Taxicab distance - Revision history

Overthink: /* With powers of 2 and 3 taken for granted */ made format consistent

2025-11-03T02:08:23Z

With powers of 2 and 3 taken for granted: made format consistent

← Older revision		Revision as of 02:08, 3 November 2025
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	\| 64/63 \|\| 1 / 3's / 7 \|\| (7 comma)		\| 64/63 \|\| 1 / 3's / 7 \|\| (7 comma)
	\|-		\|-
	\| 729/704 \|\| 3's / 11 \|\| (11-L diesis)		\| 729/704 \|\| 3's / 11 \|\| (11 L-diesis)
	\|-		\|-
	\| 33/32 \|\| 3's * 11 \|\| (11-M diesis)		\| 33/32 \|\| 3's * 11 \|\| (11 M-diesis)
	\|-		\|-
	\| 27/26 \|\| 3's / 13 \|\| (13-L diesis)		\| 27/26 \|\| 3's / 13 \|\| (13 L-diesis)
	\|-		\|-
	\| 1053/1024 \|\| 3's * 13 \|\| (13 M-diesis)		\| 1053/1024 \|\| 3's * 13 \|\| (13 M-diesis)

TallKite: /* Triangularizing proposal */ Major rewrite and expansion

2025-08-27T07:39:18Z

Triangularizing proposal: Major rewrite and expansion

Show changes

BudjarnLambeth: /* External links */ Make generalised see also, add 1 item

2025-01-10T06:54:27Z

External links: Make generalised see also, add 1 item

← Older revision		Revision as of 06:54, 10 January 2025
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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]]

BudjarnLambeth: /* Further implications */

2024-12-30T06:07:59Z

Further implications

← Older revision		Revision as of 06:07, 30 December 2024
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	== Further implications ==		== Further implications ==
	Taxicab-2 intervals tend to show up very frequently as [[basis element]]s in [[fractional ~~subgroup~~]]s. (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 ==		== External links ==

BudjarnLambeth: Mention taxicab distance in fractional subgroups

2024-12-30T06:07:31Z

Mention taxicab distance in fractional subgroups

← Older revision		Revision as of 06:07, 30 December 2024
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	\| 531441/512000 \|\| 3's / 5 / 5 / 5 \|\| (125 L-diesis)		\| 531441/512000 \|\| 3's / 5 / 5 / 5 \|\| (125 L-diesis)
	\|}		\|}

			== Further implications ==
			Taxicab-2 intervals tend to show up very frequently as [[basis element]]s in [[fractional subgroup]]s. (And of course, by definition, all basis elements in integer subgroups are taxicab-1.)

	== External links ==		== External links ==

FloraC: Recategorize

2023-11-26T11:14:35Z

Recategorize

← Older revision		Revision as of 11:14, 26 November 2023
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	[[Category:Math]]		[[Category:Math]]
	[[Category:Interval complexity ~~measure~~]]		[[Category:Interval complexity measures]]

FloraC: Recategorize

2023-08-29T13:48:16Z

Recategorize

← Older revision		Revision as of 13:48, 29 August 2023
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	* [[Wikipedia: Hamming distance]]		* [[Wikipedia: Hamming distance]]

	[[Category:~~Complexity~~]]		[[Category:Math]]
			[[Category:Interval complexity measure]]

Fredg999: Categories

2023-02-26T06:25:04Z

Fredg999: Fredg999 moved page Commas by taxicab distance to Taxicab distance: WP:CONCISE

2023-02-26T06:24:41Z

Fredg999 moved page Commas by taxicab distance to Taxicab distance: WP:CONCISE

← Older revision	Revision as of 06:24, 26 February 2023
(No difference)

Fredg999: Add Wikipedia box, improve lead section, categories, misc. edits

2023-02-26T06:22:09Z

Add Wikipedia box, improve lead section, categories, misc. edits

← Older revision		Revision as of 06:22, 26 February 2023
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	~~One~~ [[~~measurement~~]] of ~~the complexity of the~~ [[~~comma~~]] ~~could be~~ the number of [[prime factor]]s it has, regardless of their magnitude, but counting repetitions.		{{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.

	~~When~~ combined with excluding the smallest primes, this measurement can give an idea of how many "strange harmonic moves" a comma is comprised of.		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 ~~'''~~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		81/80 = 2^-4 * 3^4 * 5^-1
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	=== 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]].

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	=== 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.)

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	\| 531441/512000 \|\| 3's / 5 / 5 / 5 \|\| (125 L-diesis)		\| 531441/512000 \|\| 3's / 5 / 5 / 5 \|\| (125 L-diesis)
	\|}		\|}

	~~== See also ==~~

	* [[Height]]

	== External links ==		== External links ==
			* [[Wikipedia: Hamming distance]]
	* [~~https://en.wikipedia.org/wiki/Taxicab_geometry Taxicab geometry -~~ Wikipedia]
	* [https:~~//en.wikipedia.org/wiki/Hamming_distance~~ Hamming distance ~~- Wikipedia~~]

	[[Category:Commas]]		[[Category:Commas]]
	[[Category:Math]]		[[Category:Math]]
	[[Category:Measure]]		[[Category:Measure]]

← Older revision		Revision as of 06:25, 26 February 2023
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	* [[Wikipedia: Hamming distance]]		* [[Wikipedia: Hamming distance]]

	[[Category:~~Commas]]~~		[[Category:Complexity]]
	~~[[Category:Math]]~~
	~~[[Category:Measure~~]]