
Just
Intonation and Whole Number Frequency Ratio Systems. 
Just Intonation is a tuning system which operates on the
assumption that musical harmonics only occur at frequencies
which are at small integer ratios to the fundamental pitch.
The ratios selected for Just Intonation seem to aim at producing
"beatless" music; which to my mind is a futile, a folly. see
Pitch, Pi..... Chapter One.

The diagram below
shows where some of these ratios occur in the first octave.
(i.e. 0  1200 cents)
The pattern shown begins with 3/2 at 702 cents, and ascends
via the series 4/3, 5/3, 5/4, 6/5,...... .
The numbers 0  1200 in 100 cent increments are the positions
used in 12tET as a reference. Remember that the graph represents
cents. It is on a log scale, (like a fretboard, or slide rule),
so that the midpoint in frequency (pitch*1.5 = 702 cents) is
at the 58.5% (7.02/1200) position across the graph.
The positions shown in blue are the result of the integer ratios
using all integers up to and including 8 for the divisor.
The positions shown in white use the integers from 9 to 16 as
the divisor. 

You may notice some
significant and interesting patterns in this graph:
1. As the integers increase, many "new" values will fall at
ratios which were previously hit. Eg. 6/4 = 3/2 etc.
2. There is an absence of "hits" close to values and their multiples,
which are frequently hit. Notice the gaps around 3/2 (702 cents),
5/3 (884 cents) ....... These are the ratios which are believed
to be particularly significant in Just Intonation. If you are
looking for a pitch as the Vth (i.e. around 700 cents) for your
music; there is only one choice.
3. There are no hits near the ends of the graph, as to hit these
locations requires large divisors.
4. The hits each side of 702 cents are "mirror images".
To compare graph above to first 44 steps of fourth and fifths
using LucyTuning. 

Many advocates of
Just Intonation claim that lower integer ratios produce more
consonant intervals, than higher integer ratios. The exact size
of intervals is determined by their position in a scale. Eg.
from C to D (One Large interval (L) in LucyTuning), may be of
different sizes in Just Intonation, dependent upon what note
is considered to be the tonic of the scale in which it is used.
This can result in what is know as in JI circles, "wandering
tonics"., and creates retuning requirements during modulation
and transposition. I consider JI to be a simplistic, paradoxical,
naive, single dimensional and static mapping system for tuning,
although many "diehards", are currently attempting to resuscitate
it. 
Four
fourths plus one third: should they equal two octaves? 
Note 
Position 
Just
Intonation

Pythagorean
Tuning

Interval
L & s to next

C 
I 
4/3 
4/3 
2L+s 
F 
IV 
4/3 
4/3 
2L+s 
Bb 
bVII 
4/3 
4/3 
2L+s 
Eb 
bX 
4/3 
4/3 
2L+s 
Ab 
bXIV 
5/4 
81/64 
2L 
C 
XVI 
Product
is
(256/81)*(5/4)=

Product
is
(256/81)*(81/64)=

__________ 
Total
Ratio

2 Octaves
4.0

Less
than 2 Octaves
3.9506173

2 Octaves
4.0

10L+4s
4.0

Note 
Position 
Just
Intonation

Pythagorean
Tuning

Interval
L & s



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