In this article, I wish to explore the exciting topic of equal temperament progressively, starting with the Pythagorean system, showing its limitations, and then arriving at an explanation of what this tuning method is and why its introduction was necessary.

First, clarifying what the word “temperament” means is helpful. In a sense, one could use it as a synonym for “tuning” (and, sometimes, it will happen), but, to be precise, tempering is that process of adjusting individual frequencies after a global tuning. We could call it “fine-tuning,” but, in practice, it was necessary precisely because the Pythagorean system, as we shall see, introduced errors, and some notes had to be slightly modified to avoid glaring out-of-tune.

### Step one: the discovery of the octave, or the cyclicity of notes

Around 500 B.C., Pythagoras and his disciples began experimenting with a monochord, a taut string mounted over a sound box with one end fixed and the other movable. An example of a monochord is shown in the figure, where the C bridge is movable and allows the string to be shortened.

Pythagoras’ first discovery was the inverse proportionality relationship between wire length and resonant frequency. Although he did not yet have the tools derived from Newton’s discoveries, Pythagoras understood that if the tension remained constant, a shorter wire would oscillate faster than a longer one. It is possible to observe the phenomenon within certain limits even with the naked eye, so it should not surprise us that a man of such insight came to this conclusion. We now know that the exact relationship (referring to the monochord shown above) is:

The term under square root is equal to the ratio of the tension W to a constant λ that depends on the physical nature of the string. However, Pythagoras dealt exclusively with relationships between lengths and between frequencies, so if we call L_{1} the first length (e.g., the whole string) and L_{2} the second length, the relationship between the relative frequencies will depend only on the lengths:

Having made this premise, we can assume what Pythagoras did to conduct his experiments. He took two identical monochords and fixed the length of the first one. As explained, neither length nor absolute frequency matters, as he worked exclusively with ratios. Therefore, he began to reduce the length of the second monochord and play the two notes simultaneously.

#### The love of whole numbers

At this point, it is good to make a premise. Pythagoras and the philosophical school had a robust preference for numerical elements that showed perfection. In other words, they regarded whole and rational numbers (i.e., fractions of integers) as “privileged.” At the same time, despite having perfect knowledge of them, they disliked irrational numbers (e.g., √2) since, having infinite decimal digits and not being able to be expressed as ratios between integers, they appeared as “necessary accidents” (his celebrated theorem churns out one after another, the golden section is irrational, π does its share, etc.), but, if possible, to be avoided.

#### Pythagoreans and numerology

In addition, the Pythagoreans were attached to a numerology with an esoteric character and particularly liked specific numbers to which they attributed symbolic meaning. For example, the Tetraktys could be summarized in the identity 1 + 2 + 3 + 4 = 10, in which the first four numbers added together led back to an origin (10 ⇒ 1 + 0 = 1) of a higher spiritual level. The number 4 symbolizes the Earth, with its elements air, water, earth, and fire, all represented graphically with triangles (i.e., with the number 3).

On the other hand, the number 3 symbolizes the threefold human nature: physical, psychic, and spiritual. Therefore, 7 represented the perfect fulfillment of the human and cosmos union. We could continue this disanimation at length, but the above is sufficient to understand that Pythagoras primarily analyzed integer ratios between lengths and was careful not to consider arbitrary lengths.

#### The octave: the perfect consonance par excellence

On the other hand, it is pretty likely that, at first, he evaluated the sound effect of the two notes by sliding the movable bridge of the second monochord until he noticed that when it reached half the length (i.e., creating a 1:2 or 2:1 ratio), the two sounds had a robust affinity. The correct term to describe this condition is that of *consonance*. In the case of the ratio in question, this consonance was not only perfect (i.e., the most pleasing that could be heard by sliding the movable bridge) but also had an additional character.

Pythagoras, of course, limited himself to the observation (an already significant result). Still, we now know that humans, from a neurophysiological point of view, tend to recognize two sounds in a 1:2 ratio as having the exact nature, even though the former is more severe than the latter. This phenomenon is more an object of study in cognitive science and neuroscience than music.

Nonetheless, one cannot help but note that if a name is given to the sound at the base frequency, the same name (albeit with a pitch distinction) must be given to the sound with double frequency. In a nutshell, Pythagoras discovered what we now call the “octave” and, at the same time, realized that sounds had a cyclic nature.

