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=== Prime subgroups ===
=== 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.
Pure-[[pentave]]s 26ed5 is incompatible with [[prime limit]] tuning. Of all primes up to 41, 5 and 41 are the only two it approximates well. 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.


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.
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, with similar error to [[5edo]], but it’s a huge improvement on 26ed5. And the 5, 11, 17, 23 and 43 are genuinely solid approximations. Other tunings which are almost identical to 46ed17, and so provide those same benefits, are [[8ed18/11]] and [[20ed24/7]].


[[29ed6]] is a compressed version of 26ed5, compressing 5/1 by roughly 6 cents, but it is not enough to bring many primes into line. Further compression than that is required.
If one attempts to [[Octave stretch|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.
{{Harmonics in equal
| steps = 26
| num = 5
| denom = 1
| intervals = prime
| collapsed = 1
| start = 1
| title = Prime harmonics 2 to 31 (26ed5)
}}
{{Harmonics in equal
| steps = 26
| num = 5
| denom = 1
| intervals = prime
| collapsed = 1
| start = 12
| title = Prime harmonics 37 to 79 (26ed5)
}}


[[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.
=== Composite subgroups ===
If one does not restrict to primes and allows all integers, pure-pentaves 26ed5 can instead be used as a strong tuning for the giant [[subgroup]]:


=== Composite subgroups ===
'''5.6.12.22.32.44.49.52.56'''
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:
'''63.81.91.98.104.117.126'''
* 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 ====
Or it can be a strong tuning for any smaller subgroup that is contained within that group.
{{Harmonics in equal
{{Harmonics in equal
| steps = 26
| steps = 26
Line 28: Line 42:
| denom = 1
| denom = 1
| intervals = integer
| intervals = integer
| collapsed = 1
| start = 1
| start = 1
| title = Harmonics 2 to 12 (26ed5)
| title = Integer harmonics 2 to 12 (26ed5)
}}
}}
{{Harmonics in equal
{{Harmonics in equal
Line 38: Line 53:
| collapsed = 1
| collapsed = 1
| start = 12
| start = 12
| title = Harmonics 13 to 23 (26ed5)
| title = Integer harmonics 13 to 23 (26ed5)
}}
}}
{{Harmonics in equal
{{Harmonics in equal
Line 47: Line 62:
| collapsed = 1
| collapsed = 1
| start = 23
| start = 23
| title = Harmonics 24 to 34 (26ed5)
| title = Integer harmonics 24 to 34 (26ed5)
}}
}}
{{Harmonics in equal
{{Harmonics in equal
Line 56: Line 71:
| collapsed = 1
| collapsed = 1
| start = 34
| start = 34
| title = Harmonics 35 to 45 (26ed5)
| title = Integer harmonics 35 to 45 (26ed5)
}}
}}
{{Harmonics in equal
{{Harmonics in equal
Line 65: Line 80:
| collapsed = 1
| collapsed = 1
| start = 45
| start = 45
| title = Harmonics 46 to 56 (26ed5)
| title = Integer harmonics 46 to 56 (26ed5)
}}
}}
{{Harmonics in equal
{{Harmonics in equal
Line 74: Line 89:
| collapsed = 1
| collapsed = 1
| start = 56
| start = 56
| title = Harmonics 57 to 68 (26ed5)
| title = Integer harmonics 57 to 68 (26ed5)
| columns = 12
| columns = 12
}}
}}
=== Fractional subgroups ===
Fractional subgroups are another approach to taming 26ed5. Once can use any of the JI ratios approximated by its individual intervals as [[basis element]]s 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
==== ''N''th-prime subgroups ====
These are some [[Half-prime subgroup|''n''th-prime subgroups]] which 26ed5 approximates well:
* 14/11.15/11.16/11.17/11.18/11.23/11.28/11.55/11.66/11 subgroup '''(11th-prime)'''
* 9/7.10/2.12/2.20/7.23/14.24/7.29/7.31/7.33/7 subgroup '''(14th-prime)'''
* 7/4.9/4.10/2.12/2.13/4.17/16.19/8.49/16 subgroup '''(16th-prime)'''
* 10/2.12/2.11/3.16/15.17/15.21/10.22/5.23/15.25/6.28/15 subgroup '''(30th-prime:''' 5th, third & half''')'''
* 7/4.9/4.10/2.12/2.11/3.13/9.19/9.25/6.31/9.35/9 subgroup '''(36th-prime:''' 9th & quarter''')'''
* 7/4.9/4.10/2.12/2.13/4.18/17.28/17 subgroup '''(68th-prime:''' 17th & quarter''')'''
* 7/4.9/4.10/2.12/2.13/4.14/11.15/11.16/11.17/11.18/11.19/8.23/11.28/11 subgroup '''(88th-prime:''' 11th & 8th''')'''
* 10/2.11/3.12/2.13/9.16/15.17/15.19/9.21/10.22/5.23/15.25/6.28/15.31/9.35/9 subgroup '''(90th-prime:''' 9th, 5th & quarter''')'''
* 7/4.9/4.9/7.10/2.12/2.13/4.17/16.19/8.20/7.23/14.24/7.29/7.31/7.33/7.49/16 subgroup '''(112th-prime:''' 16th & 14th''')'''
* 6/5.10/2.12/2.21/10.22/5.24/13.27/10.29/13.39/10 subgroup '''(130th-prime:''' 13th & 5th & half''')'''


