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Alas, all of the explanations of Kornerup's Golden Scale that have come my way are incomprehensible. Even Kornerup's own explanation is turgid and nearly impossible to understand.
However, the idea behind Kornerup's Golden Section tuning is both simple and profound. And, as it turns out, very easy to explain in extremely simple terms. --
Thorwald Kornerup was a Danish music theorist born in the late 19th century. He was an early advocate of non-12 equal temperaments, but he became disenchanted with the lack of a universal way of measuring tunings. Kornerup realized around the 1920s that no one had established a purely mathematical method for judging the characteristics of tunings. For thousands of years, this or that music theorist had judged varous tunings by comparing it to some other tuning which was set up as a "standard."
However, Kornerup realized that this didn't provide an impartial yardstick. J. Murray Barbour, for instance, describes a huge array of meantone and just intonations in his 1951 book "On Tuning and Temperament." However, Barbour measures these ji and meantone tunings by calculating the difference of their fifths and thirds from 12-tone equal temperament!
According to Barbour's measurement method, a tuning is called "good" if its fifths and thirds are close to those of 12-TET, while Barbour calls a tuning "bad" if its fifths and thirds are far from those of 12-TET.
But this doesn't really answer any useful questions about the tunings, unless you believe that 12-TET is the ultimate ideal to which all intonation should aspire. (And presumably most of us on this tuning forum do not buy that canard.)
Donald Hall makes a similar gaffe when he uses the harmonic series as a yardstick for measuring various equal temperaments. Hall calculates the difference of this or that fifth and third in this or that division of the octave, and calls a division of the octave "Good" if its fifths and thirds are close to the 3/2 and 5/4, while he calls a division of the octave "bad" if its fifths and thirds are far from the 3/2 and 5/4.
However, this doesn't really answer any useful questions about equal temperaments either.
For example, one of the more pleasant sounding intervals is the neutral third formed by the geometric mean twixt the 6/5 and the 5/4. This turns out to the be a very messy real number, since it's the square root of 3/2 = 1.224744871. According to the theory of consonance by small numbers, the square root of 3/2 should sound horribly dissonant...but of course, that's not the case at all. the 350.9775 cent interval is a very pleasant interval to listen to. To these old ears it sounds more consonant than the 6/5, and much more consonant than the 4/3. Yet another example that the theory of consonance by small numbers is utter hogwash.
The point is that if you measure intervals in an equal-tempered scale ONLY according to the harmonic series, you'll miss entirely consonant triads with a near-neutral third in 'em. Thus Easley Blackwood, by using the harmonic series as a yardstick, completely missed the most consonant and beautiful- sounding harmony in the 17-tone equal tempered scale, namely the neutral triad formed by the 17-TET fifth and the neutral 17-TET third formed by 5 steps of the 17-TET scale.
So we can see that the harmonic series is not any more of a reliable or useful yardstick for musical scales than is 12-TET.
This leaves us with a question:
As we all know, the Golden Section is a proportion with some interesting properties. Rectangles whose long side is 1.6180339 times the short side have a peculiar property--you can subdivide such a rectangle into another smaller rectangle whose area is 1/1.6180339 of the larger rectangle's area, and the small rectangle turns out also to have sides in the proportion of 1.6180339. (This happens because the golden section = 1 + 1/[golden section])
Kornerup realized that this infinite self-similar subdivision of rectangles could also apply to musical intervals.
The Golden Section is given by 1 + [(sqrt(5) - 1)/2].
So Kornerup devised an infinitely self-similar musical scale. He turned the ratio 1.6180339 into a musical interval: [ln(1.6180339)/ln(2)]*1200 = 833.0902963 cents.
Kornerup then asked: "What is the size of the perfect fourth which is the Golden Section major sixth as the GS major sixth is the octave?"
Answer: the golden section fourth is 8.330902963/1.6180339 = 514.8781188 cents.
Then Kornerup found the minor third whose proportion to the golden section fourth was the same as that of the golden section fourth to the golden section major sixth:
The process can also be continued on the large end, with octave reduction:
833.0902963 cents*1.6180339 = 1347.968415 cents = 147.9684152 cents.
