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Note that any subset of any of these subgroup elements is still a valid ''n''th-prime subgroup. So one can remove as many basis elements as desired to simplify the subgroup down, if they so wish.
Note that any subset of any of these subgroup elements is still a valid ''n''th-prime subgroup. So one can remove as many basis elements as desired to simplify the subgroup down, if they so wish.


Of all subgroup interpretations of 26ed5, be they integer or fractional, the 60th-prime subgroup interpretation might be the most useful, as it includes more simple, small-numeral [[consonance]]s than any other interpretation. It includes a 6/5, 7/4, 9/4, 13/4, 11/3 and of course 5/1.
Of all subgroup interpretations of 26ed5, be they integer or fractional, the ''loose 60th-prime subgroup interpretation'' might be the most useful, as it includes more simple, small-numeral [[consonance]]s than any other interpretation. It includes a 6/5, 7/4, 9/4, 13/4, 11/3 and of course 5/1.


== Intervals ==
== Intervals ==

Revision as of 06:31, 22 December 2024

← 25ed5 26ed5 27ed5 →
Prime factorization 2 × 13
Step size 107.166 ¢ 
Octave 11\26ed5 (1178.83 ¢)
Twelfth 18\26ed5 (1928.99 ¢) (→ 9\13ed5)
Consistency limit 3
Distinct consistency limit 3

26 equal divisions of the 5th harmonic (abbreviated 26ed5) is a nonoctave tuning system that divides the interval of 5/1 into 26 equal parts of about 107 ¢ each. Each step represents a frequency ratio of 51/26, or the 26th root of 5.

Theory

Prime subgroups

Pure-octaves 26ed5 is incompatible with prime limit tuning. Of all primes up to 37, 5 is the only one it approximates well.

Many of 26ed5’s 'near-miss' primes are tuned sharp, so 26ed5 can be made to work more normally by compressing 26ed5’s equave, making 5/1 slightly flat but still okay and the other primes more in-tune.

A good compressed tuning of 26ed5 is 46ed17, which transforms 26ed5 from a 5.41 tuning to a 3.5.11.17.23.43 tuning. The 3/1 in 46ed17 isn’t that good, comparable to 5edo, but it’s a huge improvement on 26ed5. And the 5, 11, 17, 23 and 43 are genuinely solid approximations.

If one attempts to stretch 26ed5 instead of compress, one will not find any tunings that approximate primes well until reaching 11edo, so only compression is viable, not stretching.

Composite subgroups

If one ignores primes and focuses on integers in general, 26ed5 can instead be used as a strong tuning for the obscure subgroup 5.6.12.22.32.34.41.44.46.49.53.56.59.63.67.

One can also use any subset of that subgroup for example:

  • Only numbers below 40: 5.6.12.22.32.34
  • Only numbers below 50: 5.6.12.22.32.34.44.46.49
  • Only 5 and the composite numbers: 5.6.12.22.32.34.44.46.49.53.56.63
  • Only 6 and the primes: 5.6.41.59.67

Tables of harmonics

Harmonics 2 to 12 (26ed5)
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) -21.2 +27.0 -42.3 +0.0 +5.9 -46.7 +43.6 -53.1 -21.2 +28.2 -15.3
Relative (%) -19.8 +25.2 -39.5 +0.0 +5.5 -43.6 +40.7 -49.6 -19.8 +26.3 -14.3
Steps
(reduced)
11
(11)
18
(18)
22
(22)
26
(0)
29
(3)
31
(5)
34
(8)
35
(9)
37
(11)
39
(13)
40
(14)
Harmonics 13 to 23 (26ed5)
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) -46.7 +39.3 +27.0 +22.5 +24.7 +32.9 +46.5 -42.3 -19.7 +7.0 +37.2
Relative (%) -43.6 +36.7 +25.2 +21.0 +23.0 +30.7 +43.3 -39.5 -18.3 +6.5 +34.7
Steps
(reduced)
41
(15)
43
(17)
44
(18)
45
(19)
46
(20)
47
(21)
48
(22)
48
(22)
49
(23)
50
(24)
51
(25)
Harmonics 24 to 34 (26ed5)
Harmonic 24 25 26 27 28 29 30 31 32 33 34
Error Absolute (¢) -36.5 +0.0 +39.3 -26.1 +18.1 -42.6 +5.9 -50.9 +1.3 -52.0 +3.5
Relative (%) -34.1 +0.0 +36.6 -24.3 +16.9 -39.8 +5.5 -47.5 +1.2 -48.5 +3.3
Steps
(reduced)
51
(25)
52
(0)
53
(1)
53
(1)
54
(2)
54
(2)
55
(3)
55
(3)
56
(4)
56
(4)
57
(5)
Harmonics 35 to 45 (26ed5)
Harmonic 35 36 37 38 39 40 41 42 43 44 45
Error Absolute (¢) -46.7 +11.7 -35.7 +25.3 -19.7 +43.6 +0.9 -40.8 +25.6 -14.2 -53.1
Relative (%) -43.6 +10.9 -33.3 +23.6 -18.4 +40.7 +0.8 -38.1 +23.9 -13.2 -49.6
Steps
(reduced)
57
(5)
58
(6)
58
(6)
59
(7)
59
(7)
60
(8)
60
(8)
60
(8)
61
(9)
61
(9)
61
(9)
Harmonics 46 to 56 (26ed5)
Harmonic 46 47 48 49 50 51 52 53 54 55 56
Error Absolute (¢) +16.0 -21.2 +49.5 +13.8 -21.2 +51.7 +18.1 -14.9 -47.2 +28.2 -3.0
Relative (%) +14.9 -19.8 +46.2 +12.9 -19.8 +48.3 +16.9 -13.9 -44.1 +26.3 -2.8
Steps
(reduced)
62
(10)
62
(10)
63
(11)
63
(11)
63
(11)
64
(12)
64
(12)
64
(12)
64
(12)
65
(13)
65
(13)
Harmonics 57 to 68 (26ed5)
Harmonic 57 58 59 60 61 62 63 64 65 66 67 68
Error Absolute (¢) -33.7 +43.4 +13.8 -15.3 -43.9 +35.1 +7.4 -19.9 -46.7 +34.0 +8.0 -17.7
Relative (%) -31.4 +40.5 +12.9 -14.3 -41.0 +32.7 +6.9 -18.6 -43.6 +31.7 +7.4 -16.5
Steps
(reduced)
65
(13)
66
(14)
66
(14)
66
(14)
66
(14)
67
(15)
67
(15)
67
(15)
67
(15)
68
(16)
68
(16)
68
(16)

