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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-octaves 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, comparable 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]].
 
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.
 
Other tunings which are almost identical to 46ed17, and so provide those same benefits, are [[8ed18/11]] and [[20ed24/7]].


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.
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.
Line 24: Line 20:
Or it can be a strong tuning for any smaller subgroup that is contained within that group.
Or it can be a strong tuning for any smaller subgroup that is contained within that group.


Just some examples of possible smaller subgroups, not exhaustive:
=== Tables of harmonics ===
*Only [[basis element]]s within the 50 [[integer limit]]: 5.6.12.22.32.44.49
*Only 6 and odd basis elements: 5.6.49.63.81.91.117
*Only [[7-limit]] basis elements: 5.6.12.32.49.56.63.81.98.126
* Only [[11-limit]] basis elements: 5.6.12.22.32.44.49.56.63.81.98.126
====Tables of harmonics====
{{Harmonics in equal
{{Harmonics in equal
| steps = 26
| steps = 26
Line 85: Line 76:
}}
}}


===Fractional subgroups===
== Intervals ==
Fractional subgroups are another approach to taming 26ed5. One 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, for example the '''5.6.7/4.11/3.13/4''' subgroup, the '''5.6.7/4.9/4.9/7.11/3.13/4.13/7.13/9''' subgroup, etc.
 
====''N''th-prime subgroups====
These are some [[Half-prime subgroup|''n''th-prime subgroups]]{{idiosyncratic}} which 26ed5 approximates well:
 
{| class="wikitable mw-collapsible"
|+''N''th-prime subgroups
! Family
!Most distinctive related families
!Subgroup basis elements
!Optional extra elements (sprinkle in any 1 or 2 of these)
|-
! 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- & 8th-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
|-
!140th-basis
|14th-, 10th- & quarter-basis
|7/4.13/4.23/14.29/7.31/7
|29/20.61/20
|}
 
====''N''th-basis subgroups====
These are some [[Half-prime subgroup|''n''th-basis subgroups]]{{idiosyncratic}} which 26ed5 approximates well.
 
{| class="wikitable mw-collapsible"
|+''N''th-basis subgroups
!Family
!Most distinctive related families
!Subgroup basis elements
!Optional extra elements (sprinkle in any 1 or 2 of these)
|-
!11th-basis
|
|14/11.15/11.16/11.55/11.66/11
|<small>17/11.18/11.28/11.23/11</small>
|-
!14th-basis
|7th- & half-basis
|9/7.10/2.12/2.20/7.24/7
|<small>23/14.29/7.31/7.33/7</small>
|-
!16th-basis
|8th-, quarter- & half-basis
|7/4.9/4.10/2.12/2.13/4
|<small>17/16.19/8</small>
|-
!18th-basis
|9th- & 6th-basis
|10/2.12/2.11/3.25/6.35/9
|<small>13/9.19/9.23/18.31/9</small>
|-
!30th-basis
|15th- & 10th-basis
|10/2.11/3.12/2.16/15.21/10
|<small>17/15.22/5.23/15.25/6.28/15.38/15.41/30.34/15.47/10</small>
|-
!60th-basis
|15th-, 10th- & quarter-basis
|7/4.9/4.10/2.11/3.12/2
|<small>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</small>
|-
!68th-basis
|17th- & quarter-basis
|7/4.9/4.10/2.12/2.13/4
|<small>18/17.28/17.41/34.43/17.63/34.67/34.75/17.80/17</small>
|-
!88th-basis
|11th- & 8th-basis
|7/4.9/4.10/2.12/2.14/11.15/11.16/11
|<small>13/4.17/11.18/11.19/8.23/11.25/22.28/11.47/44.53/44.63/22.67/22</small>
|-
!90th-basis
|15th-, 10th-, 9th- & 6th-basis
|10/2.11/3.12/2.16/15.21/10
|<small>13/9.17/15.19/9.22/5.23/15.23/18.25/6.28/15.31/9.35/9.38/15.41/30.43/15.47/19.49/30</small>
|-
! 140th-basis
|14th-, 10th- & quarter-basis
|7/4.9/4.9/7.10/2.12/2.20/7
|<small>13/4.23/14.24/7.29/7.29/20.31/7.33/7.61/20</small>
|}
 
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 ''n''th-basis subgroup. So one can remove as many basis elements as desired to simplify the subgroup down, if they so wish.
 
