# Pythagorean tempered scale

The frequency of a note is not always strictly specified. Notes are tuned to each other and so, depending on which note we start with, what frequency we choose for it, and how we tune we may end up with various frequencies for the remaining notes.

The Pythagorean tempered scale, also known as "Pythagorean tuning", is a way of computing the frequencies of a twelve semitone scale (the chromatic scale) starting from an arbitrary base note with arbitrarily chosen frequency and using strictly the ratio 3:2 to build the just fifth of each note.

The following is an example of a Pythagorean tempered scale built starting with A = 440 Hz.

 Eb Base * ((3/2)^6) / 8 313.24 E Base * ((3/2)^1) / 2 330 F Base * ((2/3)^4) * 4 347.65 F# / Gb Base * ((3/2)^3) / 4 371.25 G Base * ((2/3)^2) * 2 391.11 G# / Ab Base * ((3/2)^5) / 8 417.66 A Base frequency, arbitrary 440 A# / Bb Base * ((2/3)^5) * 8 463.54 B Base * ((3/2)^2) / 2 495 C Base * ((2/3)^3) * 4 521.48 C# / Db Base * ((3/2)^4) / 4 556.88 D Base * ((2/3)^1) * 2 586.67 D# Base * ((2/3)^6) * 8 618.05

Starting with A = 440 Hz and multiplying A = 440 Hz by three gives as the closest odd order harmonic of A, which is E = 1,320 Hz. Dividing by two takes E one octave down to E = 660 Hz. Thus, E = 440 * (3 / 2) = 660 Hz, which is the fifth of A = 440 Hz, seven semitones higher. Alternatively, as in the table above, we can take E even further an octave down to get E = 330 Hz. The fifth of E = 330 Hz, seven semitones higher, is B = 330 * 3 / 2 = 495 Hz. B can also be computed directly from A as B = 440 * ((3/2)^2) / 2. We can continue in this direction to compute the just fifth of B, which is F# / Gb = 440 * ((3/2)^3) / 4 = 371.25 (this one taken an octave down from the just fifth of B at 495 Hz).

Given that the fifth of a tone is seven semitones higher or five semitones lower depending on the octave is a nice property with the octave being split into 12 semitones. Since 12 and 7 do not have common divisors the process above can be continued to compute all notes on the 12 semitone scale. If we want to work with close harmonics of the base frequency though, rather than just going up with multiples of 3/2, we should also start from A = 440 Hz and try to go down with multiples of 2/3. For example, A is the fifth of D and so D = 440 * 2 / 3. Taking the result one octave higher will give us D = 440 * (2 / 3) * 2 = 586.67 Hz. D is also the fifth of G = 391.11 Hz, and so on.

The problem with going in two directions is that at some point we can compute Eb = 440 * ((3/2)^6) / 8 = 313.24 and D# = 440 * ((2/3)^6) * 8 = 309.03 (D# in the table above is given an octave higher). Thus, in Pythagorean tempered tuning D# and Eb are two different notes. In fact, the difference is approximately equal to 23.46 cents (1200 * log2 (313.24 / 309.03) = 23.46), which is almost a quarter tone interval (25 cents). If D# and Eb differ, than the interval around those notes is out of tune. Such interval is known as a wolf interval. The appearance of a wolf interval is just one example of where some just tempered scales can create problems. Such problems here depend on how the 3/2 ratio is applied. If we chose to always compute A an octave higher as equal to A multiplied by two and if we always tune up the octave starting from A until the next A, then the problem is not that great (but that is not really Pythagorean tuning).

In any case, the Pythagorean tempered scale is a just tempered scale, because it uses rational numbers to build all notes and hence it uses nice harmonics of the base note frequency. Thus, the computed frequencies differ from an equal tempered scale as it does not split the interval from the base note (A = 440 Hz in this example) to the same note one octave higher (A = 880 Hz in this example) evenly. An equal tempered scale would compute A# = 466.16 Hz, for example, whereas in the example above A# = 469.86 Hz. As with other just tempered scales, the Pythagorean tempered scale have notes that sound well together as they are harmonics of each other, but music performed using this scale may sound different transposed as the semitones of the scale are not the same between each two adjacent notes.

A computation of the Pythagorean tempered scale is also presented in the topic on Frequency of notes.

Scale, Scale (index)

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