Tones and semitones: those bricks that never fit in the right holes

Music appears to be an art based on precision and rigor. Anyone who has studied solfeggio knows this well and will remember the emphasis on tones and semitones. Woe to name E# or F♭! But are things really in those terms? Unfortunately, that is not the case. In this short article, I will highlight the existing problems and have no solution!

Is it fair to say that B# is different from C?

I will preface this by saying that some statements may be debatable. I certainly do not intend to proclaim dogma, so those who disagree can (and should) maintain their position without any problem. Having made this necessary introduction, I can get to the heart of the matter by providing two arguments that should help to understand how things are.

The names of the notes are conventional.

Whatever one may say, the names of the notes are conventional, and, to tread a bit, the frequencies are too. Only in 1939, for example, it was decided to adopt 440 Hz A as the universal standard. Still, before that date, music proliferated in a jungle of tuning forks at frequencies scattered between 415 Hz and 435 Hz.

As for names, it was determined that the sharps sign (#) had the task of “moving forward,” a note one step called precisely semitone (flat had the opposite task). Of course, someone might ask, but what are tones and semitones? The answer to this question is discussed in the next section. For now, let us assume that the semitone is the height of a step and that the notes are placed along a 12-step scale starting at C and ending at the next C.

Why the steps are precisely 12 is a consequence of the initial choice, made by Pythagoras, to have 7 notes in what is appropriately called a diatonic scale. Again, in the next paragraph, everything will become more evident. For now, it is sufficient to assume that we have decided to name the notes in a certain way (different from region to region) and that the structure of the diatonic scale is made by a sequence: tone – tone – semitone – tone – tone – tone – semitone.

So, for example, C and D are two steps apart, and the middle step must necessarily be called C# or D♭. In contrast, E and F are one step apart, so E# = F and F♭ = E. Similarly with B# and C. If we decided to twist reality by changing names and intervals, no problem would arise. We would have a structure where, perhaps, B and C are 3 semitones apart and C and D only a semitone apart. If you are wondering if I have gone mad, I can assure you that I have not, and as you continue reading, you will realize my arguments more clearly.

Tones and semitones: How high are these steps only?

If you are unfamiliar with equable temperament, I invite you to read my article. What is important here, however, is to know that Pythagoras constructed the diatonic scale based on the rational ratio 3/2 (corresponding to the Pythagorean fifth) and arrived at a structure where the tone was 9/8 = 1.125 broad and the semitone 256/243 = 1.0535.

As explained in the article, this had rather disastrous consequences since it was impossible to obtain an octave (ed. recall that if the base frequency is f, the upper octaves will have frequencies equal to 2f, 4f, 8f, 16f, etc.) through, for example, overlapping fifths. So after giving generic names to the 7 notes, what was possible was to move from fifth to fifth and arrive at an octave (e.g., from C – G – D – A – E – B – F – C), but numerically the final value did not coincide with a power of 2 of the base frequency.

Although this example is present in the cited article, I repeat it here with a frequency of the first C = 1 Hz. The last C will be 4 octaves higher and have a frequency of 16 Hz. Because we moved with 7 fifths, the amplitude equals (3/2⁾⁷ ≈ 17.086 Hz. This system has an obvious problem, the consequences of which were nefarious for the tuning of instruments.

But let’s do another brief experiment. Always starting from C = 1 Hz, with 5-fifths, we get a B with a frequency approximately equal to 7.59 Hz. Since the Pythagorean semitone equals 1.0535, the note following B and higher by one semitone has a frequency of about 7.59 × 1.0535 = 8.00002 Hz. There is a slight error, but it seems that Si# is equal to “more or less” to C.

Let’s start from C at 16 Hz and reduce it by one semitone, 16 / 1.0535 = 15.18747. Let’s lower it by an octave, 15.18747 / 2 = 7.59374. We continue to the first octave and get Si = 1.89844 Hz. It is easy to see that raising it to one Pythagorean semitone gives the next C with a tiny error. But we know that starting from C = 1 Hz, G equals 1.5 Hz, and the interval G – B corresponds to the modern major third that Pythagoras identified with the interval of 5/4 = 1.25.

Thus, if we start from G and apply that range, we get a B = 1.875 Hz, which is very different from the previous value! Raising this B by one Pythagorean semitone, we get 1.875 × 1.0535 = 1.97531, which is not equal to 2! So now, B# is different from C. It seems like a paradox: Why don’t things work perfectly as well as mathematics leads us to think? The reason is straightforward and can be explained by two arguments.

Pythagoras and his love of whole numbers

The first reason for this discrepancy is choosing to work with proportions based on ratios between integers and rational numbers. Pythagoras liked regular ratios because he associated music with the harmony of the spheres, which, according to his thinking, could not be based on irrational numbers (i.e., with infinite decimal places).

For his diatonic scale, he listened for consonances between two monochords, of which the first played the fundamental note and the second was shortened progressively, but always according to ratios such as 2/3, 3/4, 4/5, etc. He observed that the consonance corresponding to frequencies in the ratio of 1: 3/2 (e.g., C – G) was extraordinarily pleasing, and after various analyses, he chose precisely that ratio to find the scale.

In making this choice, he “condemned” the other reports that they could no longer be present accurately. In the previous example, we used a ratio of 5/4, which led to disaster. This should come as no surprise: Pythagoras elected only 3/2 as a foundational ratio, effectively eliminating all other ratios and making it almost impossible to use them through movements of tones and semitones.

Who determined that the right fifth should have a ratio of 3/2?

This is the crucial question that should be attempted to be answered. The only possible and reasonable answer is Pythagoras and his “craving” for whole numbers. In equable temperament, the octave is divided into 12 semitones of equal amplitude equal to the twelfth root of 2, or about 1.05946. This number is irrational and, therefore, an investor in Pythagoras, but at the same time, it is, in fact, the solution to all problems.

To clarify the context, suppose we are working with middle C on the piano at a frequency of 261.6 Hz. According to the Pythagorean system, the next G should have a frequency of 261.6 × (3/2) = 392.4 Hz. Let us admit that he, endowed with an overbearing ear, has observed excellent consonance, but what about equable temperament? In this case, G is 7 semitones away from C. Therefore, the resulting frequency is about 391.95713 Hz! The discrepancy is about 0.5 Hz, and I highly doubt that even Mozart, under the influence of nootropics and psychedelics, would be able to notice the difference!

This is the crux of the matter. Using equable temperament, B# is always equal to C, and as explained in the above article, tuning and tempering of instruments becomes extraordinarily simple. Even the immense Bach wanted to “praise” this method by composing the “Well-Tempered Clavier.” So why continue to think that Pythagoras must be correct and that the relationships must be strictly rational?

This makes no logical sense and is based only on a questionable aesthetic criterion, considering also that the Pythagoreans considered music only from a theoretical point of view and despised those who delighted in composing small pieces.

And finally, the longed-for answer to the initial question

The answer is No! Or, to be exact, it’s “No” if a person specifies that she wants to adopt a more comfortable and rational equable temperament. Suppose someone were to think that Pythagoras is a kind of demigod, so his word is unquestionable well. In that case, he is free to tune his instruments using the Pythagorean scale, but this in no way authorizes him to think that the consequences of such a choice should become universal law.

For further exploration of the relationship between music and mathematics.

Music by the Numbers: From Pythagoras to Schoenberg

• 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 2024-07-01 / Affiliate links / Images from Amazon Product Advertising API

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