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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
| 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, 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).  
 
[[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]].


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]].  


== 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'''.  


Line 27: Line 30:
! rowspan="2" | 2.5.7 intervals !! colspan="3" | Intervals of extensions
! rowspan="2" | 2.5.7 intervals !! colspan="3" | Intervals of extensions
|-
|-
! Tridecimal didacus !! [[Hemithirds]] !! Hemiwürschmidt
! Tridecimal didacus !! [[Luna and hemithirds#Intervals|Hemithirds]] !! Hemiwürschmidt
|-
|-
| 0
| 0
Line 103: 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 117: 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 131: Line 134:
| 77/64, 110/91
| 77/64, 110/91
| 25/21
| 25/21
| 6/5, 115/96
| 6/5
|-
|-
| 15
| 15
Line 168: Line 171:
| 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 183: 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 209: 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 218: 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 223: 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 228: Line 347:
|  
|  
| 189.474
| 189.474
| ↓ ''[[Spell]]'' (49/48)
| 19e val
| 19e val
|-
|-
Line 233: Line 353:
| [[125/112]]
| [[125/112]]
| 190.115
| 190.115
|
| Full-comma
| Full-comma
|-
|-
Line 238: Line 359:
|  
|  
| 190.909
| 190.909
|
| 44dee val
| 44dee val
|-
|-
Line 243: Line 365:
|  
|  
| 192.000
| 192.000
| ↑ Spell <br> ↓ [[Luna and hemithirds#Tuning spectrum|Hemithirds]] (1029/1024)
|  
|  
|-
|-
Line 248: Line 371:
|  
|  
| 192.593
| 192.593
|
| 81ee val
| 81ee val
|-
|-
Line 253: Line 377:
|  
|  
| 192.857
| 192.857
|
| 56e val
| 56e val
|-
|-
Line 258: Line 383:
|  
|  
| 193.103
| 193.103
|
| 87e val
| 87e val
|-
|-
Line 263: 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 268: Line 395:
|  
|  
| 193.220
| 193.220
|
| 118ee val
| 118ee val
|-
|-
Line 273: 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 288: Line 419:
|  
|  
| 193.789
| 193.789
|
| 161e val
| 161e val
|-
|-
Line 293: Line 425:
|  
|  
| 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 303: Line 443:
|  
|  
| 194.012
| 194.012
|
| 167e val
| 167e val
|-
|-
Line 308: Line 449:
|  
|  
| 194.118
| 194.118
|
|  
|  
|-
|-
Line 313: 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 318: Line 461:
|  
|  
| 194.286
| 194.286
|
|  
|  
|-
|-
Line 323: Line 467:
|  
|  
| 194.366
| 194.366
|
|  
|  
|-
|-
Line 328: Line 473:
| [[11/8]]
| [[11/8]]
| 194.591
| 194.591
|
|  
|  
|-
|-
Line 333: Line 479:
|  
|  
| 194.595
| 194.595
| ↑ Hemiwurschmidt
|  
|  
|-
|-
Line 338: Line 485:
| [[196/125]]
| [[196/125]]
| 194.678
| 194.678
|
| 1/4-comma
| 1/4-comma
|-
|-
Line 343: Line 491:
|  
|  
| 194.872
| 194.872
|
| 117d val
| 117d val
|-
|-
Line 348: Line 497:
|  
|  
| 195.000
| 195.000
|
|  
|  
|-
|-
Line 353: Line 503:
| [[11/10]]
| [[11/10]]
| 195.001
| 195.001
|
|  
|  
|-
|-
Line 358: Line 509:
|  
|  
| 195.349
| 195.349
|
|  
|  
|-
|-
Line 363: 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 373: Line 527:
| [[28/25]]
| [[28/25]]
| 196.198
| 196.198
|
| Untempered tuning
| Untempered tuning
|-
|-
Line 378: Line 533:
|  
|  
| 196.364
| 196.364
|
| 55de val
| 55de val
|-
|-
Line 383: 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
 
== Images ==
[[File:Didacus.png|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.


[[Category:Temperaments]]
[[Category:Didacus| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Subgroup temperaments]]
[[Category:Hemimean clan]]
[[Category:Hemimean clan]]

Latest revision as of 20:19, 15 July 2025

Didacus
Subgroups 2.5.7, 2.5.7.11
Comma basis 3136/3125 (2.5.7);
176/175, 1375/1372 (2.5.7.11)
Reduced mapping ⟨1; 2 5 9]
Edo join 6 & 25
Generator (POTE) ~28/25 = 193.772 ¢
MOS scales 1L 5s, 6L 1s, 6L 7s, 6L 13s, 6L 19s
Ploidacot diseph
Minimax error (7-odd limit) 1.22 ¢;
(11-odd limit) 4.13 ¢
Target scale size (7-odd limit) 13 notes;
(11-odd limit) 19 notes

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 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.

It also has a simple extension to prime 11 - undecimal didacus, by tempering out 176/175, the valinorsma, so that (5/4)2 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)3, and therefore the 3rd harmonic 15 generators down; and hemiwürschmidt (31 & 37) tempers out 2401/2400 so that (5/4)8 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.

