First edition April 2001

Revised and extended second edition April 2015

Copyright © 2011, 2015 Peter A. Frazer.

In most web browsers you can **listen** to the intervals and tunings described here. Wherever there is slight or distinct colour in a table you can click on the cell to hear a note or notes. Right click (two finger tap on a MacBook, touch and hold on Android, etc.) for an alternative. For intervals the alternate click plays a chord whilst the primary click plays a note sequence. For single notes the alternate click may render the note in a different octave and for note sequences the sequence is reversed. Try it!

This feature is known to work in Firefox and Chrome on the PC, MacBooks, and Android on a Samsung Galaxy. Unfortunately, due to a lesser degree of audio support, it does not work in Internet Explorer, Safari on a PC or on older iPads. You will need good quality headphones or speakers; all of the examples use pure tones and some use low frequencies so you may not hear it all on laptop speakers or inner ear headphones.

Advanced readers may like to know that the reference pitch for each page can be changed. By default the reference pitch is A = 440 Hz but this can be changed by appending to the end of the URL in the address bar, for example `?A=432`

or `?C=256`

. Where a note other than A is used the offset from A will be calculated using equal temperament. You need to do this for each page.

Technical specialists may wish to turn on the browser debug log to see the actual frequencies of notes being synthesized.

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

Appendix A: COLLECTION OF TUNING ANALYSIS TABLES

Appendix B: COLLECTION OF EXPLANATORY TABLES

Appendix C: VISUALIZATION OF TUNING INTERVALS AS LISSAJOUS FIGURES

T 1.1.1 A pure tone with a frequency of 110 cycles per second

T 1.1.2 Beating sound between two similar frequencies

T 1.2.1 A series of harmonically related tones

T 1.3.1 Perception of pitch intervals is related to ratios between frequencies

T 1.3.2 The interval of an octave

T 1.4.1 Notes of the same pitch class

T 1.5.1 Transposing up an octave

T 1.5.2 Transposing down an octave

T 1.7.1 Derivation of a fifth from the third harmonic

T 1.8.1 Subtracting a fifth from an octave to yield a fourth

T 1.8.2 A fourth exists between the 3rd and 4th harmonics

T 1.9.1 A major third exists between the 4th and 5th harmonics

T 1.9.2 A minor third exists between the 5th and 6th harmonic

T 1.9.3 A major third and a minor third constitute a fifth

T 2.3.1 Interpretation of detail from *School of Athens*

T 2.4.1 Greater Perfect System

T 2.5.1 Ancient Greek Modes

T 2.6.1 Intervals of the Greater Perfect System

T 2.7.1 Descending and Ascending Fifths

T 2.7.2 Sorted Descending and Ascending Fifths

T 2.7.3 Intervals of Pythagorean Tuning - Ancient Greek Phrygian Mode (Descending)

T 2.9.1 Pentatonic Scale

T 2.9.2 Intervals of the Pentatonic Scale

T 3.2.1 Series of Fifths

T 3.2.2 Sorted Series of Fifths

T 3.3.1 Medieval Modes

T 3.5.1 Extended Modes

T 3.6.1 Diatonic Major Scale - Pythagorean Tuning

T 3.6.2 Analysis of Pythagorean Tuning

T 4.2.1 Pythagorean Diatonic Tuning with Added Bb and F#

T 4.2.2 Early Pythagorean Chromatic Tuning (Eb x G#)

T 4.2.3 Analysis ofEarly Pythagorean Chromatic Tuning (Eb x G#)

T 4.2.4 Late Pythagorean Chromatic Tuning (F# x B)

T 4.2.5 Analysis of Late Pythagorean Chromatic Tuning (F# x B)

T 4.3.1 Just Tuning - Zarlino

T 4.3.2 Analysis of Just Diatonic Tuning

T 4.4.1 Chromatic Just Tuning

T 4.4.2 Analysis of Just Chromatic Tuning

T 4.6.1 Sestina

T 5.1.1 Extended Series of Fifths

T 5.1.2 Sorted Series of Fifths

T 5.2.1 Quarter Comma Mean Tone Temperament

T 5.2.2 Chromatic Quarter Comma Mean Tone Temperament

T 5.4.1 Common Tuning Intervals

T 5.5.1 Analysis of Quarter Comma Mean Tone Temperament

T 5.6.1 Introduction of Sharps

T 5.6.2 Introduction of Flats

T 5.7.1 Intervals and their Inversions

T 5.8.1 Andreas Werckmeister Temperament III

T 5.8.2 Analysis of Andreas Werckmeister Temperament III

T 5.9.1 Equal Temperament

T 5.9.2 Analysis of Equal Temperament

T 5.10.1 Numerical Comparison of Tuning Systems

F 0.1.1 Echo-harmonizer unit for electric guitar using ZX81 microcomputer and custom electronics.

F 2.3.1 Detail from Raphael's *School of Athens*

F 2.6.1 Frontispiece to *Theorica Musice*, Franchino Gafurio, 1492

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.

F 0.1.1 Echo-harmonizer unit for electric guitar using ZX81 microcomputer and custom electronics.

Made by Brian T. Sanderson and Peter A. Frazer, 1984.

Phil Riddell (electric guitar) experiments with the ZX81 based effects unit, 1984.

(Phil Riddell and Brian Sanderson speaking)

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 was the **Midicode Synthesizer** a Windows XP program available on the internet for over a decade with a few loyal users.

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.

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 references 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 contact me.