Taxicab distance: Difference between revisions

Line 9:

81/80 = 2^-4 * 3^4 * 5^-1

|-4| + |4| + |-1| = 9

This corresponds to an interval's unweighted [http://en.wikipedia.org/wiki/Lp_space L1] distance on a prime-factor lattice, as opposed to the more common weighted L1 metric, corresponding to the log of Tenney/Benedetti Height.

Line 15:

Line 14:

If you discard powers of both 2 and 3, you get an understanding of commas relevant to [[Sagittal_Corner|Sagittal notation]], which notates higher-prime-limit ratios in terms of their deviation from a cycle of fifths. In this sense, 81/80 has a taxicab distance of 1, as it contains only a single instance of 5, which is why in Sagittal notation it is called the "5-comma".

[[KiteGiedraitis|Kite Giedraitis]] has proposed triangularizing the taxicab distance, analogous to the triangularized 5-limit lattice in which 5/3 and 5/4 are both one step away from 1/1. The rationale is that a root movement of 5/3 is just as easy as one of 5/4. Likewise a move by 7/5 or 7/6 is as easy as one by 7/4. First factor the numerator and denominator into prime numbers and discard all twos. Then allow each prime to cancel out a smaller prime on the other side of the ratio. Thus 81/80 reduces to 27/5, 1 five-step and 3 three-steps, as in the common I - VIm - IIm - V - I. Given a choice, cancel out as high a prime as possible. 15/14 reduces to 3/7, e.g. Ih7 - ryIVs7 - ryIh7.

=With powers of 2 taken for granted=

@@ Line 9: / Line 9: @@
 /80 = 2^-4 * 3^4 * 5^-1
 |-4| + |4| + |-1| = 9
 This corresponds to an interval's unweighted [http://en.wikipedia.org/wiki/Lp_space L1] distance on a prime-factor lattice, as opposed to the more common weighted L1 metric, corresponding to the log of Tenney/Benedetti Height.
@@ Line 15: / Line 14: @@
 If you discard powers of both 2 and 3, you get an understanding of commas relevant to [[Sagittal_Corner|Sagittal notation]], which notates higher-prime-limit ratios in terms of their deviation from a cycle of fifths. In this sense, 81/80 has a taxicab distance of 1, as it contains only a single instance of 5, which is why in Sagittal notation it is called the "5-comma".
+[[KiteGiedraitis|Kite Giedraitis]] has proposed triangularizing the taxicab distance, analogous to the triangularized 5-limit lattice in which 5/3 and 5/4 are both one step away from 1/1. The rationale is that a root movement of 5/3 is just as easy as one of 5/4. Likewise a move by 7/5 or 7/6 is as easy as one by 7/4. First factor the numerator and denominator into prime numbers and discard all twos. Then allow each prime to cancel out a smaller prime on the other side of the ratio. Thus 81/80 reduces to 27/5, 1 five-step and 3 three-steps, as in the common I - VIm - IIm - V - I. Given a choice, cancel out as high a prime as possible. 15/14 reduces to 3/7, e.g. Ih7 - ryIVs7 - ryIh7.
 =With powers of 2 taken for granted=