 15
| 516.4
| 516.4
|  
|  
| 35/26, 88/65
| '''4/3'''
| '''4/3'''
| 75/56
| 75/56
Line 127: Line 145:
| 16
| 16
| 710.8
| 710.8
| 121/80
|  
| 98/65
| 112/75
| 112/75
| '''3/2'''
| '''3/2'''
Line 134: Line 153:
| 905.3
| 905.3
|  
|  
| 22/13
| 5/3
| 5/3
| 42/25
| 42/25
Line 139: Line 159:
| 18
| 18
| 1099.7
| 1099.7
| 121/64
|  
| 49/26
| 28/15
| 28/15
| '''15/8'''
| '''15/8'''
Line 146: Line 167:
| 94.1
| 94.1
|  
|  
| 55/52
| 25/24
| 25/24
| 21/20
| 21/20
|}
|}
<nowiki />* In [[CWE]] undecimal didacus
<nowiki/>* In [[CWE]] undecimal didacus
 
=== 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, 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 center-all left-1"
|-
! Steps of 6edo
| '''Unison'''
| '''Wholetone'''
| '''Ditone'''
| '''Tritone'''
| '''Tetratone'''
| '''Pentatone'''
| '''Hexatone'''
|- style="background-color: #DFDFDF;"
! "Augmented" interval
| 33.89
| 228.24
| 456.49
| 650.84
| 845.19
| 1039.54
|
|-
! JI intervals represented
| 50/49, 56/55, 65/64
| 8/7, 25/22
| 13/10, 64/49
| 16/11
| 13/8
| 20/11, 64/35
|
|- style="background-color: #DFDFDF;"
! "Major" interval
|
|
| 422.60
| 616.95
| 811.30
|
|
|-
! JI intervals represented
|
|
| 14/11, 32/25
| 10/7
| 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
| ''2/1''
|- style="background-color: #DFDFDF;"
! "Minor" interval
|
|
| 388.70
| 583.05
| 777.40
|
|
|-
! JI intervals represented
|
|
| 5/4, 44/35
| 7/5
| 11/7, 25/16
|
|
|- style="background-color: #DFDFDF;"
! "Diminished" interval
|
| 160.46
| 354.81
| 549.16
| 743.51
| 971.76
| 1166.11
|-
! JI intervals represented
|
| 11/10, 35/32
| 16/13, 49/40
| 11/8
| 20/13, 49/32
| 7/4, 44/25
| 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 ==
[[File:Didacus.png|thumb|alt=Didacus.png|600x560px|A chart of the tuning spectrum of didacus, showing the offsets of prime harmonics 5, 7, and 11, as a function of the generator; all edo tunings are shown with vertical lines whose length indicates the edo's tolerance, i.e. half of its step size in either direction of just, and some small edos supporting the temperament are labeled.]]
=== Optimized tunings ===
=== Optimized tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
Line 164: Line 300:
|-
|-
! Equilateral
! Equilateral
| (2.5.7) CEE: ~28/25 = 193.681¢
| (2.5.7) CEE: ~28/25 = 193.681¢<br>(12/29-comma)
(12/29-comma)
|-
|-
! Tenney
! Tenney
Line 188: Line 323:


=== Tuning spectrum ===
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
The below tuning spectrum assumes undecimal didacus.
! EDO<br />generator
 
