Kite's thoughts on fifthspans - Revision history

TallKite: TallKite moved page Fifthspan to Kite's thoughts on fifthspans: Editor VectorGraphics has repeatedly edited pages I've written about my own research and inserted misinformation. On discord he is openly hostile to me. This move is necessary to avoid a toxic work environment for me.

2025-08-20T01:41:20Z

TallKite moved page Fifthspan to Kite's thoughts on fifthspans: Editor VectorGraphics has repeatedly edited pages I've written about my own research and inserted misinformation. On discord he is openly hostile to me. This move is necessary to avoid a toxic work environment for me.

← Older revision	Revision as of 01:41, 20 August 2025
(No difference)

Fredg999: Text replacement - "Ups and Downs Notation" to "Ups and downs notation"

2025-08-20T01:07:17Z

Text replacement - "Ups and Downs Notation" to "Ups and downs notation"

← Older revision		Revision as of 01:07, 20 August 2025
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	After finding F, there are two ways to proceed. The first way is easier if using a spreadsheet or other software. Multiply F by X and reduce it modulo N. If the number is greater than N/2, further reduce it by subtracting N. For example, the fifthspan of 8\17 is (-5 ⋅ 8) mod 17 = 11, which reduces to -6.		After finding F, there are two ways to proceed. The first way is easier if using a spreadsheet or other software. Multiply F by X and reduce it modulo N. If the number is greater than N/2, further reduce it by subtracting N. For example, the fifthspan of 8\17 is (-5 ⋅ 8) mod 17 = 11, which reduces to -6.

	The second way is easier to calculate in one's head, especially for larger edos. It uses the name of the interval in [[Ups and ~~Downs Notation~~\|ups and downs notation]]. One up has a fifthspan of F. The fifthspans of any ups or downs are added onto the fifthspan of the un-upped/downed interval. Again, If the number is greater than N/2, subtract N. For example, 8\17 is an up-4th. The fifthspan of a 4th is -1, and the fifthspan of ^1 is -5, and -1 + -5 = -6. Thus in any single-ring edo, the fifthspan of vM2 is 2-F, and the fifthspan of ^^4 is 2F-1.		The second way is easier to calculate in one's head, especially for larger edos. It uses the name of the interval in [[Ups and downs notation\|ups and downs notation]]. One up has a fifthspan of F. The fifthspans of any ups or downs are added onto the fifthspan of the un-upped/downed interval. Again, If the number is greater than N/2, subtract N. For example, 8\17 is an up-4th. The fifthspan of a 4th is -1, and the fifthspan of ^1 is -5, and -1 + -5 = -6. Thus in any single-ring edo, the fifthspan of vM2 is 2-F, and the fifthspan of ^^4 is 2F-1.

	== The fifthspan mapping ==		== The fifthspan mapping ==

TallKite: /* Applications of the fifthspan mapping */

2024-08-11T21:48:48Z

Applications of the fifthspan mapping

← Older revision		Revision as of 21:48, 11 August 2024
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	See also: [[Antipodes]], [[Uniform solfege]]		See also: [[Antipodes]], [[Uniform solfege]]

	The term fifthspan was coined by [[Kite Giedraitis]],		The term fifthspan was coined by [[Kite Giedraitis]].

	[[Category:Fifth]]		[[Category:Fifth]]
	[[Category:Mapping]]		[[Category:Mapping]]

TallKite: /* Applications of the fifthspan mapping */

2024-08-11T21:48:33Z

Applications of the fifthspan mapping

← Older revision		Revision as of 21:48, 11 August 2024
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	See also: [[Antipodes]], [[Uniform solfege]]		See also: [[Antipodes]], [[Uniform solfege]]

			The term fifthspan was coined by [[Kite Giedraitis]],

	[[Category:Fifth]]		[[Category:Fifth]]
	[[Category:Mapping]]		[[Category:Mapping]]

TallKite at 08:21, 1 June 2023

2023-06-01T08:21:37Z

← Older revision		Revision as of 08:21, 1 June 2023
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	== Other edos ==		== Other edos ==

	12-edo's best approximation of [[3/2]] is 7\12. Since 7 and 12 are co-prime, 12-edo is single-ring, meaning that 12-edo has only one circle of fifths. Other edos are multi-ring, or "ringy". For example, [[15edo\|15-edo's]] best approximation of 3/2 is 9\15. Since the [[wikipedia:Greatest_common_divisor\|GCD]] of 9 and 15 is 3, 15-edo is a triple-ring edo. The concept of fifthspan doesn't apply to multi-ring edos. Using an alternative approximation of 3/2 affects the ringiness: 18-edo is not multi-ring, but 18b-edo is.		The concept of fifthspan doesn't apply to [[Ring number\|multi-ring]] edos. Using an alternative approximation of 3/2 affects the ringiness: 18-edo is not multi-ring, but 18b-edo is.