Given, in fact, the arbitrariness of the base frequency f, the same sound was found for every doubled value 2f. Starting from 2f, the phenomenon occurred at the frequency of 4f, according to an exponential law of the type ^{2nf}. Having clarified this point, it is good to ask another obvious question: why is this interval called “octave”?

### Pythagoras invents the diatonic scale.

For reasons that should be obvious, Pythagoras and the Pythagoreans looked at music more from a speculative than a practical side. In other words, they were interested in studying and defining the so-called “Harmony of the Spheres” rather than creating a system for musical composition (which, indeed, they considered an even deplorable practice).

In experimenting with the monochord, the philosopher had noticed another consonance that was very pleasing to the ear, which corresponded to a division of 3/2 (remember, to avoid confusion, that the frequency of the base note is equal to 1, while the octave consonance corresponded to 2, so all the intervals we are going to consider have a ratio a/b ∈ [1, 2]). Pythagoras, who, as already explained, used only ratios of whole numbers, found other pleasing intervals, for example, with the ratio 5/4.

Before moving on, it is good to remember a small rule of thumb: the “sum” of two intervals is obtained by multiplying the ratios. On the other hand, if we call the “fifth” (i.e., its final name) the interval of 3/2 and we want to calculate the fifth of the fifth, we must first divide the chord into 3 parts and take 2, then divide the 2-part long section into 3 and retake 2. Consequently, the fifth of the fifth will be given by the ratio (3/2) × (3/2) = 9/4.

So Pythagoras had the fundamental note (i.e., whole chord), the octave (i.e., halved chord), and at least a couple of intermediate consonant intervals. A breakdown based on these few notes would have been unsatisfactory and numerologically unacceptable. Considering the celestial bodies then known, he divided the interval (1, 2) into 7 parts, whereby the repetition of the first note was precisely the octave (we can now state this with full knowledge).

Here, I will not go into an explanation of the procedure he followed (available in the reference text given at the end of the article) but suffice it to say that he used the ratio of 3/2 and all ratios derivable from it by power elevations (i.e., repeated multiplications). After some simple mathematical manipulation, he obtained a result that is still known today as the diatonic scale and coincides (at least formally) with our major scale.

Starting from the fundamental note, he found a very regular sequence of relationships between successive sounds:

The first note had a frequency f, the second (9/8)f, and so on. Since 9/8 = 1.125 and 256/243 = 1.0535, Pythagoras called the first ratio “tone” and the second “semitone.” However, considering what was said just now, this terminology could be misleading because semitone does not coincide with halftone, that is:

The tone is slightly larger than two semitones, but this detail is unimportant. The fact remains that the gap between the first and second notes is more significant than that between the third and fourth notes.

#### A quick jump forward to about the year 1000

One helpful detail that has not yet been defined (and was never defined by Pythagoras) concerns the name of the notes. Since it is helpful for our purposes to be able to mention them, it is worth mentioning that the first to give well-defined names to the seven notes was Guido d’Arezzo, who was also credited with the invention of a musical notation that, with some variations, has come down to us. He based himself on some verses well known in ecclesiastical circles, taking the initials of each word and obtaining the sequence:

Ut – Re – Mi – Sol – La – Si

Except for the first note, this is the modern solfeggio system in use even in the Anglo-Saxon world (where notes are defined by letters of the alphabet). Guido was so good at teaching that he even thought of a practical method for memorizing and singing notes based on the phalanges of the hand.

#### The modern diatonic scale

We have all the elements needed to construct the “modern” diatonic scale, starting with C. This is a choice of convenience; if we introduce the signs for sharps (#) and flats (♭), saying that the former raises a note by a semitone while the latter lowers it by the same value, we can construct the diatonic scale of each fundamental note. For our purposes, however, it is much easier to work with unaltered notes:

C – D – E – F – G – A – B – (C)

All intervals are one tone except those between E – F and B – C, which is one semitone. Of course, even if Pythagoras did not explicitly define the names of the notes, our reasoning remains valid, but clearly, they are more convenient. Now, let us do some quick calculations to see the ratios of the most critical intervals.

#### Second Pythagorean

The interval C – D (second major) consists of one tone and does not need much work. It is equal to 9/8.

#### Third Pythagorean

The interval C – E is composed of two tones, and we get:

In this case, those with some rudiments of music theory may notice an “oddity.” Although the relationship is between whole numbers, it is improbable that Pythagoras divided the string into 81 parts, making only 64 parts vibrate. Indeed, this is the first consequence of his choice to base the scale on the 3/2 ratio. He had realized, for example, that the 5/4 ratio was also quite agreeable, but, in the end, he opted for the former.

