Kite-valved Brass Instruments
A simple horn without any valves or slides is only capable of playing the harmonic series. A conventional brass instrument (with the exception of the trombone, which is naturally microtonal) has valves that lengthen the tubing and lower the pitch by small amounts, typically a minor 2nd, a major 2nd and a minor 3rd. By selecting the proper combination of valves, the player can fill the gap between the various harmonics and play a complete 12-edo scale. The intervals don't add up precisely because they add a fixed length rather than multiply the length by a fixed ratio. The player must compensate for this.
A possible Kite-tuned brass instrument would have 4 valves that lower the pitch by 2\41, 4\41, 8\41 and 16\41. These valves are analogous to the frets of a Kite guitar. Each fret is 2\41, so these valves are for 1, 2, 4 or 8 frets respectively. Various valve combinations lower the pitch by 1 to 15 frets. (A 1\41 valve is actually physically difficult to make, since the tube length is so short.)
The individual harmonics in the harmonic series are analogous to the strings of a Kite guitar. The instrument proposed here would have especially long tubing, so that the harmonics 3-9 are easily played, and harmonics 1 and 2 are not and would be unused. The reasoning is that the interval between two harmonics should be an odd number of edosteps, which 3/2 is not. This is so that the harmonic will fill in the gaps of the lower harmonic. Also, each harmonic needs to be at most 16 frets (32 edosteps) from the harmonic two above it, since that is the range of the valves. Thus the ratio must be at most 12/7 = 32\41. Harmonics 3, 4, 5, 6, 7 and 9 meet these requirements. The 8th harmonic is like an extra string that overlaps the 7th harmonic's notes and provides an alternate path to certain notes. Harmonics 10 and 11 overlap the 9th harmonic, so 9 is a good stopping point.
The fingering chart below assumes the valves run in order with 1st = 1 fret, 2nd = 2 frets, 3rd = 4 frets and 4th = 8 frets. ○ = open, ● = closed. It also assumes the lowest note of the instrument is F (with all valves closed, and playing the 1st harmonic). Notes that are duplicated by a higher harmonic are in parentheses.
| fret | valves | harmonics | ||||||
|---|---|---|---|---|---|---|---|---|
| 3 | 4 | 5 | 6 | 7 | 8 | 9 | ||
| 0 | ●●●● | C | F | vA | C | ^^D/vEb | F | G |
| 1 | ○●●● | ^^C | ^^F/vGb | ^A | ^^C | D#/^Eb | ^^F/vGb | ^^G/vAb |
| 2 | ●○●● | C#/^Db | F#/^Gb | vA#/Bb | C#/^Db | vE | F#/^Gb | G#/^Ab |
| 3 | ○○●● | vD | vG | ^A#/vvB | vD | ^E | vG | vA |
| 4 | ●●○● | ^D | ^G | B | ^D | (F) | ^G | ^A |
| 5 | ○●○● | vD#/Eb | vG#/Ab | vC | vD#/Eb | (^^F/vGb) | vG#/Ab | vA#/Bb |
| 6 | ●○○● | ^D#/vvE | ^G#/vvA | ^C | ^D#/vvE | (F#/^Gb) | ^G#/vvA | ^A#/vvB |
| 7 | ○○○● | E | A | vC#/Db | E | (vG) | A | B |
| 8 | ●●●○ | vF | ^^A/vBb | ^C#/vvD | vF | (^G) | ^^A/vBb | vC |
| 9 | ○●●○ | ^F | A#/^Bb | D | ^F | (vG#/Ab) | A#/^Bb | ^C |
| 10 | ●○●○ | vF#/Gb | vB | (^^D/vEb) | vF#/Gb | (^G#/vvA) | vB | vC#/Db |
| 11 | ○○●○ | ^F#/vvG | ^B | (D#/^Eb) | ^F#/vvG | (A) | ^B | ^C#/vvD |
| 12 | ●●○○ | G | (C) | (vE) | (G) | (^^A/vBb) | C | D |
| 13 | ○●○○ | ^^G/vAb | (^^C) | (^E) | (^^G/vAb) | (A#/^Bb) | ^^C | ^^D/vEb |
| 14 | ●○○○ | G#/^Ab | (C#/^Db) | (F) | (G#/^Ab) | (vB) | C#/^Db | D#/^Eb |
| 15 | ○○○○ | (vA) | (vD) | (^^F/vGb) | (vA) | (^B) | vD | vE |
The next chart shows each harmonic's 16 notes plotted against the full 41edo gamut of notes. There is over 2 octaves total range. Note the 7 missing notes in the lowest part of the range, and the 3 missing notes in the highest part. The next two harmonics, 10 and 11, merely extend the pitches of harmonic 9 without supplying any missing notes.
