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Musical Temperament

From the 10th to the 19th century, the art of musical temperament was a vital and essential part of European musical practice. During the last two centuries of this period, learned and often vituperative debate centred around certain tempering schemes, and a large body of now historical literature attests to the importance in which the topic was held. Yet today it is all but unknown outside academic circles. Even more curious, at a time when the sheer quantity of music is overwhelming, it must be said that this lack of understanding of temperament has constrained musical expression in the modern world within narrow and crippling limits. This statement will undoubtedly strike many as somewhat absurd, even incredible; but such, indeed, is the case.

Musical temperament is a topic arising from considerations that,whilst initially very simple, demonstrate a complexity and range of possibilities when developed beyond a certain point that is fascinating to the academic mind, delightful to the artist, and dismaying to the layman. An introduction to the topic should therefore focus on basic principles, clarify its importance, and indicate how complexities arise from detailed considerations without following them too far.

The New Grove Dictionary of Music defines temperament as "Tunings of the scale in which some or all of the concords are made slightly impure in order that few or none will be left distastefully so." In simple, practical terms, what this means is that the pitches of notes in a musical scale are raised or lowered by small amounts to achieve certain artistic effects. Those who have learnt instruments such as the violin or guitar will be well aware of the importance of accurate tuning. Those who have only played instruments such as the flute and recorder will have less concern with tuning, since the notes of these instruments are fixed during manufacture, but can be varied by small amounts during performance by blowing and fingering techniques. Those whose musical experience is limited to keyboard instruments will quite likely think it strange that the pitch of notes should be varied at all - they usually regard their instruments as being either "in tune" or "out of tune", and, like most laymen, may wonder why other tunings should be considered.

The modern musical instrument keyboard
In order to clarify these ideas, let us examine the modern Western music keyboard as shown in Figure 1. As most people know, it consists of seven white and five black keys named in a repeating sequence using the first seven letters of the alphabet, but starting with C instead of A. The black keys are named with reference to their adjacent white keys, either sharp if they lie above, or flat if below. This naming scheme may seem irrational and capricious, but it does derive from centuries of tradition, and there are valid reasons for it that need not concern us here.

What is of immediate interest is that the white keys from C to C' form the most common scale in modern Western music, the major scale. The upper C note, named C' here, is an octave above the lower C or root note. The word octave comes from the Italian word for eight, since there are eight notes in modern scales. Technically, the interval of an octave denotes a doubling or halving of frequency. The pitch of a note is determined largely by its frequency - that is, the number of vibrations per second of the string or air column that produces it. However, where frequency is a physical measurement, pitch is a subjective sensation - that is, how we hear the note - and this can be influenced by several factors such as the timbre or quality of the sound. For example, notes from a flute generally have a pure and gentle sound, whereas those from a trumpet are often harsh and strident by comparison. Such differences can cause us to hear intervals slightly differently from what might be expected from simple measurements.

Table 1. Common musical intervals
RatioIntervalConsonance
1:1UnisonMost consonant
2:1Octave:
3:2Perfect FifthConsonant
4:3Perfect Fourth:
5:4Major Third:
6:5Minor Third:
7:4Minor Seventh:
9:8Major Tone:
10:9Minor ToneDissonant
16:15Major Semitone:
17:16Minor Semitone Most dissonant

Musical notes, intervals, and scales
Understanding the ratios of the frequencies of notes is the key to understanding the basics of temperament. We have stated that an octave is a doubling or halving of frequency, so the note C' in Figure 1 will have a frequency double that of the note C. A musical interval is the pitch difference between two notes, in other words, the ratio of their frequencies. An octave therefore represents a frequency ratio of 2:1. It is a fascinating fact of musical theory that two notes with simple integer ratios, when sounded together, produce a sound that is pure, sweet and harmonious to our ears, whereas those with complex ratios sound harsh, discordant and inharmonious. The technical terms for this difference are consonance and dissonance, the former meaning pure and harmonious, the latter strident or discordant. Thus we can draw up Table 1 showing the relative consonance of common musical intervals. Once again, the names given to the intervals derive from centuries-old usage, and do have valid historical reasons behind their choice.

We now come to one the most interesting aspects of musical theory, one which formed a basic consideration for composers of earlier epochs, and still does for those working in traditional music - the choice of a scale upon which to construct a melody. In today's Western music there are really only two choices, the major and minor scales. In earlier times the choice was much wider. Let us see why.

Table 2. Frequency ratios in violin tuning
StringNoteRatio to previousRatio to lowestSingle octave
HighestE3:227:827:16
3A3:29:49:8
2D3:23:23:2
LowestG---

Tuning a violin
The first thing to do is to put aside theory and turn to practical matters. In order to sound the notes of a scale on an instrument, it must first be tuned. Using a violin as an example, the strings are tuned a Perfect Fifth apart. The lowest string is G, and the next lowest is D, since these two notes have a frequency ratio of 3:2, as in Table 2 (which should be read from bottom to top). The next string will be A, since A:D is again 3:2, and the top string will be E, E:A again being 3:2. This gives us four notes of our scale, and the topic of tempering begins to appear if we calculate the various ratios of these notes, convert them to a single octave, and write down the results. In Table 2 we can see that the ratios get more complicated as the notes get higher, but are still integer ratios. In order to move the notes within a single octave, we double the second number of each ratio (because the octave is a 2:1 ratio) until the result lies between 1 and 2.

