User:2^67-1/Nonoctave stuff

I am quite pissed that most temperaments documented here are octave-repeating, and even in nonoctave equal temperaments the pages are most often stubs, with minor exceptions. Hence I want to work on nonoctave stuff, therefore this explains the Moremajorthanmajor decodings. But I can probably only do that collaboratively with you guys if I am on Discord, and if you can give me other alternatives, I would very much like to accept it - like sending me emails.

I believe that the most fundamental temperaments are exotemperaments, as they can be detempered into scales or be made into functional analysis frameworks, among many other uses. Hence I think we should document exotemperaments in various nonoctave subgroups (whichever ones you want to, it's up to you, and you can feel free to put them in my talk page or this page's discussion page).

Anyway, that's all I want to say for now.

Expanded subgroups

Define the notation (x).y where y is a subgroup and x, the equivalence interval, is defined as an interval in y where two notes at that distance are taken to be equivalent. This is called an expanded subgroup.

There are an infinite number of expanded subgroups for any given y.

5ed7

See 5ed7.

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 it's step size is 674 cents.