I think I've read that even very ancient cultures were able to discern that an octave difference corresponded to a pipe of twice the length, and so on. But at what point were musicians and composers able to understand that each note in a scale of fixed notes represents a certain number of Hz?

(I'm interested in facts relating to various areas - if it was known earlier in Vanuatu than in Vienna, I'm interested in that level of detail!)

  • Just an insight: in a vibrating string, it's easier to see that higher pitches correspond to faster vibration
    – Emilio
    Dec 19 '16 at 16:20
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    This is probably a book-length topic. But I'm willing to bet there's not much information available. Dec 19 '16 at 16:22

If you have access to a good academic library, then the following article appears to be on point regarding the Western tradition:

S. Dostrovsky, Early Vibration Theory: Physics and Music in the Seventeenth Century. Archive for History of Exact Sciences, Vol. 14, No. 3 (5.XII.1975), pp. 169–218.

Here are a few pertinent pasages; I have bolded the TL;DR quote:

That pitch can be identified with frequency was a major discovery of the seventeenth century, and this identification made possible very precise measurements of relative frequencies. ...

Following his father Vincenzo, Galileo was critical of the traditional use of the ratios based on string lengths to describe musical intervals. He went on to connect pitch with frequency and to explain clearly that frequency ratios correspond to intervals.

After the above quote, Dostrovsky quotes extensively from Galileo's Discourse on Two New Sciences (1638). In particular, Galileo describes scraping a sharp metal stylus across a metal plate so that it squeaks, and examining the spacing of the scratches left behind on the plate. He found that higher-pitched squeaks corresponded to scratches that were closer together (and thus a higher frequency of scratches.)

Finally, other folks were thinking the same way around the same time:

A number of Galileo's contemporaries recognized the central importance of frequency even though they did not discuss it as clearly and as explicitly as he did. Isaac Beeckman had associated pitch with frequency already by 1615, when he had tried to derive the inverse proportionality between frequency and length of a vibrating string. Although Descartes had not emphasized frequency in his early Compendium Musicae, in L'Homme, written in 1632, he referred to it as the source of pitch: "les petites secousses composeront un son que l'âme jugera... plus aigu ou plus grave, selon qu'elles seront plus promptes à s'entresuivre, ou plus tardives." ... Mersenne often dwelt on reasons for the musical ratios. In Harmonie Universelle he gave a variety of possible ratios for the octave and a variety of reasons for the particular ratio 2:1. Finally he wrote that it is "entirely necessary" to use this ratio: "sound being nothing other than the movement of the air, and this movement finding itself always double in the octave, and never quadruple or octuple, it follows that the two sounds of the octave are in the same ratio as these movements." ...

To summarize: according to the picture that was around by the end of the first third of the seventeenth century, sound consisted of a succession of pulses (percosse for Galileo, secousses for Descartes, battemens for Mersenne and, later, Huygens.) ... The identification of pitch with frequency became categorically accepted, in spite of the fact that it was only possible to measure relative frequencies and these only by using the identification itself. The problem of determining frequency absolutely was to challenge natural philosophers throughout the seventeenth century.

(Translation of the Descartes quote: "The small jolts compose a sound that the soul will judge... higher or lower, according to whether they are quicker or slower to follow each other.")

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    (I'm not 100% happy with "jolt" as the translation for "secousse" in the quote above, BTW; if anyone has a better translation I'm open to suggestions.) Dec 19 '16 at 20:23
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    This is fascinating, but of course it still doesn't really show (what would?) how widespread the knowledge of the math behind music was amongst musicians. Remember what Leibniz said: Musica est exercitium arithmeticae occultum nescientis se numerare animi. That is: Music is a hidden arithmetical exercise; the soul does not know it is counting. Dec 20 '16 at 13:23
  • @ScottWallace: it did occur to me that this didn't really answer the question as asked; it's just a necessary condition. It'd be interesting to know how widespread the knowledge is today — if you asked 100 musicians what a sound actually is, how it travels, or what makes a higher pitch different from a lower pitch, how many of them would be able to answer correctly? (Particularly those musicians without formal musical schooling.) Dec 20 '16 at 15:45
  • I wonder that too. I doubt that it's been formally studied, but who knows? Dec 20 '16 at 19:39

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