Fractional subgroups

Fractional subgroups are another approach to taming 26ed5. One can use any of the JI ratios approximated by its individual intervals as basis elements for a subgroup.

There are dozens of possible combinations, here is a small sampling of possible ones:

  • 5.6.7/4.11/3.13/4 subgroup
  • 5.6.7/4.9/4.9/7.11/3.13/4.13/7.13/9 subgroup
  • 5.6.7/4.11/3.13/4.17/11.19/8.23/11.29/7.31/7 subgroup

Nth-prime subgroups

These are some nth-prime subgroups which 26ed5 approximates well:

Loose nth-prime subgroups (non-prime numerators allowed)
Family Most distinctive related families Subgroup basis elements
11th-prime 14/11.15/11.16/11.17/11.18/11.23/11.28/11.55/11.66/11
14th-prime 7th- & half-prime 9/7.10/2.12/2.20/7.23/14.24/7.29/7.31/7.33/7
16th-prime 8th-, quarter- & half-prime 7/4.9/4.10/2.12/2.13/4.17/16.19/8
18th-prime 9th- & 6th-prime 10/2.12/2.11/3.13/9.19/9.23/18.25/6.31/9.35/9
30th-prime 15th- & 10th-prime 10/2.11/3.12/2.16/15.17/15.21/10.22/5.23/15.25/6.28/15.38/15.41/30.34/15.47/10
60th-prime 15th-, 10th- & quarter-prime 7/4.9/4.10/2.11/3.12/2.13/4.16/15.17/15.21/10.22/5.23/15.25/6.28/15.29/20.38/15.41/30.43/15.47/10.49/30.61/20.69/20
68th-prime 17th- & quarter-prime 7/4.9/4.10/2.12/2.13/4.18/17.28/17.41/34.43/17.63/34.67/34.75/17.80/17
88th-prime 11th- & eighth-prime 7/4.9/4.10/2.12/2.13/4.14/11.15/11.16/11.17/11.18/11.19/8.23/11.25/22.28/11.47/44.53/44.63/22.67/22
90th-prime 15th-, 10th-, 9th- & 6th-prime 10/2.11/3.12/2.13/9.16/15.17/15.19/9.21/10.22/5.23/15.23/18.25/6.28/15.31/9.35/9.38/15.41/30.43/15.47/19.49/30
112th-prime 16th- & 14th-prime 7/4.9/4.9/7.10/2.12/2.13/4.17/16.19/8.20/7.23/14.24/7.29/7.31/7.33/7
130th-prime 13th- & 10th-prime 6/5.10/2.12/2.21/10.22/5.24/13.27/10.29/13.35/13.39/10.47/10.54/13


Strict nth-prime subgroups (only prime numerators allowed)
Family Most distinctive related families Subgroup basis elements
16th-prime 8th-, quarter- & half-prime 7/4.13/4.17/16.19/8
18th-prime 9th- & 6th-prime 11/3.13/9.19/9.23/18.31/9
30th-prime 15th- & 10th-prime 11/3.17/15.23/15.41/30.47/10
60th-prime 15th-, 10th- & quarter-prime 7/4.11/3.13/4.17/15.23/15.29/20.41/30.43/15.47/10.61/20
68th-prime 17th- & quarter-prime 7/4.13/4.41/34.43/17.67/34
88th-prime 11th- & eighth-prime 7/4.13/4.17/11.19/8.23/11.47/44.53/44.67/22
90th-prime 15th-, 10th-, 9th- & 6th-prime 11/3.13/9.17/15.19/9.23/15.23/18.31/9.41/30.43/15.47/19.49/30
112th-prime 16th- & 14th-prime 7/4.13/4.17/16.19/8.23/14.29/7.31/7.33/7


Note that 5/1 = 10/2 = 55/11, & 6/1 = 12/2 = 66/11.

