Harmony/Structure of Pitch/Just Intonation/Preface

A Guide to Playing in Tune

JUST INTONATION is habitually defined as any system of tuning in which  all of the intervals can be represented by ratios of whole numbers, with a  strongly-implied preference for the smallest numbers compatible with a given  musical purpose.

Within this broad conglomeration of tuning systems we will concentrate on elaborating what we call Functional Tuning, in which the tuning is so intimately associated with the function which a note plays that we are almost faced with a chicken/egg dilemma of deciding which comes first.

Functional Tuning
- is limited to the ratios 2/1 (the octave), 3/2 (the perfect fifth), and 5/4 (the major third),
- will seem the most natural and satisfying to the ear, and
- will apply to music of all styles and periods.

This does not mean that the music of all styles and periods will be tuned in the same fashion,
     but rather that this system of Functional Tuning can tune music of all styles and periods,
          each in its own fashion.

This presentation is divided into 4 sections -

Diagrams in which we study the birth of the World of Pitch.

Tables in which we list the precise positions and distances in a variety of units.

Performance in which, mostly for string instruments,
     we apply this concept of Functional Tuning to performance.

Natural Harmonics in which we examine their pitch.

It might be preferable to see them in this order, but certainly not obligatory.
There will be constant reference from each one to the others.
Performers might wish to start
with section 3 (for string players) or with section 4 (for brass players)
and then refer to the other sections as need occurs.



Functional Tuning is music itself, and cannot be studied without constantly refering to the structure of Melody and  Harmony (for they are not tuned or structured the same way).

Equi-tempered Tuning
We have been living with Equi-tempered Tuning for so long, and now, with electronic MIDI keyboards, so extensively, that we no longer realize how out-of-tune it is.

How else can Functional Tuning be defined but to say -


Wherever a keyboard instrument is not present, in choirs, orchestras, chamber groups, this is the tuning which the ear normally chooses and prefers.


Aim of This Presentation

Our aim is to present Functional Tuning in such a way that it will be clear, comprehensible, applicable, and hopefully, agreeable. When a performer knows what is going on, what notes to tune a little higher, or lower, how much, and why, (s)he is in a far better position to hear the precise pitch better and to develop healthy automatic responses to given conditions. We are well aware of the size of this endeavor and hope that you will assist us with your comments.


We now suggest that you now go to the
of Just Intonation