Xenharmonic Wiki

Kite's thoughts on fifthspans: Difference between revisions

← Older edit
VisualWikitext
TallKite (talk | contribs)
autopatrolled, Sysops
3,990 edits
m changed ringy to multi-ring
TallKite (talk | contribs)
autopatrolled, Sysops
3,990 edits
m 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.
 
(9 intermediate revisions by 3 users not shown)
Line 2: Line 2:


== 12-edo example ==
== 12-edo example ==
For every interval of [[12-edo]], the interval's fifthspan is simply the shortest distance one must travel around 12-edo's [[circle of fifths]] to reach the interval. Traveling in a fifthward direction creates a positive fifthspan, and traveling fourthward creates a negative fifthspan.
For every interval of [[12-edo]], the interval's fifthspan is simply the shortest distance one must travel around 12-edo's [[circle of fifths]] to reach the interval. Traveling in a fifthward direction creates a positive fifthspan, and traveling fourthward creates a negative fifthspan.
{| class="wikitable" style="text-align:center"
 
|+The fifthspan of 12-edo intervals
{| class="wikitable center-all"
!edostep
|-
|0
|+ The fifthspan of 12-edo intervals
|1
|-
|2
! edostep
|3
| 0
|4
| 1
|5
| 2
|6
| 3
|7
| 4
|8
| 5
|9
| 6
|10
| 7
|11
| 8
|12
| 9
| 10
| 11
| 12
|-
|-
!interval
! interval
|P1
| P1
|m2
| m2
|M2
| M2
|m3
| m3
|M3
| M3
|P4
| P4
|A4/d5
| A4/d5
|P5
| P5
|m6
| m6
|M6
| M6
|m7
| m7
|M7
| M7
|P8
| P8
|-
|-
!fifthspan
!fifthspan
|0
| 0
| -5
| -5
|2
| 2
| -3
| -3
|4
| 4
| -1
| -1
|6
| 6
|1
| 1
| -4
| -4
|3
| 3
| -2
| -2
|5
| 5
|0
| 0
|}
|}
The fifthspan of 6\12 could be either 6 or -6. To ensure a unique fifthspan, the positive value is chosen over the negative one. In practice, the fifthspan of an N-edo interval is always subject to reduction modulo N, so the choice is inconsequential.
The fifthspan of 6\12 could be either 6 or -6. To ensure a unique fifthspan, the positive value is chosen over the negative one. In practice, the fifthspan of an N-edo interval is always subject to reduction modulo N, so the choice is inconsequential.


Line 55: Line 60:


== 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.
 
{| class="wikitable" style="text-align:center"
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.
|+The fifthspan of [[17edo|17-edo]] intervals
 
!edostep
{| class="wikitable center-all"
|0
|-
|1
|+ The fifthspan of [[17edo|17-edo]] intervals
|2
|-
|3
! edostep
|4
| 0
|5
| 1
|6
| 2
|7
| 3
|8
| 4
|9
| 5
|10
| 6
|11
| 7
|12
| 8
|13
| 9
|14
| 10
|15
| 11
|16
| 12
|17
| 13
| 14
| 15
| 16
| 17
|-
|-
!interval
! interval
|P1
| P1
|m2
| m2
|~2
| ~2
|M2
| M2
|m3
| m3
|~3
| ~3
|M3
| M3
|P4
| P4
|^4
| ^4
|v5
| v5
|P5
| P5
|m6
| m6
|~6
| ~6
|M6
| M6
|m7
| m7
|~7
| ~7
|M7
| M7
|P8
| P8
|-
|-
!fifthspan
! fifthspan
|0
| 0
| -5
| -5
|7
| 7
|2
| 2
| -3
| -3
| -8
| -8
|4
| 4
| -1
| -1
| -6
| -6
|6
| 6
|1
| 1
| -4
| -4
|8
| 8
|3
| 3
| -2
| -2
| -7
| -7
|5
| 5
|0
| 0
|}
|}


== 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:
 
{| class="wikitable" style="text-align:center"
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:
|+The chain of fifths
 
!interval
{| class="wikitable center-all"
|...
|-
|d4
|+ The chain of fifths
|d8
|-
|d5
! interval
|m2
| ...
|m6
| d4
|m3
| d8
|m7
| d5
|P4
| m2
|P1/P8
| m6
|P5
| m3
|M2
| m7
|M6
| P4
|M3
| P1/P8
|M7
| P5
|A4
| M2
|A1
| M6
|A5
| M3
|...
| M7
| A4
| A1
| A5
| ...
|-
|-
!fifthspan
! fifthspan
|...
| ...
| -8
| -8
| -7
| -7
Line 154: Line 167:
| -2
| -2
| -1
| -1
|0
| 0
|1
| 1
|2
| 2
|3
| 3
|4
| 4
|5
| 5
|6
| 6
|7
| 7
|8
| 8
|...
| ...
|}
|}
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.  
 
