This is Part 2 of the presentation that was delivered at the Skunkworks Master Class 2021 Series. The presentation explains the fundamental physics of music and some of the practical aspects of how this applies to playing a musical instrument.

•Ionian Greek Philosopher (traditionally credited with the discovery of the right angle triangle rule)

•Pythagorus may have devised the doctrine of musica universalis, which holds that the planets move according to mathematical equations and thus resonate to produce an inaudible symphony of music.

•Scholars debate whether Pythagoras developed the numerological and musical teachings attributed to him, or if those teachings were developed by his later followers, particularly Philolaus of Croton.

As brass players we all know that our open instruments can play a series of notes from pedal C upwards which is called a harmonic series. This harmonic series are the notes (or frequencies) that naturally resonant in the column of air contained within the tubing of our instrument. The frequency (pitch) is related to the length of tubing (or air column) as this dictates the allowable wave lengths at which the air column will vibrate (called resonance). This is the harmonic series as discovered by Pythagoras c 500 BC.

Pythagoras found that the following ratios made harmonious sounds and today we call these intervals an octave, perfect fifth etc.:

2:1 (octave)

3:2 (perfect fifth)

4:3 (perfect fourth)

5:4 (major third)

6:5 (minor third)

This musical discovery lined up with his theory on the nature of numbers and whole number ratios. He also believed that the orbits of the planets were governed by these ratios, an idea which carried through to the end of the Renaissance, hence the term "Music of the Spheres".

•By using these natural harmonic ratios you can build a scale;

•There are many ways to do this, in this case we will use the 4th, 5th and 6th harmonics to create an 8 note scale beginning with the fourth harmonic C.

•As the harmonic series numbers 1 to 8 correspond to the relative frequencies of the notes we will use these numbers to build our scale as multiples of the frequency of the tonic of the scale (i.e. C, the first note of the scale).

•Then we can obtain the frequency of each note by multiplying the note's ratio number by the frequency we choose for the tonic, say, middle C which is 261.6 cycles per second (Hertz or Hz).

The natural harmonic series discovered by Pythagoras are the natural resonances that will vibrate on a string etc. The frequency of the upper harmonics are all integer multiples of the fundamental frequecy, eg. 2, 3 ,4, 5 ...

So, the first note of the scale (tonic) is C in the middle of the stave. It's relative frequency in the harmonic series is 4 however, as we want to relate every other note in the scale to this note, we want it's value to be 1. We can easily achieve this by dividing by 4. So 4/4=1=C. But now we also have to divide our other notes by 4. So the 5th harmonic E=5/4 and the 6th harmonic G=6/4=3/2. We also know that if we double our tonic we will get it's octave so top C = 2. So our scale so far looks like this (see above).

We will now begin to fill in the blank spots in our scale. We can use the same three harmonics in the series to do this i.e. 4th, 5th and 6th harmonics. The interval between the 4th and 5th harmonic is a major third and the ratio in frequency is 5:4. Therefore if we multiply our G by 5/4 (a major third) we will get a B. 3/2 x 5/4 = 15/8 = B. Similarly, to get the D we will need to go up a perfect fifth from the G. The ratio for a perfect fifth is 6/4 or 3/2 as seen from our harmonic series. So D = 3/2 x 3/2 = 9/4. Now this note is more than an octave above our tonic so we need to lower it by an octave which can be simply done by dividing by 2. 9/4 divided by 2 = 9/8. Our scale now looks like this (see above).

We will now fill in the blanks for F and A using the same technique as for D and B except we will go down from C instead of up from G. Down a minor third from C is A. The ratio for a minor third is 6/5 (5th and 6th harmonics) however this ratio raises the note a minor third. To go down a minor third we use the reciprocal of the this ratio i.e. 5/6. So using our top C in the scale we obtain A = 2 x 5/6 = 10/6 = 5/3. Similarly, F is a perfect fifth below C so F = 2 x 2/3 = 4/3. So now we have our complete 8 note scale.

This particular type of scale is called a Just Intonation scale or temperament. It is not the type of scale/temperament regularly used today which is equal temperament.

If you were to play a scale using Just intonation you would notice that it does not sound the same as a scale in "equal" temperament. The intervals between the notes are slightly different. However, it was one of many temperaments used before equal temperament was widely accepted.

