throwing this out there for you trigonometrists

[bloke]

If you're smart enough to be a trigonometry expert, you probably already understand what a three valve compensating system is, and how it works on a brass instrument.

So here's the puzzle:

Were someone too build a BB flat instrument and install a three valve compensating piston set. What length of inline valve circuit - in inches - would offer the largest number of fairly close to In tune usable pitches below low e natural, with the combination of either any of the three individual pistons (which only compensate when depressed in combination with one or more of the other of the three pistons, but do not compensate when depressed as individual pistons).

To rephrase:
What separate non-compensating fourth valve circuit length would offer the most number of usable pitches - from double low E flat down to double low B natural - when installed inline and activated in conjunction with a three valve compensating system?

[gocsick]

If you target the middle of your missing 3 valve notes Db at being perfectly in tune with 4+2+3 that gives you almost a spot on tritone.. sqrt 2 times the open bugle length..

But that puts Eb at almost 50 cents sharp and B0 almost 50 cents flat.

[bloke]

If you target the middle of your missing 3 valve notes Db at being perfectly in tune with 4+2+3 that gives you almost a spot on tritone.. sqrt 2 times the open bugle length..

But that puts Eb at almost 50 cents sharp and B0 almost 50 cents flat.

It's obvious what would give me E flat, but it was there anything that would give me a pretty good E flat and a pretty good d? None of us have much call for anything lower than that other than home practice.
Thank you sir.
... Maybe something to put with 12 for E-flat that would also work for putting with 13 to play D...??
As a reminder, 13 is like a spot on regular fourth valve, and 123 is nearly spot on in tune for the next half step down (rather than being badly sharp) with a three valve compensating system.

[gocsick]

Ok. I think I have a solution for you.. I wrote up a quick brief force optimization loop in Python (Well co-pilot assisted.. because that is how the world works these days).

Instead of assuming standard low-register fingerings, I treated the 4th valve length as a free design variable and simply tried everything. For each candidate design, I first forced one chosen note (for example C below the staff) to be exactly in tune using a specific 4 + (1–3) fingering. That uniquely fixes how long the 4th-valve slide must be. With that 4th-valve length set, I then checked every possible fingering that uses the 4th valve for the other nearby low notes and picked the fingering that came closest to correct pitch for each note. I repeated this process for all reasonable “anchor” choices (E♭, D, D♭, C, B as the in-tune reference, using different valve combinations) and compared the results. The “best” solution was the one that gave the largest number of usable low notes within a practical tuning tolerance, rather than making any single note perfect at the expense of the rest.

The best answer is that 4th valve should be in-between a tritone and a perfect 5th ... 40 cents flat tritone or a circuit length of 96.54 inches

Best-fit 4th-valve design (summary)
4th-valve added length:
Pitch drop of 4 alone:
Resulting low-register behavior (best fingering chosen per note)

Target note. Best fingering. Cents error

E♭1 4 + 2. −9.3

D1. 4 + 1. +19.7

D♭1. 4 + 2 + 3. −25.6

C1. 4 + 1 + 3 (anchor). 0.0

B0. 4 + 1 + 2 + 3. +24.5


Worst-case error: ~25–26 cents


If anyone is actually interested.. I will post the logic as another post and I can post the otton code as well.

[gocsick]

This is the logic model I have co-pilot to put together the code.

BRUTE-FORCE OPTIMIZATION METHOD FOR A NON-COMPENSATING 4TH VALVE
(WITH EXPLICIT CALCULATIONS)

---

1. DEFINE THE TARGET NOTES

Choose a set of low notes to optimize.
In this case:

E♭1 = 7 semitones below open
D1 = 8 semitones below open
D♭1 = 9 semitones below open
C1 = 10 semitones below open
B0 = 11 semitones below open

Define “usable” as absolute tuning error ≤ 20 cents.
---

2. DEFINE THE BASE BUGLE

Let L0 = open bugle length (for a BB♭ tuba, about 216 inches).
All valve lengths are expressed as fractions of L0 so the result scales to any horn.

