Kite's thoughts on 41edo brass instruments

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

Background - subharmonic trumpets

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 intervals, 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 cents of the intervals don't add up precisely because they add a fixed length rather than multiply the total length by a fixed ratio. The player must compensate for this.

36edo example

The standard uncoiled length of a trumpet is 48 inches. Consider 3 valves that add 1, 2 and 4 inches of length. The 1st valve lengthens the tube to 49 inches, and thus increases its length by 49/48. Since the pitch of the horn is inversely proportional to the length, the 1st valve lowers the pitch by 49/48 (about 36¢). The 2nd valve lengthens the tube to 50 inches and lowers the pitch by 50/48 = 25/24. The 1st and 2nd valves combined increase the length to 51 inches and lower the pitch by 51/48 = 17/16. If the 3rd valve were to add 3 inches, it would be redundant. Instead the 3rd valve adds 4 inches, so that the 3 valves in various combinations add between 1 and 7 inches. The scale this produces is the subharrmonic series 8-note fragment 55::48. Ratios such as 7/6 and 10/9 occur, thus this is a just intonation scale.

The central step of 55::48 is 52/51 = 33.6¢, which is almost exactly 1/3 of a 12edo semitone, or 1\36. The other 6 steps are only slightly larger or smaller. Therefore this subharmonic fragment approximates 7 steps of 36edo quite well.

Consider the gap between the 6th and 7th harmonics = 7/6. Since 7/6 is about 8\36, there are 7 intermediate edosteps. Thus the 7 steps are just enough to fill the gap. The gap between the 7th and 8th harmonics, and 8th to 9th, are both smaller so those gaps are easily filled. But 36edo mistunes the 10th and 11th harmonics, so the useful harmonics are 6-9. Thus a horn with a fundamental 3 octaves below middle-C could play from G below middle-C up to D a 12th higher.

To fill the gaps between the lower harmonics, we can add a 4th valve that adds 8 inches. This gives us 63::48. The central step is now 56/55 = 31.2¢, closer to 1\39. There are 16 notes, and 16\39 = 492¢, nearly filling the gap between the 3rd and 4th harmonics. The horn would need especially long tubing, so that harmonics 3-9 are easily played, and harmonics 1 and 2 are not and would be unused.

Other edos

The valve increments of 1, 2 and 4 inches were chosen purely to make the calculations simple. One could have valves of 1/N, 2/N and 4/N times the total length for any whole number N. This generates the subharmonic series N+7::N. N can be chosen so that N+4/N+3 closely approximates 1 edostep. N needn't even be a whole number, making the approximation even more accurate.

But there's a technical issue. A horn valve has to add a certain minimum length, and less than 50¢ is physically impossible. So instead of adding 1, 2, 4 and 8 edosteps of length, in larger edos we must start with 2 edosteps. From here there are two approaches. The first one uses skip-valving, analogous to the skip-fretting of a Kite guitar, and has valves of 2, 4, 8 and 16 edosteps. The second one has valves of 2, 3, 4 and 8 edosteps. Why 8? Because combining 2, 3 and 4 supplies 2-7 and 9, and 8 is needed to fill in the gap. Adding 8 to 2-7 and 9 supplies 10-15 and 17. Thus these 4 valves lower by 2-15 and 17 edosteps.

A third approach for even larger edos has valves of 3, 4, 5 and 6 edosteps, supplying 3-15 and 18, with 9 present as both 3+6 and 4+5.

The next section uses the second approach, and the final section uses the first approach.

Unskipped 41edo instruments

The best delta-1ratio approximation of 2\41 is 28/27. This gives us these valve lengths:

valve	edosteps	ratio	over-54	cents	as 41edo
1st	2	28/27	56/54	63.0	2\41 + 4.4¢
2nd	3	19/18	57/54	93.6	3\41 + 5.8¢
3rd	4	29/27	58/54	123.7	4\41 + 6.6¢
4th	8	31/27	62/54	239.2	8\41 + 5.0¢

These valves make the subharmonic series 15-note fragment 71::54, with 70 and 55 missing. In the chart below, ○ is a closed/released valve, and ● is an open/depressed valve. Notes are for a Bb trumpet.

The 2 largest errors are bolded. Adjusting 56/54 slightly will improve the tuning, but not by much.

The intervals are descending so 63.0¢ means 63.0¢ below the unvalved note. Thus the error is the opposite sign from what one expects. For example, the 1st valve lowers by 63¢, 4.4¢ wider than 2\41, thus the note is 4.4¢ flat of 41edo.

Note that 69/54 or 23/18 is almost exactly midway between 15\41 and 14\41. And 71/54 is 17\41 - 23.7¢, which is actually 16\41 + 5.5¢. Thus 2+3+4+8 = 16! This is fortuitous because 4/3 is 17\41 and 16 edosteps neatly fills the gap between the 3rd and 4th harmonics.

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# Eb	^G# vvA	^C	^D# vvE	F# ^Gb	^G# vvA	^A# vvB
6	2+4	●○●○	60	10/9	182.4	-6.8		D# vEb	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

Here are the full 2 octaves produced by a Bb trumpet. Note that the bolded notes for the two largest errors often have alternative notes.

Full 2 octave range (low note middle-C)

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

Skip-valved instruments

Skip-valves omit every other step of 41-edo. The intonation isn't nearly as good as full-41 instruments.

A skip-valved Kite brass instrument has 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