The Mathematics of Tuning and Temperament
With audio examples

Contents

1.   Introduction

2.   General Mathematical Basis

3.   The Principle Tuning Systems

3.1  Pythagorean Tuning

3.2  Ptolemaic Tuning: "just intonation"

3.3  Mean-tone Temperament

3.4  Equal Temperament

4.   Harmonic Comparison

4.1  Perfect 5ths

4.2  Major 3rds

4.3  Major 6ths

5.   Summary of sample sounds

1. Introduction

Almost all Western music is based upon the diatonic scale, having seven notes within an octave. The octave is the fundamental interval for which the notes are related in frequency exactly by the ratio of 2:1. This simple ratio means that to the ear they sound as if they are almost "the same note". Virtually all modern Western music divides the octave into 12 equal semi-tones on which the music is based. The system is a compromise, and the octave has not always been divided this way. Over the ages there have been four principle tuning systems in wide usage for Western music. This paper takes a look at these four systems with illustrative audio clips.

This description of the mathematical basis is very abridged. It is not intended to be a definitve work on the subject, but rather a practical illustration of the various aspects of different systems. Various audio samples are included to illustrate a few of the key points. These are encoded as MPEG Layer 3 audio at 16kbps. For more detailed information on the subject the reader is referred to the many excellent references, historical, musical and mathermatical, that may be found.

2. General Mathematical Basis

For pure monophonic (one note at a time) musical melody, it does not matter a great deal what scale or tuning method is used. However, could you imagine modern music with only one note being played at a time? Almost all modern music is polyphonic (many notes sounding simultaneously). For the music to sound consonant (in tune) the different notes being sounded simultaneously need to be related in a way which is pleasing to the ear. The relationship between the frequencies of notes of the important musical intervals is what makes them sound pleasing. The important ratios are as follows: All four of the main tuning systems considered here were devised around some or most of these intervals.

3. The Principle Tuning Systems

NOTE: The values given as "cents" represent the intervals where the octave is divided into 1200 cents. This is the modern way of representing pitch in such a way that each semi-tone comprises 100 cents.

3.1 Pythagorean Tuning

Western music is generally considered to have started from Pythagoras, the ancient Greek. Pythagoras devised a system based on mathematical principles. He defined the scale around the ratios of the fifth, being in the ratio 3:2 exactly, and the fourth being 4:3 exactly. The difference between these two was then 9:8, which he defined as the tone, or whole step. He then divided the octave so that there were the seven notes, just as we have today, but to get the mathematics to add up he was left with two semi-tones which he defined as 256:243.

Thus the Pythagorean scale has the following intervals: 

Cumulative Intervals:    1      9:8    81:64    4:3     3:2    27:16  243:128    2

Note:                    C       D       E       F       G       A       B       C 

Incremental Intervals:      9:8     9:8   256:243   9:8     9:8     9:8   256:243

Cents:                      204     204     90      204     204     204     90 
It is interesting to note that Pythagoras did not recognise the major third, which is distictly sharp at 81:64 compared with the ideal of 5:4.

The chromatic Pythagorean scale is formed by inserting semitones equal to 114 cents in such a way as to keep all perfect fifths true, except for the interval G#-Eb, which needs to be adjusted so that the intervals add up mathematically. This difference is known as the "comma of Didymus".

3.2 Ptolemaic Tuning

Ptolemaic Tuning is generally referred to as "just intonation". To create perfect major third intervals, this system alters one of the fifths, D-A. (And thus the major sixths are also perfect, except for F-D.) This makes the triad D-F-A quite unusable, although the others are perfectly in tune. The Just Intonation scale employs two different sized tones in the ratios 9:8 and 10:9, and thus it can hardly be considered satisfactory even for purely melodic music. 
Cumulative Intervals:    1      9:8     5:4     4:3     3:2     5:3    15:8      2

Note:                    C       D       E       F       G       A       B       C 

Incremental Intervals:      9:8    10:9    16:15    9:8    10:9     9:8    16:15

Cents:                      204     182    112      204     182     204    112 
The above comments hold true for scales and intervals based solely on the "white notes". Adding the "black notes" to give a full chromatic Just Intonation scale creates more perfect fifths, but major thirds and sixths which are not true. In fact the result is three different sizes of semi-tones. (16:15, 135:128 and 256:243.) It is thought that this system, although considerably debated, was not used much.

