Stephen Fox

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Practical Mathematics

This page gives some simple mathematical information and practical formulas which may be of help to musicians and technicians in making adjustments to woodwind instruments.

The following material is just a start; more will be added later.  No attempt is made here to give comprehensive theoretical background; that can be found in various books.

Frequency

Frequency is measured in "Hertz" (Hz), also known as "cycles per second";
mathematically, 1 Hz = 1 / (1sec)

To find the frequency of a particular note of the (equal tempered) scale, start with concert A at whatever pitch prevails (e.g, 440Hz or 442Hz for modern instruments, 415Hz for Baroque instruments, 430Hz for Classical period instruments, etc.), then:

for each octave up / down, the frequency is multiplied / divided by 2;
for each semitone up / down, the frequency is multiplied / divided by by the twelfth root of 2 (approximately 1.05946)

Example:
low C on a Bb clarinet is one semitone above low B (concert A), which is an octave below A440;
so, the frequency is (440Hz / 2) x 1.05946 = 233.1Hz

For very small differences in pitch, we use "cents";
1 cent is one hundredth of a semitone, corresponding to a frequency ratio of one twelve hundredth root of 2 (about 1.00057779)

Acoustical length

The sounding or acoustical tube length for a given note of the scale on a clarinet-type instrument (overblowing at the twelfth) can be found from:

fundamental register:  (acoustical length) = (speed of sound) / (4 x frequency)
second register:  (acoustical length) = (3 x speed of sound) / (4 x frequency)
lower altissimo register:  (acoustical length) = (5 x speed of sound) / (4 x frequency)
etc.

With instruments that overblow at the octave (flute, saxophone, oboe, bassoon, etc.):

fundamental register:  (acoustical length) = (speed of sound) / (2 x frequency)
second register:  (acoustical length) = (speed of sound) / (frequency)
lower altissimo register:  (acoustical length) = (3 x speed of sound) / (2 x frequency)
etc.

The speed of sound inside a wind instrument at normal temperatures can be taken as 345000mm/s (but this is quite sensitive to temperature; see below).

Example:
the acoustical length of low C on a Bb clarinet is (345000mm/s) / (4 x 233.1Hz) = 370.0mm

Change in length vs. change in pitch

The change in sounding length necessary to achieve a given change in pitch (e.g., for pulling out/pushing joints or shortening/lengthening a barrel) can be calculated from:

(change in length) = - (acoustical length) x (pitch change in cents) / (1731 cents)

This formula is an approximation assuming proportionally small changes in length, but it is accurate enough for most practical purposes.

Example:

to flatten low C on a Bb clarinet by 10 cents, the joints need to be pulled out or the barrel lengthened by
(370mm) x (10 cents) / (1731 cents) = 2.1mm

the same increase in length will flatten throat Bb (acoustical length 208mm) by 17 cents, but middle B natural (using the sounding length of bottom E, 588mm) only by 6 cents

Effect of temperature on pitch

The speed of sound in air, and hence the frequency of a note produced by an air column of a given length, is directly proportional to the square root of the absolute temperature.

Absolute temperature is measured from absolute zero, which is approximately minus 273°C.  The units are Kelvin (K); 1K is the same size as 1°C.

Example:
the frequency ratio between notes played at 20°C and 24°C is
the square root of ([273 + 24 = 297K] / [273 + 20 = 293K]) = 1.00680...
so, 440Hz at 20°C becomes (440Hz x 1.00680) = 443Hz at 24°C

Note that the temperature of the air inside a wind instrument is a complex blend of the temperature of the room and that inside the player's body; that's why the pitch rises as the instrument "warms up".

Equivalent acoustical length of clarinet mouthpiece

As long as we stay in the lower registers, the equivalent acoustical length of a clarinet mouthpiece, when played on a particular clarinet, can be found very simply and neatly from the following principle:  it is the length of a cylinder whose diameter is the main bore diameter of the clarinet, and whose volume is the same as the physical internal volume of the mouthpiece.

The internal volume of the mouthpiece can be measured by putting waterproof tape over the windway, filling the mouthpiece up with water a large number of times, collecting the water and measuring the total volume, and dividing by the number of trials.

If the mouthpiece volume is V and the central bore diameter of the clarinet is D, the equivalent length L is then calculated from

L = (4/pi) x (V/D²)

Example:

if the measured internal volume of a mouthpiece is 12.5ml, and the central bore diameter of the clarinet is 14.6mm, then the equivalent length of the mouthpiece on that clarinet is
L = 4 / 3.14159 x 12,500 / (14.6²) = 74.7mm

whereas the same mouthpiece on a clarinet with a bore diameter of 15.0mm has an equivalent length of
L = 4 / 3.14159 x 12,500 / (15.0²) = 70.7mm

so using the same mouthpiece, the 15.0mm bore clarinet will require a barrel 3.0mm longer than the 14.6mm bore clarinet, all other factors being equal (BUT the relationship changes when we get into the upper registers)

This is where the mathematics starts to get more cumbersome, and the derivation of the formulas less obvious...

Change in tone hole size vs. change in pitch

The acoustical length for a particular note includes a portion which extends past the tone hole which primarily controls that note.  This is called the "open hole end correction"; it depends on a number of factors, including the diameter of the tone hole, the diameter of the bore, the distance to the next open hole, the thickness of the body wall, and the height of the pad over the hole.

