User:2^67-1/Nonoctave stuff: Difference between revisions

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See [[5ed7]].

5ed7 is a remarkable exotemperament. ~~Despite it's large step size~~ of ~~674 cents~~ it is ~~able~~ to ~~approximate~~ the ~~first five prime~~ harmonics ~~accurately for it's size~~: less than 25% relative error. ~~Thus~~

5ed7 is a remarkable exotemperament. It has several properties worthy of mention:

* Firstly, it can be represented somewhat accurately as an over-2 and over-4 scale. The former is important because it shows that the scale, when tempered, can be played using a koncovka on the root note. The scales are thus 2:3:4:6:9:14 or 2:3:4:6:10:14, and 4:6:9:13:19:28. The latter is important because one can get an extremely accurate 5ed7 approximation for its simplicity.

* Secondly, the harmonics from 1 to 16 are monotonic. Following the patent val mapping of 2 (2 steps), 3, (3 steps), 5 (4 steps), 7 (5 steps), 11 (6 steps) and 13 (7 steps) the harmonics are:

** 1 (0 steps)

** 2 (2 steps)

** 3 (3 steps)

** 4 (2+2=4 steps)

** 5 (4 steps)

** 6 (2+3=5 steps)

** 7 (5 steps)

** 8 (2+2+2=6 steps)

** 9 (3+3=6 steps)

** 10 (2+4=6 steps)

** 11 (6 steps)

** 12 (2+2+3=7 steps)

** 13 (7 steps)

** 14 (2+5=7 steps)

** 15 (3+4=7 steps)

** 16 (2+2+2+2=8 steps)

* (Note that prime 17 is mapped to 7 steps, breaking the monotonically increasing sequence.)

* Thirdly, 5ed7 has less than 25% relative error for primes 2, 3, 5, 7, and 11, which as stated before correspond to 2, 3, 4, 5, and 6 steps. This is interesting considering its step size is 674 cents.

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 See [[5ed7]].
-ed7 is a remarkable exotemperament. Despite it's large step size of 674 cents it is able to approximate the first five prime harmonics accurately for it's size: less than 25% relative error. Thus
+ed7 is a remarkable exotemperament. It has several properties worthy of mention:
+* Firstly, it can be represented somewhat accurately as an over-2 and over-4 scale. The former is important because it shows that the scale, when tempered, can be played using a koncovka on the root note. The scales are thus 2:3:4:6:9:14 or 2:3:4:6:10:14, and 4:6:9:13:19:28. The latter is important because one can get an extremely accurate 5ed7 approximation for its simplicity.
+* Secondly, the harmonics from 1 to 16 are monotonic. Following the patent val mapping of 2 (2 steps), 3, (3 steps), 5 (4 steps), 7 (5 steps), 11 (6 steps) and 13 (7 steps) the harmonics are:
+** 1 (0 steps)
+** 2 (2 steps)
+** 3 (3 steps)
+** 4 (2+2=4 steps)
+** 5 (4 steps)
+** 6 (2+3=5 steps)
+** 7 (5 steps)
+** 8 (2+2+2=6 steps)
+** 9 (3+3=6 steps)
+** 10 (2+4=6 steps)
+** 11 (6 steps)
+** 12 (2+2+3=7 steps)
+** 13 (7 steps)
+** 14 (2+5=7 steps)
+** 15 (3+4=7 steps)
+** 16 (2+2+2+2=8 steps)
+* (Note that prime 17 is mapped to 7 steps, breaking the monotonically increasing sequence.)
+* Thirdly, 5ed7 has less than 25% relative error for primes 2, 3, 5, 7, and 11, which as stated before correspond to 2, 3, 4, 5, and 6 steps. This is interesting considering its step size is 674 cents.