User:Overthink/Miscellaneous ideas

This is a page containing ideas that are not (yet) developed enough to get its own page.

Meantone-superpyth fifth pairs

Let the perfect fifth in a tuning be [math]\displaystyle{ f/o }[/math] octaves. If a fifth is flat*, the corresponding sharp fifth is [math]\displaystyle{ (2o-2f)/(2o-f) }[/math] octaves, or [math]\displaystyle{ (2o-2f) }[/math] steps of [math]\displaystyle{ (2o-f) }[/math]-edo, and the same formula is used to obtain the corresponding flat fifth of a sharp one. As an example, the perfect fifth of 12edo is [math]\displaystyle{ 7/12 }[/math] octaves, so [math]\displaystyle{ f=7 }[/math] and [math]\displaystyle{ o=12 }[/math]. The corresponding sharp fifth is [math]\displaystyle{ (2\cdot12-2\cdot7) }[/math] steps of [math]\displaystyle{ (2\cdot12-7) }[/math]-edo, or 10 steps of 17edo. We apply the same formula on 10\17 to get [math]\displaystyle{ (2\cdot17-2\cdot10)/(2\cdot17-10)=14/24=7/12 }[/math] octaves.

We can check that the formula can be applied twice to get back the original fifth as follows: If our original fifth was [math]\displaystyle{ f_1/o_1 }[/math] octaves, the corresponding fifth will be [math]\displaystyle{ (2o_1-2f_1)/(2o_1-f_1) }[/math] octaves. We plug this into the formula with [math]\displaystyle{ f_2=2o_1-2f_1 }[/math] and [math]\displaystyle{ o_2=2o_1-f_1 }[/math], and we get [math]\displaystyle{ (2(2o_1-f_1)-2(2o_1-2f_1))/(2(2o_1-f_1)-(2o_1-2f_1))=(2f_1)/(2o_1)=f_1/o_1 }[/math] octaves.

This formula can also work if the fifth is not a rational fraction of the octave; if we let the fifth be [math]\displaystyle{ a }[/math] octaves, the corresponding fifth is [math]\displaystyle{ (2-2a)/(2-a) }[/math] octaves.

Some examples of corresponding fifth pairs are below.

4\7 ~ 3\5

15\26 ~ 22\37

26\45 ~ 19\32

11\19 ~ 16\27

40\69 ~ 29\49

29\50 ~ 42\71

18\31 ~ 13\22

25\43 ~ 36\61

32\55 ~ 23\39

7\12 ~ 10\17

38\65 ~ 27\46

31\53 ~ 44\75

(Just 3/2) 701.955 ¢ ~ 703.932 ¢

24\41 ~ 17\29

*By flat, this means being flat of the fifth of argent tuning, or 702.944 cents, rather than the just fifth. Note that in the formula, the argent fifth corresponds to itself.

Miscellaneous thoughts

The 10-form structure (temperaments include pajara and miracle) should really be explored more, and also the 14-form (temperaments include octacot, squares, and harry) and possibly also 15-form (temperaments include tritikleismic and valentine).