The frequency of a note is not exact, but depends on what note you start with, what frequency you assign to it, and how you tune other notes to it. For example, starting with A = 440 Hz (we could just as easily choose something else, but A = 440 Hz seems to be commonly accepted) and choosing an equal tempered scale will give us B = 493.88 Hz, because the equal tempered tuning splits the frequency interval between A = 440 Hz and A = 880 Hz (an octave higher) in twelve equal steps. Choosing Helmoltz’s just tempered or the Pythagorean just tempered tuning will give us B = 495 Hz, because these tunings do not split the frequency interval between the two As evenly, but rather attempt to find the closest harmonics of A. Thus, the frequency of a note depends on what base note is chosen and what tuning is used to get from that base note to other notes.

Here is a Microsoft Excel file with computations for the frequencies of notes. These frequencies are computed assuming that middle A is 440 Hz. There are three examples in the file: 1) equal tempered tuning; 2) Helmholtz’s just tempered tuning; and 3) Pythagorean just tempered tuning.

Starting from A, Pythagorean tuning allows us to compute the note D# / Eb in two separate ways: 1) going up from A with multiples of 3/2 (((3/2)^6)/8 for this note); or 2) going down from A using multiples of 2/3 (((2/3)^6)*8 for this note). First, D# is the just fifth of of G#, which is the just fifth of C#, which is the just fifth of F#, which is the just fifth of B, which is the just fifth of E, which is the just fifth of A. Second, going backwards we get Eb, of which Bb is the just fifth, of which F is the just fifth, of which C is the just fifth, of which G is the just fifth, of which D is the just fifth, of which A is the just fifth. Thus, in Pythagorean tuning D# and Eb do not necessarily compute to the same thing. Both frequencies are provided in the spreadsheet. An interval that contains these notes is out of tune. Such an interval is called a wolf interval and hence, while D# is placed within the Pythagorean tempered tuning column, the Eb is placed next to it and is marked with the "wolf alternative". Technically, both are a part of the Pythagorean scale.

See also:

Scale (index)

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