Talk:Kite's thoughts on fifthspans: Difference between revisions

It occurred to me that it should be possible to define fifthward and fourthward distances for multi-ring EDOs if fractional distances are allowed.  For instance, 34EDO has 2 rings of fifths, so with only integer fifthwards and fourthwards distances, the fifthwards/fourthwards antipode is undefined. But if we divide the perfect fifth (20\34) in 4 (5\34), we get the odd number of 1\34 increments we need to move between rings of fifths.  Then we can define a fifthward distance for any interval in 34EDO, for instance its best approximation to 5/4, which is 11\34:  Moving upwards by 9 quarter-fifths (9/4 fifths) gives us 45\34, which octave-reduces to 11\34.  Since Tetracot divides the (tempered) perfect fifth into 4 equal parts, this provides a reason why the Tetracot generalized/isomorphic keyboard mapping is a good one for making use o the excellent 5-limit harmony of 34EDO.