| 41-edo
gamut |
harmonics | ||||||
|---|---|---|---|---|---|---|---|
| 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
| C | C | ||||||
| ^^C | |||||||
| C#/^Db | |||||||
| vD | |||||||
| D | |||||||
| ^D | |||||||
| vD#/Eb | |||||||
| ^D#/vvE | |||||||
| E | E | ||||||
| vF | |||||||
| F | F | ||||||
| ^F | |||||||
| ^^F/vGb | |||||||
| vF#/Gb | |||||||
| F#/^Gb | |||||||
| ^F#/vvG | |||||||
| vG | |||||||
| G | G | ||||||
| ^G | |||||||
| ^^G/vAb | |||||||
| vG#/Ab | |||||||
| G#/^Ab | |||||||
| ^G#/vvA | |||||||
| (vA) | vA | ||||||
| A | A | ||||||
| ^A | |||||||
| ^^A/vBb | |||||||
| vA#/Bb | |||||||
| A#/^Bb | |||||||
| ^A#/vvB | |||||||
| vB | |||||||
| B | B | ||||||
| ^B | |||||||
| vC | |||||||
| C | (C) | C | |||||
| ^C | |||||||
| (^^C) | ^^C | ||||||
| vC#/Db | |||||||
| (C#/^Db) | C#/^Db | ||||||
| ^C#/vvD | |||||||
| (vD) | vD | ||||||
| D | D | ||||||
| ^D | |||||||
| (^^D/vEb) | ^^D/vEb | ||||||
| vD#/Eb | |||||||
| (D#/^Eb) | D#/^Eb | ||||||
| ^D#/vvE | |||||||
| (vE) | vE | ||||||
| E | E | ||||||
| (^E) | ^E | ||||||
| vF | |||||||
| F | (F) | (F) | F | ||||
| ^F | |||||||
| (^^F/vGb) | (^^F/vGb) | ^^F/vGb | |||||
| vF#/Gb | |||||||
| (F#/^Gb) | F#/^Gb | ||||||
| ^F#/vvG | |||||||
| (vG) | vG | ||||||
| G | (G) | G | |||||
| (^G) | ^G | ||||||
| (^^G/vAb) | ^^G/vAb | ||||||
| (vG#/Ab) | vG#/Ab | ||||||
| (G#/^Ab) | G#/^Ab | ||||||
| (^G#/vvA) | ^G#/vvA | ||||||
| (vA) | vA | ||||||
| A | (A) | A | |||||
| ^A | |||||||
| (^^A/vBb) | ^^A/vBb | ||||||
| vA#/Bb | |||||||
| (A#/^Bb) | A#/^Bb | ||||||
| ^A#/vvB | |||||||
| (vB) | vB | ||||||
| B | B | ||||||
| (^B) | ^B | ||||||
| vC | |||||||
| C | C | ||||||
| ^C | |||||||
| ^^C | |||||||
| vC#/Db | |||||||
| C#/^Db | |||||||
| ^C#/vvD | |||||||
| vD | |||||||
| D | D | ||||||
| ^^D/vEb | |||||||
| D#/^Eb | |||||||
| vE | |||||||