==Intervals==
{| class="wikitable mw-collapsible"
{| class="wikitable mw-collapsible"
|+
|+
Line 212: Line 82:
! rowspan="2" |Step
! rowspan="2" |Step
! rowspan="2" |Cents
! rowspan="2" |Cents
! colspan="7" |Just intonation approximation
! colspan="2" |Just intonation approximation
|-
|-
!60th-basis
!<small>5.6.12.22… subgroup (described above)</small>
!68th-basis
!<small>5.6.12.22… subgroup;</small> <br><small>with ratios simplified</small>
!88th-basis
!90th-basis
!140th-basis
!Integer <br>(5.6.12.22.32... <br>as above)
!Integer <br>(simplified)
|-
|-
!1
!1
!107.2
!107.2
|16/15
|36/34, 34/32, 32/30
|18/17
|18/17, 17/16, 16/15
|47/44
|16/15
|
|36/34, 34/32
|18/17, 17/16
|-
|-
!2
!2
!214.3
!214.3
|17/15
|
|25/22
|17/15
|
|34/30, 25/22
|34/30, 25/22
|17/15, 25/22
|17/15, 25/22
Line 244: Line 99:
!3
!3
!321.5
!321.5
|6/5
|41/34
|53/44
|6/5
|
|6/5
|6/5
|6/5
|6/5
Line 254: Line 104:
!4
!4
!428.7
!428.7
|
|56/44, 63/49
|
|14/11, 9/7
|14/11
|23/18
|9/7
|63/49
|9/7
|-
|-
!5
!5
!535.8
!535.8
|41/30
|
|15/11
|41/30
|19/14
|(6/5)x(25/22)
|(6/5)x(25/22)
|15/11
|15/11
Line 274: Line 114:
!6
!6
!643.0
!643.0
|29/20
|52/36, 32/22
|
|13/9, 16/11
|16/11
|13/9
|29/20
|32/22
|16/11
|-
|-
!7
!7
!750.2
!750.2
|23/15
|
|17/11
|23/15
|54/35
|34/22
|34/22
|17/11
|17/11
Line 294: Line 124:
!8
!8
!857.3
!857.3
|49/30
|28/17
|18/11
|49/30
|23/14
|(6/5)x(15/11)
|(6/5)x(15/11)
|18/11
|18/11
Line 304: Line 129:
!9
!9
!964.5
!964.5
|7/4
|7/4
|7/4
|26/15
|7/4
|56/32
|56/32
|7/4
|7/4
Line 314: Line 134:
!10
!10
!1071.7
!1071.7
|28/15
|63/34
|
|28/15
|13/7
|104/56
|104/56
|13/7
|13/7
Line 324: Line 139:
!11
!11
!1178.8
!1178.8
|
|67/34
|
|
|
|49/25
|49/25
|49/25
|49/25
Line 334: Line 144:
!12
!12
!1286.0
!1286.0
|21/10
|21/10
|23/11, 21/10
|21/10, 19/9
|21/10
|(6/5)x(56/32)
|(6/5)x(56/32)
|21/10
|21/10
Line 344: Line 149:
!13
!13
!1393.2
!1393.2
|9/4
|9/4
|9/4
|
|9/4
|81/36
|81/36
|9/4
|9/4
Line 354: Line 154:
!14
!14
!1500.3
!1500.3
|
|
|19/8
|
|
|12/5
|12/5
|12/5
|12/5
Line 364: Line 159:
!15
!15
!1607.5
!1607.5
|38/15
|30/12
|43/17
|5/2
|28/11
|38/15
|
|56/22, 30/12
|28/11, 5/2
|-
|-
!16
!16
!1714.7
!1714.7
|27/10
|27/10
|27/10
|27/10
|27/10
|81/30
|81/30
|27/10
|27/10
Line 384: Line 169:
!17
!17
!1821.8
!1821.8
|43/15
|160/56
|
|63/22
|43/15
|20/7
|20/7
|63/22
|63/22
|-
|-
!18
!18
!1929.0
!1929.0
|61/20
|
|67/22
|
|61/20
|
|
|
|
Line 404: Line 179:
!19
!19
!2036.2
!2036.2
|13/4
|13/4
|13/4
|
|13/4
|104/32
|104/32
|13/4
|13/4
Line 414: Line 184:
!20
!20
!2143.3
!2143.3
|69/20
|
|
|31/9
|24/7
|
|
|
|
Line 424: Line 189:
!21
!21
!2250.5
!2250.5
|11/3
|
|
|11/3
|
|22/6
|22/6
|11/3
|11/3
Line 434: Line 194:
!22
!22
!2357.7
!2357.7
|39/10
|39/10
|39/10
|35/9, 39/10
|39/10
|
|
|
|
Line 444: Line 199:
!23
!23
!2464.8
!2464.8
|25/6
| 25/6
|25/6
|25/6
|29/7, 25/6
|25/6
|25/6
|25/6
|25/6
Line 454: Line 204:
!24
!24
! 2572.0
! 2572.0
|22/5
|75/17
|
|22/5
|31/7
|22/5
|22/5
|22/5
|22/5
Line 464: Line 209:
!25
!25
!2679.1
!2679.1
|47/10
|80/17
|
|47/10
|33/7
|
|
|
|
Line 474: Line 214:
!26
!26
! 2786.3
! 2786.3
|5/1
|5/1
| 5/1
|5/1
|5/1
| 5/1
| 5/1
|5/1
|5/1
Line 489: Line 224:
Inspired by the [[elevenplus]] scale of [[22edo]], the '''13ed5plus scale''' is [[13ed5]] plus a step of 26ed5 in between two of its steps.
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 2
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.
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.
Line 504: Line 239:
!Step
!Step
!Cents
!Cents
!JI approximation  
!JI approximation <br><small>(5.6.12.22… subgroup;</small> <br><small>ratios simplified)</small>
<small>(5.6.12.22… subgroup;</small>
 
<small>ratios simplified)</small>
|-
|-
|1
|1
|107.2
|107.2
|18/17, 17/16
|18/17, 17/16, 16/15
|-
|-
|3
|3
Line 539: Line 271:
|15
|15
|1607.5
|1607.5
|28/11, 5/2
|5/2
|-
|-
|17
|17
|1821.8
|1821.8
|63/22
|20/7
|-
|-
|19
|19

Revision as of 05:29, 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-octaves 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, comparable 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.

Composite subgroups

If one ignores primes and focuses on integers in general, 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.

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

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 81/30 27/10
17 1821.8 160/56 20/7
18 1929.0
19 2036.2 104/32 13/4
20 2143.3
21 2250.5 22/6 11/3
22 2357.7
23 2464.8 25/6 25/6
24 2572.0 22/5 22/5
25 2679.1
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.

Table

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
26 2786.3 5/1
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