== Intervals ==
== Intervals ==
Line 106: Line 99:
! rowspan="2" |Step
! rowspan="2" |Step
! rowspan="2" |Cents
! rowspan="2" |Cents
! colspan="7" |Just intonation approximation
! colspan="2" |Just intonation approximation
|-
|-
!68th-prime
!<small>5.6.12.22… subgroup <br>described above</small>
!88th-prime
!<small>5.6.12.22… subgroup <br>(with ratios simplified)</small>
!90th-prime
!112th-prime
!130th-prime
!Integer (5.6.12.22.32... as above)
!Integer (simplified)
|-
|-
!1
!1
!107.2
!107.2
|18/17
|36/34, 34/32, 32/30
|47/44
|18/17, 17/16, 16/15
| 16/15
|17/16
|
| 36/34, 34/32
|18/17, 17/16
|-
|-
! 2
!2
!214.3
!214.3
|
|34/30, 25/22
|25/22
|17/15, 25/22
|17/15
|
|
|25/22
|25/22
|-
|-
!3
!3
!321.5
!321.5
|41/34
|53/44
|6/5
|6/5
|
|6/5
|6/5
|6/5, 41/34
|6/5, 41/34
|-
|-
!4
!4
! 428.7
!428.7
|
|56/44, 63/49
| 14/11
|14/11, 9/7
|23/18
|9/7
|
| 63/49
| 9/7
|-
|-
! 5
!5
!535.8
!535.8
|
|(6/5)x(25/22)
|15/11
|15/11
|41/30
|19/14
|
|
|
|-
|-
!6
!6
!643.0
!643.0
|
|52/36, 32/22
|16/11
|13/9, 16/11
|13/9
|
|
|32/22
|16/11
|-
|-
!7
!7
!750.2
!750.2
|
|17/11
|23/15
|
|
|34/22
|34/22
|17/11
|17/11
Line 188: Line 141:
!8
!8
!857.3
!857.3
|28/17
|(6/5)x(15/11)
|18/11
|18/11
|
|23/14
|
|
|
|-
|-
!9
!9
!964.5
!964.5
|7/4
|7/4
|
|7/4
|
|56/32
|56/32
|7/4
|7/4
Line 208: Line 151:
!10
!10
!1071.7
!1071.7
|63/34
|104/56
|
|28/15
|13/7
|13/7
|24/13
|63/34
|63/34
|-
|-
!11
!11
!1178.8
!1178.8
|67/34
|
|49/25
|
|
|49/25
|49/25
|49/25
|49/25
Line 228: Line 161:
!12
!12
!1286.0
!1286.0
|
|(6/5)x(56/32)
|23/11
|21/10, 19/9
|
|21/10
|21/10
|46/22
|23/11
|-
|-
!13
!13
!1393.2
!1393.2
|81/36
|9/4
|9/4
|9/4
|
|9/4
|29/13
|
|
|-
|-
!14
!14
!1500.3
!1500.3
|
|12/5
|19/8
|12/5
|
|19/8
|
|
|
|-
|-
!15
!15
!1607.5
!1607.5
|43/17
|30/12
|28/11
|5/2
|38/15
|
|