More on Just Intonation: https://en.wikipedia.org/wiki/Just_intonation

A temperament is a system of tuning that specifies the intervals between notes in a scale. Over the last 2500 years many temperaments have been developed starting with Pythagoras' pentatonic scale.

The pentatonic scale can be easily constructed from a cycle of fifths e.g. C, G, D, A, E.

However, as you can imagine, you could not create the complexities of modern music with a five note scale. Equal temperament is now almost universally used in Western music and is probably the most important development in Western music in the last 400 years (equal temperament uses 12 semitones in an octave with the octave being the 13th note).

Western music as we know it today began with unison singing and then simple harmonies (such as a drone note). As music became more complex, different temperaments were developed to enable different parts to be sung or played together.

Temperaments such as Just Intonation, Mean tone, Kellner's Bach, Kirnberger III, Shifted Vallotti/Young, Werkmeister III were all developed to try to ameliorate the fundamental problem posed by the natural harmonic series. This problem is known as the Pythagorean comma (because it was discovered by Pythagoras).

The Pythagorean comma becomes evident when constructing a cycle of fifths using the natural harmonic ratio for a perfect fifth of 3:2;

Thus, in this cycle we must go up in perfect fifths 12 times to get back to the original note (C). As a perfect fifth is 3/2 x the frequency of the previous note, we multiply 3/2 by itself 12 times and obtain 129.7463. This is the relative frequency of the last C to the first C.

By always increasing by a perfect fifth we have raised the pitch by approximately 7 octaves in the process.

So to compare the first and last C in the cycle lets us raise the first C by 7 octaves. We do this by doubling the frequency 7 times. So 1, 2, 4, 8, 16, 32, 64, 128. So you can see that the first C when raised by 7 octaves is a different frequency to the C raised by twelve 5ths (although ostensibly they are both the same note);

This ratio, 129.7563 : 128 is the *Pythagorean comma. i.e. *12 perfect fifths do not equal up to 7 perfect octaves.

This leads to all sorts of problems for any scale or temperament built from the natural harmonic series.

The numbers underneath the scale above represent the relative frequencies of each note to the tonic (i.e. the first C in the scale which equals 1). Instead of comparing each note to the tonic let us look at the relative frequencies between each note in the scale i.e. the intervals.

To calculate the interval between each note we divide the value for the current note (ie the relative frequency value) by the value of the previous note in the scale.

For example, the interval between D and C is (9/8)/1 = 9/8

The interval between E and D is (5/4)/(9/8) = 40/36 = 10/9

and so on.

We can see from this example that for Just Intonation the intervals from C to D and D to E, both ostensibly whole tones, are two different values. The same occurs between F, G, A and B. The two semitones in this temperament, E to F and B to C, are the same value.

This intonation will give you an "in tune" major triad C, E , G and G, B, D and F, A, C.

This presents a problem when you want to play in different keys and/or play together with different instruments pitched in different keys. For instance, in C major for Just intonation the interval between the tonic and the next note is 9/8 however in D major the interval between the tonic and the next note is 10/9 and therefore the D major scale won't sound the same as C major. (different intervals in different places).

Similarly with Pythagorean intonation (constructed from the circle of 5ths) the whole tones and semitones are equal but the major third C to E, G to B and F to A are so out of tune so you cannot play major triads with this intonation).

i.e. Natural major third = 5/4 = 2.25

Major third using Pythagorean intonation above = 1 x 9/8 x 9/8 = 2.2656

Many temperaments were developed over hundreds of years to try and solve this problem, aiming to find a temperament that worked (to lesser or greater extent) for as many keys as possible. It was all a matter of compromise. A certain temperament would sound good in this key but sound horrible in another key. Particularly awful notes/intervals were called "wolf tones" as the dissonance would howl.

Equal Temperament is believed to have been invented in 1639 by the French monk and Mathematician Marin Mersenne. It wasn't until about 1750 however that equal temperament became widely accepted.

Equal temperament is a temperament that allows you to play in any key and with any other instrument. However, it achieves this goal by compromising on the tuning of every note.

For equal temperament the 12 semitones in an octave are divided into 12 equal intervals.

