Kite's thoughts on 41edo brass instruments

This page discusses the Kite trumpet, Kite flugelhorn, Kite french horn, Kite tuba, etc.,

Full 41edo instruments

The intervals don't add up precisely because they add a fixed length rather than multiply the length by a fixed ratio. But the ratios do add up in a sense.

The 4 valves can lower the pitch by various JI ratios:

valve	edosteps	ratio	over-54	cents	41edo
1st	2	28/27	56/54	63	2\41 + 4¢
2nd	3	19/18	57/54	94	3\41 + 6¢
3rd	4	29/27	58/54	124	4\41 + 7¢
4th	8	31/27	62/54	239	8\41 + 5¢

Consider a trumpet of 54 units of length (54 cm, 54 inches, whatever). Each valve adds 2, 3, 4 or 8 cm of length. When two valves are opened, these additions are summed. For example the first two valves add 2+3 = 5 cm of length. Thus 56/54 "plus" 57/54 equals 59/54. Subharmonic series 69::54

edosteps to lower by		valves	subhar- monics	as a ratio	cents	error from 41		harmonics
								3	4	5	6	7	8	9
0	0	○○○○	54	1/1	0.0	0.0		F	vA# Bb	^C# vD	F	^^G vAb	vA# Bb	C
2	2	●○○○	56	28/27	63.0	4.4		^E	^A	C# ^Db	^E	G	^A	^B
3	3	○●○○	57	19/18	93.6	5.8		E	A	vC# Db	E	vG	A	B
4	4	○○●○	58	29/27	123.7	6.6		vE	vA	^^C vDb	vE	^F# vvG	vA	vB
5	2+3	●●○○	59	59/54	153.3	7.0		^D# vvE	^G# vvA	^C	^D# vvE	F# ^Gb	^G# vvA	^A# vvB
6	2+4	●○●○	60	10/9	182.4	6.8		D# ^Eb	G# ^Ab	C	D# ^Eb	vF# Gb	G# ^Ab	A# ^Bb
7	3+4	○●●○	61	61/54	211.0	6.1		vD# Eb	vG# Ab	vC	vD# Eb	^^F vGb	vG# Ab	vA# Bb
8	8	○○○●	62	31/27	239.2	5.0		^^D vEb	^^G vAb	^B	^^D vEb	^F	^^G vAb	^^A vBb
9	2+3+4	●●●○	63	7/6	266.9	3.5		^D	^G	B	^D	F	^G	^A
10	2+8	●○○●	64	32/27	294.1	1.5		D	G	vB	D	vF	G	A
11	3+8	○●○●	65	65/54	321.0	-1.0		vD	vG	^A# vvB	vD	^E	vG	vA
12	4+8	○○●●	66	11/9	347.4	-3.8		^C# vvD	^F# vvG	A# ^Bb	^C# vvD	E	^F# vvG	^G# vvA
13	2+3+8	●●○●	67	67/54	373.4	-7.0		C# ^Db	F# ^Gb	vA# Bb	C# ^Db	vE	F# ^Gb	G# ^Ab
14	2+4+8	●○●●	68	34/27	399.1	-10.7		vC# Db	vF# Gb	^^A vBb	vC# Db	^D# vvE	vF# Gb	vG# Ab
15	3+4+8	○●●●	69	23/18	424.4	-14.7		^^C vDb	^^F vGb	^A	^^C vDb	D# ^Eb	^^F vGb	^^G vAb
16*	2+3+4+8	●●●●	71	71/54	473.8	5.5		^C	^F	A	^C	vD# Eb	^F	^G

*71/54 is 17\41 - 23.7¢ which is actually 16\41 + 5.5¢

Full 2 octave range (low note middle-C)