{| class="wikitable" style="text-align:center"
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.  
|+The genchain for pergen (P8, P4/3)
 
!interval
{| class="wikitable center-all"
|...
|-
|m7
|+ The genchain for pergen (P8, P4/3)
|^m6
|-
|v5
! interval
|P4
| ...
|^m3
| m7
|vM2
| ^m6
|P1/P8
| v5
|^m7
| P4
|vM6
| ^m3
|P5
| vM2
|^4
| P1/P8
|vM3
| ^m7
|M2
| vM6
|...
| P5
| ^4
| vM3
| M2
| ...
|-
|-
!genspan
! genspan
|...
| ...
|6
| 6
|5
| 5
|4
| 4
| 3
| 3
| 2
| 2
| 1
| 1
|0
| 0
| -1
| -1
| -2
| -2
Line 200: Line 217:
| -5
| -5
| -6
| -6
|...
| ...
|}
|}


== 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.
 
{| class="wikitable" style="text-align:center"
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.
|+The fifthspan of one edostep for edos 12-24
 
!edo
{| class="wikitable center-all"
|12
|-
|13
|+ The fifthspan of one edostep for edos 12-24
|14
|-
|15
! edo
|16
| 12
|17
| 13
|18
| 14
|19
| 15
|20
| 16
|21
| 17
|22
| 18
|23
| 19
|24
| 20
| 21
| 22
| 23
| 24
|-
|-
!fifthspan
! fifthspan
| -5
| -5
|5
| 5
|N/A
| N/A
|N/A
| N/A
| -7
| -7
| -5
| -5
|5
| 5
|7
| 7
|N/A
| N/A
|N/A
| N/A
| -5
| -5
| -7
| -7
|N/A
| N/A
|}
|}
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 ==
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).
{| class="wikitable" style="text-align:center"
 
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
|-
!
!
!prime 2
! prime 2
!prime 3
! prime 3
!prime 5
! prime 5
!prime 7
! prime 7
!prime 11
! prime 11
!prime 13
!solfege string
|-
|-
![[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
|RuThaShaTho
|-
|-
![[31-edo]]
![[31-edo]]
|0
| 0
|1
| 1
|4
| 4
|10
| 10
| -13
| -13
|15
|MaLuShoSi
|-
|-
![[41-edo]]
![[41-edo]]
| 0
| 1
|  -8
|  -14
|  -18
|20
|FoDeFlePi
|-
![[53edo|53-edo]]
|0
|0
|1
|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:
 
{| class="wikitable" style="text-align:center"
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:
|+12-edo as a special case of (P8, P5)
 
{| class="wikitable center-all"
|-
|+ 12-edo as a special case of (P8, P5)
|-
!
!
!2/1
! 2/1
!3/1
! 3/1
!5/1
! 5/1
!7/1
! 7/1
|-
|-
!period = 2/1
! period = 2/1
|1
| 1
|1
| 1
|0
| 0
|4
| 4
|-
|-
!generator = 3/2
! generator = 3/2
|0
| 0
|1
| 1
|4
| 4
| -2
| -2
|}
|}


== Applications of the fifthspan mapping ==
== Applications of the fifthspan mapping ==
Many [[Microtonal Keyboards|microtonal keyboards]] use chains of fifths to generate the layout. Such keyboards are called [[wikipedia:Generalized_keyboard|generalized keyboards]].
Many [[Microtonal Keyboards|microtonal keyboards]] use chains of fifths to generate the layout. Such keyboards are called [[wikipedia:Generalized_keyboard|generalized keyboards]].
* Cortex Design's [https://www.lumatone.io/ Lumatone] aka [http://terpstrakeyboard.com/ Terpstra] keyboard
* Cortex Design's [https://www.lumatone.io/ Lumatone] aka [http://terpstrakeyboard.com/ Terpstra] keyboard
Line 309: Line 361:
* 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]], [[Uniform solfege]]
The term fifthspan was coined by [[Kite Giedraitis]].


[[Category:Fifth]]
[[Category:Fifth]]
[[Category:Mapping]]
[[Category:Mapping]]
Retrieved from "https://en.xen.wiki/w/Kite%27s_thoughts_on_fifthspans"