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What are harmonics?

Harmonics are whole number multiples of the basic pitch of a note. 
Many timbres are made up entirely of harmonics of various strengths.

Sounds of a flute, recorder, or ocarina can be simulated with just two 
 or three harmonics. Other instruments will need a fair number, 
especially, string instruments have many harmonics.

A string player can selectively sound harmonics of a note by playing a 
 note, and lightly touching on the string in various places.

It's also possible to train to hear harmonics.

One can also try singing the partials to a drone, and this is a lot of 
 fun, and anyone can do it. Just play a drone, e.g. on a cello, then try 
 to sing an octave above, an octave plus a fifth and so on - one can use 
 a clip such as this one to give one guide tones to find the notes 
easily.

cello_c2_partials.mid

Here is the 'cello drone without the guide tones:

cello_drone_c2.mid

Even if you can't hear the harmonics in the timbre as such, you will 
very likely find you can sing them to the timbre as a drone, perhaps 
using a guide tone to find them the first few times. This is also a 
great way to learn to sing pure (just intonation) intervals.

One field where the recognition of partials in a timbre has been 
developed to a high degree is the craft of bell making. The more 
general term for a constituent frequency of a timbre is" partial", and 
bells have inharmonic partials - frequencies that aren't constrained to 
 simple multiples or near multiples of the basic frequency.

The 17th century carillioneur (and recorder player) Jacob Van Eyck 
pioneered modern methods of tuning bells. This is done by ear, in 
combination with measuring instruments.

Interestingly, bell tuners also use ratios to target the partials. 
They make some use of the so called sub-harmonic series (the inverses 
of the harmonics) for the first few partials of a bell, favouring the 
ratios 1/12, 1/6 1/5 1/4 1/3.

http://www.oakcroft13.fsnet.co.uk/lehr.htm

This gives the characteristic minor chord type sound of a church bell.



This is because the physics of the bell makes it hard to target a 
major third. Major third bells do exist. Can be by serendipity as for 
this bell: but one can also nowadays make major third bells to order.

Another pattern of partials that concern bell founders are doublets - 
closely spaced pairs of partials caused by a small deviation of the 
bell from perfect symmetry.

Here is a fascinating web site about bell partials, which also has an 
excellent user - friendly program, designed for bells,that one can also 
 use to find the partials in any instrument.

http://www.oakcroft13.fsnet.co.uk/index.htm

The numbers such as 9/8, 6/5 and 5/4 that one sees so often in 
definitions of scales often come straight from the harmonic series. So 
for instance, you get the 5/4 - major third, from the ratio of the 
fifth and fourth multiple of the asic note. The 6/5 is a minor third, 
3/2 is a major fifth and 9'8 is a whole tone.

10/9 is another form of the whole tone. If you play harmonics 8 9 10 
12, you get four of the five notes of the just intonation pentatonic 
scale. 8/8 = 1/1, 9/8, 10/8 = 5/4, and 12/8 = 3/2.

It is perfectly possible to play this fragment scale on the harmonics 
of a string instrument (and on other instruemnts that can be played in 
this way).

The missing note is the major sixth which is a third below the octave, 
 so is at a ratio of 8/5 (i.e. 4/5 of 2/1)

The missing note is the major sixth which is a minor third below the 
octave, so is at a ratio of 5/3 (i.e. 5/6 of 2/1). Note that this is 
alsoa 10/9 whole tone above the 3/2.

You won't find this one in the same fragment of the harmonic series as 
 it would be at a ratio of 40/3 to the fundamental.

However, if you multiply all the numbers by 3, you can then find the 
entire pentatonic scale in the harmonic series as

(8 9 10 12 40/3) times 3 = 24 27 30 36 40

These harmonics are a bit high even for a stringed instrument (can a 
skilled player play these? anyone know).

Some instruments use notes from the harmonic series when played 
normally, notably the natural trumpet of course. It is also possible 
for a flute player to play the harmonic series on a hosepipe by blowing 
 across the end, playing it as an end blown flute.

There is quite a degree of interest also in scales based on inharmonic 
 partials - i.e. using the constituent frequencies of a timbre such as a 
 bell sound to make a scale.

There is also some speculation that the Indonesian gamelan scales may 
have originally been inspired partly in this way. (Ok to say this?)