It is enough to have a calculator to see that 5/4 = 1.25 and that the Pythagorean third is slightly higher than that range. Only several centuries later, the musician Gioseffo Zarlino gave birth to the so-called “natural scale,” in which this interval is included. Since this article aims to explain equal temperament, I will not discuss this scale explicitly, but those who wish may refer to the text cited at the end.

#### Fourth Pythagorean

The interval C – F has an amplitude of two tones and one semitone:

In this case, the ratio is between relatively small integers (4/3), which is no accident. But to explain why, we must first verify that the fifth corresponds to the chosen value 3/2.

#### Pythagorean Fifth

The “privileged” C – G interval that Pythagoras chose to find the diatonic scale can be easily calculated:

We, therefore, have confirmation (ed. as was apparent) that the resulting tone and semitone intervals allow us to define a Pythagorean fifth precisely equal to 3/2. Looking at the tone-semitone succession shown above, we notice that starting from the fifth note, we have the sequence, tone-tone-semitone, or a Pythagorean fourth. However, in terms of notes, this is equivalent to saying that the G – C interval is a fourth.

This is not strange and falls under the theory of so-called “interval revolutions,” which can be studied in any harmony text. In particular, using modern terminology, we can define a straightforward rule: the inversion of a “positional” interval X (i.e., based on the ordered notes in the scale) is obtained as 9 – X. Thus, the octave, for example, will have the “first,” i.e., unison, as the inversion; the third will have the sixth, etc.

However, what changes in the revolts is the amplitude of the resulting intervals (e.g., major-minor, excess-diminished). This topic is simple enough but beyond our context, so I refer interested readers to a good, harmonious text that clearly explains this information.

However, in this particular case, with a ratio of 3/2, we are in the presence of a “right fifth.” The facing of a “right” interval is also “right,” so we can be sure that the C – F interval is precisely as wide as the G – C interval and vice versa. For this reason, when Pythagoras chose the value 3/2, he implicitly introduced the related insurgent 4/3.

### Pythagoras gets into trouble.

At this point, it is clear that, regardless of the names of the notes, there is a cyclicity based on the succession of fundamental frequencies (f, 2f, 4f, …, 2nf,…), which applies to every note in the scale. Everything could appear to be arranged correctly, framed in a mathematical system that cannot falter. Yet, without perhaps expressing it openly, Pythagoras knew well that there was a problem and also knew the reason should be sought precisely in his “eagerness” to have only whole or rational numbers.

#### The math doesn’t add up.

Without further ado, let us consider starting from a C at any octave, which we will denote by (1) and proceed by fifths. What you get is:

C(1) → G(1) → D(2) → A(2) → E(3) → B(3) → F(4) → C(5)

Beautiful. If the base frequency is f, going from fifth to fifth, we arrive at C with frequency 2^{4}f= 16f. But we also know that G(1) is (3/2)f. Now, even without knowing the numbers, we can check whether it is somehow possible that by raising 3/2 to an integer power n > 1, it is possible to obtain a power of the type 2^{k} with k > 1, that is:

With a simple manipulation, we get:

It is clear without the need for deep mathematical knowledge that the first member is always odd, just as the second member is always even, regardless of n and k. Consequently, although positional notation allows us to arrive at the starting note 4 octaves higher, this is mathematically impossible with the interval of fifths! In the specific case, if f = 100 Hz, we should arrive at 1600 Hz, but 7-fifth intervals bring us to (3/2^{)7} × 100 ≈ 1708.6 Hz, a much higher frequency than expected!

Moreover, considering ratios between integers, the final fractions will have a prime numerator and denominator. This means that if the fraction is a/b, a and b do not share factors (e.g., 3 and 2 are prime to each other, while the fraction 2/4 is not and can be reduced to 1/2 that meets this condition). Consequently, a/b division is always a decimal number.

It is straightforward to show that any ratio between integers raised to the power n > 1 can never equal any power of 2! This means that using Pythagorean intervals, it will never be possible to reach any octave unless you do an octave jump. In a different form, but still based on the same principle, Zarlino’s natural scale also presents the same problem, the solution to which lies in the hated irrational numbers that Pythagoras tried hard to avoid.

