Taxicab distance

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Taxicab distance is a measure of the complexity of a just interval by the number of prime factors 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

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:

```81/80 = 2^-4 * 3^4 * 5^-1
|-4| + |4| + |-1| = 9
```

This corresponds to an interval's unweighted L1 distance on a prime-factor lattice, as opposed to the more common weighted L1 metric, corresponding to the log of Tenney/Benedetti height.

One way to add weighting back in is to exclude powers of 2 (to assume that powers of 2 don't affect the complexity of the move). For 81/80 we then get 4+1=5.

If you discard powers of both 2 and 3, you get an understanding of commas relevant to 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".

Examples

Following tables shows some unweighted taxicab distances of ratios in prime-factor lattice without the usual octave reduction.

taxicab distance of some ratios
Ratio Split form Taxicab distance
1/1 1 0
2/1 2 1
5/3 5 / 3 2
7/4 7 / 2 / 2 3
15/14 3 * 5 / 2 / 7 4
55/42 5 * 11 / 2 * 3 * 7 5

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.)

With powers of 2 taken for granted

1-move intervals
(2's removed)
Ratio split form
5/4 5
4/3 1 / 3
7/4 7
128/127 1 / 127
2-move intervals
(2's removed)
Ratio split form
6/5 3 / 5
16/15 1 / 3 / 5
33/32 3 * 11
65/64 5 * 13
3-move commas
(2's removed)
Ratio split form
25/24 5 * 5 / 3
128/125 5 * 5 * 5
21/20 3 * 7 / 5
26/25 13 / 5 / 5
49/48 7 * 7 / 3
64/63 1 / 3 / 7 / 7
256/245 1 / 5 / 7 / 7
80/77 5 / 7 / 11
22/21 11 / 3 / 7
40/39 5 / 3 / 13
96/91 3 / 7 / 13
55/52 5 * 11 / 13
1024/1001 1 / 7 / 11 / 13
512/507 1 / 3 / 13 / 13
169/160 13 * 13 / 5
176/169 11 / 13 / 13

With powers of 2 and 3 taken for granted

The relation of powers of 3 to the other factor(s) is represented by "3's". The ones with names (all of them so far) have a corresponding sagittal accidental, though closeby commas share symbols.

1-move commas (2's and 3's removed)
Ratio split form (comma)
81/80 3's / 5 (5 comma)
32805/32768 3's * 5 (5 schisma)
64/63 1 / 3's / 7 (7 comma)
729/704 3's / 11 (11-L diesis)
33/32 3's * 11 (11-M diesis)
27/26 3's / 13 (13-L diesis)
1053/1024 3's * 13 (13 M-diesis)
2187/2176 3's / 17 (17 kleisma)
4131/4096 3's * 17 (17 comma)
513/512 3's * 19 (19 schisma)
19683/19456 3's / 19 (19 comma)
736/729 23 / 3's (23 comma)
261/256 3's * 29 (29 comma)
2-move commas (2's and 3's removed)
Ratio split form (comma)
5103/5120 3's * 7 / 5 (5:7 kleisma)
352/351 11 / 3's / 13 (11:13 kleisma)
896/891 7 / 3's / 11 (7:11 kleisma)
2048/2025 / 3's / 5 / 5 (25 comma/diaschisma)
55/54 11 * 5 / 3's (55 comma)
45927/45056 3's * 7 / 11 (7:11 comma)
52/51 3's * 13 / 17 (13:17 comma)
45/44 3's * 5 / 11 (5:11 S-diesis)
1701/1664 3's * 7 / 13 (7:13 S-diesis)
1408/1377 11 / 3's / 17 (11:17 S-diesis)
6561/6400 3's / 5 / 5 (25 S-diesis)
40/39 5 / 3's / 13 (5:13 S-diesis)
36/35 3's / 5 / 7 (35 M-diesis)
8505/8192 3's * 5 * 7 (35 L-diesis)
3-move commas (2's and 3's removed)
Ratio split form (comma)
250/243 5 * 5 * 5 / 3's (125 M-diesis)
531441/512000 3's / 5 / 5 / 5 (125 L-diesis)