1347.968415 cents*1.6180339 = 2181.0587115 cents = 981.0587115 cents
2181.0587115 cents* 1.6180339 = 3529.027127 cents = 1129.027127 cents
3529.027127 cents *1.6180339 = 910.0858381 cents 5710.085838 cents*16180339 = 839.1129646 cents and so on.
The process can obviously be carried out to ever larger intervals without limit, provided we octave-reduce afterward.
M. Joel Mandelbaum and others have pointed out that Kornerup failed in his attempt to find a "universal yardstick" for measuring musical scales. After all, Kornerup's Golden Section scale is really nothing more than another tuning--albeit arrived at by unusual mathematical means. However, there is no basis for declaring any of the intervals of the Golden Section Scale "superior" to any other intervals of any other scales. As always, the beauty of any interval remains in the ear of the listener. There remains no objective absolute method of measuring the absolute quality of any given musical interval, since (as we've seen) comparison to this or that equal temperament is obviously hopelessly subjective, and comparison to this or that member of the harmonic series is equally subjective and just as fraught with peril of ignoring intervals that sound good but don't have the right "numbers."
However, the real contribution of Kornerup's Golden Section scale is that it is the first recorded example of a self-similar non-just non-equal-tempered tuning. Thus Kornerup's Golden Section scale is the gateway to the infinite universe of n-j n-e-t tunings, and as such lays claim to a singular importance. The limitless sonic realm of the n-j n-e-t tunings is replete with self-similar tunings, of which Kornerup's Golden Section scale is only one among many. For instance, one could choose to define a tuning in which the just twelfth is to the octave as pi is to two. This is John Harrison's pi tuning, another n-j n-e-t scale with the property of infinite self- similarity.
Or one could choose to define a tuning in which the interval of 2 octaves + 7/6 is to the 1/1 as 4.6692106 (Feigenbaum's constant).
Or one could choose to define a tuning in which the octave + 11/8 is to the 1/1 as 2.718281828:1
And so on.
Because there's no limit to the amount by which we can extend the process of self-similar interval replication as long as we octave-reduce, there is by definition no limit to the number of intervals in any of these scales. As we continue the process, the scale fills with more and more different kinds of intervals, and if we continue the process long enough we can obtain an interval as close as we like to any given target interval.
Thus the general class of infinitely self-similar n-j n-e-t scales offers a boundless realm for musical exploration, and Thorward Kornerup and John Harrison (the greatest clock maker of the 18th century, and one of the great mechanical geniuses of all time) were the first to see this possibility.
--mclaren
Re Kornerup: Brian is not quite correct in his derivation of Kornerup's Golden tuning. Barbour says that the Golden Fifth is (15-sqr(5))/22 octaves or 696.214.. cents. Kornerup derived it by setting the relation of the minor third to the whole tone to PHI (1.618034 or (1+sqr(5))/2 . The minor third is 2 octaves - 3 fifths and the whole tone is 2 fifths - 1 octave. (2oct-3F)/(2F-oct) = (1+sqr(5)/2 where oct =1200 cents. By simplifying this equation, one gets (15-sqr(5))/22 oct. or 696.614474 cents for the fifth and about 504 cents for the fourth.
However, Brian's interation method does generate a novel non-JI-non-ET scale. The problem I have with most of the scales of this type is that they contain a few large intervals and a very dense section at the bottom of the gamut around 1/1. I think one might want to generate more balanced scales either by combining the results of such an iterative process with its inversion around the octave or the initial interval or else applying the process to the initial large intervals to fill them in too. This would seem to be equally quasi-fractal and melodically more interesting. I've generated some, but haven't had time to write them up for XH.
BTW, Jacques Dudon apparently independently rediscovered Kornerup's Golden tuning a few years ago. I just got a copy of a new CD of his in which he presents his optoacoustic instrument,the "Photophone," which might be described as an optical siren in that rotating patterned disks interrupt light beams to generate sound via photodiodes . The music is rather ethereal and Near Eastern in flavor. Dudon composes with 11 and 13 limit JI tetrachordal scales, gamelanish scales, etc. (The JIN or FPM may carry it, otherwise one would have to write Dudon in France.
I might add that Golden tuning, which is a variety of meantone, is not the only way to use the Golden Ratio (or section) in tuning. In my Xenharmonikon 15 column, "Notes & Comments," I said the following:
--John
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