If we use the symbol c for the open hole end correction, it can be calculated from:

c = (z/2) x [sqr(1 + (4/z)(t + hd)(D/d)²) -1]

or, more compactly, if we use Q to denote the square root,

c = (z/2) x (Q -1)

where

d = diameter of tone hole in question
D = bore diameter at tone hole in question
z = distance to next open tone hole
t = depth of tone hole
h = a numerical factor accounting for the height of the pad over the hole;
it can be taken as approximately 0.75 for an open hole with no pad,
or 1.0 for a pad at average opening height

This formula (adapted from Benade, Fundamentals of Musical Acoustics, 2nd ed., p. 450) is accurate enough to be useful as long as we stay in the low registers of the instrument's range.

If we make a proportionally small change in any of the above dimensions, we can calculate the effect on the open hole end correction, and hence the effect on the pitch of the note in question.

If we change the hole diameter (d) by a small amount, then:

(change in c) = - (change in d) x (D/d)² x (2t/d + h) / Q

Example:

for low C on a Bb clarinet, the above dimensions are approximately
D = 14.6mm
d = 6.0mm
z = 15.5mm
t = 6.0mm
h = 1.0

so, we calculate
Q = 4.40
c = 26.3mm

therefore
(change in c) = - (change in d) x 4.0

if the tone hole diameter is reduced from 6.0mm to 5.5mm (a change of  minus 0.5mm),
(change in c) = 2.0mm

this will thus increase the acoustical length of the note by about 2.0mm; as calculated in a previous section, this will flatten low C by approximately 10 cents

Optimum proportions of speaker hole (cylindrical bore instruments)

For a cylindrical bore instrument, provided that the speaker hole serves only to produce the upper registers (i.e., that it does not also have to function as a tone hole for throat Bb), its optimum diameter vs. length proportions can be calculated from (adapted from Benade, 2nd ed., p. 458):

(d/D)² x L / (t + hd) = 0.7570

where
d = diameter of speaker hole
D = main bore diameter
L = acoustical length of the note for which the speaker hole is ideally located
t = length of speaker tube
h = approximately 0.75 (assuming a healthy pad opening height, which is not always the case!)

so, rearranging the formula,

t = [(d/D)² x L / (0.7570)] - hd

Example:

for a Bb clarinet with a bore diameter of 14.6mm, and with the speaker hole located ideally for low A (which has an acoustical length of 440mm), some optimum speaker tube dimensions are:

 d = 2.1mm t = 10.5mm d = 2.2mm t = 11.5mm d = 2.3mm t = 12.7mm d = 2.4mm t = 13.9mm etc.

which agrees very well with, for example, the speaker tubes found on Reform-Boehm clarinets by German makers such as Wurlitzer; the speaker hole diameter is considerably larger on conventional clarinets, where the hole also produces throat Bb

Optimum proportions of octave hole (conical bore instruments)

A similar calculation gives us the optimum diameter vs. depth proportions for the octave holes on conical bore instruments:

(d/D)² x L / (t + hd) = 3.253

where, again,
d = diameter of octave hole
D = bore diameter at octave hole
L = acoustical length of the note for which the octave hole is ideally located
t = depth of octave hole
h = approximately 0.75

so,

t = [(d/D)² x L / (3.253)] - hd

The additional proviso with conical instruments is that the depth of the octave hole needs to be as small as possible (hence the octave tube should be cup shaped, with the hole itself drilled through a thin membrane)

Examples:

1.  The lower octave hole on an oboe operates over the range from D (sounding length 587mm) to G# (415mm), for an average sounding length of 501mm; if the bore diameter at the octave hole is 7.8mm, some optimum hole dimensions are:

 d = 0.50mm t = 0.25mm d = 0.60mm t = 0.45mm d = 0.70mm t = 0.70mm etc.

which agrees well with what we see in practice

2.  The lower octave hole on an alto saxophone also operates over the range from D (concert F, sounding length 988mm) to G# (concert B, 699mm), for an average sounding length of 844mm; if the bore diameter at the octave hole is 25.0mm, some optimum hole dimensions are:

 d = 2.3mm t = 0.5mm d = 2.5mm t = 0.7mm d = 2.7mm t = 1.0mm etc.

which does not agree very well with most saxophone octave tubes, which are usually either constant diameter or conical, rather than cup shaped

Mathematical modelling of mouthpiece facing curves

When measuring and refacing clarinet and saxophone mouthpieces, it is traditional (at least in North America) to follow prescriptive tables of measurements based on a set of standard feeler gauges.  It may be instructive to examine the curves more precisely by finding a mathematical formula which reproduces the shape of the curve, allowing us to check the standard prescriptions and to adjust the length and tip opening to any desired dimensions.

A good approximation of the facing curve would be a power function (similar to a parabola but not necessarily quadratic), starting at the point at which the curve breaks away from the flat table, with progressively increasing curvature towards the tip:

y = t (1 – x/L) ^ p

where
L = facing length
t = tip opening
y = opening at distance x from tip
and the exponent p is to be determined empirically.

A lower value of p gives a flatter curve, a higher value more “bump” in the middle of the facing.  Examination of several German clarinet mouthpieces (with long, close facings) gives p = 1.7 as the best fit.  This compares with the figure of 1.5 determined by Arthur Benade in his survey of clarinet mouthpieces, and the arbitrary choice of p = 2 used by Conn for their study of saxophone mouthpieces.  It is possible that slightly different values of p would be appropriate for various reed cuts.

To be continued...