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Kite's thoughts on 41edo brass instruments: Difference between revisions

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


== Full 41edo instruments ==
== Background - subharmonic trumpets ==
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.
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) plays harmonics 2 through 8, and sometimes higher ones as well. It 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 the valves add a fixed length rather than multiply the total length by a fixed ratio. The player must compensate for this.


The 4 valves can lower the pitch by various JI ratios:
=== 36edo example ===
Consider a trumpet with an uncoiled length of 48 inches, with 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 an 8-note subharrmonic series fragment 55::48, a 53-limit 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. Since the 2nd harmonic is unused, the horn would benefit from a somewhat longer coiled length and/or a somewhat narrower bore, so that harmonics 3-9 are more easily played.
 
=== 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¢ might be 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|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. Since 5/4 = 17\53, a 53edo trumpet could mostly cover harmonics 4-10.
 
The next section uses the second approach, and the final section uses the first approach.
 
See also [[Kite's thoughts on 31edo brass instruments]].
 
== 2-3-4-8 instruments ==
The best [[Delta-N ratio|delta-1]]<nowiki/>ratio approximation of 2\41 is 28/27. This gives us these valve lengths:
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 12: Line 33:
!over-54
!over-54
!cents
!cents
!41edo
!as 41edo
|-
|-
!1st
!1st
!2
!2
|28/27
|[[28/27]]
|56/54
|56/54
|63
|63.0
|2\41 + 4¢
|2\41 + 4.4¢
|-
|-
!2nd
!2nd
!3
!3
|19/18
|[[19/18]]
|57/54
|57/54
|94
|93.6
|3\41 + 6¢
|3\41 + 5.8¢
|-
|-
!3rd
!3rd
!4
!4
|29/27
|[[29/27]]
|58/54
|58/54
|124
|123.7
|4\41 + 7¢
|4\41 + 6.6¢
|-
|-
!4th
!4th
!8
!8
|31/27
|[[31/27]]
|62/54
|62/54
|239
|239.2
|8\41 + 5¢
|8\41 + 5.0¢
|}
|}
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
These valves make the subharmonic series 15-note fragment 71::54, with 70 and 55 missing. Slightly adjusting the cents of each valve may slightly improve the intonation. 
 
In the chart below, ○ is a closed/released valve and ● is an open/depressed valve. The fingering chart assumes the valves run in order with 1st = 2\41, 2nd = 3\41, 3rd = 4\41 and 4th = 8\41. The largest errors are '''bolded'''. Notes are for a Bb trumpet. Many notes have alternate fingerings. Notes in parentheses have an alternate fingering with markedly better intonation. 
 
The intervals are descending so 63.0¢ means 63.0¢ <u>below</u> the unvalved note. Thus the error is the opposite sign from what one expects. For example, the 1st valve lowers by 63¢, 4.4¢ <u>wider</u> than 2\41, thus the note is 4.4¢ <u>flat</u> 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 is actually 16! This is fortuitous because 4/3 is 17\41 and 16 edosteps neatly fills the gap between the 3rd and 4th harmonics.
 
The error from 41edo is the sum of the valve ratio's error and the harmonic's error. For example, 28/27 is 4.4¢ flat of 2\41 and the 5th harmonic is 5.8¢ sharp of 13\41, so the error for the 5th harmonic's C# is 1.4¢.


