Taxicab distance: Difference between revisions

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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.

[[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, and both are "five-steps". Likewise a move by 7/5 or 7/6 is as easy as one by 7/4, and all three are "seven-steps". First factor the numerator and denominator into prime numbers and discard all twos. Then allow each prime to cancel out one smaller prime on the other side of the ratio, if possible. 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, 1 three-step and 1 seven-step, e.g. Ih7 - Vh7 - ryIh7. 99/98 reduces to 33/7 because 11 cancels one 7, and the remaining 7 cancels one 3. (However, if a move by 11/7 is allowed, arguably one by 9/7 should be too.)

=With powers of 2 taken for granted=

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 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.
+[[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, and both are "five-steps". Likewise a move by 7/5 or 7/6 is as easy as one by 7/4, and all three are "seven-steps". First factor the numerator and denominator into prime numbers and discard all twos. Then allow each prime to cancel out one smaller prime on the other side of the ratio, if possible. 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, 1 three-step and 1 seven-step, e.g. Ih7 - Vh7 - ryIh7. 99/98 reduces to 33/7 because 11 cancels one 7, and the remaining 7 cancels one 3. (However, if a move by 11/7 is allowed, arguably one by 9/7 should be too.)
 =With powers of 2 taken for granted=