41-edo gamut	harmonics
	3	4	5	6	7	8	9
C
	^C
	^^C vDb
	vC# Db
	C# ^Db
	^C# vvD
	vD
D	D
	^D
	^^D vEb
	vD# Eb
	D# ^Eb
	^D# vvE
	vE
E	E
	^E
F	F
		^F
		^^F vGb
		vF# Gb
		F# ^Gb
		^F# vvG
		vG
G		G
		^G
		^^G vAb
		vG# Ab
		G# ^Ab
		^G# vvA
		vA
A		A	A
		^A	^A
			^^A vBb
		Bb	vA# Bb
			A# ^Bb
			^A# vvB
			vB
B			B
			^B
			vC
C			C
			^C	^C
			^^C vDb	^^C vDb
			vC# Db	vC# Db
			C# ^Db	C# ^Db
				^C# vvD
			vD	vD
D				D
				^D
				^^D vEb
				vD# Eb	vD# Eb
				D# ^Eb	D# ^Eb
				^D# vvE	^D# vvE
				vE	vE
E				E	E
				^E	^E
					vF
F				F	F
					^F	^F
					^^F vGb	^^F vGb
					vF# Gb	vF# Gb
					F# ^Gb	F# ^Gb
					^F# vvG	^F# vvG
					vG	vG
G					G	G
						^G	^G
					^^G vAb	^^G vAb	^^G vAb
						vG# Ab	vG# Ab
						G# ^Ab	G# ^Ab
						^G# vvA	^G# vvA
						vA	vA
A						A	A
						^A	^A
							^^A vBb
						vA# Bb	vA# Bb
							A# ^Bb
							^A# vvB
							vB
B							B
							^B
C							C
D

Skip-valved instruments

Skip-valves omit every other step of 41-edo.

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 skip-valved Kite brass instrument would have 4 valves that lower the pitch by 2\41, 4\41, 8\41 and 16\41. Skip-valved is analogous to the skip-fretting 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/depressed, ○ = closed/released. It also assumes the lowest note of the instrument is F (with all valves open, and playing the 1st harmonic). Notes that are duplicated by a higher harmonic are in parentheses. The final column shows the sharpness caused by intervals not adding up precisely.

fret	valves	harmonics							error
		3	4	5	6	7	8	9
0	●●●●	C	F	vA	C	^^D/vEb	F	G	0.0¢
1	○●●●	^^C	^^F/vGb	^A	^^C	D#/^Eb	^^F/vGb	^^G/vAb	0.0¢
2	●○●●	C#/^Db	F#/^Gb	vA#/Bb	C#/^Db	vE	F#/^Gb	G#/^Ab	0.0¢
3	○○●●	vD	vG	^A#/vvB	vD	^E	vG	vA	3.8¢
4	●●○●	^D	^G	B	^D	(F)	^G	^A	0.0¢
5	○●○●	vD#/Eb	vG#/Ab	vC	vD#/Eb	(^^F/vGb)	vG#/Ab	vA#/Bb	7.3¢
6	●○○●	^D#/vvE	^G#/vvA	^C	^D#/vvE	(F#/^Gb)	^G#/vvA	^A#/vvB	14.4¢
7	○○○●	E	A	vC#/Db	E	(vG)	A	B	24.6¢
8	●●●○	vF	^^A/vBb	^C#/vvD	vF	(^G)	^^A/vBb	vC	0.0¢
9	○●●○	^F	A#/^Bb	D	^F	(vG#/Ab)	A#/^Bb	^C	13.7¢
10	●○●○	vF#/Gb	vB	(^^D/vEb)	vF#/Gb	(^G#/vvA)	vB	vC#/Db	27.0¢
11	○○●○	^F#/vvG	^B	(D#/^Eb)	^F#/vvG	(A)	^B	^C#/vvD	43.0¢
12	●●○○	G	(C)	(vE)	(G)	(^^A/vBb)	C	D	52.7¢
13	○●○○	^^G/vAb	(^^C)	(^E)	(^^G/vAb)	(A#/^Bb)	^^C	^^D/vEb	70.8¢
14	●○○○	G#/^Ab	(C#/^Db)	(F)	(G#/^Ab)	(vB)	C#/^Db	D#/^Eb	88.5¢
15	○○○○	(vA)	(vD)	(^^F/vGb)	(vA)	(^B)	vD	vE	108.4¢

One way to correct the cents errors might be to have a 5th valve, a trombone-style one, that moves a tiny amount depending on which of the other 4 valves are open. It could be done with a microchip, 4 sensors and a very accurate stepper motor. It would run off a battery, or perhaps solar power.

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 8 missing notes in the lowest part of the range, and the 3 missing notes in the highest part, all marked with asterisks. The next two harmonics, 10 and 11, merely extend the pitches of harmonic 9 without supplying any missing notes. Likewise the previous harmonic, 2, merely extends the pitches of harmonic 3 downwards.

Full 2 octave range (low note C)

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