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

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=== 36edo example ===
=== 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.
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.  
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.  
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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.
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.
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 ===
=== 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.  
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. 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.
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|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.
The next section uses the second approach, and the final section uses the first approach.


== Full 41edo instruments ==
== Unskipped 41edo instruments ==
The best [[Delta-N ratio|delta-1]] ratio approximation of 2\41 is 28/27. This gives us these valve lengths:
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"
|+
|+
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!over-54
!over-54
!cents
!cents
!41edo
!as 41edo
|-
|-
!1st
!1st
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|8\41 + 5.0¢
|8\41 + 5.0¢
|}
|}
These valves make the subharmonic series 15-note fragment 71::54, with 70 and 55 missing. 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 in the chart below 2+3+4+8 = 16! This is fortuitous because 4/3 is 17\41 and 16 edosteps fills the gap between the 3rd and 4th harmonics.  
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 intervals are descending so 63.0¢ means 63.0¢ below the unvalved note. Thus the error is the opposite sign from what is expected. For example, the 1st valve lowers by 63¢, 4.4¢ more than 2\41, thus the note is 4.4¢ flat of 41edo.
The 2 largest errors are '''bolded'''. Adjusting 56/54 slightly will improve the tuning, but not by much.  


In the chart below, ○ is a closed/released valve, and ● is an open/depressed valve. The 2 largest errors are bolded. Notes are for a Bb trumpet.
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¢ <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 = 16! This is fortuitous because 4/3 is 17\41 and 16 edosteps neatly fills the gap between the 3rd and 4th harmonics.


{| class="wikitable center-all"
{| class="wikitable center-all"
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! rowspan="2" |error
! rowspan="2" |error
from 41
from 41
! rowspan="18" |
! rowspan="19" |
! colspan="7" |harmonics
! colspan="7" |harmonics
|-
|-
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|vA# Bb
|vA# Bb
|C
|C
|-
!
!
!
!
!
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|-
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!2
!2
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|153.3
|153.3
| -7.0
| -7.0
|^D# vvE
|^D# Eb
|^G# vvA
|^G# vvA
|^C
|^C
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|182.4
|182.4
| -6.8
| -6.8
|D# ^Eb
|D# vEb
|G# ^Ab
|G# ^Ab
|C
|C
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!23/18
!23/18
|424.4
|424.4
| '''-14.6'''
| '''14.7'''
|'''^^C vDb'''
|'''^^C vDb'''
|'''^^F vGb'''
|'''^^F vGb'''
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|^G
|^G
|}
|}
Here are the full 2 octaves produced by a Bb trumpet:
 
 
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.


*
*
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|C
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|^C
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|(missing)
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|'''^^C vDb'''
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|^B
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!B
|'''vC# Db'''
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|B
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!
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|C# ^Db
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|vB
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!
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|^C# vvD
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|^A# vvB
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|vD
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|A# ^Bb
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!D
!
|D
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|vA# Bb
|vA# Bb
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|^D
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|^^A vBb
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|^A
|^A
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!A
|vD# Eb
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|A
|A
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|D# ^Eb
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|vA
|vA
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|^D# vvE
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|^G# vvA
|^G# vvA
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!
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|vE
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|G# ^Ab
|G# ^Ab
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!E
!
|E
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|vG# Ab
|'''vG# Ab'''
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!
!
|^E
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|^^G vAb
|^^G vAb
|'''^^G vAb'''
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|(missing)
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|^G
|^G
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!F
!G
|F
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|G
|G
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|^F
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|vG
|vG
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!
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|'''^^F vGb'''
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|^F# vvG
|^F# vvG
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!
!
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|'''vF# Gb'''
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|F# ^Gb
|F# ^Gb
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!
!
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|F# ^Gb
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|vF# Gb
|'''vF# Gb'''
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!
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|^F# vvG
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|^^F vGb
|'''^^F vGb'''
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|^F
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!G
!F
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|G
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|F
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!
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|^G
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|vF
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|^^G vAb
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|^E
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!E
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|vG# Ab
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|E
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|G# ^Ab
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|vE
|vE
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|^G# vvA
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|^D# vvE
|'''^D# vvE'''
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|vA
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|D# ^Eb
|'''D# ^Eb'''
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!A
!
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|A
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|vD# Eb
|vD# Eb
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!
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|^A
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|^D
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!D
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|Bb
|vA# Bb
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|D
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|^C# vD
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|vB
|C# ^Db
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|C# ^Db
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!B
!
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|B
|vC# Db
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|'''vC# Db'''
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|^B
|^^C vDb
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|'''^^C vDb'''
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|vC
|^C
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|^C
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|vC
|^C
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|^^C vDb
|^B
|'''^^C vDb'''
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!B
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|B
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|vC# Db
|'''vC# Db'''
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|C# ^Db
|vB
|C# ^Db
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|A# ^Bb
|vD
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!D
!
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|vA# Bb
|vA# Bb
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|D
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|'''^^A vBb'''
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|vD# Eb
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|D# ^Eb
|'''D# ^Eb'''
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|'''^D# vvE'''
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|E
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!
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!F
!G
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|^^F vGb
|'''^^F vGb'''
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|F# ^Gb
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|vF# Gb
|'''vF# Gb'''
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|'''vF# Gb'''
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|F# ^Gb
|F# ^Gb
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!
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|'''^^F vGb'''
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|^F# vvG
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|vG
|vG
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!G
!F
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|G
|G
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!
!
|(missing)
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|^G
|^G
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!
!
|^E
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|^^G vAb
|^^G vAb
|'''^^G vAb'''
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|-
!
!E
|E
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|vG# Ab
|'''vG# Ab'''
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|-
!
!
|vE
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|G# ^Ab
|G# ^Ab
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!
!
|^D# Eb
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|^G# vvA
|^G# vvA
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!
!
|D# vEb
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|vA
|vA
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!A
!
|vD# Eb
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|A
|A
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!
!
|^^D vEb
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|^A
|^A
|-
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!
!
|^D
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|^^A vBb
|-
|-
!
!D
|D
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|vA# Bb
|vA# Bb
|-
|-
!
!
|vD
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|A# ^Bb
|-
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!
!
|^C# vvD
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|^A# vvB
|-
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!
!
|C# ^Db
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|vB
|-
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!B
!
|'''vC# Db'''
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|B
|-
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!
!
|'''^^C vDb'''
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|^B
|-
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!
!
|^C
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|(missing)
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!C
!C
|(missing)
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|C
|}
|}


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