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#redirect [[Hemimean clan #Didacus]]
{{Infobox Regtemp
| Title = Didacus
| Subgroups = 2.5.7, 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
| Generator = 28/25 | Generator tuning = 193.772 | Optimization method = POTE
| MOS scales = [[1L 5s]], [[6L 1s]], [[6L 7s]]
| Mapping = 1; 2 5 9
| Odd limit 1 = 7 | Mistuning 1 = ??? | Complexity 1 = 13
| Odd limit 2 = 11 | Mistuning 2 = ??? | 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]].
 
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.
 
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.
 
For technical data, see [[Hemimean clan #Didacus]].
 
== Interval chain ==
In the following table, odd harmonics and subharmonics 1–35 are labeled in '''bold'''.
 
{| class="wikitable sortable center-all right-2"
|-
! rowspan="3" | &#35; !! rowspan="3" | Cents* !! colspan="3" | Approximate ratios
|-
! rowspan="2" | 2.5.7.11 intervals !! colspan="3" | Intervals of extensions
|-
! Hemithirds !! Hemiwürschmidt
|-
| 0
| 0.0
| '''1/1'''
|
|
|-
| 1
| 194.4
| 28/25, 49/44, 55/49
|
|
|-
| 2
| 388.9
| '''5/4''', 44/35
|
| 144/115
|-
| 3
| 583.3
| 7/5
|
|
|-
| 4
| 777.7
| 11/7, '''25/16'''
|
| 36/23
|-
| 5
| 972.1
| '''7/4''', 44/25
|
| 184/105
|-
| 6
| 1166.6
| 49/25, 55/28
|
| 96/49, 45/23
|-
| 7
| 161.0
| 11/10, '''35/32'''
|
| 23/21, 126/115
|-
| 8
| 355.4
| 49/40, 121/98
| 128/105
| 60/49, 92/75
|-
| 9
| 549.9
| '''11/8'''
|
| 48/35, 63/46, 115/84
|-
| 10
| 744.3
| 49/32, 77/50
| 32/21
| 75/49, 23/15
|-
| 11
| 938.7
| 55/32, 121/70
| 128/75
| 12/7
|-
| 12
| 1133.1
| 77/40
| 40/21
| 48/25, 23/12
|-
| 13
| 127.6
| 121/112
| 16/15
| 15/14
|-
| 14
| 322.0
| 77/64, 121/100
| 25/21
| 6/5, 115/96
|-
| 15
| 516.4
|
| '''4/3'''
| 75/56
|-
| 16
| 710.8
| 121/80
| 112/75
| '''3/2'''
|-
| 17
| 905.3
|
| 5/3
| 42/25
|-
| 18
| 1099.7
| 121/64
| 28/15
| '''15/8'''
|-
| 19
| 94.1
|
| 25/24
| 21/20
|}
<nowiki />* In [[CWE]] undecimal didacus
 
== Tunings ==
=== Optimized tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Prime-optimized tunings
|-
! rowspan="2" | Weight-skew\Order !! colspan="2" | 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¢
|}
 
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | [[Delta-rational chord|DR]] and equal-beating tunings
|-
! Optimized chord !! Generator value !! Polynomial !! Further notes
|-
| 4:5:7 (+1 +2) || ~28/25 = 197.346 || ''g''<sup>5</sup> &minus; 3''g''<sup>2</sup> + 2 = 0 || Close to -1/5-comma
|-
| 5:7:8 (+2 +1) || ~28/25 = 193.829 || ''g''<sup>5</sup> &minus; ''g''<sup>2</sup> &minus; 4 = 0 || Close to 37/95-comma
|-
| 7:8:10 (+1 +2) || ~28/25 = 193.630 || ''g''<sup>5</sup> + ''g''<sup>2</sup> &minus; 3 = 0 || Close to 33/80-comma
|-
| 8:11:14 (+1 +1) || ~28/25 = 195.043 || ''g''<sup>9</sup> &minus; ''g''<sup>5</sup> &minus; 1 = 0 ||
|-
| 11:14:16 (+3 +2) || ~28/25 = 192.698 || ''g''<sup>9</sup> &minus; 5''g''<sup>5</sup> + 6 = 0 ||
|}
 
