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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]].
 
A good compressed tuning of 26ed5 is [[46ed17]], which transforms 26ed5 from a 5.41 tuning to a 3.5.11.17.23. 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 and 23 are genuinely solid approximations.


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.
{{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)
}}


=== Composite subgroups ===
=== 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'''.
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'''


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. 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, 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:
{| class="wikitable"
|+
!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
|7/4.9/4.10/2.12/2.13/4.17/16.19/8.49/16
|-
!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
|7/4.9/4.9/7.10/2.12/2.13/4.17/16.19/8.20/7.23/14.24/7.29/7.31/7.33/7.49/16
|-
!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
|}
Note that 5/1 = 10/2 = 55/11, and 6/1 = 12/2 = 66/11.
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.


== Intervals ==
== Intervals ==
Line 149: Line 99:
! rowspan="2" |Step
! rowspan="2" |Step
! rowspan="2" |Cents
! rowspan="2" |Cents
! colspan="8" |Just intonation approximation
! colspan="2" |Just intonation approximation
|-
|-
!60th-prime
!<small>5.6.12.22… subgroup <br>described above</small>
!68th-prime
!<small>5.6.12.22… subgroup <br>(with ratios simplified)</small>
!88th-prime
!90th-prime
!112th-prime
!130th-prime
!Integer (5.6.12.22.32... as above)
!Integer (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
|17/16
|
| 36/34, 34/32
|18/17, 17/16
|-
|-
! 2
!2
!214.3
!214.3
|17/15
|34/30, 25/22
|
|17/15, 25/22
|25/22
|17/15
|
|
|25/22
|25/22
|-
|-
!3
!3
!321.5
!321.5
|6/5
|6/5
|41/34
|53/44
|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, 9/7
| 14/11
|23/18
|9/7
|
| 63/49
| 9/7
|-
|-
! 5
!5
!535.8
!535.8
|41/30
|(6/5)x(25/22)
|
|15/11
|15/11
|41/30
|19/14
|
|
|
|-
|-
!6
!6
!643.0
!643.0
|29/20
|52/36, 32/22
|
|13/9, 16/11
|16/11
|13/9
|
|
|32/22
|16/11
|-
|-
!7
!7
!750.2
!750.2
|23/15
|
|17/11
|23/15
|
|
|34/22
|34/22
|17/11
|17/11
Line 239: Line 141:
!8
!8
!857.3
!857.3
|49/30
|(6/5)x(15/11)
|28/17
|18/11
|18/11
|49/30
|23/14
|
|
|
|-
|-
!9
!9
!964.5
!964.5
|7/4
|7/4
|7/4
|
|7/4
|
|56/32
|56/32
|7/4
|7/4
Line 261: Line 151:
!10
!10
!1071.7
!1071.7
|28/15
|104/56
|63/34
|
|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
Line 283: Line 161:
!12
!12
!1286.0
!1286.0
|(6/5)x(56/32)
|21/10
|21/10
|
|23/11
|21/10, 19/9
|
|21/10
|46/22
|23/11
|-
|-
!13
!13
!1393.2
!1393.2
|81/36
|9/4
|9/4
|9/4
|9/4
|
|9/4
|29/13
|
|
|-
|-
!14
!14
!1500.3
!1500.3
|
|12/5
|
|12/5
|19/8
|
|19/8
|
|
|
|-
|-
!15
!15
!1607.5
!1607.5
|38/15
|30/12
|43/17
|5/2
|28/11
|38/15
|
|
|56/22
|28/11
|-
|-
!16
!16
!1714.7
!1714.7
|27/10
|32/12, 81/30
|
|8/3, 27/10
|
|27/10
|
|35/13, 27/10
|
|
|-
|-
!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
|110/36, 104/34
|
|55/18, 52/17
|67/22
|
|49/16
|
|67/22
|67/22
|-
|-
!19
!19
!2036.2
!2036.2
|104/32
|13/4
|13/4
|13/4
|13/4
|
|13/4
|
|
|
|-
|-
!20
!20
!2143.3
!2143.3
|69/20
|(32/12)x(63/49)
|
|
|31/9
|24/7
|24/7
|
|
|
|-
|-
!21
!21
!2250.5
!2250.5
|11/3
|
|
|11/3
|
|
|22/6
|22/6
|11/3
|11/3
Line 393: Line 211:
!22
!22
!2357.7
!2357.7
|117/30
|39/10
|39/10
|
|
|35/9, 39/10
|
|39/10
|
|
|-
|-
!23
!23
!2464.8
!2464.8
|25/6
|
|
|25/6
|29/7
|54/13
|25/6
|25/6
|25/6
|25/6
|-
|-
!24
!24
!2572.0
! 2572.0
|22/5
|75/17
|
|22/5
|31/7
|22/5
|22/5
|22/5
|22/5
|22/5
Line 426: Line 226:
!25
!25
!2679.1
!2679.1
|47/10
|56/12
|80/17
|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
|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}}
Retrieved from "https://en.xen.wiki/w/26ed5"