First, intervals which we consider concordant (pleasant-sounding) have simpler ratios than discordant (harsh-sounding) ones; generally, the higher the number, the more discordant the interval and second, all these ratios are numbers which are multiples of the prime numbers two, three and five: you may also notice that these ratios are identical to certain members of the harmonic series. Right, we've got nice simple numbers describing these intervals so what's the problem? The problem is that the interval ratios between successive degrees of the scale are not equal. This means that if you play an interval in one key, it'll sound different in another. Take a fifth, for example. Play the interval C-G in this puretone temperament and it'll sound perfectly OK but play F#-C# and the interval ratio, instead of being 3:2 is now 1024:675 or almost a fifth of a semitone too sharp (divide 45:32, the augmented fourth from C, by 32:15, the Minor Second raised one octave). Now since the ear can just about hear a difference of one hundredth of a semitone, playing this interval is going to sound distinctly odd.

So, although the puretone temperament has the benefit of greater overall concordancy, because its primary interval ratios (3:2, 4:3 etc.) lie alongside those some of the harmonic series, it will only work in one key. This limits the possibilities of harmony, an essential component of Western music of the last 400 years, thus many attempts were made by various people over the years to "re-tune" or temper the interval ratios such that as many keys as possible sounded approximately the same.

We owe the basis for our present musical system to the ancient Greeks. Some 2500 years ago, Pythagoras (he of the right-angled triangles!) realised that simple intervals could be formed by dividing a stretched string such that the divisions were in whole-number ratios; divide it in half and you get an octave; divide by a ratio of 3:2 and you get a fifth and so on. He asked a very important question, "Why is consonance determined by the ratio of small whole numbers?", meaning that as the ratio numbers get larger, the interval gets more dissonant. Now at that time, the Greeks were into the natural sciences; they were probably amongst the first people to study natural phenomena and to seek an explanation of the way the universe worked through empirical observations and deductions, but they were also noted for their aesthetic appreciation of the arts. In particular, the epic poems of Aristophanes, Homer and others were probably performed by "speaking on key" rather than mere reading. Often, this would be accompanied by musical interjections and underscoring on instruments such as the lyre or aulos. A common instrument from those times was the tetrachord, a kind of four-stringed harp. These essentially open-tuned instruments spanned a fourth on the outer pair of strings with the inner pair tuned to intervals ranging from quarter tones to two tones. Pythagoras spent much time experimenting with different ways of tuning these instruments in an attempt to formalise the procedure. One of his major contributions to music was extending the range of the scale by using two tetrachords tuned a fifth apart. This then gave a range of an octave and, consequently, a much wider variation of scales became possible which were to become known as Modes.[2] Here they are mapped to our present system:

If these are remapped into the key of C we get:

Since these scales now spanned an octave, some method was needed to tune them in a consistent fashion. As I said before, Pythagoras noted that dividing a string into half and then half again produced a succession of higher and higher octaves; not really much use as the basis for a musical scale. His temperament is based, therefore, on a continuous projection of the pure fifth which, as you recall, is a ratio of 3:2. Starting with the pure fifth C-G, add the pure fifth above, D; drop this an octave.

Now our scale has C, D and G in it - OK, go back to the high D and go up another pure fifth to A; drop this an octave, go up another pure fifth, drop that appropriately and continue in like fashion until you arrive at C again (this is after 12 iterations encompassing a span of seven octaves). You should now have a chromatic scale of twelve notes with successive intervals limited to tones or half-tones and everything all bundled up nicely - or do we?

It is perhaps unfortunate for us that Pythagoras didn't proceed to divide the string into fifths, sevenths and so on otherwise music might have taken a very different course. The problem stems from Pythagoras's use of the pure fifth as the sole basis for calculating the scale. To demon- strate this, we're going to have to take a look at a device called a cent (groan! Not more maths?). Put simply, a cent is one hundredth part of an equal semitone; with twelve semitones in an octave, this gives us 1200 cents in an octave. The cent is more useful than ratios because ratios only show the order of magnitude of the interval, not the relative size.