The ideal length ratio for a note n semitones below open is:

R_target(n) = 2^(n / 12)

---

3. DEFINE VALVE BEHAVIOR

Valves 1–3 are compensating relative to each other:

Any combination of 1–3 alone produces exactly the correct total length for its semitone sum.

When valve 4 is pressed:

Valve 4 adds a fixed physical length a4 * L0.

Valves 1–3 still add the same physical lengths they would add on the open bugle.

Define m = semitone value of the chosen 1–3 combination:

m = 0, 1, 2, 3, 4, 5, or 6
---

4. CHOOSE AN ANCHOR CONDITION

Pick:

one target note with semitone drop n

one fingering that uses valve 4 and a 1–3 combination with value m

Force that fingering to be exactly in tune.

This sets the 4th valve length:

a4 = 2^(n / 12) − 2^(m / 12)

Fourth-valve slide length in inches:

ΔL4 = a4 * L0

---

5. CALCULATE ACTUAL LENGTH FOR ANY FINGERING USING 4


For any fingering that uses valve 4 and a 1–3 combination with value m:

R_actual = a4 + 2^(m / 12)

---

6. COMPUTE TUNING ERROR

Pitch is inversely proportional to tube length.

Cents error = 1200 * log2( R_target(n) / R_actual )

Positive cents = sharp
Negative cents = flat

---

7. EXHAUSTIVE SEARCH FOR EACH NOTE

For each target note n:

Test all possible values of m (0 through 6).

Compute R_actual for each fingering.

Compute cents error.

Keep the fingering with the smallest absolute error.

---

8. SCORE THE DESIGN

For the chosen anchor:

Count how many notes have |error| ≤ 20 cents.

Record the worst absolute error.

---

9. REPEAT FOR ALL ANCHORS

Repeat steps 4–8 for all reasonable anchor choices, such as:

E♭ = 4+2, 4+1, 4+1+2, 4+3, etc.
Then repeat for D, D♭, C, and B as anchors.
---

10. SELECT THE BEST SOLUTION

The optimal 4th-valve length is the one that:

maximizes the number of usable notes

minimizes the worst-case error

[bloke]

Computer programs are like instant spreadsheets in the way that they define parameters.

Thanks for participating, defining and managing this exercise with and for me.

[gocsick]

I like a good math problem. It took to the time I would usually spend on the NYT's Sudokus.

[bloke]

I think I could do pretty well with two rotors, but - if it's going to be a five valve system, I guess I might as well use a traditional five valve system with no compensating valves...
... It's just that I have a really nice clean English 3 valve compensating system sitting here in a 5 gallon bucket...
And it's the same bore size (.728") as as a four-valve front action piston set that I think I could snag for free... but the front action non-compensating is off of a Brazil made sousaphone. The build quality is functional, but I certainly wouldn't describe it as wonderful...
... I guess if you pull a slide, depress a piston, and you hear some pop noise, it's a functional piston.

Again, I really do appreciate your participation in this thread and your calculations.

[donn]

Some time back I wrote this little gimmick for some fun with "Elm", a javascript compiler programming language that's a lot like Haskell: Valve Lab.

To be honest, I didn't follow the explanation - "forced one note to be exactly in tune" etc. - but I'm probably too old and unambitious to be thinking hard enough.

The logic for valve combinations is bound to be about the same, I imagine. My value for BBb is 233 inches, which (if I remember right) I got from table of pitch wavelengths. I know that isn't the length of the tuba, but maybe it's the length we should use to calculate the cylindrical valve extension?

Anyway, I'm going to hope that you can figure out the UI without detailed instructions. Valves are denoted by nominal step length, which you can set. If the rendering is working right with your browser, flat/sharp combinations are red/blue. Lengths are in centimeters; I thought I had a pulldown to choose units, and I see it in the code at the bottom, but maybe I added that after the online version.

[edit] The notes are on 2nd partial. [/edit]

[edit] Note that this has nothing to do with trigonometry, acoustics or any science of any description. It's making some very basic assumptions about the relationship between valve tubing lengths, and note wavelengths and from there it's just algebra. These assumptions may not be valid in the real world. No guarantees, express or implied. The author is irresponsible. [/edit]

[bloke]

thnx!