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 ==

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 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"

|-

|+ Fifthspans of various primes in various edos

|+Fifthspans of various primes in various edos

|-

!

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! prime 7

! prime 11

!prime 13

!solfege string

|-

! [[19-edo]]

![[19-edo]]

| 0

| 1

| 4

| -9

| 6

| -4

|MaThoPaFla

|-

! [[22-edo]]

![[22-edo]]

| 0

| 1

| 9

| -2

| -6

| -9

|RuThaShaTho

|-

! [[31-edo]]

![[31-edo]]

| 0

| 1

| 4

| 10

| -13

|15

|MaLuShoSi

|-

! [[41-edo]]

![[41-edo]]

| 0

| 1

| -8

| -14

| -18

|20

|FoDeFlePi

|-

![[53edo|53-edo]]

|0

|1

| -8

| -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:

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* 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: [[Antipodes]], [[Uniform solfege]]

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

[[Category:Fifth]]

[[Category:Mapping]]

@@ Line 61: / Line 61: @@
 == 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.
 {| class="wikitable center-all"
@@ Line 130: / Line 130: @@
 == 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"
@@ Line 179: / Line 179: @@
 |}
-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"
@@ Line 261: / Line 261: @@
 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 ==
 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 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"
 |-
-|+ Fifthspans of various primes in various edos
+|+Fifthspans of various primes in various edos
 |-
 !
@@ Line 277: / Line 279: @@
 ! prime 7
 ! prime 11
+!prime 13
+!solfege string
 |-
-! [[19-edo]]
+![[19-edo]]
 | 0
 | 1
 | 4
-| -9
+|  -9
 | 6
+| -4
+|MaThoPaFla
 |-
-! [[22-edo]]
+![[22-edo]]
 | 0
 | 1
 | 9
-| -2
+|  -2
-| -6
+|  -6
+| -9
+|RuThaShaTho
 |-
-! [[31-edo]]
+![[31-edo]]
 | 0
 | 1
 | 4
 | 10
-| -13
+|  -13
+|15
+|MaLuShoSi
 |-
-! [[41-edo]]
+![[41-edo]]
 | 0
 | 1
+|  -8
+|  -14
+|  -18
+|20
+|FoDeFlePi
+|-
+![[53edo|53-edo]]
+|0
+|1
 | -8
 | -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"
@@ Line 340: / Line 361: @@
 * 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: [[Antipodes]], [[Uniform solfege]]
+The term fifthspan was coined by [[Kite Giedraitis]].
 [[Category:Fifth]]
 [[Category:Mapping]]