|56/22
|28/11
|-
|-
!16
!16
!1714.7
!1714.7
|
|32/12, 81/30
|
|8/3, 27/10
|27/10
|
|35/13, 27/10
|
|
|-
|-
!17
!17
!1821.8
!1821.8
|
|160/56
|63/22
|43/15
|20/7
|20/7
|
|63/22
|63/22
|-
|-
!18
!18
!1929.0
!1929.0
|
|110/36, 104/34
|67/22
|55/18, 52/17
|
|49/16
|
|67/22
|67/22
|-
|-
!19
!19
!2036.2
!2036.2
|104/32
|13/4
|13/4
|13/4
|
|13/4
|
|
|
|-
|-
!20
!20
!2143.3
!2143.3
|
|(32/12)x(63/49)
|
|31/9
|24/7
|24/7
|
|
|
|-
|-
!21
!21
!2250.5
!2250.5
|
|
|11/3
|
|
|22/6
|22/6
|11/3
|11/3
Line 328: Line 211:
!22
!22
!2357.7
!2357.7
|
|117/30
|
|35/9, 39/10
|
|39/10
|39/10
|
|
|-
|-
!23
!23
!2464.8
!2464.8
|
|
|25/6
|29/7
|54/13
|25/6
|25/6
|25/6
|25/6
|-
|-
!24
!24
!2572.0
! 2572.0
|75/17
|
|22/5
|31/7
|22/5
|22/5
|22/5
|22/5
|22/5
Line 358: Line 226:
!25
!25
!2679.1
!2679.1
|80/17
|56/12
|
|14/3
|47/10
|33/7
|47/10
|
|
|-
|-
!26
!26
!2786.3
! 2786.3
|5/1
| 5/1
|5/1
|5/1
|5/1
|5/1
|5/1
|5/1
|}
==Scales==
===13ed5plus===
[[Category:14-tone scales]]
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 [[consonance]]s while leaving out many of the less-used intervals. Making it practical to use on an instrument.
====Properties====
13ed5plus is a [[:Category:14-tone scales|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]].
====Intervals====
{| class="wikitable mw-collapsible"
|+
The 13ed5plus scale
!Step
!Cents
!JI approximation <br><small>(5.6.12.22… subgroup;</small> <br><small>ratios simplified)</small>
|-
|1
|107.2
|18/17, 17/16, 16/15
|-
|3
|321.5
|6/5
|-
|5
|535.8
|15/11
|-
|7
|750.2
|17/11
|-
|9
|964.5
|7/4
|-
|11
|1178.8
|49/25
|-
|13
|1393.2
|9/4
|-
|15
|1607.5
|5/2
|-
|17
|1821.8
|20/7
|-
|19
|2036.2
|13/4
|-
|21
|2250.5
|11/3
|-
|23
|2464.8
|25/6
|-
|25
|2679.1
|14/3
|-
|26
|2786.3
|5/1
|5/1
|}
|}
{{todo|expand}}