Here we go, in at the deep end: Cents may be calculated from ratios in the following fashion:

                C     D     E     F     G     A     B     C

Pythagoras       0    204   408   498   702   906   1110  1200

Aristoxenus      0    204   386   498   702   906   1088  1200

This has a better Major 3rd and Major 7th and is closer to the puretone temperament in that some of the the interval ratios in this scale are multiples of the prime numbers 2, 3 and 5 (the so-called 5-limit of primes); the only problem now is that the tones are of different sizes and are known historically as Major and Minor Tones. Even before Aristo- xenus, another theoretician called Archytas had proposed the use of 5:4 and 8:7 ratios as consonant intervals thus advocating the use of ratios within the 7-limit of primes.

Various other people had their two cents worth, so to speak, in order to find ways of making ensemble playing and multi-part music in general easier. In particular, it had been noted that people had a natural tendency to sing intervals that were closer to being pure rather than the intervals within the 3-limit, 2:1 and 3:2. Also, there were considerable difficulties in fitting these intervals within the limits of a keyboard and make them serve the requirements of the musical form. One major contribution to the theories of temperament was Giuseppe Zarlino, the MaŒtre de Chapelle of St. Marks in Venice, who in 1560 proposed inverting the Major and Minor tones of the upper group in Aristoxenus's scale to relieve the monotony of having two identically tuned halves of the scale. This gives us the following intervals:

        C     D     E     F     G     A     B     C

Zarlino  0    204   386   498   702   884   1088  1200

This is, effectively, the puretone scaling. He then suggested reducing every fifth by two sevenths of a comma in order to lose the comma amongst the rest of the intervals. This method of redistributing commas was further refined by Francis Salinas, a blind musician and Professor of Music in Naples, among others. Here we have the start of Meantone temperament, the purpose of which is to redistribute the intervals such that the principle ones, such as fourths and fifths remain fairly true and others retuned so as to still fulfil their function within the scale. The most common way of achieving this is to flatten the first four fifths C-G, G-D, D-A, A-E reducing them enough to produce a true third - this also has the effect of removing the difference between the Major and Minor tones producing a "Mean" whole tone between the two, hence the name. Thus the Major thirds and the Minor sixths are true whereas the fifths are a little flat. The problem with this is that only keys that have few accidentals sound OK, others less so. This allows the use of the first six major keys in the cycle of fifths and the first three minor and also allows a certain degree of modulation from key to key.

One of the most problematical intervals is the fifth G#-D#. It is way too sharp and its inversion too flat - this is known historically as the Wolf Tone, so called because the mistuning was reminiscent of the howling of wolves - there are other wolves but this one is the most disturbing. The Wolf Tones are probably the main reason that composers of the period avoided using keys with a large number of accidentals. For example, Mozart rarely, if ever, composed any works in Db, F#, Ab and B Major or C#, Eb, F, F# and G# Minor as these keys make wolf tones stick out like a sore thumb. Curiously he also avoided B Minor which is all the more odd when you consider that his favourite key was D Major, closely followed by C and Bb Major. [5]

It is worth remembering at this point that the main reason for all this tomfoolery is the burgeoning development of the keyboard. Because there is a physical limitation on the number of keys which can be used to play notes, some means had to be found to permit the tonalities demanded by developments in polyphonic music to work within this limitation. [6] Incidentally, a good orchestra will effectively play in puretone tempera- ment but will constantly adjust its intonation so as to achieve the most concordant sound (remember that concordancy or discordancy is judged solely by its perceived effect on the ear) but with the pianoforte [7] becoming a more dominant force in Western composition and composers seeking to explore this new tonal palette, some means had to be found to facilitate their requirements.

