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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
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| 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
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| 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
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| 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
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| 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
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| 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 ===
== Intervals ==
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.
{| class="wikitable mw-collapsible"
|+
Intervals of 26ed5
! rowspan="2" |Step
! rowspan="2" |Cents
! colspan="2" |Just intonation approximation
|-
!<small>5.6.12.22… subgroup <br>described above</small>
!<small>5.6.12.22… subgroup <br>(with ratios simplified)</small>
|-
!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===
[[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.


There are dozens of possible combinations, here is a small sampling of possible ones:
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.
* 5.6.7/4.11/3.13/4 subgroup
* 5.6.7/4.11/3.13/4.17/11.19/8.23/11.29/7.31/7 subgroup
* 5.6.7/4.9/4.13/4.17/16.19/8.49/16 subgroup
* 5.6.11/3.13/9.16/15.17/15.19/9.21/10.22/5.23/15.25/6.28/15.31/9.35/9 subgroup
* 5.6.9/7.20/7.23/14.24/7.29/7.31/7.33/7 subgroup
* 5.6.14/11.15/11.16/11.17/11.18/11.23/11.28/11 subgroup


== Intervals ==  
====Properties====
# 107.2 (18/17, 17/16, 16/15)
13ed5plus is a [[:Category:14-tone scales|14-tone scale]].
# 214.3 (17/15)
# 321.5 (6/5, 23/19)
# 428.7 (14/11, 9/7)
# 535.8 (19/14, 15/11)
# 643.0 (13/9, 16/11)
# 750.2 (23/15, 17/11)
# 857.3 (18/11, 23/14, 28/17)
# 964.5 (7/4)
# 1071.7 (24/13, 13/7, 28/15)
# 1178.8 (49/25)
# 1286.0 (23/11, 21/10, 19/9)
# 1393.2 (29/13, 9/4)
# 1500.3 (19/8)
# 1607.5 (28/11)
# 1714.7 (27/10)
# 1821.8 (20/7)
# 1929.0 (49/16)
# 2036.2 (13/4)
# 2143.3 (24/7, 31/9)
# 2250.5 (11/3)
# 2357.7 (35/9, 39/10)
# 2464.8 (29/7, 25/6)
# 2572.0 (22/5, 31/7)
# 2679.1 (33/7)
# 2786.3 (5/1)


== See also ==
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]].
* [[Half-prime subgroup]]


{{todo|expand}}
====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
|}
Retrieved from "https://en.xen.wiki/w/26ed5"