### Tuning and tempering with Pythagorean intervals

It should be clear by now that the Pythagorean system does not allow for accurate tuning of an instrument. Indeed, suppose a master builder works on a church organ. When adjusting reed length, he will start from a fixed reference, probably shared with his guild members and handed down from master to student, and move forward, for example, by fifths. If the reference is C(1) and is 1 meter long, it will create a barrel for G(1) of length 2/3 = 0.66667 meters, and so on.

As you may have already realized, the trouble is that each step carries a fraction of an error that will lead it, for example, to an utterly out-of-tune C(5). Knowing this problem, luthiers proceeded to what is known in the jargon as “tempering,” that is, they adjusted each note to minimize the error. This goes without saying because it is clear that this process was highly empirical and related to the ear of the individual luthier. So it could easily have been that a composition sounded good on one organ but had prominent out-of-tune tones on another.

### Dulcis in fundo: the equal temperament

The most elegant solution to this problem was accepting irrational numbers as intervallic ratios. Avoiding unnecessary preamble, the proposal for equable temperament was as simple as it was ingenious: divide the octave into equal intervals. By then, the tonal system was entirely in use, and it was well known that there were 5 altered notes in each octave, the decision was simple, 12 equal semitones per octave.

The octave corresponds to twice the frequency of the fundamental, so bearing in mind that ratios between intervals overlap by multiplying them, the amplitude of the semitone ratio s had to be such that the condition was verified:

That is, the semitone ratio had to have a constant amplitude and equal to:

If you remember, the Pythagorean semitone ratio had an amplitude of 1.0535, so the difference is relatively small. However, the n-th roots of 2 are irrational numbers (i.e., with infinite decimal places), resulting in a variation of all interval ratios. Before we do some simple experiments, it is essential to say that now, the tone t corresponds to two semitones, that is:

Therefore, we can say that the right fifth has an amplitude of 7 semitones, or about 1.49831, which is very close to the Pythagorean ratio of 3/2. Similarly, the major third equals 4 semitones, so about 1.25992 versus the Pythagorean ratio of 5/4. As can be seen, equal temperament does not introduce many variations. While it is impossible to implement without approximations (i.e., irritation numbers must necessarily be truncated), it offers the advantage of a method that can be conventionally accepted and lead to consistent intonations from instrument to instrument.

Also, as should be obvious, obtaining the octave by overlapping intervals is always possible. If we have n > 12 semitones (for n < 12, we are within the first octave) and k > 1 octaves, the equation:

It can be easily fulfilled. For example, considering the experiment done earlier with fifths, starting from C(1), we would arrive at C(5), that is, k = 4 and 2^{4} = 16, so it is sufficient to take a total of n = 48 semitones, which, considering the starting semitone, becomes 49 / 7 = 7 right fifths. Should the reason for adding a semitone is not apparent, remember that a semitone separates two notes, so if we start counting from C(1), the twelfth semitone is between the last note, B(1), and the first note of the new octave C(2).

#### The trend of frequencies and string lengths in equal temperament

In the following diagram, I have depicted the 3-octave frequency trend starting from middle C on the piano at 261.6 Hz and the corresponding string length trend by arbitrarily choosing the tension and elasticity (which, being constant factors, do not matter for the curve structure):

### Conclusions

I hope this article can help novice musicians understand the rudiments of the music system so that they can be clear about what kind of development has been conducted throughout the centuries. Of course, mine is a simplified treatment in which I have omitted many mathematical demonstrations and some details that should be taken for granted. I am sorry if this has caused any confusion, and I invite you to contact me via the form provided for questions, clarification, correction of errors, etc. I will continue this activity by covering other aspects of music theory (such as this one) and more practical ones, primarily related to the instrument I love most: the classical guitar!

### For more on the Pythagorean system and other mathematical aspects of music

- How music has influenced mathematics, physics, and astronomy from ancient Greece to the twentieth century
- Music is filled with mathematical elements
- The works of Bach are often said to possess a math-like logic, and Arnold Schoenberg, Iannis Xenakis, and Karlheinz Stockhausen wrote music explicitly based on mathematical principles
- Yet Eli Maor argues that it is music that has had the greater influence on mathematics, not the other way around
- Starting with Pythagoras, proceeding through Schoenberg, and bringing the story up to the present with contemporary string theory, Music by the Numbers tells a fascinating story of composers, scientists, inventors, and eccentrics who have played a role in the age-old relationship between music, mathematics, and the physical sciences

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Last update on 2024-07-08 / Affiliate links / Images from Amazon Product Advertising API