Both Zarlino and Salinas knew about equal temperament but disliked the severe mistuning inherent in the thirds and sixths but, like the moving hand of Omar, progress moves ever on. A French monk called Marin Mersenne was probably the first person to calculate the equal tempered semitone, the basis for equal temperament, in around 1620 although some people accredit Simon Stevin, an organ tuner at the workshops of Andreas Werckmeister with the discovery somewhat earlier in 1608. There is also evidence to suggest that a Chinese gentleman by the name of Chu Tsai-yu worked it out several years before its calculation in the West. Since the guiding principle for equal temperament is the redistribution of the comma amongst all the intervals of the scale and not just certain ones as most variations on meantone temperament seek to do, the best way to do this is to find the twelfth root of 2, i.e., that number when multiplied by itself twelve times equals 2: this number is 1.059463094; this interval ratio is the equal tempered semitone. Composers now had, at least in theory, complete freedom to modulate to any key without hearing wolf tones - but at a price.

It has been suggested that J.S. Bach wrote "The Well-Tempered Clavier" for equal temperament but this is erroneous. Research by the American musicologist John Barnes in the Seventies shows that what Bach probably used was a variation on meantone temperament devised independently by Francescantonio Vallotti and Thomas Young. It is almost certain that Bach knew of the existence of equal temperament but would have never used it himself as it would have been impractical to tune a clavichord this way since its pitch alters depending on how hard the keys are struck; in extreme circumstances, the pitch can vary by up to a Minor 3rd.

It took nearly two centuries for equal temperament to find universal acceptance by the musical world; the first pianos to be tuned this way were produced by Broadwoods in the middle of the 19th century and by the beginning of the 20th, virtually all pianos were tuned this way. While equal temperament has its disadvantages, it has lead the way for the full development of harmonic music and the rich variety of musical styles which has grown up in the last one hundred and fifty years. One fact to note that the figures used to calculate all these scales are based on the theo~ retical values. In practice, even equally tempered instruments, such as the piano, sound flat in their upper octaves when they are tuned in strict accordance with the equal tempered scale. Piano-tuners employ a trick called, 'brightening the treble' or 'stretch tuning', which means that the top one and a half to two octaves are sharpened slightly; the low bass octaves are also lowered in a similar fashion.

The need for this technique seems to arise partly from anomalies with our aural perception at high and low frequencies and partly with mechanical inadequacies in the piano itself; outside of the range 64 Hz to 4100 Hz, we lose our ability to distinguish intervals correctly. On a purely subjective note, I have noticed that my perception of pitch alters at high sound pressure levels; music sounds "flatter" when played very loud [8] as well as altering the band of frequencies at which intervals can be distinguished so this is another good reason for keeping monitor levels down and slapping the client's hand (gently) when he reaches for that volume pot and tries to blast your compression drivers through the back wall of the studio and out into the carpark!!

One final point to bear in mind is the fact that the pitch standard has varied considerably over the years. In fact, before the 15th century, the pitch standard had ranged from a' 504.2 Hz to a' 377 Hz. The Mean Pitch, proposed by Michael Praetorius in 1619 set the reference at 424.2 Hz. This standard more or less lasted for over two centuries and agreed closely with Handel's own fork (422.5 Hz in 1751) and that of the London Philharmonic (423.3 in 1820). In 1859 a French Government Commission was set up to establish a new standard as earlier in the century, with the development of brass instruments, pitch standards had been steadily climbing because of the increased brilliance of tone these instruments displayed at higher tunings - in 1858 the standard at the Paris Opera was 428 Hz and in Vienna 456.1 Hz. The Commission settled on 435 Hz which was embodied by Lissajous in a standard fork 'diapason normal' of 435.4 Hz which remains to this day as the only legal standard. Over the years pressure from the military bands once again forced up the standard and with the increase of broadcasting and the consequent need to maintain consistency from orchestra to orchestra, an International Conference in 1939 finally nailed the lid shut on the debate and set a reference of a' 440 Hz @ 20 C. This reference gives us the following frequencies for the tempered heptatonic scale:

c'      261.6256   
d'      293.6648   
e'      329.6276
f'      349.2282 
g'      391.9954
a'      440.0000
b'      493.8833
c" 523.2511

