De site over het instrument alpenhoorn
Pipes and frequencies

Application to the alphorn

We will now investigate how useful the previous models are to explain the frequencies produced by an alphorn.
For this purpose I have chosen my own alphorn, in E, build by myself.
Specifications:
Length of the conical tube included the mouthpiece: 337 cm
Inner diameters of the conical tube: 1,3 cm and 5,5 cm.
Length of the bell: 60 cm.
Inner diameter at the end of the bell: 23 cm.
Anyhow, an alphorn is neither a complete cone nor a truncated cone! However, if we put aside the bell of the alphorn, then we get a truncated conical tube and this tube we can test.
Specifications of this truncated conical tube:
Length L: 337 cm
Diameters d1 and d2: 1,3 cm and 5,5 cm
a = d1L / (d2-d1) = 1,3 x 337 / (5,5 - 1,3) ≈ 104 (cm), so the apex a is not small and therefor we may expect that model 4 will better apply than model 3.

To keep the testing process simple, I made use of a chromatic tuner (SEIKO SAT500) to identify the produced notes.
This means that the measuring-unity is not hertz but semitone.

Results of this test:

 Frequencies according to model 4   L = 337 d1 = 1,3 d2 = 5,5 Notes according to model 4 Frequencies according to model 3   L= 337 d1 = 0 d2 = 5,5   f = 343 / 2L' Notes according to model 3 Produced notes f1 = 44 (Hz) F1 f = 51 (Hz) G#1 f2 = 86 F2 2f = 101 G#2 E2 / F2 f3 = 133 C3 + 3f = 152 D#3 C3 + f4 = 182 F#3 4f = 202 G#3 - F#3 - f5 = 232 A#3 5f = 253 C4 - A#3 f6 = 282 C#4 + 6f = 304 D#4 - C#4 f7 = 332 E4 + 7f = 354 F4 / F#4 E4 f8 = 382 F#4 / G4 8f = 405 G4 / G#4 F#4 f9 = 432 A4 - 9f = 455 A#4 A4 -- f10 = 483 B4 - 10f = 506 B4 + B4 -- f11 = 533 C5 / C#5 11f = 557 C#5 C5 f12 = 584 D5 12f = 607 D5 / D#5 D5

The results are clear. Model 4 gives a good prognosis for the produced notes. The frequencies of model 3 are to high.

This result enables us to conclude that model 4 will also give a good prognosis for the produced notes, is we should extend the tested truncated conical tube with length 337 cm to a truncated conical tube with length 397 cm.
After that we will be in the position to compare these notes with the real notes of our alphorn in E and to see what the effect of the bell is.

An easy calculation shows that the diameter at the end of the extended truncated cone is 6,25 cm.

Results of this comparison:

 Extended truncated cone (no bell)   L = 397 d1 = 1,3 d2 = 6,25   Frequencies according to model 4 Notes according to model 4 Alphorn in E (with bell)     L = 397         Produced notes f1 = 40 (Hz) D#1 / E1 f2 = 74 D2 D#2 / E2 f3 = 114 A#2 - B2 f4 = 155 D#3 E3 f5 = 197 G3 G#3 f6 = 239 A#3 / B3 B3 f7 = 282 C#4 + D4 - f8 = 325 E4 - E4 f9 = 367 F#4 F#4 f10 = 410 G#4 - G#4 f11 = 453 A4 / A#4 A4 f12 = 496 B4 B4

The second and third column show the well-know effect of the bell:
the bell raises progressively the lower frequencies.
The result is a full harmonic series with the fundamental missing!