AN INTRODUCTION TO TEMPERAMENTS AND THE CIRCLE OF FIFTHS
In the western world, we tend to take for granted our musical scale, formed of whole tone and half tone steps. These steps are arranged in two ways: the major scale and the minor. In the major scale, we have a series of intervals, using a mixture of whole tone and half tone steps; for example, beginning on middle C we have:
C- -D- -E-F- -G- -A- -B-C
Two dashes indicate a whole tone interval between notes; a single dash: half a tone. In the key of C sharp, we have:
C# --D# - -E# -F# - - G# - - A#- - B# -C#
On a modern piano, F is E# and C is B#. However, this rule did not always apply with some keyboard instruments, as we shall see later. Now let us look at the minor scale(s) of C.
C- -D-Eb- -F- -G-Ab- -Bb- -C (natural minor scale)
C- -D-Eb- -F- -G- -A- -B-C (melodic minor scale)
C- -D-Eb- -F- -G- Ab- - -B-C (harmonic minor scale)
In the harmonic minor scale, there are three half steps ( 1 ½ tones) between Ab and B.
These scales have in fact been in use for the last four hundred years. Prior to this, the popular scales were the dorian and lydian modal scales, often associated with church music. This is all very well you might say, but how is a scale formed, and how are the intervals calculated? How does a piano tuner tune a piano? How does a violinist tune a violin?
To begin at the beginning, we have octaves: two notes sounding the same but one octave apart. If we take middle C at 261.63 cycles per second ( the number of sound waves that pass through the air every second), C one octave higher is exactly twice the frequency: 523.26 cycles per second. An octave above this C is 1046.52 c.p.s. . If we go one octave below middle C, the frequency is 130.815 c.p.s.
Now we can tune two strings or pipes to exactly one octave apart by adjusting one string or pipe until there are no beats. The octaves are then in unison.
Having produced our octave notes, there are many different ways in which we can calculate the different notes of the scale. These different systems are called temperaments.
The Greeks discovered that natural intervals exist; these intervals having existed since creation. These ratios are commonly called: Just Intonation. These are a series of simple, whole number ratios. their values can be demonstrated by setting up a piece of piano wire, secured at each extremity. By introducing a bridge at exactly half its length, the pitch will be one octave higher: the same note but double the frequency. Suppose our wire gives us middle C at 261.63 cycles "beats" per second. If we half its length, the frequency becomes: 261.63 X 2= 523.26 c.p.s., which is C above middle C. The other ratios of the diatonic major scale are produced by applying the bridge to different points along the wires length:
8/9ths of the string raises the pitch one whole tone. D
4/5ths raises the pitch two whole tones: this is called a major 3rd. E
3/4ths raises the pitch 3 ½ tones: perfect 4th. F
2/3rds raises the pitch 4 ½ tones: perfect 5th. G
3/5ths raises the pitch 5 ½ tones: major 6th. A
8/15ths raises the pitch 6 ½ tones: major 7th. B
½ the length raises the pitch 7 tones: octave. c
The resultant pitches are:
The pitches or frequencies are given in cycles per second, otherwise known as Hertz.
Now that we know the intervals for the major scale, we can add the ratios for the chromatic notes, beginning on C#.
15/16ths raises the pitch of our wire a semitone to C#
5/6ths raises the pitch 1 ½ tones to Eb, a minor 3rd.
45/32 raises the pitch 3 whole tones to F#, an augmented 4th.
8/5raises the pitch 4 whole tones to Ab, a diminished 6th.
16/9 raises the pitch 5 whole tones to Bb, a diminished 7th.
The resultant pitches are:
We now have all the notes for the just intonation chromatic scale. But not quite!
The whole tone steps are produced from two different ratios: 9/8 which is a major whole tone, and 10/9 which is a minor whole tone.
A diminished 5th is different from an augmented 4th, although on a piano, F# and Gb are the same. The ratio of a diminished 5th is 64/45; in other words, our string is shortened to 45/64ths of its original length. The frequency is: 372.1 c.p.s., so Gb and F# differ by 4.18 c.p.s.
The remaining ratios are:
7/4 an augmented 6th: A#
9/5 a diminished 7th. Bb
If by now, your head is swimming with ratios and confusing intervals, do not despair! The system of notes and cents makes intervals much easier to understand.
The cent is derived from taking an equal tempered semitone and dividing it by a hundred. 100 cents therefore equal a semitone and 200 cents equal one equally tempered whole tone. 1200 cents equal one octave. Cent values can be employed for any interval in any tuning system.
Now let us look at the cent values of just intonation tuning.
What is clear, is that our octave can be divided into many different intervals; indeed, far more than the twelve semitones to an octave that most musicians and listeners take for granted. If just intonation provides perfect tuning, why is it not the standard method of tuning instruments?
The problem with tuning any fixed pitch instrument to just intonation, is the fact that with only twelve notes to an octave, the intonation is very uncompromising. If for example, we tune a piano to just intonation, based upon C, we can play in perfect intonation within the key of C, and near perfect intonation in a few other keys, although some contain single interval errors. If we stray far from C, to say: Ab major, the intonation sounds appalling, with many notes considerably out of tune to even the most unpractised of ears.
In order to tune a keyboard instrument, in which all 24 major and minor keys (from C major to B minor inclusive) are virtually in perfect tune, requires a minimum of 31 notes to the octave! Our perfect tuning in one key, is obtained at the expense of good or even tolerable tuning in most other keys. In order to keep the keyboard as simple as possible, and to avoid making instruments which are of mammoth size and expense, the decision was taken several centuries ago, to adhere generally to 12 notes to the octave, and to find ways of compromising the tuning, in order to provide several playable keys, which all sounded relatively well in tune.Next page