Helmholtz tempered scale

The Helmholtz tempered scale, also known as "Helmholtz tuning", is a just tempered scale (or "just tempered tuning") that defines the frequency of notes starting from a base note frequency (for example A = 440 Hz as in the table below) and using the ratio multiples provided in the following table.

A Base frequency, arbitrary 440.00
A# / Bb Base * 25 / 24 458.33
B Base * 9 / 8 495.00
C Base * 6 / 5 528.00
C# / Db Base * 5 / 4 550.00
D Base * 4 / 3 586.67
D# / Eb Base * 45 / 32 618.75
E Base * 3 / 2 660.00
F Base * 8 / 5 704.00
F# / Gb Base * 5 / 3 733.33
G Base * 9 / 5 792.00
G# / Ab Base * 15 / 8 825.00
A Base * 2 880.00

This scale is a "just tempered scale" because it uses rational numbers to build all notes. In other words, it uses harmonics of the base note frequency. As with other just tempered scales the frequencies computed above differ from those in an equal tempered scale as these frequencies do 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, for example, would compute A# = 466.16 Hz, whereas in the example above A# = 458.33 Hz. An equal tempered scale of these twelve notes will split the octave in twelve equal semitones. This is not true for the Helmholtz tempered scale. In the example above the semitone between A# and A is equal to 1200 log2(458.33 / 440) = 71 cents approximately, whereas the semitone between B and A# is equal to 1200 log2(495 / 458.33) = 133 cents approximately.

As with other "just tempered scales", the advantage of the Helmholtz tempered scale compared to an equal tempered scale is that the Helmholtz scale has notes that sound well together as they are harmonics of each other. A disadvantage is that music performed using this tuning may sound different when transposed as the semitones used to construct the intervals between adjacent notes on the scale are not all the same.

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

See also:
Scale, Scale (index)

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