	{\| class="wikitable center-all"		{\| class="wikitable center-all"
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	== Rank-2 temperaments ==		== Rank-2 temperaments ==

	Unlike edos, which have one or more finite circles of 5ths, rank-2 temperaments have one or more infinite chains of 5ths. If the temperament's [[pergen]] is unsplit, i.e. is (P8, P5), there is only one chain, and an interval's fifthspan is the distance one must travel along this chain to reach the interval. The fifthspan can be derived directly from the pythagorean name, using this chart:		Unlike edos, which have one or more finite circles of 5ths, rank-2 temperaments have one or more infinite chains of 5ths. If the temperament's [[pergen]] is unsplit, i.e. is (P8, P5), there is only one chain, and an interval's fifthspan is the distance one must travel along this chain to reach the interval. The fifthspan is simply the interval's [[Monzo\|prime-3-count]]. It can be derived directly from the pythagorean name, using this chart:

	{\| class="wikitable center-all"		{\| class="wikitable center-all"
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	\|}		\|}

	Examples of unsplit pergens include [[Meantone]], [[Schismatic\|Layo ~~aka~~ Schismatic]], and [[Archy\|Ru ~~aka~~ Archy]]. 3-limit just intonation, also known as pythagorean tuning, is simply a special case of the unsplit pergen. The concept of fifthspan doesn't apply to split pergens. If the pergen is split but the octave is unsplit, the concept may be generalized to genspan, the distance along the genchain, or chain of generators.		Examples of unsplit pergens include [[Meantone]], [[Schismatic\|Layo/Schismatic]], and [[Archy\|Ru/Archy]]. 3-limit just intonation, also known as pythagorean tuning, is simply a special case of the unsplit pergen. The concept of fifthspan doesn't apply to split pergens. If the pergen is split but the octave is unsplit, the concept may be generalized to genspan, the distance along the genchain, or chain of generators.

	{\| class="wikitable center-all"		{\| class="wikitable center-all"

TallKite: /* The fifthspan mapping */ added some more

2023-01-09T03:04:32Z

The fifthspan mapping: added some more

← Older revision		Revision as of 03:04, 9 January 2023
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	If N-edo's best approximation of a prime P is X edosteps, or X\N, then P's fifthspan is the fifthspan of X\N. Just as an edomapping or [[patent val]] assigns an edostepspan to each prime, a fifthspan mapping assigns a fifthspan to each prime. Prime 2's fifthspan is always 0, and prime 3's fifthspan is always 1. For example, in 12-edo, 5/4 is best approximated by 4\12, which is a major 3rd, which has fifthspan 4. 7/4 is a minor 7th, fifthspan -2. Thus 12edo's fifthspan mapping of 2.3.5.7 is (0 1 4 -2).		If N-edo's best approximation of a prime P is X edosteps, or X\N, then P's fifthspan is the fifthspan of X\N. Just as an edomapping or [[patent val]] assigns an edostepspan to each prime, a fifthspan mapping assigns a fifthspan to each prime. Prime 2's fifthspan is always 0, and prime 3's fifthspan is always 1. For example, in 12-edo, 5/4 is best approximated by 4\12, which is a major 3rd, which has fifthspan 4. 7/4 is a minor 7th, fifthspan -2. Thus 12edo's fifthspan mapping of 2.3.5.7 is (0 1 4 -2).

	The fifthspan mapping can be expressed very concisely as a ~~series~~ of [[uniform solfege]] syllables.		The fifthspan mapping can be expressed very concisely as a string of [[uniform solfege]] syllables. Primes 2 and 3 are always Da and Sa, so those two primes can be omitted. The remaining primes are listed in order. All meantone edos start with Ma, all archy edos have Tha as the 2nd syllable, etc. Two edos can have the same mapping. For example both 19edo and 26edo are MaThoPaFla.

	{\| class="wikitable center-all"		{\| class="wikitable center-all"
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	! prime 11		! prime 11
	!prime 13		!prime 13
	!solfege		!solfege string
	\|-		\|-
	![[19-edo]]		![[19-edo]]
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	\| -6		\| -6
	\| -9		\| -9
	\|~~RiThaShaTho~~		\|RuThaShaTho
	\|-		\|-
	![[31-edo]]		![[31-edo]]

