Loudest of all-
p 4 \
Next loudest.
2p 8 \
p+q 9 \
p-q 1 \
Next loudest.
3p 12 \
p+q+r 15 \
2p-q 3 second difference tones.
Science
and Music
Of course, the differential tones produce further differential tones, and the summation
tones, still further summation tones, and this goes on to infinity.
Music for All of Us.
"In compound musical sounds, or clangs, we generally hear, along with the fundamental
n, the overtones or partial tones 2n, 3n, 4n, etc., each of which corresponds to
simple pendular vibrations. If two such musical sounds, to the fundamentals of which the rates of vibration n and m correspond, be melodically or harmonically combined,
there may result, if certain relations n and m are satisfied, a partial coincidence
of the harmonics, whereby in the first case the relationship of the two sounds is rendered perceptible, and in the second a diminution of beats is effected".
(Footnote: The p th harmonic of n coincides with the q th of m when pn = qm, that
is m = (p/q) n, where p and q are whole numbers").
Harmony and discord are, however, not determined by beats alone. In melodic as well
as harmonic combinations, notes whose rates of vibrations bear to one another some
simple ratio, are distinguished (1) by their agreeableness and (2) by a sensation
characteristic of
this ratio. As for agreeable quality, there is no denying this is partly explained
by the coincidence of the overtones and, in the case of harmonic combination, by
the consequent effacement of the beats, resulting always where the ratios of the
numbers representing the vibrations satisfy certain definite conditions"...
" Let us now turn to the second point, the characteristic sensation corresponding
to each interval, and ask if this can be explained on our present theory. If a
fundamental n can be melodically or harmonically combined with its 3rd m, the
5th harmonic of the first note (5n) will coincide with the forth of the second note (4m).
This, according to the theory of Helmholtz, is the common feature characterizing
all third combinations".
See
Hetrodyning and Powers of Phi
The Analysis of Sensations
... Euler develops an arithmetical rule for calculating the degree of harmoniousness
of an interval or a chord from the ratios of the periods of the vibrations which
characterize the intervals. The Unison belongs to the first degree, the Octave
to the second, the Twelfth and Double Octave to the third, the Fifth to the forth, the
Forth to the fifth, the Major Tenth and Eleventh to the sixth, the Major Sixth
and Major Third to the seventh, the minor Sixth and minor Third to the eighth,
the subminor Seventh 4:7 to the ninth, and so on..."
Sensations of Tone.
Sensations of Tone.
Sound
67.10.1
(Note:The following features were found to be linked to the musical scale intervals
in a sometimes uncanny manner as you may also note).
67.13
67.15
67.17
67.18 Universal Consonants C-E-G from Rameau
67.20 Ratio 192 / 243 or 8 / 9 ^2/3
67.21
67.23.2
67.24 Ratio 6561 / 8192
67.27.1
67.27.2
7.29
67.35
67.38
67.39 Newton's 'Rings' Experiment and the Dorian Scale
67.46 "The Gap" (The Real Mediator between "This Side" and the "Other Side").
"Newton took the mean of each pair of alternative divisions, and found on measuring
the length of the final divisions of the image of the spectrum that they were in
approximately the same ratios as the divisions of a string capable of sounding the
notes in an octave".
"Newton was clearly fond of this analogy, in its newly extended form, for when he
published his treatise 'Opticks' in 1704 he repeated the material on the octave
of color in much the same way. What is more, when he came to summarize his measurements of the diameters of what we now call "Newton's rings", he again used the musical
scale to do so. Remember that at first he did not produce the rings by monochromatic
light, and that the rings were therefore colored. It was natural enough, under
these circumstances, that he should extend his musical comparison. What he does is
calculate the thickness of the wedge of air between the glasses at those points
where his rings are made by his 7 spectral colours. He finds that these thicknesses
are in the ratio of the lengths of a string yielding the notes of the octave, raised
to the power of 2/3".
67.47
(Footnote: Newton used the Dorian mode. The thicknesses are then proportional to the
following numbers":
1, (8/9)^2/3, (5/6)^2/3, (3/4)^2/3, (2/3)^2/3, (3/5)^2/3, (9/16)^2/3,
(1/2)^2/3.
(Note: Fullers calculations of the number of vector edges:
1, 8/9 x 3-2, 5/6 x 3-2, 3/4 x 3-2, etc.
