Didacus: Difference between revisions

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Created page with "{{Infobox regtemp | Title = Didacus | Subgroups = 2.5.7 | Comma basis = 3136/3125 (2.5.7) | Edo join 1 = 6 | Edo join 2 = 25 | Mapping = 1; 2 5 9 | Generators = 28/25 | Generators tuning = 194.4 | Optimization method = CWE | MOS scales = 1L 5s, 6L 1s, 6L 7s, 6L 13s, 6L 19s | Odd limit 1 = 2.5.7 7 | Mistuning 1 = 1.22 | Complexity 1 = 13 }}"
 
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| Odd limit 1 = 2.5.7 7 | Mistuning 1 = 1.22 | Complexity 1 = 13
| Odd limit 1 = 2.5.7 7 | Mistuning 1 = 1.22 | Complexity 1 = 13
}}
}}
'''Didacus''' is a temperament of the [[2.5.7 subgroup]], tempering out [[3136/3125]], 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.
[[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]].
== Interval chain ==
=== 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" | # !! rowspan="3" | Cents* !! colspan="4" | Approximate ratios
|-
! rowspan="2" | 2.5.7 intervals !! colspan="3" | Intervals of extensions
|-
! Tridecimal didacus !! [[Luna and hemithirds#Intervals|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
|}
<nowiki/>* In [[CWE]] undecimal didacus

Revision as of 02:03, 9 March 2026

Didacus
Subgroups 2.5.7
Reduced mapping ⟨1; 2 5 9]
ET join 6 & 25
Generators (CWE) ~28/25 = 194.4¢
MOS scales 1L 5s, 6L 1s, 6L 7s, 6L 13s, 6L 19s
Ploidacot diseph
Comma basis 3136/3125 (2.5.7)
Minimax error 2.5.7 7-odd-limit: 1.22¢
Target scale size 2.5.7 7-odd-limit: 13 notes

Didacus is a temperament of the 2.5.7 subgroup, tempering out 3136/3125, 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.

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.

Interval chain

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

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