3.3 Mean-tone Temperament

In this scheme the major thirds are made exact. This results in the fifths becoming slightly flattened but in such a way that the error of the Ptolemaic system is spread out over four fifths. This reduces the dissonance and makes the fifths more acceptable. The whole tones are also all equal in size, being half the major third. Melodically the Mean-tone scale is more acceptable than the Ptolemaic scale. 
Cumulative Intervals:    1              5:4                                      2

Note:                    C       D       E       F       G       A       B       C 

Cents:                      193     193    117      193     193     193    117 
The chromatic Mean-tone scale has semi-tones of two very different sizes: wide 117 cent semi-tones in the diatonic scale, with 76 cent semi-tones balancing the whole-tone interval of 193 cets. Mean-tone termperament was designed for keyboard instruments and it was an acceptable compromise as long as the "black notes" beyond Eb or G# were not used. The G#:Eb fifth was so bad as to be unusable - it was often given the name "the wolf".

3.4 Equal Temperament

In equal temperament the octave is divided into 12 equal semi-tones each of 100 cents. The intervals are then built from the semi-tones. For example a fifth is 7 semi-tones and a third 5. The Equal temperament scale is universally used today in Western music. 
Cumulative Intervals:    1                                                       2

Note:                    C       D       E       F       G       A       B       C 

Cents:                      200     200    100      200     200     200    100 
The chromatic Equal temperament scale with all semi-tones = 100 cents means that perfectly tuned intervals have been totally eliminated. However, the mistuning on fifths is only 2 cents and on thirds 14 cents which the ear does not appear to mind. The big advantage is that all keys are equally usable.

4. Harmonic Comparison

The differences between the tuning systems are most evident in polyphonic music, particularly when chords and intervals incorporating the "black notes" are used. To illustrate this samples of the basic two-note chords for some of the 5th, 3rd and 6th intervals have been constructed. Each of the audio samples consists of a sequence of the chord based on: Pythagorean, Just intonation, Mean-tone temperament and Equal intonation played directly one after the other.

4.1 Perfect Fifths

The basic diatonic 5th interval is the Perfect 5th C:G. The deviation from perfect tuning is 0, 0, -5, -2 cents respectively.

In all systems the 5ths are perfectly tuned, or very nearly so, with the exception of two intervals. The Perfect 5th D:A is -22 cents in just intonation, and the enharmonic fifth 5th G#:Eb (the Wolf) is -24, -2, +35, -2 cents out respectively.

4.2 Major Thirds

The basic diatonic 3rd interval is the Major 3rd C:E. The deviation from perfect tuning is +22, 0, 0, +14 cents respectively. This illustrates the fact that the Major third was not recognised in Pythagorean tuning.

Apart from the three Major third intervals contained within the diatonic scale, all other thirds become sharp in Just intonation, and some of the enharmonic intervals become unusable in Mean-tone tuning as well. An example of this is the Major 3rd C#:F which has mis-tuning errors of -2, +20, +42, +14 cents respectively.

4.3 Major Sixths

The basic diatonic 6th interval is the Major 6th C:A. The deviation from perfect tuning is +22, 0, +6, +16 cents respectively.

In the Just intonation system all sixths incorporating one or more of the notes not in the diatonic scale and the sixth F:D become sharp by 22 cents. Some of the enharmonic intervals become unusable in Mean-tone temperament. An example is the Major 6th C#:Bb which has tuning errors of -2, +20, +49, +16 cents respectively.

5. Summary of sample sounds

Here is a complete list of audio clips available:
Pythagorean scale
Just intonation scale
Mean-tone scale
Equal temperament scale
Chromatic Pythagorean scale
Chromatic Just intonation scale
Chromatic Mean-tone scale
Chromatic Equal temperament scale
Perfect fifth C:G
Perfect fifth D:A
Perfect fifth G#:Eb
Major third C:E
Major third C#:F
Major sixth C:A
Major sixth C#:Bb
Copyright © 1996 -1998 David Bartlett
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