{| class="wikitable center-all"
{| class="wikitable center-all"
Line 54: Line 83:
ratio
ratio
! rowspan="2" |cents
! rowspan="2" |cents
! rowspan="2" |error
! colspan="5" |error from 41edo
from 41
! rowspan="19" |
! rowspan="18" |
! colspan="7" |harmonics
! colspan="7" |harmonics
|-
|-
!3, 6
!4, 8
!5
!7
!9
!3
!3
!4
!4
Line 73: Line 106:
!1/1
!1/1
|0.0
|0.0
| -0.5
|0.0
|0.0
| 5.8
| 3.0
| -1.0
|F
|F
|Bb
|vA# Bb
|vD
|(vD)
|F
|F
|vAb
|^^G vAb
|Bb
|vA# Bb
|C
|C
|-
!
!
!
!
!
|
|
|
|
|
|
|
|
|
|
|
|
|
|-
|-
!2
!2
Line 88: Line 144:
!28/27
!28/27
|63.0
|63.0
|4.4
| -4.9
| -4.4
|1.4
| -1.5
| -5.4
|^E
|^E
|^A
|^A
Line 103: Line 163:
!19/18
!19/18
|93.6
|93.6
|5.8
| -6.3
| -5.8
|0.0
| -2.8
| -6.8
|E
|E
|A
|(A)
|vC# Db
|vC# Db
|E
|E
Line 118: Line 182:
!29/27
!29/27
|123.7
|123.7
|6.6
| -7.1
| -6.6
| -0.8
| -3.7
| -7.6
|vE
|vE
|vA
|vA
Line 133: Line 201:
!59/54
!59/54
|153.3
|153.3
|7.0
| -7.4
|^D# vvE
| -7.0
| -1.1
| -4.0
| -7.9
|^D# Eb
|^G# vvA
|^G# vvA
|^C
|^C
Line 148: Line 220:
!10/9
!10/9
|182.4
|182.4
|6.8
| -7.3
|D# ^Eb
| -6.8
| -1.0
| -3.8
| -7.8
|D# vEb
|G# ^Ab
|G# ^Ab
|C
|C
Line 163: Line 239:
!61/54
!61/54
|211.0
|211.0
|6.1
| -6.6
| -6.1
| -0.3
| -3.2
| -7.1
|vD# Eb
|vD# Eb
|vG# Ab
|vG# Ab
Line 178: Line 258:
!31/27
!31/27
|239.2
|239.2
|5.0
| -5.5
| -5.0
|0.8
| -2.1
| -6.0
|^^D vEb
|^^D vEb
|^^G vAb
|^^G vAb
Line 193: Line 277:
!7/6
!7/6
|266.9
|266.9
|3.5
| -3.9
| -3.5
| 2.4
| -0.5
| -4.4
|^D
|^D
|^G
|^G
Line 208: Line 296:
!32/27
!32/27
|294.1
|294.1
|1.5
| -1.9
| -1.5
| 4.4
| 1.5
| -2.4
|D
|D
|G
|G
Line 223: Line 315:
!65/54
!65/54
|321.0
|321.0
| -1.0
|0.5
| 1.0
| 6.8
| 3.9
|0.0
|vD
|vD
|vG
|vG
Line 238: Line 334:
!11/9
!11/9
|347.4
|347.4
| -3.8
|3.3
| 3.8
| 9.6
|6.8
|2.8
|^C# vvD
|^C# vvD
|^F# vvG
|^F# vvG
Line 253: Line 353:
!67/54
!67/54
|373.4
|373.4
| -7.0
|6.6
| 7.0
|'''12.9'''
|10.0
|6.1
|C# ^Db
|C# ^Db
|F# ^Gb
|F# ^Gb
|vA# Bb
|'''(vA# Bb)'''
|C# ^Db
|(C# ^Db)
|vE
|vE
|F# ^Gb
|F# ^Gb
Line 268: Line 372:
!34/27
!34/27
|399.1
|399.1
| -10.7
|'''10.2'''
|vC# Db
| '''10.7'''
|vF# Gb
|'''16.5'''
|^^A vBb
|'''13.6'''
|vC# Db
|'''9.7'''
|^D# vvE
|'''vC# Db'''
|vF# Gb
|'''vF# Gb'''
|vG# Ab
|'''^^A vBb'''
|'''(vC# Db)'''
|'''(^D# vvE)'''
|'''(vF# Gb)'''
|'''(vG# Ab)'''
|-
|-
!15
!15
Line 283: Line 391:
!23/18
!23/18
|424.4
|424.4
| -14.7
|'''14.2'''
|^^C vDb
| '''14.7'''
|^^F vGb
|'''20.5'''
|^A
|'''17.6'''
|^^C vDb
|'''13.7'''
|D# ^Eb
|'''^^C vDb'''
|^^F vGb
|'''^^F vGb'''
|^^G vAb
|'''(^A)'''
|'''(^^C vDb)'''
|'''(D# ^Eb)'''
|'''(^^F vGb)'''
|'''(^^G vAb)'''
|-
|-
!16
!16
!2+3+4+8
!2+3+4+8
!●●●●
!●●●●
!70
!71
!35/27
!71/54
|449.3
|473.8
| -19.0
| -6.0
| -5.5
| 0.3
| -2.6
| -6.5
|^C
|^C
|^F
|^F
|A
|A
|^C
|(^C)
|vD# Eb
|vD# Eb
|^F
|^F
Line 308: Line 424:
|}
|}


Here are the full 2 octaves produced by a Bb trumpet. The low vF and the high vC are missing.