Table 3. Notes of a Pythagorean scale
#NoteRatio to previousRatio to lowest
8 G - 2:1
7 F# 9:8 243:128
6 E 9:8 27:16
5 D 9:8 3:2
4 C - (4:3)
3 B 9:8 81:64
2 A 9:8 9:8
1 G - -

If this process is continued, we end up with the result in Table 3 (again read from bottom to top), the seven notes of a scale constructed from sequential Perfect Fifths. This is known as the Pythagorean scale in honour of the Greek philosopher who first remarked it.

It can be seen that the sequences in Table 3 are incomplete, and this is where tempering enters as a consideration. The intervals B-C and F#-G are not assigned ratios, since in using them it is necessary to choose between several options. If we continue the sequence of 3:2 ratios, that for the note C is 177,147:131,072. This awkward ratio is tolerably close to 4:3, and since 4:3 is the inverse of 3:2 (3:4 is an octave below 3:2), practical composers often settled for it as a workable solution. This is the reason for the caution given earlier; we started out with something apparently simple - a sequence or 3:2 ratios - and are now faced with numbers growing increasingly large and daunting. Herein lies some of the fascination of musical theory.

Equal temperament
We have now come far enough in our discussion to understand what equal temperament is, and why it appeared so elegant a solution to musical theorists of the 18th century. The following explanation has been much simplified in order not to introduce the sort of mathematics that has driven so many schoolchildren to despair. Those with courage and a scientific calculator may choose to explore the topic further.

As far back as the 15th century, musical theorists realized that if the octave could be divided into twelve equal parts, the resulting twelve notes would be very close to the notes of a natural scale - that is, one having simple integer ratios between the notes. However, calculating this interval had to await the invention of logarithms in the 17th century, since the number required is the twelfth root of two. If we denote this number by s, then multiplying it twelve times gives the result of two, the interval of an octave:

s x s x s x s x s x s x s x s x s x s x s x s = 2

Anyone with a modern calculator can verify that the value of s is 1.05946..., and that multiplying it twelve times gives the expected result. In the 17th century this was done by dividing the logarithm of two by twelve, and finding the antilogarithm of the result, a procedure worthy of a genius at the time. Of the many different tempering schemes then in use, one of the most popular was just temperament. Table 4 compares the pitches of notes in the equal tempered and just scales, again reading bottom to top. Frequencies in this table are given in cents for ease of comparison. Those unfamiliar with cents should simply note that values in the two columns are all different except for the first and last, and therefore all frequencies differ except first and last. (An explanation of cents is given in another document on this site.)

Table 4. Comparison of equal and just temperaments
Equal temper Just temper
Note Ratio to previous Cents Cents Ratio to previous Note
C's1200120016:15C'
B2s110010889:8B
A2s90088410:9A
G2s7007029:8G
Fs50049816:15F
E2s40038610:9E
D2s2002049:8D
C-00-C

Even without a detailed understanding of exactly what the numbers mean, Table 4 demonstrates the crucial difference between the two tempering schemes. The intervals between the notes of the just scale are all simple integer ratios, and these ratios are found in Nature as fundamental relationships in the structure of matter, of living creatures, and of energetic mechanisms. By contrast, the ratios in the equal-tempered scale are all identical, and this ratio does not exist in the natural world. Whilst the numbers in the cents columns may have no immediate meaning to those unfamiliar with them, they show that none of the intervals in the equal-tempered scale are pure, whereas most are in the just scale. This is why equally tempered music can never achieve the same consonance and power of emotional evocation as can naturally tempered music.

The howling wolves
Because equal-tempered intervals are impure, they create a wavering effect caused by beats in the resulting sound. All who have flown in a multi-engined aircraft will remember the pulsing sound of the engines during takeoff, when each engine is running at a slightly different speed and their individual sounds beat together. As the aircraft gains altitude, the pilot trims engine speeds to bring them into synchronization, and the beats disappear. This effect is used by wolves when hunting. Upon finding a herd of prey, the wolf pack splits into two or three groups. The groups move apart, and each sets up a howl at slightly different frequencies. This produces an eerie, ululating quaver in the sound that strikes fear into the hearts of their prey. The herd begins to run, then to panic, and the wolves move in for the kill.

Medieval monks, who devoted their lives to mastering perfect intervals in the chants and hymns they sung, so detested the impure intervals of bad tempering that they named them "the howling wolves". These musical experts possessed, not only far more acute hearing than modern people, but also a much greater emotional sensitivity to music and sound. Their opinions have since been vindicated by the discovery that the beat frequencies in modern musical scales are in the same range as the electrical signals in the human brain (see Table 3).

Today, the wolves howl day and night, but we have become so deafened by the clamour of our mechanical nightmare world that none can hear them wailing. Unless we soon regain a humane level of emotional sensitivity and a proper respect for Nature, the wolves will eventually devour us.