Re: Draft FAQ on harmonics

Here is a draft of a newbie faq - assumes a bit, but not that much,
and leads naturally into a number of central areas.

Origins was a conversation today about bell partials with a friend who is
not a musician and had visited a bell foundry in France.

$.$.$

What are harmonics, and how do they lead to the ratios notation?

If you sound a single frequency, it is very dull, just a kind of a hum,
like the sound you get when you put a microphone close to its speaker.
So, to make a lively and interesting sound, one adds harmonics - multiples
of the basic frequency of the notes.

The same technique is sometimes used to simulate instrument sounds.

Sounds of a flute, recorder, or ocarina can be simulated
with just two or three harmonics. Other instruments will need
a fair number, for instance if you want to simulate the sound
of any of the members of the violin family.

The other way round, a string player can selectively sound some of the
harmonics of a note by playing it, and then lightly touching on the string
in various places.

It's also possible to train to hear some of the constituent harmonics of
an instrument timbre, perhaps most easily if one can play the instrument
oneself.

One field where this 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, but cab be of any frequency.

Bells are tuned by ear, in combination with measuring instruments,
with the 17th century carillioneur (and recorder player) Jacob Van
Eyck pioneering the field of tuning bells.

Initerestingly, bell tuners also use ratios. 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

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).

You won't find this one in the same fragment of the harmonic series
as it would be at a ratio of 64/5 to the fundamental.

However, if you multiply all the numbers by 5, you can then find the
entire pentatonic scale in the harmonic series as

(8 9 10 12 64/5) times 5
= 40 45 50 60 64

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 to play virtuoso tunes on a hosepipe by sounding
harmonics, and a skilled player can get a most interesting
pure fluty sound in this way.

This simple example introduces some of the ideas that will be
needed for more advanced studies of the harmonic series and
of scales in ratios notation.

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?)

Robert

Correction:

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 also
a 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

Robert