To put the same thing in another way, we have just identified the frequency ratio 1.5 with the interval of a [perfect] fifth, although our table gave the value as 1.4983. The difference is only small – 1.13 parts in a thousand – but by the time we have taken the twelve steps needed to pass completely around the clock-face, it has been multiplied twelvefold into the difference of 13.6 parts in a thousand, which represents the aforesaid difference in pitch of almost a quarter of a semitone. When this is allowed for, the true clock-face is that shown in fig. 55; it extends to infinity in both directions and all simplicity has disappeared.” Sir James Jeans book Science and Music, (Dover Publications, 1968), p. 166
So Sir James Jeans is DIVIDING 1.5 by 1.4983 to get 1.001335 which is 1.33 parts in a thousand in contrast to 1.0136 being 13.6 parts in a thousand. But this is assuming that 2/3 is also NOT the Perfect Fifth. In other words the Ditonic Comma assumes you can not just multiply but also "halve" the Perfect Fifth back into the octave - thereby "losing" or hiding this multiplying factor!!
So as Chinese Nobel Physicist C.N. Yang points out about Hertz - the noncommutative phase at zero energy was eliminated from the math:
Throughout his lifetime, Maxwell always wrote his equations with the vector potential A playing key role. After his death, Heaviside and Hertz gleefully eliminated A. But we know with Quantum Mechanics that A has physical meaning. It cannot be eliminated (E.g., the Aharanov-Bohm experiment). C.N. Yang youtube lecture,
The Aharonov–Bohm effect, sometimes called the Ehrenberg–Siday–Aharonov–Bohm effect, is a quantum mechanical phenomenon in which an electrically charged particle is affected by an electromagnetic potential (V, A), despite being confined to a region in which both the magnetic field B and electric field E are zero.So as Alain Connes emphasizes - this is about a mathematical structure as a ratio - NOT about particular numbers. It is just from music theory that 2 and 3 are the simplest numbers to use to demonstrate this noncommutative phase structure with a geometric dimension of zero.
With the exception of B#, or the ditonic comma, ratio 531441/524288, Table 11.2 shows that the Chinese and Pythagorean progression of ratios are identical,That website clarifies that what you erroneously call the Pythagorean Comma is actually the Ditonic Comma. Philolaus identified the komma [531441 : 524288 aka the Ditonic Comma] with the unit, 1
note that all is geometric magnitude so that 12:8 (3/2) plus 8:6 (4/3) = 2/1 as geometric magnitude from the double octave. This has to use 0 to 8 as one "root tonic" for 6/8 wavelength as 4/3 frequency and then 0 to 12 as the other root tonic for 12/8 frequency as 3/2 with 8/12 wavelength. This is called the "phantom tonic" in music theory since the Perfect Fourth can not be created from the harmonic series as 3 does not go into 2 from 1 as the root tonic denominator. In other words the root tonic changes due to complementary opposites.
And so the use of "zero" is to create the geometric mean as logarithm by covering up the noncommutative phase. So at "zero" energy there is STILL a noncommutative phase quantum energy. Professor Richard McKirahan:
So you have 8 to the 6th power and 9 to the 6th power (531441) so that the 9 is equated to 3/2 squared as 9/4 halved to 9/8 with the 9 being 3 squared and so 9 to the 6th = 3 to the 12th.
So instead of taking 12:9, which is 3/4 of 12, we take 8:6, which is 3/4 of 8. And so by adding the length 12 to 8 [as geometric magnitude not wavelength!!] with the length 8 to 6, [as geometric magnitude, not wavelength!!] we get the length 12 to 6, which corresponds to the ratio 2:1.
We are told that 6:8:9:12 is just multiplying (1:4/3:3/2:2) times 6 but in actually 9/8 is assuming that 3/2 is also the geometric mean squared.
So then you get 6 whole tones as 9/8 - being greater than the octave by the so-called Pythagorean Comma. But again this is not the original Pythagorean Comma! This already assumes the octave is a geometric mean squared definition - not the number 2 as a doubling.
So you have 8 to the 6th power and 9 to the 6th power (531441) as the secret of the Ditonic Comma as the fake Pythagorean Comma due to geometric mean. From Euclid and earlier from Philolaus and Archytas as 8 to the 6th compared to 9 to the 6th which assumes that the starting root tonic "frequency" as the 1 is 8 to the 6th which is already from assuming the Perfect Fifth is squared!
262144 was then used by Aristotle (and Philolaus) as the starting root tonic value so that the octave was then 524288 as 8 to the 6th with the difference to the Perfect Fifth as 3 to the 12th being 531441 from 9 to the 6th. So that is the diatonic scale of 6 notes above the root tonic but with 12 notes - the error keeps multiplying for each note, as Sir James Jeans points out - so you have a quarter note error.
The 9/8 value assumes that the 5th being multiplied out is then "halved" back into the scale and so the noncommutative phase does not matter.
The Chinese system is based on the mathematical method of working back and forth by taking 3/2 of a base frequency, 3/4 of the frequency so produced, and so forth. Once a series of 12 frequencies is produced, they use each of those 12 frequencies as the fundamental frequency for a new scale. The result is 144 frequencies.
Notice that the ratios of this pentatonic scale originate in the first five tones of the up-and-down principle of scale generation; that is, 1/1 or do is kung, 3/2 or sol is chih, 9/8 or re is shang, 27/16 or la is yü, and 81/64 or mi is chiao
So Philolaus used 256/243 as the half tone =
while the 12th root of 2 as the half tone=
Pretty good Pre-Socratic Logarithmic math!!
1.423828125 x 1.05349794239 = 1.5
That is Philolaus stating 9/8 to the 3rd x 254/243= the Perfect Fifth.
