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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 3.5.11.17.23.43 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.
{{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 giant [[subgroup]]:
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'''
'''5.6.12.22.32.44.49.52.56'''
Line 23: Line 37:


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:
*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 35: 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 45: 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 54: 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 63: 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 72: 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 81: 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===
== 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 99:
! 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 <br>described above</small>
!68th-basis
!<small>5.6.12.22… subgroup <br>(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 116:
!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 121:
!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 131:
!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 141:
!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 146:
!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 151:
!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 156:
!11
!11
!1178.8
!1178.8
|
|67/34
|
|
|
|49/25
|49/25
|49/25
|49/25
Line 334: Line 161:
!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 166:
!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 171:
!14
!14
!1500.3
!1500.3
|
|
|19/8
|
|
|12/5
|12/5
|12/5
|12/5
Line 364: Line 176:
!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
|32/12, 81/30
|27/10
|8/3, 27/10
|27/10
|27/10
|27/10
|81/30
|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
|
|61/20
|
|
|-
|-
!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 201:
!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 434: Line 211:
!22
!22
!2357.7
!2357.7
|117/30
|39/10
|39/10
|39/10
|39/10
|35/9, 39/10
|39/10
|
|
|-
|-
!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 221:
!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 226:
!25
!25
!2679.1
!2679.1
|47/10
|56/12
|80/17
|14/3
|
|47/10
|33/7
|
|
|-
|-
!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 241:
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 498: Line 250:
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]].
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====
====Intervals====
{| class="wikitable mw-collapsible"
{| class="wikitable mw-collapsible"
|+
|+
Line 504: Line 256:
!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 288:
|15
|15
|1607.5
|1607.5
|28/11, 5/2
|5/2
|-
|-
|17
|17
|1821.8
|1821.8
|63/22
|20/7
|-
|-
|19
|19
Line 559: Line 308:
|25
|25
|2679.1
|2679.1
|
|14/3
|-
|-
|26
|26
Retrieved from "https://en.xen.wiki/w/26ed5"