! [[Eigenmonzo|Eigenmonzo<br />(unchanged interval)]]*
{| class="wikitable center-all left-4 left-5"
! EDO<br>generator
! [[Eigenmonzo|Eigenmonzo<br>(unchanged interval)]]*
! Generator (¢)
! Generator (¢)
! Extension
! Comments
! Comments
|-
|-
Line 197: 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 202: 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 207: Line 347:
|  
|  
| 189.474
| 189.474
| ↓ ''[[Spell]]'' (49/48)
| 19e val
| 19e val
|-
|-
Line 212: Line 353:
| [[125/112]]
| [[125/112]]
| 190.115
| 190.115
|
| Full-comma
| Full-comma
|-
|-
Line 217: Line 359:
|  
|  
| 190.909
| 190.909
|
| 44dee val
| 44dee val
|-
|-
Line 222: Line 365:
|  
|  
| 192.000
| 192.000
| ↑ Spell <br> ↓ [[Luna and hemithirds#Tuning spectrum|Hemithirds]] (1029/1024)
|
|-
| [[81edo|13\81]]
|
| 192.593
|  
|  
| 81ee val
|-
|-
| [[56edo|9\56]]
| [[56edo|9\56]]
|  
|  
| 192.857
| 192.857
|
| 56e val
| 56e val
|-
|-
Line 232: Line 383:
|  
|  
| 193.103
| 193.103
|
| 87e val
| 87e val
|-
|-
Line 237: Line 389:
| [[5/4]]
| [[5/4]]
| 193.157
| 193.157
| 1/2-comma
|
|-
| 1/2-comma, '''lower bound of (2.5.7.11) 7- and 11-odd-limit diamond tradeoff'''
|-  
| [[118edo|19\118]]
| [[118edo|19\118]]
|  
|  
| 193.220
| 193.220
|
| 118ee val
| 118ee val
|-
|-
Line 247: Line 401:
|  
|  
| 193.548
| 193.548
| ↑ Hemithirds <br> ↓ ''[[Hemiwurschmidt]]'' (2401/2400)
|  
|  
|-
|
| [[35/32]]
| 193.591
|
| 3/7-comma
|-
|-
|  
|  
| [[7/4]]
| [[7/4]]
| 193.765
| 193.765
|
| 2/5-comma
| 2/5-comma
|-
| [[161edo|26\161]]
|
| 193.789
|
| 161e val
|-
|-
| [[130edo|21\130]]
| [[130edo|21\130]]
|  
|  
| 193.846
| 193.846
|
| 130e val
| 130e val
|-
|
| [[49/40]]
| 193.917
|
| 3/8-comma
|-
|-
| [[99edo|16\99]]
| [[99edo|16\99]]
|  
|  
| 193.939
| 193.939
|
|  
|  
|-
|-
Line 267: Line 443:
|  
|  
| 194.012
| 194.012
|
| 167e val
| 167e val
|-
|-
Line 272: Line 449:
|  
|  
| 194.118
| 194.118
|
|  
|  
|-
|-
Line 277: Line 455:
| [[7/5]]
| [[7/5]]
| 194.171
| 194.171
| 1/3-comma
|
| 1/3-comma, '''upper bound of (2.5.7) 7-odd-limit diamond tradeoff'''
|-
|-
| [[105edo|17\105]]
| [[105edo|17\105]]
|  
|  
| 194.286
| 194.286
|
|  
|  
|-
|-
Line 287: Line 467:
|  
|  
| 194.366
| 194.366
|
|  
|  
|-
|-
Line 292: Line 473:
| [[11/8]]
| [[11/8]]
| 194.591
| 194.591
|
|  
|  
|-
|-
Line 297: Line 479:
|  
|  
| 194.595
| 194.595
| ↑ Hemiwurschmidt
|  
|  
|-
|-
Line 302: Line 485:
| [[196/125]]
| [[196/125]]
| 194.678
| 194.678
|
| 1/4-comma
| 1/4-comma
|-
|-
Line 307: Line 491:
|  
|  
| 194.872
| 194.872
|
| 117d val
| 117d val
|-
|-
Line 312: Line 497:
|  
|  
| 195.000
| 195.000
|
|  
|  
|-
|-
Line 317: Line 503:
| [[11/10]]
| [[11/10]]
| 195.001
| 195.001
|
|  
|  
|-
|-
Line 322: Line 509:
|  
|  
| 195.349
| 195.349
|
|  
|  
|-
|-
Line 328: Line 516:
| 195.623
| 195.623
|  
|  
| '''Upper bound of (2.5.7.11) 11-odd-limit diamond tradeoff'''
|-
|-
| [[49edo|8\49]]
| [[49edo|8\49]]
|  
|  
| 195.918
| 195.918
|
|  
|  
|-
|-
Line 337: Line 527:
| [[28/25]]
| [[28/25]]
| 196.198
| 196.198
|
| Untempered tuning
| Untempered tuning
|-
|-
Line 342: Line 533:
|  
|  
| 196.364
| 196.364
|
| 55de val
| 55de val
|-
|-
Line 347: 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'''
|}
|}
<nowiki />* Besides the octave
<nowiki/>* Besides the octave


=== Other tunings ===
=== Other tunings ===
* [[DKW theory|DKW]] (2.5.7): ~2 = 1\1, ~28/25 = 194.061
* [[DKW theory|DKW]] (2.5.7): ~2 = 1200.000, ~28/25 = 194.061
 


[[Category:Temperaments]]
[[Category:Didacus| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Subgroup temperaments]]
[[Category:Hemimean clan]]
[[Category:Hemimean clan]]
Retrieved from "https://en.xen.wiki/w/Didacus"