His math is even BETTER than logarithmic tuning - but that's why he still gets the Pythagorean Comma.
His value for the octave is 1.99999999999
His value for the Perfect Fourth is rounded up to 1.333334 whereas 4/3 is NOT rounded up and instead just 1.33333333
Fabio Bellissma DECIPHERS the Boethius image that is at the top of this blog!! googlebook review
This musical property is the counterpart of the principle mathematical characteristic of the Pythagorean diatonic, very Pythagorean indeed, constituted by the fact that each interval of the scale is expressed by the ratios of type 2 to the m divided by 3 to the n OR 3 to the m divided by 2 to the n.
The chromatic Pythagorean scale is:The notes F and c found in the table presented before, 131072:177147 = 0,73990527641 and 262144:531441 = 0,49327018427, respectively are slightly different from 3:4 and 1:2, which are those of the Pythagorean scale. as mentioned before, a do (c) was considered the initial note, so this result means that the fourth and the octave obtained by the aforementioned process are not
exactly the same obtained in the discovery with the monochord, attributed to Pythagoras, of the numerical ratios corresponding to the main intervals of the musical scale. One can put it down to the fact that the aforementioned cycles of the octaves and of fifths don’t meet, as it will be shown later and that the best approximation for this meeting occurs with 7 octaves and 12 fifths. So it is plausible that the ratios determined by the fourth and by the octave in the Pythagorean scale result from the experiment of the monochord, namely, from the musical perception that such ratios produce these intervals, they were the perfect consonances in the context of Greek music and which were capable of being produced by ratios composed by small numbers, being easier to relate to the intervals using only the ear....
... Pythagorean tuning were first considered by musical theorists [445,7] in order to answer the need of a new musical language provided by polyphony. anachronistically speaking, this means that, supposing both these both cycles meet, there would be m and n integers such that (2:3)n = (1:2)m, that is, 3n = 2m+n, which is impossible, since the left term is odd and the right is even.
A further reference to the problem of 'cutting the tone' can be found in Plutarchus (1962, De Iside et Osiride, 367 e-f). Plutarchus ascribed it to the Pythagoreans, but there he was speaking about Osiris and the unlucky 17, which was the natural 'candidate' to cut the tone (between 8 and 9, i.e between 16 and 18).....
In Book III Aristoxenus' theory is criticized from a Pythagorean point of view. The crucial question is 'cutting the tone', i.e. finding a middle proportional between 8 and 9. It is easy to see that 17 is not a middle proportional between 16 and 18. In III,5 and III,8 Boethius describes Philolaus' attempt to cut the tone. Philolaus set out to solve the problem starting from purely 'numeric' considerations by displaying an idea of 'ratio' as a generic relation between two numbers and by analyzing the music intervals as ratios between two numbers or as differences between the same numbers or as single numbers. There we can find the statement of the problem in terms of dichotomy of the tone, of the comma and of the diesis (Diels and Kranz 1964, 44 A26, B6).
The only argument I know to support the antiquity (at least, to the second half of the 5th century) of the 'geometrical' approach is provided by Theaetetus’ words in Theaetetus 147c- 148b (Plato 1964): "Theodorus was proving for us via diagrams something about powers, in particular about the 3-foot and the 5-foot, demonstrating that these are not commensurable in length with the 1-foot, and selecting each power individually in this way up to the 17-foot, but in this one for some reason he encountered difficulty". No doubt this proposition seems to be embedded into a geometrical approach. There has been a great debate about the ‘geometric’ reason of the ‘difficulty’ encountered by Theodorus with the 17-foot (Knorr 1975). The narrated episode had to happen in 399 B.C. but the dialogue was written probably around 367 B.C. Hence, when he wrote this dialogue, Plato had credibly a good acquaintance with the well established 'geometric' theory of incommensurability, most of all after Theaetetus, and probably he remembered something about its 'state of art' in his youth, taught in the lessons of Theodorus, and 'some difficulty' with 17. But I can guess that he could have been wrong in the details. The trouble with 17 could be connected not to some geometrical construction, but to the role this number played in the Pythagorean arithmetic, where it was called the 'obstacle', because it "broke the proportion of 9/8 in not equal intervals" (Plutarchus 1962, De Iside et Osiride, 367 f). This hypothesis was considered even by an anonymous commentator (Diels, Schubart, Heiberg 1905), but ignored by Knorr (1975, note 79) for the 'inappropriateness for the context': the geometrical context was unsuitable for a musical interpretation, but this was clear also for the commentator. In order to put forward that musical interpretation credibly the commentator had good reasons to think that a contamination between music and geometry was possible. There is also a second oddity in the episode above.
Why did Theaetetus mention 3-foot and 5-foot among the ‘powers not commensurable in lengths with the 1-foot’, but ignored 2-foot, the most important one? (Burnyeat 1976). If my hypoythesis is not wrong the answer is easy: 3 and 5 are the natural candidates as arithmetic means for the dichotomy of the octave (1:2) and of the fifth (2:3). In other words 3, 5 and 17 are the most natural values for cutting the most important musical ratios, whereas 2 plays no role in this musical problem. Last but not least, my interpretation could even make clear a third problem in Plato’s text: the ambiguous employment of the word dÚn©mij and its derived terms. According to the geometrical interpretation, within few lines they can mean ‘square’, ‘side of the square’ and ‘possibility’ (Knorr 1975, 62-68, Szabo 1978, 36-45), i.e. heterogeneous magnitudes and concepts. I have to remark that in a musical interpretation all the magnitudes are homogeneous, and dÚn©mij could have consistently meant ‘possible musical value’, i.e. tone that can or cannot cut an interval.