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m Remove all the fractional subgroup stuff because it turned out to be redundant and needlessly complicated: integers work better for this specific scale
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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"
|+
|+
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! 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
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