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 The Harmonic Series
Author: Matt74 
Date:   2020-11-14 04:11
Attachment:  375px-Harmonic_partials_on_strings.svg.png (23k)
Attachment:  pythagoras.jpeg (92k)

Many of you know this, just ignore it.

However, the post about reeds made me realize that not everyone was fortunate enough to take a class on how the harmonic series works. The standard text is by Boethius, but he is reporting on Pythagoras.

Everything here applies to wind instruments, but strings are easier to understand.

1.

Take a chord stretched between two posts and pluck it (C for example).

If the same cord is stretched at the same tension between two different posts, that are only 1/2 as far apart, it will sound an octave higher (C')

If it's 1/3 the length it will sound an octave and a fifth higher (G')

1/4 = two octaves (C")

1/5 = two octaves and a third (E")

1/6 = two octaves and a fifth (G")

The series is: C, C', G', C", E", Ab", C"', D'", E'", Gb/F"', G"', and the intervals keep getting closer.

2.

THE SAME THING HAPPENS if you very lightly touch the first string and then pluck it - at 1/2 it's length, at 1/3 it's length, and so on. The places you touch are called "nodes" and the notes are called "overtones". A standing wave is formed. See picture attached.

3.

The frequency of the notes all have simple numerical relationships in the same proportion as the parts of the string. 1 to 2, or A = 440, A' = 880, etc.

Only the octaves are exactly "in tune" with the piano, because the piano is out of tune with nature. The octave is not divided into 12 equal parts naturally, this is where "historical" temperaments come from. Back in the day you could tune a keyboard so that it sounded great in one key, but if you did this it sounded bad in another key. Dividing the octave into 12 equal parts allows all the keys to sound the same, but it also means that everything but the octaves is little out of tune. (I think this is why pianos are good at mopey film tunes.)

4.

When you pluck the whole string all of the overtones are present in the sound you hear, but the first tone or "fundamental" predominates. When you play the first overtone the fundamental is no longer there, but the higher ones are.

5.

THE SAME THING HAPPENS AGAIN if you buzz into a trumpet, or blow into a flute and overblow, or overblow a reed instrument. The register keys are placed at, or close to a node. Altissimo fingerings leave fingers off of some of these nodes. Depending on the instrument some overtones are stronger than others and this is what gives it its characteristic sound or timbre. Sometimes you can even "split" the tone and hear both the fundamental and the octave, or another overtone, at the same time.

6.

Depending on the instrument, some of these overtones cannot be produced. The clarinet only sounds the odd harmonics because the even ones are cancelled out. Some instruments may not be able to produce the fundamental, or very high overtones, etc. A block of wood may not be able to produce any audible overtones, but other things do. (I used to see a street performer who could play very nice melodies on a gianormus wrench....)

7.

The acoustic design of an instrument can prevent certain overtones from being heard, or can bring others out, but it is the natural vibration of the string or air column itself that produces the overtones. Some resonators (like a guitar body) amplify certain overtones, but they do not create them.

More here https://en.wikipedia.org/wiki/Harmonic_series_(music)

- Matthew Simington


Post Edited (2020-11-14 04:21)

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