Taxicab distance: Difference between revisions
m →With powers of 2 taken for granted: Ensure that for the three narrow tables, the labeling is consistently two lines |
Add Wikipedia box, improve lead section, categories, misc. edits |
||
| Line 1: | Line 1: | ||
{{Wikipedia|Taxicab distance}} | |||
'''Taxicab distance''' is a measure of the [[complexity]] of a [[just interval]] by the number of [[prime factor]]s it has, regardless of their magnitude, but counting repetitions. | |||
In particular, when combined with excluding the smallest primes, this measurement can give an idea of how many "strange harmonic moves" a comma is comprised of. | |||
Taxicab distance is not a [[height]] because there are infinitely many elements with the same taxicab distance, unless considering only the intervals with a given [[prime limit]]. | |||
== How to calculate taxicab distance on a prime-number lattice == | == How to calculate taxicab distance on a prime-number lattice == | ||
To calculate the taxicab distance between 1/1 and any interval, take the sum of the absolute values of the exponents of the prime factorization. For the example of 81/80: | |||
To calculate the | |||
81/80 = 2^-4 * 3^4 * 5^-1 | 81/80 = 2^-4 * 3^4 * 5^-1 | ||
| Line 17: | Line 19: | ||
=== Examples === | === Examples === | ||
Following tables shows some unweighted taxicab distances of ratios in prime-factor lattice ''without'' the usual [[octave reduction]]. | Following tables shows some unweighted taxicab distances of ratios in prime-factor lattice ''without'' the usual [[octave reduction]]. | ||
| Line 38: | Line 39: | ||
=== Triangularizing proposal === | === Triangularizing proposal === | ||
[[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.) | [[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.) | ||
| Line 182: | Line 182: | ||
| 531441/512000 || 3's / 5 / 5 / 5 || (125 L-diesis) | | 531441/512000 || 3's / 5 / 5 / 5 || (125 L-diesis) | ||
|} | |} | ||
== External links == | == External links == | ||
* [[Wikipedia: Hamming distance]] | |||
* [ | |||
[[Category:Commas]] | [[Category:Commas]] | ||
[[Category:Math]] | [[Category:Math]] | ||
[[Category:Measure]] | [[Category:Measure]] | ||