{| class="wikitable mw-collapsible mw-collapsed center-all"
{| class="wikitable mw-collapsible mw-collapsed center-all"
|+
|+
Full 2 octave range (low note C)
Full 2 octave range (lowest note is an octave below middle-C)
! rowspan="2" |41-edo
! rowspan="2" |41-edo
gamut
gamut
Line 332: Line 451:
|
|
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|C
|-
|-
!*
!
|
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|
Line 342: Line 460:
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|
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|(missing)
|-
|-
!
!
Line 350: Line 469:
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|^B
|-
|-
!*
!B
|
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|
Line 360: Line 478:
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|B
|-
|-
!
!
Line 368: Line 487:
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|vB
|-
! *
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|-
|-
!
!
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|^A# vvB
|-
!D *
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|-
|-
!
!
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|A# ^Bb
|-
!*
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|-
|-
!
!
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|vA# Bb
|
|vA# Bb
|-
!*
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|-
|-
!
!
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|^^A vBb
|-
!*
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|-
!E
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|-
!*
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|-
|-
!
!
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|^A
|
|^A
|-
|-
!F
!A
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|A
|A
|-
|-
!
!
Line 493: Line 549:
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|vA
|
|vA
|-
|-
!
!
Line 502: Line 558:
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|^G# vvA
|
|^G# vvA
|-
|-
!
!
Line 511: Line 567:
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|
|G# ^Ab
|
|G# ^Ab
|-
|-
!
!
Line 520: Line 576:
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|
|vG# Ab
|
|'''(vG# Ab)'''
|-
|-
!
!
Line 528: Line 584:
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|
|^^G vAb
|
|^^G vAb
|
|'''(^^G vAb)'''
|-
|-
!
!
Line 538: Line 594:
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|^G
|
|^G
|-
|-
!G
!G
Line 546: Line 602:
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|G
|
|G
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|-
|-
Line 555: Line 611:
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|vG
|
|vG
|
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|-
|-
Line 564: Line 620:
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|^F# vvG
|
|^F# vvG
|
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|-
|-
Line 573: Line 629:
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|F# ^Gb
|
|F# ^Gb
|
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|-
|-
Line 582: Line 638:
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|vF# Gb
|
|'''(vF# Gb)'''
|
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|-
|-
Line 591: Line 647:
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|^^F vGb
|
|'''(^^F vGb)'''
|
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|-
|-
Line 600: Line 656:
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|^F
|
|^F
|
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|-
|-
!A
!F
|
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|F
|F
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|
Line 618: Line 674:
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|vF
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Line 626: Line 682:
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|^E
|
|^E
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|-
|-
!
!E
|
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|E
|E
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Line 644: Line 700:
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|vE
|
|vE
|
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|
Line 653: Line 709:
|
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|^D# vvE
|
|'''(^D# vvE)'''
|
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Line 662: Line 718:
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|D# ^Eb
|
|'''(D# ^Eb)'''
|
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|-
|-
!B
!
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|vD# Eb
|vD# Eb
|
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Line 680: Line 736:
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|^^D vEb
|
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Line 689: Line 745:
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|^D
|
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|-
|-
!C
!D
|
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|D
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Line 706: Line 762:
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|(vD)
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|vD
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Line 716: Line 772:
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|^C# vvD
|
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Line 724: Line 780:
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|C# ^Db
|
|(C# ^Db)
|
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Line 733: Line 789:
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|vC# Db
|
|'''(vC# Db)'''
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Line 742: Line 798:
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|^^C vDb
|
|'''(^^C vDb)'''
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Line 751: Line 807:
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|^C
|
|(^C)
|
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|-
|-
!D
!C
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|C
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Line 769: Line 825:
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|vC
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Line 778: Line 834:
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|^B
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Line 784: Line 840:
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|-
|-
!
!B
|
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|B
|
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Line 796: Line 852:
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|vB
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Line 805: Line 861:
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|^A# vvB
|
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Line 814: Line 870:
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|A# ^Bb
|
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Line 820: Line 876:
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|-
|-
!E
!
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|
|vA# Bb
|'''(vA# Bb)'''
|
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Line 832: Line 888:
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|'''^^A vBb'''
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Line 840: Line 896:
!
!
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|^A
|
|'''(^A)'''
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Line 847: Line 903:
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|-
|-
!F
!A
|
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|(A)
|A
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Line 858: Line 914:
!
!
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|vA
|
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Line 867: Line 923:
!
!
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|^G# vvA
|
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Line 876: Line 932:
!
!
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|G# ^Ab
|
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Line 885: Line 941:
!
!
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|vG# Ab
|
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Line 894: Line 950:
!
!
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|^^G vAb
|
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Line 903: Line 959:
!
!
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|^G
|
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Line 912: Line 968:
!G
!G
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|G
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Line 921: Line 977:
!
!
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|vG
|
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Line 930: Line 986:
!
!
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|^F# vvG
|
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Line 939: Line 995:
!
!
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|F# ^Gb
|
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Line 948: Line 1,004:
!
!
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|'''vF# Gb'''
|
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Line 957: Line 1,013:
!
!
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|'''^^F vGb'''
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Line 966: Line 1,022:
!
!
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|^F
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Line 973: Line 1,029:
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|-
|-
!A
!F
|
|F
|
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Line 983: Line 1,039:
|-
|-
!
!
|
|(missing)
|
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== Skip-valved instruments ==
== Skip-valved instruments ==
Skip-valves omit every other step of 41-edo.
Skip-valves omit every other step of 41-edo. The intonation isn't nearly as good as a 2-3-4-8 instrument.
 
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|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.)
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|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 individual harmonics in the harmonic series are analogous to the strings of a Kite guitar. 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 fingering for 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.  
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.  
Retrieved from "https://en.xen.wiki/w/Kite%27s_thoughts_on_41edo_brass_instruments"