TallKite: /* The fifthspan mapping */ added solfege names, added 53edo example

2023-01-09T02:37:35Z

The fifthspan mapping: added solfege names, added 53edo example

← Older revision		Revision as of 02:37, 9 January 2023
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	If N-edo's best approximation of a prime P is X edosteps, or X\N, then P's fifthspan is the fifthspan of X\N. Just as an edomapping or [[patent val]] assigns an edostepspan to each prime, a fifthspan mapping assigns a fifthspan to each prime. Prime 2's fifthspan is always 0, and prime 3's fifthspan is always 1. For example, in 12-edo, 5/4 is best approximated by 4\12, which is a major 3rd, which has fifthspan 4. 7/4 is a minor 7th, fifthspan -2. Thus 12edo's fifthspan mapping of 2.3.5.7 is (0 1 4 -2).		If N-edo's best approximation of a prime P is X edosteps, or X\N, then P's fifthspan is the fifthspan of X\N. Just as an edomapping or [[patent val]] assigns an edostepspan to each prime, a fifthspan mapping assigns a fifthspan to each prime. Prime 2's fifthspan is always 0, and prime 3's fifthspan is always 1. For example, in 12-edo, 5/4 is best approximated by 4\12, which is a major 3rd, which has fifthspan 4. 7/4 is a minor 7th, fifthspan -2. Thus 12edo's fifthspan mapping of 2.3.5.7 is (0 1 4 -2).

			The fifthspan mapping can be expressed very concisely as a series of [[uniform solfege]] syllables.

	{\| class="wikitable center-all"		{\| class="wikitable center-all"
	\|-		\|-
	\|+ Fifthspans of various primes in various edos		\|+Fifthspans of various primes in various edos
	\|-		\|-
	!		!
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	! prime 7		! prime 7
	! prime 11		! prime 11
			!prime 13
			!solfege
	\|-		\|-
	! [[19-edo]]		![[19-edo]]
	\| 0		\| 0
	\| 1		\| 1
	\| 4		\| 4
	\| -9		\| -9
	\| 6		\| 6
			\| -4
			\|MaThoPaFla
	\|-		\|-
	! [[22-edo]]		![[22-edo]]
	\| 0		\| 0
	\| 1		\| 1
	\| 9		\| 9
	\| -2		\| -2
	\| -6		\| -6
			\| -9
			\|RiThaShaTho
	\|-		\|-
	! [[31-edo]]		![[31-edo]]
	\| 0		\| 0
	\| 1		\| 1
	\| 4		\| 4
	\| 10		\| 10
	\| -13		\| -13
			\|15
			\|MaLuShoSi
	\|-		\|-
	! [[41-edo]]		![[41-edo]]
	\| 0		\| 0
	\| 1		\| 1
			\| -8
			\| -14
			\| -18
			\|20
			\|FoDeFlePi
			\|-
			![[53edo\|53-edo]]
			\|0
			\|1
	\| -8		\| -8
	\| -14		\| -14
	\| ~~-18~~		\|23
			\|20
			\|FoDeRiyuPi
	\|}		\|}

	For unsplit rank-2 temperaments, the fifthspan mapping is identical to the 2nd row of the temperament's mapping matrix. Mathematically, the edo's fifthspan mapping is derived by treating the edo as a special case of a specific rank-2 temperament. The 2nd row of this temperament's mapping matrix is the fifthspan mapping. The first row is easily found, it ~~simpy~~ octave-reduces the stacked 5ths. For 12-edo, the temperament is [[Meantone family\|Gu & Ru aka Dominant Meantone]]. Here is the full mapping matrix for 12-edo:		For unsplit rank-2 temperaments, the fifthspan mapping is identical to the 2nd row of the temperament's mapping matrix. Mathematically, the edo's fifthspan mapping is derived by treating the edo as a special case of a specific rank-2 temperament. The 2nd row of this temperament's mapping matrix is the fifthspan mapping. The first row is easily found, it simply octave-reduces the stacked 5ths. For 12-edo, the temperament is [[Meantone family\|Gu & Ru aka Dominant Meantone]]. Here is the full mapping matrix for 12-edo:

	{\| class="wikitable center-all"		{\| class="wikitable center-all"
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	See also the [[wikipedia:Wicki–Hayden_note_layout\|Wicki-Hayden]] layout and the [[wikipedia:Generalized_keyboard\|Bosanquet]] keyboard. When playing these instruments, one might want to locate a specific ratio on the keyboard. The dot product of the ratio's monzo with the edo's fifthspan mapping, reduced modulo N, gives the ratio's fifthspan, and hence its location on the instrument. For example, the fifhspan of 7/5 in 31-edo is (0 0 -1 1) ⋅ (0 1 4 10) = -4 + 10 = 6. Whereas in 41-edo, it's (0 0 -1 1) ⋅ (0 1 -8 -14) = -6. Note that this location is based on the indirect (consistent) mapping, not the direct (possibly inconsistent) mapping. The consistent mapping is arguably of greater value on an isomorphic keyboard.		See also the [[wikipedia:Wicki–Hayden_note_layout\|Wicki-Hayden]] layout and the [[wikipedia:Generalized_keyboard\|Bosanquet]] keyboard. When playing these instruments, one might want to locate a specific ratio on the keyboard. The dot product of the ratio's monzo with the edo's fifthspan mapping, reduced modulo N, gives the ratio's fifthspan, and hence its location on the instrument. For example, the fifhspan of 7/5 in 31-edo is (0 0 -1 1) ⋅ (0 1 4 10) = -4 + 10 = 6. Whereas in 41-edo, it's (0 0 -1 1) ⋅ (0 1 -8 -14) = -6. Note that this location is based on the indirect (consistent) mapping, not the direct (possibly inconsistent) mapping. The consistent mapping is arguably of greater value on an isomorphic keyboard.

	See also: [[Antipodes]]		See also: [[Antipodes]], [[Uniform solfege]]

	[[Category:Fifth]]		[[Category:Fifth]]
	[[Category:Mapping]]		[[Category:Mapping]]

TallKite: added a link to the antipodes page

2022-07-16T06:11:28Z

added a link to the antipodes page

← Older revision		Revision as of 06:11, 16 July 2022
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	* Piers van der Torren's [https://www.striso.org/ Striso board]		* Piers van der Torren's [https://www.striso.org/ Striso board]
	See also the [[wikipedia:Wicki–Hayden_note_layout\|Wicki-Hayden]] layout and the [[wikipedia:Generalized_keyboard\|Bosanquet]] keyboard. When playing these instruments, one might want to locate a specific ratio on the keyboard. The dot product of the ratio's monzo with the edo's fifthspan mapping, reduced modulo N, gives the ratio's fifthspan, and hence its location on the instrument. For example, the fifhspan of 7/5 in 31-edo is (0 0 -1 1) ⋅ (0 1 4 10) = -4 + 10 = 6. Whereas in 41-edo, it's (0 0 -1 1) ⋅ (0 1 -8 -14) = -6. Note that this location is based on the indirect (consistent) mapping, not the direct (possibly inconsistent) mapping. The consistent mapping is arguably of greater value on an isomorphic keyboard.		See also the [[wikipedia:Wicki–Hayden_note_layout\|Wicki-Hayden]] layout and the [[wikipedia:Generalized_keyboard\|Bosanquet]] keyboard. When playing these instruments, one might want to locate a specific ratio on the keyboard. The dot product of the ratio's monzo with the edo's fifthspan mapping, reduced modulo N, gives the ratio's fifthspan, and hence its location on the instrument. For example, the fifhspan of 7/5 in 31-edo is (0 0 -1 1) ⋅ (0 1 4 10) = -4 + 10 = 6. Whereas in 41-edo, it's (0 0 -1 1) ⋅ (0 1 -8 -14) = -6. Note that this location is based on the indirect (consistent) mapping, not the direct (possibly inconsistent) mapping. The consistent mapping is arguably of greater value on an isomorphic keyboard.

			See also: [[Antipodes]]

	[[Category:Fifth]]		[[Category:Fifth]]
	[[Category:Mapping]]		[[Category:Mapping]]

TallKite: /* Finding the fifthspan of an edo interval */ replaced scale tree with stern-brocot tree, added a wikipedia link.

2021-01-14T23:54:32Z

Finding the fifthspan of an edo interval: replaced scale tree with stern-brocot tree, added a wikipedia link.

← Older revision		Revision as of 23:54, 14 January 2021
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	== Finding the fifthspan of an edo interval ==		== Finding the fifthspan of an edo interval ==

	To find the fifthspan of X\N, first find F, the fifthspan of 1\N. F is the smaller ancestor of N on the [[~~scale~~ tree]]. For example, 17-edo appears on the scale tree as 10\17, and its smaller ancestor is 3\5. Since 10\17 is to the left of 3\5 in the scale tree, F is negative, and F = -5. And in fact, 1\17 is a minor 2nd, fifthspan -5.		To find the fifthspan of X\N, first find F, the fifthspan of 1\N. F is the smaller ancestor of N in the [[wikipedia:Stern–Brocot_tree\|Stern–Brocot tree]]. For example, 17-edo appears on the scale tree as 10\17, and its smaller ancestor is 3\5. Since 10\17 is to the left of 3\5 in the scale tree, F is negative, and F = -5. And in fact, 1\17 is a minor 2nd, fifthspan -5.

	{\| class="wikitable center-all"		{\| class="wikitable center-all"

Xenwolf: improve wikitext readability

2021-01-14T17:05:55Z

improve wikitext readability

Show changes