The 12 semitone intervals are calculated as the 12th root of 2 (2 representing an octave or doubling of frequency).

12th root of 2 = 1.05946………

The following table gives the intervals for equal temperament compared with the natural harmonics.

Also try out the different temperaments on the keyboard app below. You will notice the most difference in tuning on the notes E, A and B. (in this app the keyboard is tuned to middle C = 261.63 Hz, not A = 440 Hz. Note the difference this makes to A in Just and Pythagorean temperaments).

All valved brass instruments require some form of compensation to play in tune when the valves are used in combination.

This compensation may be in the form of "lipping" the note in tune, using a trigger or hand to pull a valve slide out, additional valves (to the standard three valves) with slides of differing length or with a built in compensating system (e.g. the Blaikleycompensating system which is used on a lot of brass band tubas and euphoniums).

By depressing a valve we add a length of tubing to the instrument (and hence lengthen the wave length of the standing wave). The length of the valve slide is calculated to be a certain ratio to the total length of the instrument with no valves depressed. e.g. a first valve slide is typically 1/8 of the total length of the instrument, 2nd valve slide = 1/15, 3rd valve slide = 1/5. (Manufacturers may make slight variations in these lengths depending on a particular design).

These additional lengths of tubing will lower the pitch of the instrument by the following intervals when used singularly;

The third valve lowers the pitch of the instrument by a minor third (one and a half tones). So, if you say the instrument without valves depressed is a unit length of 1 or 5/5, adding 1/5 will make it 6/5 in length, the same as Pythagoras' ratio for a minor third.

This is fine for single valve use, however, if you want to lower the pitch another tone you would add the first valve in combination with the third. This means that you need to add another 1/8 to the total length of the instrument to lower it another tone which is fine if the instrument length is 1 however the instrument length is now 6/5 (with the third valve depressed) and therefore the 1st valve slide is not long enough to lower it another whole tone and the resulting note will sound sharp.

You must compensate to keep the note in tune either by lipping it down, using a trigger to increase the length(s) of the valve slides or some other system of compensation.

As the instrument gets longer with the depressing of valves the situation gets worse so, for instance, when depressing all three valves it is advisable to operate your triggers (on cornets etc.) or pull slides on the larger instruments which don't have triggers or other compensating system.

The characteristic tone or timbre of a musical isntrument is due to the complex wave form that it produces.

Any complex wave is made up of a series of pure sine waves.

The frequency of the series of pure sine waves is in line with the harmonic series ie. 1, 2 ,3 4, 5, .... etc times the fundamental frequency of the complex wave.

In music these higher harmonics in a complex wave form are called overtones. It is the relative strength (amplitude) of each overtone that gives an instrument it's charateristic sound.

The complex wave to the left is made up of a series of pure sine waves (such as above) with frequencies equal to 1, 2, 3, 4, 5 ....etc times the fundamental frequency of the complex wave and at varying amplitudes for each harmonic.

The sine waves above when added together form an approximate square wave.

The amplitude and the frequency of each sine wave determine the shape of the square wave.

In this example the relative amplitudes and frequencies are:

1st harmonic: freq = 1, amplitude = 1

2nd harmonic: 2nd and all other even harmonics are not present in a square wave.

3rd harmonic: freq = 3, amplitude = 1/3

5th harmonic: freq = 5, amplitude= 1/5

To make the complex wave form on the right squarer you would keep adding further odd harmonics in the pattern established above.

i.e. amplitude of 1/7th of the 7th harmonic, 1/9th of the 9th harmonic etc.

Any periodic complex waveform can be constructed from the combination of multiple sine waves of various frequencies and amplitudes;

Similarly, any periodic complex waveform can be broken down into it’s component sine waves;

The mathematical technique used to do this is called Fourier Analysis or Fourier Transformation.

The different complex waveforms produced by different instruments determines the characteristic sound/tone/timbre of an instrument. This characteristic sound is due to the harmonics that make up the complex waveform. Try these waveforms as an example.

Whenever two sounds are close in frequency as shown with the yellow and magenta sine waves above, you can often hear the resulting sound pulsating. In music this is called “beats” and is the constructive and destructive interference of the sound waves(see diagram at right).

Counting the number of beats per second will give you the frequency difference between the two sounds;

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