Harmonics & Beats are better with JI

In the thread "Re: Equal divisions -- of what?" it was pointed out that
there are equal linear steps in the JI scale, specifically...

JI scale 24 27 30 32 36 40 45 48
Step 3 3 2 4 4 5 3

so there are several places where equal steps occur and more still when
larger steps are used. Indeed the linear series 1 2 3 4 5 6 (7) 8 9 10
can be fully formed in the JI scale [(7) needs a special blues flat].

This is something from my WWW site which investigates the steps between
notes which are also of course the beats...

Harmonics & Beats are better with Just Intonation.

One of the really nice things about the just intonation scale is that
the beats between notes are in the scale, whereas for 12T equitempered
tuning they miss by quite a lot. I am using "beats" here a bit loosely
because sometimes these beats are in the range of frequencies of notes,
not the wa-wa-wa sound we associate with beats.

First the equitempered scale to show why it doesn't work in this way:

note do re mi fa so la ti do
Freq. 1.0000 1.1225 1.2599 1.3348 1.4983 1.6818 1.8877 2.0000
Beats 0.1225 0.1374 0.0749 0.1625 0.1835 0.2049 0.1123
Beats*8 0.980 1.099 1.198* 1.300 1.468 1.639 1.797*
[* means Beats*16]
The reason for *8 or *16 is to move the note up 3 or 4 octaves to see
where it falls in the scale.

In this case the beats are not generally in tune with the key. When
beats are considered for notes that are 2 or more apart in the scale the
situation is even worse. The same will apply to 19TET etc.

Now the just intonation scale

note do re mi fa so la ti do
Freq. 1.0000 1.1250 1.2500 1.3333 1.5000 1.6667 1.8750 2.0000
Beats 0.1250 0.1250 0.0833 0.1667 0.1667 0.2083 0.1250
Beats*8 1.000 1.000 1.333* 1.333 1.333 1.667 1.000
Note do do fa fa fa la do

In this case all the beats are perfectly in tune with the scale itself.
When notes 2 or more apart are considered then the situation is still
very good. This is more easily expressed by using the ratios of
frequencies as shown below:

note do re mi fa so la ti do re mi fa so la ti
Ratio 24 27 30 32 36 40 45 48 54 60 64 72 80 90
Beats-1 3 3 2 4 4 5 3 6 6 4 8 8 10
Beats-2 6 5 6 8 9 8 9 12 10 12 16 18
Beats-3 8 9 10 13 12 14 15 16 18 20 26
Beats-4 12 13 15 16 18 20 19 24 26 30
Beats-5 16 18 18 22 24 24 27 32 36
Beats-6 21 21 24 28 28 32 35 42
Beats-7 24 27 30 32 36 40 45

What is shown is the relative beat frequency between notes, for example
in the "Beats-3" line the 10 is the beats (or difference) between mi=30
and la=40. The figures are always up the diagonals to the top line.

Now what is noticeable is that in most cases the beats are actually
perfectly in the scale as follows:

note do re mi fa so la ti do re mi fa so la ti
Beats-1 do do fa fa fa la do ...
Beats-2 do la do fa so fa so ...
Beats-3 fa so la do# do mib mi ...
Beats-4 do do# mi fa so la so# ...
Beats-5 fa so so la# do do re ...
Beats-6 tib tib do mib mib fa sob ...

Where the ... means that the cycle repeats for the next octave. All the
notes are perfectly in the key except the ones labelled: do# mib sob so#
la# tib. These all fall in the places where there are black notes and
the "b" (flat) ones are correct in my view as to what the flat
frequencies should be. The "#" (sharp) ones are unusual ratios but then
again there is no real meaning to these notes in the key.

They do have the unusual ratios including 11, 13 and 19. Some of these
are produced by the cases like fa-ti (careful how you pronounce that one
:-) which is a dreadful chord anyway, and others by the cases like re-la
which is our old friend D-A when in C, that is 27-40, which wants to be
2-3.

I would suggest that it would be possible to make some tempered tunings
which are not too far from equitempered tunings, but which have beats
that are reasonably well in tune. Has anyone ever investigated this?
It might be a fruitful exercise.

-- Ray Tomes -- rtomes@kcbbs.gen.nz -- Harmonics Theory --
http://www.kcbbs.gen.nz/users/rtomes/rt-home.htm

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