Taxicab distance
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.
A crude form of weighting 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 microtonal notations such as Sagittal notation, which notates higher-prime-limit ratios in terms of their deviation from 3-limit ratios. In this sense, 81/80 has a taxicab distance of 1, as it contains only a single instance of 5. In color notation the no-2s-no-3s taxicab distance is called the color depth.
Examples
Following tables shows some unweighted taxicab distances of ratios in prime-factor lattice without the usual octave reduction.
| 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. He proposes weighted distances, thus a "three-step" (moving by 3/2 or 4/3) is shorter than a "five-step" (moving by 5/4 or 8/5). Prime 2 is excluded, thus a "two-step" has distance zero. He proposes weighting by Benedetti height not Tenney height, to avoid decimal places.
The rationale for weighting the distances is that in Western music, root movements by perfect 4ths or 5ths (three-steps) are more common than movements by 3rds or 6ths (five-steps). The rationale for triangularizing is that a root movement of 5/3 is about as "strange" as one of 5/4, thus both are a single five-step. Likewise a move by 7/5 or 7/6 is about as strange as one by 7/4, and all three moves are a single seven-step.
First factor the numerator and denominator into prime numbers and discard all twos. Then find the largest prime in the ratio. Use it to cancel out a smaller prime on the other side of the ratio. If given a choice, cancel out as high a prime as possible. Cancelling removes only the 2nd prime; the 1st prime remains but is "used up" in the sense that it can't cancel out any more primes. Proceed similarly with the next largest prime until all primes are removed or used up.
For example, consider 441/440 = 3*3*7*7 / 2*2*5*11. Discard the twos to get 3*3*7*7 / 5*11. The highest prime is the 11. It cancels a 7 to make 3*3*7*7 / 5*(11), where the parentheses indicate that 11 is used up. The highest prime which isn't yet cancelled or used up is the remaining 7. It cancels a 5 to make 3*3*7*(7) / 5*(11). No more cancelling is possible. The ratio becomes 3*3*7 / 11, and the taxicab distance is 2 three-steps plus a seven-step plus an eleven-step.
| prime limit | ratio | factorize | remove 2s | cancellations | distance | |
|---|---|---|---|---|---|---|
| 3 | 9/8 | 3*3 / 2*2*2 | 3*3 | 3 * 3 = 9 | ||
| 5 | 6/5 | 2*3 / 5 | 3 / 5 | 5 | ||
| 10/9 | 2*5 / 3*3 | 5 / 3*3 | (5) / |
5 * 3 = 15 | ||
| 15/8 | 3*5 / 2*2*2 | 3*5 | 3 * 5 = 15 | |||
| 27/25 | 3*3*3 / 5*5 | 3*3*3 / 5*5 | 3 * 5 * 5 = 75 | |||
| 81/80 | 3*3*3*3 / 2*2*2*2*5 | 3*3*3*3 / 5 | 3 * 3 * 3 * 5 = 135 | |||
| 7 | 7/5 | 7 / 5 | 7 / 5 | (7) / |
7 | |
| 21/20 | 3*7 / 2*2*5 | 3*7 / 5 | 3*(7) / |
3 * 7 = 21 | ||
| 25/21 | 5*5 / 3*7 | 5*5 / 3*7 | 5 * 7 = 35 | |||
| 225/224 | 3*3*5*5 / 2*2*2*2*2*7 | 3*3*5*5 / 7 | 3*3* |
3 * 3 * 5 * 7 = 315 | ||
| 11 | 35/33 | 5*7 / 3*11 | 5*7 / 3*11 | 5* |
(5)* |
5 * 11 = 55 |
Thus 81/80 reduces to 1 five-step and 3 three-steps, as in the common Iy - yVIg - yIIg - yVy - yIy, where the five-step is the first root movement.
Kite notes that taxicab distance can be applied to JI chords, scales and chord progressions. A chord's distance is the largest of the distance of each of its intervals. For example, an 8:10:12:15 (maj7) chord's triangularized distance is the same as that of 15/8, which is 15. A 12:15:18:20 (maj6) chord has the same distance as 9/5, which is also 15. By the same logic, the Zarlino scale's most distant interval is 45/32, and its distance is 45. An 81/80 comma pump (in JI, so the tonic actually drifts) has a distance of 135.
With powers of 2 taken for granted
| Ratio | split form |
|---|---|
| 5/4 | 5 |
| 4/3 | 1 / 3 |
| 7/4 | 7 |
| 128/127 | 1 / 127 |
| Ratio | split form |
|---|---|
| 6/5 | 3 / 5 |
| 16/15 | 1 / 3 / 5 |
| 33/32 | 3 * 11 |
| 65/64 | 5 * 13 |
| 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.
| 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) |
| 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) |
| Ratio | split form | (comma) |
|---|---|---|
| 250/243 | 5 * 5 * 5 / 3's | (125 M-diesis) |
| 531441/512000 | 3's / 5 / 5 / 5 | (125 L-diesis) |
Further implications
Taxicab-2 intervals tend to show up very frequently as basis elements in fractional subgroups. (And of course, by definition, all basis elements in integer subgroups are taxicab-1.)