The Development of Musical Tuning Systems

Peter A. Frazer

April 2001

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Contents

0 INTRODUCTION

1 PHYSICAL ACOUSTICS OF TUNING SYSTEMS
1.1 Oscillations, Period and Frequency
1.2 Fundamentals, Pitch, Harmonics and Harmonic Series
1.3 Frequency and Intervals
1.4 Pitch Class and Octaves
1.5 Transposing by an Interval
1.6 Simple and Compound Intervals
1.7 The 3rd Harmonic and the Interval of a Fifth
1.8 Inversion of an Interval and the Interval of a Fourth
1.9 The 5th Harmonic and the Intervals of a Third
1.10 Consonance, Dissonance and Harmonic Intervals

2 ANCIENT GREEK ORIGINS OF THE WESTERN MUSICAL SCALE
2.1 Proportion and Harmony
2.2 Tetrachord
2.3 Diatonic Division of an Octave
2.4 Greater Perfect System
2.5 Ancient Greek Modes
2.6 Intervals in the Greater Perfect System
2.7 Pythagorean Tuning
2.8 Ptolemy
2.9 Pentatonic Scale

3 MEDIEVAL THEORY AND PRACTICE
3.1 The Greek Heritage
3.2 Medieval Pythagorean Tuning
3.3 Modes
3.4 Guido of Arezzo
3.5 Henricus Glareanus
3.6 Pythagorean Tuning of the Diatonic Major Scale

4 TUNING INTO THE RENAISSANCE
4.1 The Emergence of Polyphony
4.2 Chromatic Scale
4.3 Just Intonation
4.4 Chromatic Just Tuning
4.5 Arithmetic and Harmonic Means
4.6 The Common Chord and Sestina
4.7 Subharmonics

5 TEMPERAMENT
5.1 Full Circle of Fifths
5.2 Mean Tone Temperament
5.3 Cents
5.4 Some Common Tuning Intervals Measured in Cents
5.5 Analysis of Mean Tone Temperament
5.6 The Common Chord and Key Modulation
5.7 Diatonic Intervals
5.8 Well Temperament
5.9 Equal Temperament
5.10 Numerical Comparison of Tuning Systems

REFERENCES AND BIBLIOGRAPHY

Tables

Greater Perfect System
Ancient Greek Modes
Intervals of the Greater Perfect System
Descending and Ascending Fifths
Intervals of Pythagorean Tuning - Ancient Greek Phrygian Mode (Descending)
Pentatonic Scale
Intervals of the Pentatonic Scale

Series of Fifths
Sorted Series of Fifths
Medieval Modes
Extended Modes
Diatonic Major Scale - Pythagorean Tuning
Analysis of Pythagorean Tuning

Pythagorean Diatonic Tuning with Added Bb and F#
Early Pythagorean Chromatic Tuning (Eb x G#)
Late Pythagorean Chromatic Tuning (F# x B)
Just Tuning - Zarlino
Analysis of Just Diatonic Tuning
Chromatic Just Tuning
Analysis of Just Chromatic Tuning

Extended Series of Fifths
Sorted Series of Fifths
Quarter Comma Mean Tone Temperament
Chromatic Quarter Comma Mean Tone Temperament
Common Tuning Intervals
Analysis of Quarter Comma Mean Tone Temperament
Introduction of Sharps
Introduction of Flats
Intervals and their Inversions
Andreas Werckmeister Temperament III
Analysis of Andreas Werckmeister Temperament III
Equal Temperament
Numerical Comparison of Tuning Systems

0 INTRODUCTION

When I was learning elementary music theory my teacher told me that the intervals of the diatonic major scale go "tone, tone, semitone, tone, tone, tone semitone" and I asked the question "Why?".  With the benefit of hind sight I am grateful that I did not receive a satisfactory answer.  My subsequent search for the origins of the diatonic scale has led to a life long interest in the fascinating subject of musical scale structures and tuning systems.

I returned to the subject in 1984 when I was using a Sinclair ZX81 microcomputer and a Memotech 64K RAM pack together with some custom electronics to make an echo-harmonizer for electric guitar.  My problem at that time was figuring out why the device I had constructed did not transpose all intervals with the accuracy I wanted to hear.  That prompted the question of what exactly were the intervals I wanted to hear.  My search led me to Zarlino's system of just intonation.  Although I was not aware of Zarlino's progression of harmonic numbers in the range 90 to 180 I discovered that all the intervals of his just intonation could all be expressed as exact numbers of 360ths, the number of degrees in a circle.  Seen as angles on a circle the intervals of just intonation provided all the points required to construct regular 3, 4, 5 and 6 sided figures.  At the time this seemed to be of mystical significance, perhaps in much the way that the initial discovery of proportion in musical consonance struck Pythagoras or the discovery of both arithmetic and harmonic means in musical intervals struck Zarlino.  No doubt anyone who delves into the subject sufficiently deeply will be rewarded with such apparent insights even though, like the Madlebrot Set, they arise purely from the properties of numbers.

Again I returned to the subject in 1996 when as a mature student I had the opportunity to complete a masters degree in software engineering.  For my thesis I undertook the construction of a mathematical model of musical scale structure and tuning systems using the formalisms of software engineering mathematics.  Subsequently I pursued this further by constructing a computer program which made it possible to compare the various tunings.  Initially the software played only pure tones so I extended it first to include an additive harmonic oscillator working something like a drawbar organ and then added dual AM and FM synthesis with four independent drawbar operators each having their own envelope.  The result is the Midicode Synthesizer presented on the home page of this web site.

In this essay on tuning systems I seek to share what I have learnt of the subject.  I trace the development of western tuning systems from their origins in ancient Greece and Babylon up to the development of equal temperament.  I have not ventured into modern tuning systems using the seventh and higher harmonics nor into division of the tonal spectrum other than into octaves of 12 semitones (though my software supports such features).

There are many other excellent web sites on the subject of microtonal tuning systems and I have included links to just some of these on my links page.  It is often the case that a different form of words can clarify some aspect of this sometimes complex subject and I hope that my contribution may help.  I have included some elementary theory of both music and physical acoustics so that no prior knowledge is assumed.  As a teacher, I have tried to make the subject as accessible as possible to students and those new to the subject.

My approach is from a mathematical perspective and I have given derivations for several of the major tuning systems.  Wherever possible I present information in tabular form and have included tabular analysis of all possible diatonic and chromatic intervals for some of the classic tunings.

I hope to extend this web site in the future.  In the mean time I have other partially complete material and unreleased software for analysing tuning systems.  If you do not find what you are looking for, or wish to point out any errors, please email me.