Note that any subset of any of these subgroup elements is still a valid nth-prime subgroup. So one can remove as many basis elements as desired to simplify the subgroup down, if they so wish.

Of all subgroup interpretations of 26ed5, be they integer or fractional, the loose 60th-prime subgroup interpretation might be the most useful, as it includes more simple, small-numeral consonances than any other interpretation. It includes a 6/5, 7/4, 9/4, 13/4, 11/3 and of course 5/1.

Intervals

Intervals of 26ed5 (loose subgroups used)
Step Cents Just intonation approximation
60th-prime 68th-prime 88th-prime 90th-prime 112th-prime 130th-prime Integer (5.6.12.22.32... as above) Integer (simplified)
1 107.2 16/15 18/17 47/44 16/15 17/16 36/34, 34/32 18/17, 17/16
2 214.3 17/15 25/22 17/15 25/22 25/22
3 321.5 6/5 41/34 53/44 6/5 6/5 6/5, 41/34 6/5, 41/34
4 428.7 14/11 23/18 9/7 63/49 9/7
5 535.8 41/30 15/11 41/30 19/14
6 643.0 29/20 16/11 13/9 32/22 16/11
7 750.2 23/15 17/11 23/15 54/35 34/22 17/11
8 857.3 49/30 28/17 18/11 49/30 23/14
9 964.5 7/4 7/4 7/4 26/15 7/4 56/32 7/4
10 1071.7 28/15 63/34 28/15 13/7 24/13 63/34 63/34
11 1178.8 67/34 49/25 49/25
12 1286.0 21/10 21/10 23/11, 21/10 21/10, 19/9 21/10 21/10 46/22, (6/5)x(56/32) 23/11, 21/10
13 1393.2 9/4 9/4 9/4 9/4 29/13
14 1500.3 19/8 19/8
15 1607.5 38/15 43/17 28/11 38/15 56/22 28/11
16 1714.7 27/10 27/10 27/10 27/10 27/10 35/13, 27/10
17 1821.8 43/15 63/22 43/15 20/7 63/22 63/22
18 1929.0 61/20 67/22 49/16 67/22 67/22
19 2036.2 13/4 13/4 13/4 13/4
20 2143.3 69/20 31/9 24/7
21 2250.5 11/3 11/3 22/6 11/3
22 2357.7 39/10 39/10 39/10 35/9, 39/10 39/10 39/10
23 2464.8 25/6 25/6 25/6 25/6 29/7, 25/6 54/13, 25/6 25/6 25/6
24 2572.0 22/5 75/17 22/5 31/7 22/5 22/5 22/5
25 2679.1 47/10 80/17 47/10 33/7 47/10
26 2786.3 5/1 5/1 5/1 5/1 5/1 5/1 5/1 5/1

Scales

13ed5plus

Inspired by the elevenplus scale of 22edo, the 13ed5plus scale is 13ed5 plus a step of 26ed5 in between two of its steps.

In other words, 13ed5plus is all of the odd-numbered steps of 26ed5, plus step 26.

The scale is useful because it includes most of 26ed5’s consonances while leaving out many of the less-used intervals. Making it practical to use on an instrument.

Properties

13ed5plus is a 14-tone scale.

As a MOS scale, it is an example of the scale 13L 1s (5/1-equivalent). The 2/1-equivalent version would be 13L 1s.

Table

The 13ed5plus scale
Step Cents Just intonation approximation
60th-prime 68th-prime 88th-prime 90th-prime 112th-prime Integer (5.6.12.22.32... as above) Integer (simplified)
1 107.2 16/15 18/17 47/44 16/15 17/16 36/34, 34/32 18/17, 17/16
3 321.5 6/5 41/34 53/44 6/5 6/5, 41/34 6/5, 41/34
5 535.8 41/30 15/11 41/30 19/14
7 750.2 23/15 17/11 23/15 54/35 34/22 17/11
9 964.5 7/4 7/4 7/4 26/15 7/4 56/32 7/4
11 1178.8 67/34 49/25 49/25
13 1393.2 9/4 9/4 9/4 9/4
15 1607.5 38/15 43/17 28/11 38/15 56/22 28/11
17 1821.8 43/15 63/22 43/15 20/7 63/22 63/22
19 2036.2 13/4 13/4 13/4 13/4
21 2250.5 11/3 11/3 22/6 11/3
23 2464.8 25/6 25/6 25/6 25/6 29/7, 25/6 25/6 25/6
25 2679.1 47/10 80/17 47/10 33/7
26 2786.3 5/1 5/1 5/1 5/1 5/1 5/1 5/1