Theory

Interval chain

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

# Cents* Approximate ratios
2.5.7 intervals Intervals of extensions
Tridecimal didacus Hemithirds Hemiwürschmidt
0 0.0 1/1
1 194.4 28/25, 125/112 49/44, 55/49
2 388.9 5/4 44/35 144/115
3 583.3 7/5 128/91
4 777.7 25/16 11/7 36/23
5 972.1 7/4 44/25, 160/91 184/105
6 1166.6 49/25, 125/64 55/28, 128/65 96/49, 45/23
7 161.0 35/32 11/10, 100/91 23/21, 126/115
8 355.4 49/40 16/13 128/105 60/49, 92/75
9 549.9 175/128 11/8 48/35, 63/46, 115/84
10 744.3 49/32 20/13, 77/50 32/21 23/15, 75/49
11 938.7 55/32, 112/65 128/75 12/7
12 1133.1 25/13, 77/40 40/21 23/12, 48/25
13 127.6 14/13 16/15 15/14
14 322.0 77/64, 110/91 25/21 6/5
15 516.4 35/26, 88/65 4/3 75/56
16 710.8 98/65 112/75 3/2
17 905.3 22/13 5/3 42/25
18 1099.7 49/26 28/15 15/8
19 94.1 55/52 25/24 21/20

* 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.

Steps of 6edo Unison Wholetone Ditone Tritone Tetratone Pentatone Hexatone
"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
"Major" interval 422.60 616.95 811.30
JI intervals represented 14/11, 32/25 10/7 8/5, 35/22
"Perfect" interval 0.00 194.35 1005.65 1200.00
JI intervals represented 1/1 28/25 25/14 2/1
"Minor" interval 388.70 583.05 777.40
JI intervals represented 5/4, 44/35 7/5 11/7, 25/16
"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 and 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

Didacus.png
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

Prime-optimized tunings
Weight-skew\Order Euclidean
Constrained Destretched
Tenney (2.5.7) CTE: ~28/25 = 193.650¢ (2.5.7) POTE: ~28/25 = 193.772¢
Equilateral (2.5.7) CEE: ~28/25 = 193.681¢
(12/29-comma)
Tenney (2.5.7.11) CTE: ~28/25 = 194.246¢ (2.5.7.11) POTE: ~28/25 = 194.556¢
DR and equal-beating tunings
Optimized chord Generator value Polynomial Further notes
4:5:7 (+1 +2) ~28/25 = 197.346 g5 − 3g2 + 2 = 0 Close to -1/5-comma
5:7:8 (+2 +1) ~28/25 = 193.829 g5 − g2 − 4 = 0 Close to 37/95-comma
7:8:10 (+1 +2) ~28/25 = 193.630 g5 + g2 − 3 = 0 Close to 33/80-comma
8:11:14 (+1 +1) ~28/25 = 195.043 g9 − g5 − 1 = 0
11:14:16 (+3 +2) ~28/25 = 192.698 g9 − 5g5 + 6 = 0

Tuning spectrum

The below tuning spectrum assumes undecimal didacus.

EDO
generator 		Eigenmonzo
(unchanged interval)* 		Generator (¢) Extension Comments
1\7 171.429 7dee val, lower bound of (2.5.7) 7-odd-limit diamond monotone
2\13 184.615 13e val, lower bound of (2.5.7.11) 11-odd-limit diamond monotone
3\19 189.474 ↓ Spell (49/48) 19e val
125/112 190.115 Full-comma
7\44 190.909 44dee val
4\25 192.000 ↑ Spell
↓ Hemithirds (1029/1024)
13\81 192.593 81ee val
9\56 192.857 56e val
14\87 193.103 87e val
5/4 193.157 1/2-comma, lower bound of (2.5.7.11) 7- and 11-odd-limit diamond tradeoff
19\118 193.220 118ee val
5\31 193.548 ↑ Hemithirds
↓ Hemiwurschmidt (2401/2400)
35/32 193.591 3/7-comma
7/4 193.765 2/5-comma
26\161 193.789 161e val
21\130 193.846 130e val
49/40 193.917 3/8-comma
16\99 193.939
27\167 194.012 167e val
11\68 194.118
7/5 194.171 1/3-comma, upper bound of (2.5.7) 7-odd-limit diamond tradeoff
17\105 194.286
23\142 194.366
11/8 194.591
6\37 194.595 ↑ Hemiwurschmidt
196/125 194.678 1/4-comma
19\117 194.872 117d val
13\80 195.000
11/10 195.001
7\43 195.349
11/7 195.623 Upper bound of (2.5.7.11) 11-odd-limit diamond tradeoff
8\49 195.918
28/25 196.198 Untempered tuning
9\55 196.364 55de val
1\6 200.000 Upper bound of (no-threes) 7- and 11-odd-limit diamond monotone

* Besides the octave

Other tunings

  • DKW (2.5.7): ~2 = 1200.000, ~28/25 = 194.061
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