67.48.1 (Correspondences 193/243 see Fuller: Synergetics 223.17).
67.48.2
67.48.3 (Correspondences 2/3 see Fuller: Synergetics 620.03, 973.30, 982.31, 1009.98).
67.48.4 (Correspondences, Dorian mode).
"These thicknesses he subsequently equated with what he calls "the Intervals of
the following Fits of easy reflexion and easy Transmission". The explanation of
the rings offered by Newton on the basis of his theory of fits of easy reflexion
and Transmission". I have mentioned so many details in the course of my account
of Newton's analogy between light and sound that the shifting character of the
analogy has probably been lost to view. I will summarize the six examples I have
now given, three from 1672 and three from 1675:
N(1) Correspondences (some would call these analogies) are set up (or implied) between
the following concepts:
The Universal Frame
© Copyright. Robert Grace. 1999.
"The
principle mark of distinction between strings and other bodies which vibrate sympathetically,
is that different vibrating forms of strings give simple tones corresponding to the harmonic upper partial tones of the prime tone, whereas the secondary
simple tones of membranes, bells, rods &c., are inharmonic with the prime tone,
and the masses of air resonators have generally only very high upper partial tones, also chiefly inharmonic with the prime tone, and not capable of being much reinforced
by the resonator".
Sensations of Tone.
© Copyright. Robert Grace. 1999
To take a concrete instance, lets suppose that the three original frequencies p,
q, r are those of the notes c', e', g', the 4th, 5th and 6th harmonics of CC. Then
the full table will stand as follows:
q 5 the fundamental tone.
r 6 /
2q 10 second harmonics of the foregoing.
2r 12 /
q+r 10 first summation tones.
p+r 11 /
q-r 1 first difference tones.
p-r 2 /
3q 15 the third harmonics.
3r 18 /
2p+q 13 \
2q+p 14 \
2q+r 16 second summation tones.
2r+q 17 /
2r+p 16 /
2p+r 14 /
p-r-q, 7 etc.
By Sir James Jeans
M 781.1 J346 pg. 236
67.2 Summation Tones
By Leopold Stokowski
780. ST67
67.3 Diagram. Harmonic Table to the 12th
Earnst Mach
67.4 Harmonics
67.5 Diagram. Harmonics
By Dr Ernst Mach
152. M18
..."resolution into partial tones, mathematically expressed, is effected by Fourier's
law, which shews how any periodically variably magnitude, whatever be its nature,
can be expressed by a sum of the simplest periodic magnitudes. Footnote: Namely
magnitudes which vary as sines and cosines".
67.6 Table. The Unevenly Numbered Harmonics
of C 66 up to the 63rd
Helmholtz
67.7 Table. Number of any Interval, not exceeding the Tritone, contained in an Octave
Helmholtz
67.8 Diagram. Lissajous Figures and Relative Frequencies (1:2, 2:3) of the Components
By Arther Taber Jones
534. J71
67.9 Table. Determination of Tone
Helmholtz
67.10 Table of Intervals not exceeding an Octave
Helmholtz
67.11 Features Linked to Helmholtz 67.10 Table:
67.12 Log base 12sq.rt2
67.19
67.22 "The Mediator"
67.23.1
67.25 (Corr. Vibration of Hydrogen).
67.26 "This Side"
67.30 This "Gap" spans musical notes e flat to f flat in Helmholtz complete Determination
of Tone Table. It represents the gaps in Newtons prism light experiments
wherein the gaps between the prism colors were found to be in Dorian scale ratio, raised to the 2/3 power.
67.31 Ratio 25 / 32
67.32 (Corr. 1280 (2^8 x 5)).
67.33 (Corr. Fuller, Synergetics 953.60).
67.40 "The Gap" (The Real Mediator between "This Side" and the "Other Side").
67.42 (Corr, Helmholtz Tone Determination Table- Vibration Hydrogen 6561 / 8192)
67.44 (Corr. Dorian mode, Color and Music(1891)).
© Copyright. Robert Grace.1999
67.45 Newton's 'Rings' Experiment and the Dorian Scale
and also between the following:
By J.D. North
QB 29. N67
(Note: Summations:
67.49 Harmonics in Strings and Membranes, Bells and Rods