The twelve-note chromatic scale, of which there are two, is calculated by either multiplying the frequencies of the seven notes of the scale by 1.0417, which gives us the notes C#, D#, E#, F# etc or by multiplying by 0.96, which results in Cb, Db, Eb, Fb etc: these scales together are called the enharmonic scale. Notes in other octaves are obtained by multiplying or dividing by 2 an appropriate number of times - simple, eh?

Those of you who wish to pursue this fascinating subject further can find instant gratification in the following mighty tomes:

On The Sensations of Tone

H. Helmholtz, Dover

Tuning & Temperament

J.M. Barbour, Michigan State College Press

Intervals, Scales & Temperaments

L.S. Lloyd - Hugh Boyle, MacDonald & Jane's

Tuning In

Scott R. Wilkinson, Hal Leonard Books

Fundamentals of Musical Acoustics

A.H. Benade, Dover

Introduction to the Psychology of Music

G. R‚v‚sz, Longmans, Green & Co

Psychology of Music

C. Seashore, Dover

The Physics of Musical Sounds

C.A. Taylor, EUP

Musical Interval & Temperament

R.H.M. Bosanquet, MacMillan

Anecdotal History of Sound

D.C. Miller, MacMillan

Feedback and comments can be addressed to:

Chas Stoddard
Flat 1, 64 High Street
Glastonbury
SOMERSET
BA6 9DY
UK
E-mail: chas.stoddard@ukonline.co.uk

1
An interesting consideration is the phenomenon of the octave. Why is it, when we consider the audible frequency range from 20Hz to 20 KHz, we perceive a series of points along this scale that we can consider as having the same "quality" while patently being a different note? Part of the explanation may be that if we take a bi-lateral cross- section through the cochlea, that part of the ear's mechanism responsible for converting acoustic energy into electrical impulses, it reveals a spiral shape which can be described mathematically by a Fibonacci Series; the same maths govern the principles of the harmonic series. Neuro-pathology of the ear shows that octaves are decoded at the same point in each layer of the spiral. Some experts maintain that if the cochlea was a straight cone, rather than a tightly-wound spiral, we would have no perception of the octave at all; all we would hear would be a series of successively rising tones.

2
Note that these names are the original Greek ones and not the ecclesiastical names in use today. These would be Locrian, Ionian, Dorian, Phrygian, Lydian, Myxolydian and Aeolian.

3
Incidentally, if K = 300 then we have the Savart, an obsolete device much used in French literature

4
There are other devices, such as limmas, apotomes, diesises and schismas but their discussion here is beyond the scope of this article.

5
One possible explanation for his avoidance of B minor, at least for his keyboard works, is that in its ascending melodic mode, the scale throws up a G# which, while not exactly clashing, would sound odd in Meantone. Many organs of the period split the back half of the G# key to produce two separate keys sounding G# and Ab.

6
Several attempts have been made over the years, particularly in the 19th century, to develop keyboards with extra digitals, the most famous of which was a 53-note keyboard developed by R.H.M. Bosanquet, which can be viewed in the Science Museum, London.

7
OK, OK! Music historians get a gold star. At this point in history, composers were using the predecessor of our modern instrument, the fortepiano. This has a much lighter tone than today's piano and was strung differently using a wooden frame. Mozart's piano music, for example, tends to sound "bottom-heavy" on a piano whereas on the original instrument, the tone has more clarity.

8
In case you don't believe me, try this simple experiment. Find a piece of music that has a long sustained chord. Play this as loud as you can stand and then half-way through the chord, rapidly reduce the volume to a point when it's just audible - you'll notice the "pitch" of the quiet section apparently rise.

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