Xenharmonic Wiki

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

← Older editNewer edit →
VisualWikitext
TallKite (talk | contribs)
autopatrolled, Sysops
4,192 edits
WIP
TallKite (talk | contribs)
autopatrolled, Sysops
4,192 edits
done
Line 1: Line 1:
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.  
 
== 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.
 
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.
 
=== 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. 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.
 
The next section uses the second approach, and the final section uses the first approach.


== Full 41edo instruments ==
== 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 best [[Delta-N ratio|delta-1]] ratio approximation of 2\41 is 28/27. This gives us these valve lengths:
 
The 4 valves can lower the pitch by various JI ratios:
{| class="wikitable"
{| class="wikitable"
|+
|+
Line 18: Line 35:
|28/27
|28/27
|56/54
|56/54
|63
|63.0
|2\41 + 4¢
|2\41 + 4.4¢
|-
|-
!2nd
!2nd
Line 25: Line 42:
|19/18
|19/18
|57/54
|57/54
|94
|93.6
|3\41 + 6¢
|3\41 + 5.8¢
|-
|-
!3rd
!3rd
Line 32: Line 49:
|29/27
|29/27
|58/54
|58/54
|124
|123.7
|4\41 + 7¢
|4\41 + 6.6¢
|-
|-
!4th
!4th
Line 39: Line 56:
|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. But 71/54 is 17\41 - 23.7¢ which is actually 16\41 + 5.5¢. Thus in the chart below 2+3+4+8 = 16! Note that 69/54 or 23/18 is almost exactly midway between 15\41 and 14\41.
 
The cents are descending cents, which means positive numbers indicate flatness and negative numbers indicate sharpness. For example, the 1st valve lowers by 63¢, 4.4¢ more than 2\41, thus the note is 4.4¢ flat of 41edo.
 
○ = a closed/released valve, ● = an open/depressed valve. Notes are for a Bb trumpet.


{| class="wikitable center-all"
{| class="wikitable center-all"
Line 292: Line 313:
|'''^^G vAb'''
|'''^^G vAb'''
|-
|-
!16*
!16
!2+3+4+8
!2+3+4+8
!●●●●
!●●●●
Line 307: Line 328:
|^G
|^G
|}
|}
Here are the full 2 octaves produced by a Bb trumpet:


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


{| class="wikitable mw-collapsible mw-collapsed center-all"
{| class="wikitable mw-collapsible mw-collapsed center-all"
Line 470: Line 492:
|-
|-
!
!
|
|(missing)
|
|
|
|
Line 1,061: Line 1,083:
|
|
|
|
|
|(missing)
|-
|-
!C
!C
Line 1,071: Line 1,093:
|
|
|C
|C
|-
!
|
|
|
|
|
|
|
|-
!
|
|
|
|
|
|
|
|-
!
|
|
|
|
|
|
|
|-
!
|
|
|
|
|
|
|
|-
!
|
|
|
|
|
|
|
|-
!
|
|
|
|
|
|
|
|-
!D
|
|
|
|
|
|
|
|-
!
|
|
|
|
|
|
|
|-
!
|
|
|
|
|
|
|
|-
!
|
|
|
|
|
|
|
|-
!
|
|
|
|
|
|
|
|-
!
|
|
|
|
|
|
|
|-
!
|
|
|
|
|
|
|
|}
|}


== 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 full-41 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 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 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.
Retrieved from "https://en.xen.wiki/w/Kite%27s_thoughts_on_41edo_brass_instruments"