Kite's thoughts on fifthspans: Difference between revisions

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== 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.

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== 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:

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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.

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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.

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 ==

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

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

[[Category:Fifth]]

[[Category:Mapping]]

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 == Other edos ==
--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.
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 == 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:
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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.
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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.
-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 ==
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 See also: [[Antipodes]], [[Uniform solfege]]
+The term fifthspan was coined by [[Kite Giedraitis]].
 [[Category:Fifth]]
 [[Category:Mapping]]