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There is a paragraph where author talks about harmonic and inharmonic spectra, and he points out that the harmonic spectra have partials that deviate from the precise ratios (for example the term "stretched" octaves resulting from deviations in the piano spectrum)

What does he mean by saying "precise ratios"? Does he mean the ideal interval ratios of two frequencies?

  • What author, and what paragraph in what book? Surely you can edit your question to include the precise bibliographical reference. – user1044 Jan 18 '16 at 0:57
  • Real live piano strings vibrating on a real piano do not behave exactly according to pure mathematical formulae. Neither does anything else in the real world of physics. By "precise ratios" he is referring to those derived from mathematical formulae that describe theoretically ideal piano strings, which again, don't actually exist. – user1044 Jan 18 '16 at 0:59
  • "Stretched octaves" refers to the practice by piano tuners of deliberately altering strict mathematical tuning ratios to accommodate the particular design of the individual piano they are tuning. Look up some resources on piano tuning if you want to learn about "stretch tuning". – user1044 Jan 18 '16 at 1:00
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The presumption is that a pitched sound consists of partials that have frequencies that are integer multiples of the fundamental frequency, so that a note with fundamental frequency f (e.g. 100Hz) has partials at f, 2f, 3f (100, 200, 300 Hz) and so on - or in terms of ratios, 1:1, 1:2, 1:3 and so on. It's these ratios that the deviation is from in an inharmonic sound - for example, this spectrum of a bell:

enter image description here

(Picture by user Hyacinth from page https://en.wikipedia.org/wiki/Inharmonicity)

We can see here that the ratios are 1:1, 1:2.23, 1:3.73, and so on, so they could be said to deviate from the presumed 1:1, 1:2, (1:3, for which a near-equivalent is missing), 1:4... rations of a pitched note.

If a sound is inharmonic, it's only when considering it in the frequency domain like this that we can actually see a fundamental frequency. Viewed in the time domain, a sound that has non-integer-ratio partials will not actually be periodic; you could therefore consider that its actual fundamental frequency is less well-defined than for a sound that consists of only harmonic partials.

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The article or book you are asking about is talking about pianos because of the stiffness of the strings and how that makes the piano behave different from how idealized physics says it should behave. The partials of notes created by most musical instruments have integer ratios to the fundamental frequency of the note. For instance, the third partial of a note is theoretically three times the frequency of the fundamental. Because of the properties of the stings of a piano, for the high and low notes on the piano, the frequency ratios for the partials are off.

Our ears use partials quite a lot to determine the pitches of the notes we are hearing. If the partials are off, the notes will sound out of tune. To compensate for this, pianos are often tuned with "stretch" tuning. That means the high and low notes are tuned based on their partials, rather than their fundamentals in order to make sure the piano sounds like it's in-tune, when actually the fundamentals are slightly out of tune.

The answer to your exact question is this: The second partial of a note is theoretically double the frequency of the fundamental (the lowest frequency that makes up a note). The third partial is theoretically three times, the fourth, four times, the fifth, five times, etc.

See this Wikipedia article for more.

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    Just a terminology note: the fundamental is the first partial, so your numbers are off in the answer. The third partial is three times the fundamental frequency and the first partial is one times the fundamental frequency. Your description would be correct if you used the term "overtone" however. – Pat Muchmore Jan 17 '16 at 15:55

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