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Taxicab distance: Difference between revisions

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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 height|Tenney]]/[[Benedetti height]].
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 height|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.
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 [[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".
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 [[Kite's color notation|color notation]] the no-2s-no-3s taxicab distance is called the color depth.


=== Examples ===
=== Examples ===
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| 5/3 ||  5 / 3 || 2
| 5/3 ||  5 / 3 || 2
|-
|-
| 7/4 || 7 / 2 / 2 || 3
| 7/4 || 7 / 2*2 || 3
|-
|-
| 15/14 || 3 * 5 / 2 / 7 || 4
| 15/14 || 3*5 / 2*7 || 4
|-
|-
| 55/42 || 5 * 11 / 2 * 3 * 7 || 5
| 55/42 || 5*11 / 2*3*7 || 5
|}
|}


=== 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. 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*<s>7</s>*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*<s>7</s>*(7) / <s>5</s>*(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.
{| class="wikitable"
|+Examples of computing the triangularized taxicab distance, weighted by Benedetti height
!prime limit
!ratio
!factorize
!remove 2s
! colspan="2" |cancellations
!distance
|-
!3
|9/8
|3*3 / 2*2*2
|3*3
|
|
|3 * 3 = 9
|-
! rowspan="5" |5
|6/5
|2*3 / 5
|3 / 5
|<s>3</s> / (5)
|
|5
|-
|10/9
|2*5 / 3*3
|5 / 3*3
|(5) / <s>3</s>*3
|
|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
|<s>3</s>*3*3 / (5)*5
|<s>3</s>*<s>3</s>*3 / (5)*(5)
|3 * 5 * 5 = 75
|-
|81/80
|3*3*3*3 / 2*2*2*2*5
|3*3*3*3 / 5
|<s>3</s>*3*3*3 / (5)
|
|3 * 3 * 3 * 5 = 135
|-
! rowspan="4" |7
|7/5
|7 / 5
|7 / 5
|(7) / <s>5</s>
|
|7
|-
|21/20
|3*7 / 2*2*5
|3*7 / 5
|3*(7) / <s>5</s>
|
|3 * 7 = 21
|-
|25/21
|5*5 / 3*7
|5*5 / 3*7
|<s>5</s>*5 / 3*(7)
|<s>5</s>*(5) / <s>3</s>*(7)
|5 * 7 = 35
|-
|225/224
|3*3*5*5 / 2*2*2*2*2*7
|3*3*5*5 / 7
|3*3*<s>5</s>*5 / (7)
|
|3 * 3 * 5 * 7 = 315
|-
!11
|35/33
|5*7 / 3*11
|5*7 / 3*11
|5*<s>7</s> / 3*(11)
|(5)*<s>7</s> / <s>3</s>*(11)
|5 * 11 = 55
|}
Thus 81/80 reduces to 1 five-step and 3 three-steps, as in the common [[Kite's color notation#Chord progressions, keys, scales and modulations|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 ==
== With powers of 2 taken for granted ==
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Taxicab-2 intervals tend to show up very frequently as [[basis element]]s in [[subgroup|fractional subgroups]]. (And of course, by definition, all basis elements in integer subgroups are taxicab-1.)
Taxicab-2 intervals tend to show up very frequently as [[basis element]]s in [[subgroup|fractional subgroups]]. (And of course, by definition, all basis elements in integer subgroups are taxicab-1.)


== External links ==
== See also ==
* [[Wikipedia: Hamming distance]]
* [[Wikipedia: Hamming distance]]
* [[List of taxicab-2 intervals]]


[[Category:Math]]
[[Category:Math]]
[[Category:Interval complexity measures]]
[[Category:Interval complexity measures]]
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