Dialogue between humans and nature

José Rodríguez Alvira

The meantone temperament is described by Francis Salinas in *De Musica Libri Septem* (1577). In the meantone temperament major thirds are the same size as those in the harmonic series (1.25) and are divided in two equal size tones (or *mean* tone).

We will build the major scale using the meantone temperament. We begin by multiplying the frequency of C by 1.25 (the major third according to the harmonic series):

Now we find D by dividing the third in two tones of equal size, how?

The mathematical operation that allows us to find the size of the tone is the square root. Taking the square root of the size of the major third we get the size of the tone:

Let's see if it's true:

The square root of 1.25 = 1.118033988749895. This should be the size of the tone:

- Find D by multiplying C by the size of the tone.

D = 263 x 1.118033988749895 = 294.04 - Find E by multiplying D by the size of the tone.

E = 294.04 x 1.118033988749895 = 328.75

The result is the same as previously calculated, confirming that we have correctly calculated the size of the tone.

Now we need to calculate F that is a semitone above E. An octave consists of five tones and two semitones: C to D, D to E, E to F#, F# to G# and G# to A#, followed by two semitones A# to B and then B to C.

Mathematically this is expressed:

t * t * t * t * t * s * s (t = tone, s = semitone)

Or:

t^{5} * s^{2}

Therefore we get to this formula (2 is the size of the octave):

t^{5} * s^{2} = 2

We know the size of a tone, so we can write:

Our purpose is to find the value of the semitone (s). If we divide both sides of the equation by the square root of 1.25 raised to the fifth power we get:

Finally, taking the square root of both sides gives us the size of the semitone:

The size of the semitone is then:

s = 1.06998448796228

Finally we find F:

328.75 x 1.06998448796228 = 352.18

We can now complete the scale. The G is a tone above F, so we multiply F by the size of tone. A is a major third above F and B is a major third above G, we calculate both notes using the size of the major third:

The meantone temperament was effective while the number of keys in use was limited. Listening to the *Prelude in A flat major BWV 862* by J. S. Bach played using the meantone temperament makes the problem evident to our ears:

The problem is related to the meantone fifth that is smaller than the acoustic fifth. If we follow a cycle of meantone fifths the last fifth would be too big. The fifth that closes the circle is called the *wolf fifth*.

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José Rodríguez Alvira.