Fifthspan

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The fifthspan is an attribute of every interval in certain edos and rank-2 temperaments.

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 5ths to reach the interval. Traveling in a fifthward direction creates a positive fifthspan, and traveling fourthward creates a negative fifthspan.

The fifthspan of 12-edo intervals
edostep 0 1 2 3 4 5 6 7 8 9 10 11 12
interval P1 m2 M2 m3 M3 P4 A4/d5 P5 m6 M6 m7 M7 P8
fifthspan 0 -5 2 -3 4 -1 6 1 -4 3 -2 5 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.

Adding or subtracting octaves doesn't alter the fifthspan. For intervals greater than an octave, the fifthspan is the same as that of the octave-reduced interval.

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, 15-edo's best approximation of 3/2 is 9\15. Since the GCD of 9 and 15 is 3, 15-edo is a triple-ring edo. The concept of fifthspan doesn't apply to ringy edos. Using an alternative approximation of 3/2 affects the ringiness: 18-edo is not ringy, but 18b-edo is.

The fifthspan of 17-edo intervals
edostep 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
interval P1 m2 ~2 M2 m3 ~3 M3 P4 ^4 v5 P5 m6 ~6 M6 m7 ~7 M7 P8
fifthspan 0 -5 7 2 -3 -8 4 -1 -6 6 1 -4 8 3 -2 -7 5 0

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:

The chain of fifths
interval ... d4 d8 d5 m2 m6 m3 m7 P4 P1/P8 P5 M2 M6 M3 M7 A4 A1 A5 ...
fifthspan ... -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 ...

Examples of unsplit pergens include Meantone, Layo aka Schismatic, and 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.

The genchain for pergen (P8, P4/3)
interval ... m7 ^m6 v5 P4 ^m3 vM2 P1/P8 ^m7 vM6 P5 ^4 vM3 M2 ...
genspan ... 6 5 4 3 2 1 0 -1 -2 -3 -4 -5 -6 ...

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.

The fifthspan of one edostep for edos 12-24
edo 12 13 14 15 16 17 18 19 20 21 22 23 24
fifthspan -5 5 N/A N/A -7 -5 5 7 N/A N/A -5 -7 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.

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. 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 non-ringy 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 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).

Fifthspans of various primes in various edos
prime 2 prime 3 prime 5 prime 7 prime 11
19-edo 0 1 4 -9 6
22-edo 0 1 9 -2 -6
31-edo 0 1 4 10 -13
41-edo 0 1 -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 Gu & Ru aka Dominant Meantone. Here is the full mapping matrix for 12-edo:

12-edo as a special case of (P8, P5)
2/1 3/1 5/1 7/1
period = 2/1 1 1 0 4
generator = 3/2 0 1 4 -2

Applications of the fifthspan mapping

Many microtonal keyboards use chains of fifths to generate the layout. Such keyboards are called generalized keyboards.

See also the Wicki-Hayden layout and the 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.