Didacus: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
mNo edit summary
Line 179: Line 179:
|-
|-
! 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
| 67.79
Line 195: Line 195:
| 1039.54
| 1039.54
|  
|  
|-
|-  
! JI intervals represented
! JI intervals represented
| 26/25
| 26/25
Line 204: Line 204:
| 20/11, 64/35
| 20/11, 64/35
|  
|  
|-
|- style="background-color: #DFDFDF;"
! "Major" interval
! "Major" interval
| 33.89
| 33.89
Line 213: Line 213:
| 1005.65
| 1005.65
| ''1200.00''
| ''1200.00''
|-
|-  
! JI intervals represented
! JI intervals represented
| 50/49, 56/55, 65/64
| 50/49, 56/55, 65/64
Line 222: Line 222:
| 25/14
| 25/14
| ''2/1''
| ''2/1''
|-
|- style="background-color: #DFDFDF;"
! "Minor" interval
! "Minor" interval
| ''0.00''
| ''0.00''
Line 231: Line 231:
| 971.76
| 971.76
| 1166.11
| 1166.11
|-
|-  
! JI intervals represented
! JI intervals represented
| 1/1
| ''1/1''
| 28/25
| 28/25
| 5/4, 44/35
| 5/4, 44/35
Line 240: Line 240:
| 7/4, 44/25
| 7/4, 44/25
| 49/25, 55/28
| 49/25, 55/28
|-
|- style="background-color: #DFDFDF;"
! "Diminished" interval
! "Diminished" interval
|  
|  
Line 249: Line 249:
| 937.86
| 937.86
| 1132.21
| 1132.21
|-
|-  
! JI intervals represented
! JI intervals represented
|  
|  

Revision as of 12:24, 1 May 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 & 19
Generator (POTE) ~28/25 = 193.772 ¢
MOS scales 1L 5s, 6L 1s, 6L 7s
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 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).

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, 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 75/49, 23/15
11 938.7 55/32, 112/65 128/75 12/7
12 1133.1 25/13, 77/40 40/21 48/25, 23/12
13 127.6 14/13 16/15 15/14
14 322.0 77/64, 110/91 25/21 6/5, 115/96
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. Below are the intervals of the symmetric mode of Didacus[25] (6L 19s) in undecimal CEE tuning.

Steps of 6edo Unison Wholetone Ditone Tritone Tetratone Pentatone Hexatone
"Augmented" interval 67.79 262.14 456.49 650.84 845.19 1039.54
JI intervals represented 26/25 64/55, 65/56 13/10, 64/49 16/11 13/8 20/11, 64/35
"Major" interval 33.89 228.24 422.60 616.95 811.30 1005.65 1200.00
JI intervals represented 50/49, 56/55, 65/64 8/7, 25/22 14/11, 32/25 10/7 8/5, 35/22 25/14 2/1
"Minor" interval 0.00 194.35 388.70 583.05 777.40 971.76 1166.11
JI intervals represented 1/1 28/25 5/4, 44/35 7/5 11/7, 25/16 7/4, 44/25 49/25, 55/28
"Diminished" interval 160.46 354.81 549.16 743.51 937.86 1132.21
JI intervals represented 11/10, 35/32 16/13, 49/40 11/8 20/13, 49/32 55/32 25/13

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 (¢) 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
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
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
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
Retrieved from "https://en.xen.wiki/index.php?title=Didacus&oldid=194864"