What are the rules for generating the Pythagorean scale? How is the circle of fifths made?

Starting from a note chosen arbitrarily as the fundamental, one can proceed either by ascending fifths (which is equivalent to multiplying the frequencies by 3/2) or by descending fifths (which is equivalent to dividing the frequencies by 3/2). The result does not change.

Choosing, as an example, 1000 Hz as the frequency of the starting note, and the ascending direction, the following notes will have frequencies in Hz of 1500, 2250, 3375, 5062.5, etc. To bring these frequencies back within a single octave (that is, between 1000 and 2000 Hz in our example), they must be divided by 2 as many times as necessary. In our example 1000 and 1500 are already fine (because they are below 2000); 2250 and 3375 must be divided by 2, giving 1125 and 1687.5 respectively, while 5062.5 must be divided by 2 twice, giving 1265.625. So one proceeds by multiplying by 3/2 n times and dividing by 2 m times.

This is the only generating rule. Another rule is now needed to limit the number of notes that must appear in the octave.

This is precisely the origin of the problems of the Pythagorean scale: there is no way to obtain exactly an interval of (a multiple of) an octave by repeatedly multiplying or dividing by 3/2 (see the proof in the next question). So, in principle, one can go on forever, filling the scale with infinitely many notes very close to one another. Since a scale with infinitely many notes makes no sense, Pythagoras' recipe is to stop at the magic number of 11 fifths, that is, 12 notes.

This choice is dictated by the observation that multiplying by 3/2 twelve times (and dividing by 2 seven times to bring the frequency back into the octave) gives a ratio of 1.013643265, very close to 1.00000000, though not identical. Put simply, the twelfth note in the progression is the first to be very close to the starting note.

See also the pages From the monochord to musical scales, Pythagorean scale and Comparisons between the different scales.

Does the fifth correspond to the ratio 3:2 or 2:3?

To both: it depends on which quantities are compared and in which order.

• 3:2 is the ratio between the frequency of the higher note and that of the lower note. For example, an E at 330 Hz is a just fifth above an A at 220 Hz, because 330:220 = 3:2.
• 2:3 is the ratio between the lengths of the strings producing the two notes on the monochord. At equal tension the frequency of a string is inversely proportional to its length, so the higher note is produced by a string 2/3 as long as the other: if a 90 cm string gives the A, a 60 cm string gives the E (60:90 = 2:3). The Pythagorean tradition and many texts derived from it express intervals precisely as ratios between lengths. Sometimes, moreover, the frequency ratio is simply written with the lower note first.

The interval is the same. What matters is knowing whether frequencies or lengths are meant and in which order they are written. On this site, unless otherwise stated, ratios are always between frequencies, with the higher note in the numerator (see Pythagorean scale).

Why is it not possible to close the circle of fifths using only natural fifths?

Closing the circle of fifths means obtaining an interval of (a multiple of) an octave by proceeding only by natural fifths. That is, for example, starting from a given note, going up (or down) by natural fifths until a note an octave above (or below) the starting note is produced.

Let us now translate our intention into a formula. Suppose ${\displaystyle f_0}$ is the starting frequency. Proceeding by natural fifths means multiplying a starting frequency only by the ratio 3/2. We would therefore like that, by multiplying ${\displaystyle f_0}$ by 3/2 n times, with n an unknown integer, we could obtain an octave of ${\displaystyle f_0}$. This octave will have a frequency of 2^mf₀, with m an unknown integer. Indeed, the first ascending octave of ${\displaystyle f_0}$ has frequency ${\displaystyle 2f_0=2^1{f}_0}$, the second ${\displaystyle 4f_0=2^2{f}_0}$, and so on. For descending octaves we have ${\displaystyle f_0/2=2^{-1}{f}_0}$, ${\displaystyle f_0/4=2^{-2}{f}_0}$, etc.

The equation we want to solve for integer n and m is therefore

${\displaystyle {\left(\frac{3}{2}\right)}^n=2^m}$.

Multiplying both sides by 2ⁿ we obtain

${\displaystyle 3^n=2^{m+n}}$.

This equation, however, has no solution.

To see this, note first of all that n and m must have the same sign. Indeed, if ${\displaystyle n>0}$ the fifths are ascending, so the multiple of the octave must also be ascending (${\displaystyle m>0}$). Similarly, if ${\displaystyle n<0}$ the fifths are descending, and the octave must also be descending (${\displaystyle m<0}$)

The equation is the same in both cases, so we can examine just the case ${\displaystyle n,m>0}$.