Latest revision as of 23:46, 5 January 2025

← 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-pentaves 26ed5 is incompatible with prime limit tuning. Of all primes up to 41, 5 and 41 are the only two 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, with similar error to 5edo, but it’s a huge improvement on 26ed5. And the 5, 11, 17, 23 and 43 are genuinely solid approximations. Other tunings which are almost identical to 46ed17, and so provide those same benefits, are 8ed18/11 and 20ed24/7.

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.

Prime harmonics 2 to 31 (26ed5)
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) -21.2 +27.0 +0.0 -46.7 +28.2 -46.7 +24.7 +46.5 +37.2 -42.6 -50.9
Relative (%) -19.8 +25.2 +0.0 -43.6 +26.3 -43.6 +23.0 +43.3 +34.7 -39.8 -47.5
Steps
(reduced) 		11
(11) 		18
(18) 		26
(0) 		31
(5) 		39
(13) 		41
(15) 		46
(20) 		48
(22) 		51
(25) 		54
(2) 		55
(3)
Prime harmonics 37 to 79 (26ed5)
Harmonic 37 41 43 47 53 59 61 67 71 73 79
Error Absolute (¢) -35.7 +0.9 +25.6 -21.2 -14.9 +13.8 -43.9 +8.0 +14.8 -33.3 +44.2
Relative (%) -33.3 +0.8 +23.9 -19.8 -13.9 +12.9 -41.0 +7.4 +13.8 -31.1 +41.3
Steps
(reduced) 		58
(6) 		60
(8) 		61
(9) 		62
(10) 		64
(12) 		66
(14) 		66
(14) 		68
(16) 		69
(17) 		69
(17) 		71
(19)

Composite subgroups

If one does not restrict to primes and allows all integers, pure-pentaves 26ed5 can instead be used as a strong tuning for the giant subgroup:

5.6.12.22.32.44.49.52.56

63.81.91.98.104.117.126

Or it can be a strong tuning for any smaller subgroup that is contained within that group.

Integer 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)
Integer 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)
Integer 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)
Integer 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)
Integer 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)
Integer 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

Intervals of 26ed5
Step Cents Just intonation approximation
5.6.12.22… subgroup
described above 		5.6.12.22… subgroup
(with ratios simplified)
1 107.2 36/34, 34/32, 32/30 18/17, 17/16, 16/15
2 214.3 34/30, 25/22 17/15, 25/22
3 321.5 6/5 6/5
4 428.7 56/44, 63/49 14/11, 9/7
5 535.8 (6/5)x(25/22) 15/11
6 643.0 52/36, 32/22 13/9, 16/11
7 750.2 34/22 17/11
8 857.3 (6/5)x(15/11) 18/11
9 964.5 56/32 7/4
10 1071.7 104/56 13/7
11 1178.8 49/25 49/25
12 1286.0 (6/5)x(56/32) 21/10
13 1393.2 81/36 9/4
14 1500.3 12/5 12/5
15 1607.5 30/12 5/2
16 1714.7 32/12, 81/30 8/3, 27/10
17 1821.8 160/56 20/7
18 1929.0 110/36, 104/34 55/18, 52/17
19 2036.2 104/32 13/4
20 2143.3 (32/12)x(63/49) 24/7
21 2250.5 22/6 11/3
22 2357.7 117/30 39/10
23 2464.8 25/6 25/6
24 2572.0 22/5 22/5
25 2679.1 56/12 14/3
26 2786.3 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.

Intervals

The 13ed5plus scale
Step Cents JI approximation
(5.6.12.22… subgroup;
ratios simplified)
1 107.2 18/17, 17/16, 16/15
3 321.5 6/5
5 535.8 15/11
7 750.2 17/11
9 964.5 7/4
11 1178.8 49/25
13 1393.2 9/4
15 1607.5 5/2
17 1821.8 20/7
19 2036.2 13/4
21 2250.5 11/3
23 2464.8 25/6
25 2679.1 14/3
26 2786.3 5/1
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