26ed5: Difference between revisions

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 == Theory ==
-=== Subgroup interpretation ===
+=== Prime subgroups ===
-ed5 is a weak tuning for [[prime limit]] tuning. It 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'''.
+Pure-octaves 26ed5 is incompatible with [[prime limit]] tuning. Of all primes up to 37, 5 is the only one 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.
+[[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.
+[[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 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:
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 * Only 6 and the primes: '''5.6.41.59.67'''
-Fractional subgroups might also be an option for 26ed5.
+==== Tables of harmonics ====
-=== Equave stretch ===
-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.
-[[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.
-[[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.
-=== Tables of harmonics ===
 {{Harmonics in equal
 | steps = 26
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 }}
-== Intervals ==
+=== Fractional subgroups ===
-{{Interval table}}
+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.
+There are dozens of possible combinations, here is a small sampling of possible ones:
+* 5.6.7/4.11/3.13/4 subgroup
+* 5.6.7/4.11/3.13/4.17/11.19/8.23/11.29/7.31/7 subgroup
+* 5.6.7/4.9/4.13/4.17/16.19/8 subgroup
+== Intervals ==
+# 107.2 (18/17, 17/16, 16/15)
+# 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)
 {{todo|expand}}