=== Tuning spectrum ===
{| class="wikitable center-all left-4"
! EDO<br />generator
! [[Eigenmonzo|Eigenmonzo<br />(unchanged interval)]]*
! Generator (¢)
! Comments
|-
| '''[[7edo|1\7]]'''
|
| '''171.429'''
| 7dee val, '''lower bound of (2.5.7) 7-odd-limit diamond monotone'''
|-
| '''[[13edo|2\13]]'''
|
| '''184.615'''
| 13e val, '''lower bound of (2.5.7.11) 11-odd-limit diamond monotone'''
|-
| [[19edo|3\19]]
|
| 189.474
| 19e val
|-
|
| [[125/112]]
| 190.115
| Full-comma
|-
| [[44edo|7\44]]
|
| 190.909
| 44dee val
|-
| [[25edo|4\25]]
|
| 192.000
|
|-
| [[56edo|9\56]]
|
| 192.857
| 56e val
|-
| [[87edo|14\87]]
|
| 193.103
| 87e val
|-
|
| [[5/4]]
| 193.157
| 1/2-comma
|-
| [[118edo|19\118]]
|
| 193.220
| 118ee val
|-
| [[31edo|5\31]]
|
| 193.548
|
|-
|
| [[7/4]]
| 193.765
| 2/5-comma
|-
| [[130edo|21\130]]
|
| 193.846
| 130e val
|-
| [[99edo|16\99]]
|
| 193.939
|
|-
| [[167edo|27\167]]
|
| 194.012
| 167e val
|-
| [[68edo|11\68]]
|
| 194.118
|
|-
|
| [[7/5]]
| 194.171
| 1/3-comma
|-
| [[105edo|17\105]]
|
| 194.286
|
|-
| [[142edo|23\142]]
|
| 194.366
|
|-
|
| [[11/8]]
| 194.591
|
|-
| [[37edo|6\37]]
|
| 194.595
|
|-
|
| [[196/125]]
| 194.678
| 1/4-comma
|-
| [[117edo|19\117]]
|
| 194.872
| 117d val
|-
| [[80edo|13\80]]
|
| 195.000
|
|-
|
| [[11/10]]
| 195.001
|
|-
| [[43edo|7\43]]
|
| 195.349
|
|-
|
| [[11/7]]
| 195.623
|
|-
| [[49edo|8\49]]
|
| 195.918
|
|-
|
| [[28/25]]
| 196.198
| Untempered tuning
|-
| [[55edo|9\55]]
|
| 196.364
| 55de val
|-
| '''[[6edo|1\6]]'''
|
| '''200.000'''
| '''Upper bound of (no-threes) 7- and 11-odd-limit diamond monotone'''
|}
<nowiki />* Besides the octave
 
=== Other tunings ===
* [[DKW theory|DKW]] (2.5.7): ~2 = 1\1, ~~28/25 = 194.061
 


[[Category:Temperaments]]
[[Category:Temperaments]]
[[Category:Hemimean clan]]
[[Category:Hemimean clan]]

Revision as of 22:59, 31 October 2024

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 & 19
Generator (POTE) ~28/25 = 193.772 ¢
MOS scales 1L 5s, 6L 1s, 6L 7s
Ploidacot diseph
Minimax error (7-odd limit) ??? ¢;
(11-odd limit) ??? ¢
Target scale size (7-odd limit) 13 notes;
(11-odd limit) 19 notes

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

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)3, and therefore the 3rd harmonic 15 generators down; and hemiwürschmidt (31 & 37) tempers out 2401/2400 so that 5/48 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.

For technical data, see Hemimean clan #Didacus.

Interval chain

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

# Cents* Approximate ratios
2.5.7.11 intervals Intervals of extensions
Hemithirds Hemiwürschmidt
0 0.0 1/1
1 194.4 28/25, 49/44, 55/49
2 388.9 5/4, 44/35 144/115
3 583.3 7/5
4 777.7 11/7, 25/16 36/23
5 972.1 7/4, 44/25 184/105
6 1166.6 49/25, 55/28 96/49, 45/23
7 161.0 11/10, 35/32 23/21, 126/115
8 355.4 49/40, 121/98 128/105 60/49, 92/75
9 549.9 11/8 48/35, 63/46, 115/84
10 744.3 49/32, 77/50 32/21 75/49, 23/15
11 938.7 55/32, 121/70 128/75 12/7
12 1133.1 77/40 40/21 48/25, 23/12
13 127.6 121/112 16/15 15/14
14 322.0 77/64, 121/100 25/21 6/5, 115/96
15 516.4 4/3 75/56
16 710.8 121/80 112/75 3/2
17 905.3 5/3 42/25
18 1099.7 121/64 28/15 15/8
19 94.1 25/24 21/20

* In CWE undecimal didacus

Tunings

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

EDO
generator 		Eigenmonzo
(unchanged interval)* 		Generator (¢) 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 19e val
125/112 190.115 Full-comma
7\44 190.909 44dee val
4\25 192.000
9\56 192.857 56e val
14\87 193.103 87e val
5/4 193.157 1/2-comma
19\118 193.220 118ee val
5\31 193.548
7/4 193.765 2/5-comma
21\130 193.846 130e val
16\99 193.939
27\167 194.012 167e val
11\68 194.118
7/5 194.171 1/3-comma
17\105 194.286
23\142 194.366
11/8 194.591
6\37 194.595
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
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 = 1\1, ~~28/25 = 194.061
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