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 "die-hards", 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 |

Info. on Meantone tuning

Info on Pythagorean tuning

Info. on Equal temperaments

If you are still convinced that Just Intonation could be useful for
your musical endeavours; here are some links to more information on other
microtuning and associated sites so that you can see, hear, and draw your
own conclusions.
**Bookmark here first**; and come back soon!.

Links out of LucyTuning to other websites about microtuning and other
views on tuning.

Interesting
ideas on timbre and tuning from Sethares
**(recommended)**

American Festival
of Microtonality

Southeast Just Intonation
Center

FTP to Scala - a tuning program

RealTimeTuner

Bill Alves - JI information

Justonic commercial JI tuning
program

- bye-bye!