Kite's thoughts on fifthspans: Difference between revisions

== 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.

|-

|-

!fifthspan

| -5

| -3

| -1

| -4

| -2

|}

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.

== 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.

|-

|-

| -5

| -3

| -8

| -1

| -6

| -4

| -2

| -7

|}

== 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:

|-

| -8

| -7

| -2

| -1

|}

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.  

|-

| 3

| 2

| 1

| -1

| -2

| -5

| -6

|}

== Finding the fifthspan of an edo interval ==

|-

| -5

| -7

| -5

| -5

| -7

|}

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 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).

!

|-

| -9

|-

| -2

| -6

|-

| -13

|-

| -8

| -14

| -18

|}

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:

!

|-

|-

| -2

|}

== 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]].

* Cortex Design's [https://www.lumatone.io/ Lumatone] aka [http://terpstrakeyboard.com/ Terpstra] keyboard