The Spirit of Creativity: Basic Mechanisms of Creative Achievements
So this book notes that Hunke 1960 documents the Muslim introduction of 5/4 and 6/5 into Western music theory. The book erroneously calls Hunke a "he."
Sigrid Hunke - Wikipedia
In her book, "Allahs Sonne über dem Abendland" ("Allah's sun over the Occident") she asserts that "the influence exerted by the Arabs on the West was the first ...
She was a Nazi but also a pagan-influenced researcher. So maybe that's never why I've heard of Hunke before - looks like her book was not translated into English.
https://www.chrysalis-foundation.org/Ibn-Sina_Stifel_and_Zarlino.htm
Ibn Sina describes the
arithmetic and harmonic division of the “octave” by noting
The property of the two medium intervals is to compose an octave; the latter interval then contains between its two extreme terms an arithmetic mean and a harmonic mean. The ratio of the octave is, in fact, that of 4 to 2; if we introduce the number 3 between these two terms, we obtain two consecutive ratios [4:3:2] resulting from an arithmetic mean term. The ratio of the two largest terms is that of the fourth [4:3], and the ratio of the two smallest, that of the fifth [3:2]. Moreover, the ratio of 6 to 3 also constitutes an octave ratio; if between these two terms we introduce a third, i.e., 4, we obtain two consecutive ratios [6:4:3] resulting from a harmonic mean term. The ratio of the two largest terms is that of the fifth [6:4], and of the two smallest, that of the fourth [4:3]. The two ratios of the fourth and of the fifth form a replica [or form an “octave” with] one another, when they have a common degree, and when they are arranged in opposite directions.
So this is from the 9th century A.D. - exactly when the Benedictine monks were being influenced by the Islamic Platonists - as John Scotus Erigena also wrote on Platonic music theory from the Muslim influence.
So then we see here the "Bait and Switch" in action that originated from Philolaus - the octave numbers are different even though the frequencies based on the pitch of C are the same. But 3 does not go into 2 so you can't use the SAME root pitch octave based on the natural overtone series from listening.
Figure 10.35 shows that the arithmetic division of length ratio 3/2 produces a “minor third,” ratio 6/5, as the lower interval, and a “major third,” ratio 5/4, as the upper interval. In least terms, we may notate this division as interval ratios 6:5:4.
Oops! Proof that Kyle Gann and Adam Neely are wrong! The origin of 5/4 in Western music theory is not from the natural overtones but rather from Archtyas use of the arithmetic and harmonic mean for the geometric mean squared.
Arabic music theory described intervals as mathematic relationships – for example 5:4 for a major third and 6:5 for a minor third.
The Arabic Roots of Jazz and Blues......by Gunnar Lindgren..............................
citing
Sigrid Hunke, Allahs Sonne über dem Abendland (Stuttgart 1960/67)
The only problem is the author doesn't realize that the 5:4 and 6:5 originate from Archytas use of geometric mean squared - NOT from the overtone series....
"Archytas will have assigned this interval the ratio 5:4 (the nearest epimoric smaller than the ditone: (9:8)squared = 81:64 and 5:4 = 80:64)."The Monochord in Ancient Greek Harmonic Science
https://books.google.com/books?isbn=0521843243
David Creese - 2010 - History
"However, he [Archytas] noted that the product of the arithmetic mean and the harmonic mean is equal to the square of the geometric mean, so this gave a way of dividing the fifth of 3:2 into the product of 5:4 and 6:5."A Truman State University review on Scriba, Christoph J. “Mathematics and music.” (Danish) Normat 38 (1990), no. 1, 3–17, 52.
Bilingual Europe: Latin and Vernacular Cultures - Examples ...
books.google.com › books
Jan Bloemendal - 2015 - History
So the title as a whole translates as 'On the Theory of Music'. In it he defends a ... It is not the ratio 5:4 that defines the major third, but the cube root of 2 does.
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