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My piano is a half step below the standard tuning. I want to listen to and play this song. I calculated that from the 12-range of tones I should slow it down by 1/12 of its frequency but it actually doesn't work in my Audacity. Can you help me, please?

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The ratio between the frequencies of successive half-tones in a 12-tone equally tempered scale is 21/12. So to lower the frequencies by a half tone, you need to stretch the file so it is 21/12 ≈ 1.05946309 times as long.

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    Mr. Mathematician drops in to point out that you're abusing "significant digits" there :-) . Nobody does or can tune to four places, let alone eight! – Carl Witthoft Sep 28 '16 at 11:55
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    @CarlWitthoft: Perhaps, but as a theoretical physicist I prefer to think about things in terms of unrealizable platonic ideals. :-) – Michael Seifert Sep 28 '16 at 14:18
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I suggest that you tried to Change Pitch, since this process is available, rather than time-stretch the song. Change Pitch will not change the tempo of the song. If you choose to time-stretch the song, Michael gives you all the information and perfectly answers your question. As far as time-stretching is concerned, depending on the method applied, Audacity will or will not change the pitch. The details evade me as to how it is possible to change the tempo without changing the pitch (or as to how it is possible to change the pitch without changing the tempo). But it is possible. Change Tempo will change the tempo without changing the pitch, while Change Speed will change the speed, affecting tempo and pitch.

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    Tascam have had a cd player out, that does it, for well over 10 yrs . – Tim Sep 27 '16 at 16:39
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    It works, very simplified speaking, by applying a Fourier transform to obtain the frequency spectrum of short time snippets, then slowing down the sequence of these snippets (which doesn't change anything about the frequencies in each snippet) and then transforming them back to an audible signal through inverse FFT. The real trick is to avoid phase artifacts that come up if you either prolong the snippets to the gaps created when slowing down or in the overlaps when speeding up. – leftaroundabout Sep 27 '16 at 16:51
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    And it works best with small adjustments (such as this half tone down). It starts to sound stuttery with larger adjustments, as if it were slowing it down and then cutting out pieces in order to keep the time the same. – Matthew Read Sep 27 '16 at 19:21
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    To provide a simple analogy to what the FFT does: sheet music is an FFT! You read each note and "convert" the indicated frequency value into a time-sequence of amplitude variation. If you change the duration of each note, you stretch the song length. If you transpose, you shift the output frequency without changing the song length. – Carl Witthoft Sep 28 '16 at 11:57

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