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.