Setting ${\displaystyle m+n=k}$, the equation becomes ${\displaystyle 3^n=2^k}$, with n and k positive. But the left-hand side is odd for every n, while the right-hand side is even for every k, so there is no solution.

Why is an unclosed circle of fifths a problem?

The idea of building a scale arises from the need to have a finite number of basic notes with which to compose music. Building by fifths is only one way of proceeding but, if the progression by fifths does not return on itself, that is, if the circle of fifths does not close, it generates infinitely many notes, and so we are back to square one, without a scale.

Why does our tempered scale contain 12 notes?

At first sight the choice of the number of notes in the tempered scale might seem arbitrary. Indeed, since the octave is divided into equal semitones, why use exactly 12, and not 10, or 20?

In reality the choice is not so arbitrary, since 12 is the smallest possible number of notes such that all the consonant intervals of the natural scale are approximated to within 1% in the tempered scale.

We know, in fact, that temperament is a device used to close the circle of fifths so that all intervals are acceptably in tune (and therefore also acceptably out of tune) regardless of the key. The small mistunings introduced by temperament with respect to the natural scale bring the great practical advantage that fixed-tuning instruments (such as keyboard instruments) do not have to be retuned at every change of key.

To be acceptable, however, the mistuning must deviate reasonably little from the just intonation of the natural scale. This is why fewer than 12 notes cannot be used, unless deviations greater than 1% from the justly tuned intervals are accepted.

It is of course possible to use more than 12 notes, but this does not automatically guarantee a better temperament. For example, in several scales with fewer than 31 notes all the consonant intervals appear, but not with better approximations than in the 12-note scale.

To obtain a real improvement one would have to move to scales with 41 or 53 notes, but this would mean changing the construction specifications of all fixed-pitch instruments. In particular, the piano keyboard would become huge, for the same range.

For this reason tempered scales with more than 12 notes have not been widely used in Western music, while they appear in cultures where the dominant instruments do not have fixed pitch.

Can a piano be tuned in equal temperament but with just fifths and fourths?

Not in all keys. Twelve just fifths (ratio 3:2) exceed seven octaves by a small interval, the Pythagorean comma, equal to about 23.5 cents (see the previous questions on the circle of fifths). To close the circle, equal temperament narrows each fifth by about 2 cents, so the tempered fifth is slightly narrower (flat) than the just one. As a consequence the tempered fourth is slightly wider (sharp) than the just fourth. Some just fifths can be had only by distributing the comma unevenly over the others, as in historical "unequal" temperaments (for example Werckmeister's). In this case, however, the keys no longer all sound the same. The differences between scales can be heard on the page comparisons between different scales.

Can a piano be tuned with A at 432 Hz?

It can be done, but it has no particular physical justification. A at 440 Hz is an international convention (see the question on the tuning fork). Over history the reference A has varied a great deal, roughly between 400 and 465 Hz depending on place and period. Even today some orchestras tune to 442 or 443 Hz. Claims that 432 Hz is a "natural", "cosmic" or beneficial frequency have no scientific basis.

Tuning to 432 Hz simply means lowering all the notes by about 32 cents (a third of a semitone). On a piano this reduces the tension of the strings by about 4%, which is harmless, but the instrument must then be kept at that pitch. Moreover, anyone playing with other instruments must tune the same way. The timbre may change slightly, as it does with any change in the reference pitch.

How are octaves numbered in the Italian and the scientific notation?

To indicate which octave a note is in, a number is added to its name: the octave index. The conventions in use, however, differ. In the two most common ones the index changes at C: all the notes from a C to the following B have the same index. The two conventions, however, differ by one.

• In the Italian notation (also used in France and Belgium), adopted in the Italian pages of this site, the A of the tuning fork at 440 Hz is La₃ and middle C of the piano is Do₃.
• In the scientific notation (also called international or American), adopted in the English pages of this site, the same A is called A₄ and middle C is C₄.

The index of the scientific notation is therefore always one greater than that of the Italian notation:

When a frequency is given together with the name of a note, one must therefore check which convention has been used: "Do₄" means 523 Hz in the Italian notation, whereas "C₄" means 262 Hz in the scientific one. Other conventions also exist. In Helmholtz notation, common in older texts, octaves are indicated by upper- or lower-case letters and by primes: middle C is c' and the A of the tuning fork is a'. Music software does not agree either: MIDI note number 60, middle C, is called C4 by some programs and C3 by others.