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312:"pipes" for you, not only because I'm sure it was pipes in the example, but because strings also depend on tension for their pitch. Pipe pitch doesn't entirely depend on length either, but at least it's a slightly better model instrument for the experiment. (As for Vicentino, I tend to refer to meantone as a form of just intonation, which is a pretty broad term for me. Pay no mind...)
368:(with 108 zeros), and then extracting successive square roots followed by a cube root. Although it is possible to do it that way, it is easier (not to mention less fraught with arithmetical peril) to use trial and error. When one extracts square roots and especially cube roots directly, one misstep invalidates all future values. Trial and error reduces this risk considerably.
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the "exact lengths and bores of 12 pipes in correctly equal temperament." The problem of finding these lengths involves the extraction of the 12th root of 1/2, for the ratio for each semitone is 1/2^(1/12). Tsai-yu gives the approximate lengths to nine places, as 10.00000000, 9.43874312, 8.90898718, etc.
311:
I thought I might mention that you should be careful about quoting Partch on historical matters. His account in
Genesis isn't entirely accurate (you might already be aware of this). This particular factoid I think is correct, but personally I haven't checked up on it. I did change "string lengths" to
632:
This clearly shows that Chu Tsai-yu's method was not known to the author, and he made the assumption based on calculations made by the
European Mersenne. I would agree with removing this bit of trivia about the calculation from the article, for the reason that is only speculation and not noteworthy.
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While the black keys and white keys on a piano to correspond to pentatonic and diatonic scales, these two things have nothing to do with equal temperament, and equal temperament was certainly not any kind of attempt to combine the two. It's an interesting observation, but really not relevant to the
614:
Unfortunately Tsai-yu does not explain how his no doubt very laborious calculations were made. Mersenne, in solving the same problem to six places only, actually writes 2048 followed by 60 zeros as the number of which the 12th root is to be taken, to give the first fret of the viol; i.e., a number
608:
The date (1593) of Ch'eng's work is of interest in connection with the present study, as it is almost exactly contemporary with Prince Chu Tsai-yu's treatise on music, "Lu lu ching i," 1595 or 1596. Tsai-yu is honored by musicians for having been one of the first persons--if not the first--to give
581:
Of the prince's accomplishment a contemporary modern theorist reminds us that "the computation would have to begin, for certain tones, with numbers containing 108 zeros, of which the 12th root would have to be extracted, as
Mersenne did, by taking the square root twice and then the cube root. This
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This method is reasonably efficient; even proceeding strictly by hand calculations, one might be able to perform the necessary calculations within a day or so; with fractions instead of decimals, perhaps a week or two. By comparison, direct extraction of the twelfth root would take months, if not
264:
Harmonic tuning does not cause "errors" in temperament. The use of the word temperament refers to a tempering (modification) of harmonic tunings. In other words, tempering is the error, albeit an intentional and purposeful one. Furthermore harmonic tuning doesn't cause dissonance especially not in
254:
I made a few changes to the page. I found a lot of the information to be inaccurate or untrue, so a good deal of it was removed. I did try to add an explanation for the use of this particular number. If someone else can think of a more concise way of explaining it, please edit it. Here's a list of
270:
Meantone tuning was favoured in the
Renaissance, which was very much not an equal temperament. In fact, other solutions to the modulation problem were preferred, ie. Nicola Vicentino's Archicembalo with an extra manual and divided keys to provide just intonation in many more keys, or even in the
265:
the harmonic content of strings. (I'm not sure what harmonic content meant, but inharmonicity in strings is caused by tension and thickness, and doesn't have much to do with tuning, except the fact that temperaments are by nature out of tune with the harmonics of the string.)
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Thanks for researching that, Rainwarrior. You might want to work some of this information into the article (I mean let's face it, what else could we possibly put in it?), but you're right, we definitely shouldn't repeat the error and say that's what he "must have" done.
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followed by a footnote to the
Barbour source. For me, this wording is suspect, as he writes "would have to" and not "did". I'll see if I can find a source with direct commentary on Chu Tsai-Yu's actual published tables and calculation. (In a similar debate about
821:. An approximation by a ratio of two five-digit numbers makes no sense at all; you could have just given ten significant digits and would have been closer to the actual number. Who invents the need for approximations like that?
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781:, 140221/132351, 579788/547247, 720009/679598, 2019806/1906443, 2739815/2586041. (The sequence of partial quotients is 1, 16, 1, 4, 2, 7, 1, 1, 2, 2, 7, 4, 1, 2, 1.) —
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How were these ratios chosen? I whipped up a little program to find the "best" rational approximations using the method of continued fractions, and got: 1/1, 17/16,
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You're quite correct! I have updated the list using the fractions I boldfaced in your remark. You picked a great place to stop, the next partial quotient is 60. —
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may be accounted for by dividing it by 24 (the power, 12, times the approximate result, 2) and in this case subtracting it from our guess. Our next guess is then
579:(Pg 106-108), which I have not read. Whether or not this is accurate, I haven't verified myself (I asked about it earlier though.) The passage in Partch says:
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259:"Growling" should have been "howling". Growling was not the word used, historically, and it doesn't describe anything close to the sound of a wolf fifth.
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that is twice the second one of Tsai-yu's lengths, if we ignore the location of the decimal point. So we can picture Tsai-yu beginning with 108 zeros.
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Baroque, it was not uncommon for the black keys of an organ to be divded in two so that sharp and flat keys could be tuned differently.
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589:, I managed to locate an article explaining in detail his calculations. I haven't yet found such a thing for Chu Tsai-Yu.) -
294:(article needs expanding BTW) was not designed for just intonation, but extended meantone temperament, or the very similar
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I find this sentence impossible to read. I think the grammar's wrong; and I don't get the second half of the sentence.
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A SIXTEENTH CENTURY APPROXIMATION FOR PI, J.M. Barbour, The
American Mathematical Monthly, 1933 (Pp. 69-73).
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Yeah, Partch is great, but he's not always the most reliable source. I'll try to find a different one. —
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Computing the twelfth root of 2 to nine decimal places by no means requires starting with
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Follow up: I found an article by
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The passage about Prince Chu Tsai-Yu is a quote from Harry Partch's
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Its value is 1.05946309435929..., which is slightly more than
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lengthy and laborious procedure was followed without error."
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I'd strongly suggest just deleting all approximations but
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Tunings, Temperaments, and Scales
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Tunings, Temperaments, and Scales
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504:{\displaystyle x^{12}=2.012196...}
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