26ed5: Difference between revisions

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{{Stub}}
{{Infobox ET}}
{{Infobox ET}}
{{ED intro}}
{{ED intro}}


== Theory ==
== Theory ==
=== Subgroup interpretation ===
26ed5 is a weak tuning for [[prime limit]] tuning. It 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'''.
26ed5 is a weak tuning for [[prime limit]] tuning. It 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'''.


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* Only 6 and the primes: '''5.6.41.59.67'''
* Only 6 and the primes: '''5.6.41.59.67'''


Many of 26ed5’s 'near-miss' [[prime]]s are tuned sharp, so 26ed5 can be made to work more normally by [[Octave stretch|compressing]] 26ed5’s [[equave]], making [[5/1]] slightly flat but still okay and the other primes more in-tune.
Fractional subgroups might also be an option for 26ed5.


=== Harmonics ===
=== Equave stretch ===
Many of 26ed5’s 'near-miss' [[prime]]s are tuned sharp, so 26ed5 can be made to work more normally by [[Octave shrinking|compressing]] 26ed5’s [[equave]], making [[5/1]] slightly flat but still okay and the other primes more in-tune.
 
[[29ed6]] is a compressed version of 26ed5, compressing 5/1 by roughly 5 cents, but even it is not enough to bring many primes into line. Further compression than that is required.
 
[[Octave stretch|Stretching]] rather than compressing the equave is also an option. It will change a lot of [[val]]s, so the tuning may not longer be fully recognisable as 26ed5, however the right amount of stretching will improve primes.
 
=== Tables of harmonics ===
{{Harmonics in equal
{{Harmonics in equal
| steps = 26
| steps = 26
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== Intervals ==
== Intervals ==
{{Interval table}}
{{Interval table}}
{{todo|expand}}

Revision as of 23:21, 21 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

Subgroup interpretation

26ed5 is a weak tuning for prime limit tuning. It 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

Fractional subgroups might also be an option for 26ed5.

Equave stretch

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.

29ed6 is a compressed version of 26ed5, compressing 5/1 by roughly 5 cents, but even it is not enough to bring many primes into line. Further compression than that is required.

Stretching rather than compressing the equave is also an option. It will change a lot of vals, so the tuning may not longer be fully recognisable as 26ed5, however the right amount of stretching will improve primes.

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)

Intervals

Steps Cents Approximate ratios
0 0 1/1
1 107.2 18/17
2 214.3 17/15, 25/22
3 321.5 6/5, 23/19
4 428.7 23/18
5 535.8 15/11
6 643
7 750.2 17/11, 20/13, 23/15
8 857.3 18/11
9 964.5 7/4
10 1071.7 13/7, 24/13
11 1178.8
12 1286 19/9, 21/10, 23/11
13 1393.2
14 1500.3
15 1607.5
16 1714.7
17 1821.8 20/7
18 1929
19 2036.2 13/4
20 2143.3 24/7
21 2250.5 11/3
22 2357.7
23 2464.8 25/6
24 2572 22/5
25 2679.1
26 2786.3 5/1
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