As per wikipedia https://en.wikipedia.org/wiki/Pythagorean_hammers page about Pythagoras Hammers ratio discovery, there are following ratios I could find from wiki page:

C Unison 1:1, D Major Second 9:8 , F Perfect fourth 4:3, G Perfect fifth 3:2, C Octave 2:1.

now, when I look on other wiki pages and internet resources,



those internet resources have given ratios for E 5:4, A 5:3 and B 15:8 in an octave which are not in Pythagoras hammer wiki page.

Where did those ratios for E 5:4, A 5:3 and B 15:8 come from making an Octave?

Who invented or proposed them and how? I couldn't find any resources about those remaining E, A and B ratios.



1 Answer 1


Short answer: We probably will never know the full origins of ancient tuning systems

Who invented or proposed them and how? I couldn't find any resources about those remaining E, A and B ratios.

According to Wikipedia:

Gioseffo Zarlino, in the late sixteenth century, created the first justly intonated 7-tone (diatonic) scale


Whether Zarlino or someone else first created a seven-note diatonic scale, they did not invent the entire JI scale all on their own. Nor does it seem likely that they alone were displeased with Pythagorean tuning and innovated in a vacuum. Ptolemy, for one, is credited in some places with using the 5:4 ratio for thirds.

It's probably not possible to pin down one person or even a sequence of identified tuning innovators through history. The Pythagorean tuning system might have origins before recorded history, and we can't be sure whether or not individual musicians through the millennia were not playing around with different tuning methods.

Tuning with just intonation is based on the overtone series, which is a physical property that any musician with a stringed instrument and enough time on their hands can easily discover on their own. Anyone crafting a wind instrument or playing a "brass" instrument (that is, an instrument of any material where the lips are buzzed into a resonating cylinder or cone) also could discover the overtone series.

In addition, musical instruments and tuning systems underwent parallel development in different parts of the world. There's a tendency to favor the "western" lineage in many academic sources, but tuning systems were definitely developed in Asia and the Americas through antiquity.

Note that Pythagoras did not invent or discover the ratios for the second, fourth, fifth, or octave, either. The fifth (and therefore the fourth also) and octave were known to ancient Mesopotamians and Babylonians and used to tune instruments in those cultures.

  • thanks for replying, just to confirm, so all the internet is wrong about pythagoras hammer theory? and Gioseffo Zarlino created those 7 ratios? or Gioseffo created only remaining of those ratios for E A and B ? Feb 26 at 3:09
  • @administr4tor I don't know how to answer your question about "all the internet". Zarlino created a scale using the ratios. It's possible that someone used the same ratios to make a scale before then, but if they did, any record of them doing it has been lost. No one "created" any ratios. They just created scales and tuning systems based on those scales. Feb 26 at 4:28
  • @administr4tor - those ratios were not created any more than America or Australia were created. Discovered is more apt!
    – Tim
    Feb 26 at 8:06
  • @Tim discovered by Gioseffo Zarlino? Feb 26 at 17:03
  • 1
    @administr4tor from the Wikipedia page that you linked to: "The legend is, at least with respect to the hammers, demonstrably false." Also, the Wikipedia claim that Zarlino was the first to identify the just scale is unsourced, and I am highly skeptical of it. (This claim is in fact also demonstrably false if you include non-European music theorists.) In any event, Pythagoras did not recognize ratios having prime factors larger than 3; accordingly, the Pythagorean major third, also known as a ditone, has the ratio of 81:64.
    – phoog
    Feb 26 at 17:51

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