67.1 Harmonics 

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:

  1. The Fundamental Tones- Harmonics 4, 5, 6 of CC.
  2. First Difference Tones- CC (twice) and Harmonics 2 of CC.
  3. Second Harmonics- Harmonics 8, 10, 12 of CC.
  4. First Summation Tones- Harmonics 9, 10, 11 of CC.
  5. Second Difference Tones- Harmonics, 2, 3, 4, 5, 6, 7, 8 of CC.
  6. Third Harmonics- Harmonics 12, 15, 18 of CC..
  7. Second Summation Tones- Harmonics 13, 14, 15, 16, 17 of CC

Loudest of all-

p 4 \
q 5 the fundamental tone.
r 6 /

Next loudest.

2p 8 \
2q 10 second harmonics of the foregoing.
2r 12 /

p+q 9 \
q+r 10 first summation tones.
p+r 11 /

p-q 1 \
q-r 1 first difference tones.
p-r 2 /

Next loudest.

3p 12 \
3q 15 the third harmonics.
3r 18 /

p+q+r 15 \
2p+q 13 \
2q+p 14 \
2q+r 16 second summation tones.
2r+q 17 /
2r+p 16 /
2p+r 14 /

2p-q 3 second difference tones.
p-r-q, 7 etc.

Science and Music
By Sir James Jeans
M 781.1 J346 pg. 236

67.2 Summation Tones 

  1. The actual tones played- these are called the first components or fundamental tones.
  2. The higher components of the fundamental tones.
  3. The differential tones of the fundamentals.
  4. The differential tones of the components.
  5. The summation tones of the fundamentals.
  6. The summation tones of the components.
  7. The summation tones of the differentials of the fundamentals.
  8. The summation tones of the differentials of the components.

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.
By Leopold Stokowski
780. ST67

67.3 Diagram. Harmonic Table to the 12th  Earnst Mach

67.4 Harmonics 

"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").

67.5 Diagram. Harmonics 

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
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".

... 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..."

67.6 Table. The Unevenly Numbered Harmonics of C 66 up to the 63rd   Helmholtz

Sensations of Tone.

67.7 Table. Number of any Interval, not exceeding the Tritone, contained in an Octave   Helmholtz

Sensations of Tone.

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


(Note:The following features were found to be linked to the musical scale intervals in a sometimes uncanny manner as you may also note).

67.10 Table of Intervals not exceeding an Octave   Helmholtz

67.11 Features Linked to Helmholtz 67.10 Table: 

67.12 Log base 12sq.rt2 


67.14 Phi^2 x 6/5 = Pi 


.67.16 "Other Side" 


67.18 Universal Consonants C-E-G from Rameau 


67.20 Ratio 192 / 243 or 8 / 9 ^2/3 


67.22 "The Mediator" 



67.24 Ratio 6561 / 8192 

67.25 (Corr. Vibration of Hydrogen).

67.26 "This Side" 



67.28 The "Gap" is the Mediator between Phi of the "Other Side" and the hydrogen ratio 6561 on "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).

67.34 Ratio 5 / 8 


67.36 Fibonacci Ratio 

67.37 Ratio 3 / 5 


67.39 Newton's 'Rings' Experiment and the Dorian Scale 

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 

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".


(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).


67.48.3 (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:

  1. Air (a^1);
  2. Aether (a^2);
  3. Vibration in a sounding body (b^1);
  4. Vibration in a luminous body (b^2);
  5. The tone of sound (c^1);
  6. The color of light (c^2);
  7. The sensation of sound (s^1);
  8. The sensation of light (s^2);
  9. Vibration of the air (v^1);
  10. Vibration of the Aether (v^2);

and also between the following:

  1. Causation of v^1 (Vib. of the air) by b^1 (Vib. in a sounding body),
  2. Causation of v^2 (Vib. of the Aether) by b^2 (Vib. in a luminous body);
  3. Causation of s^1 (Sensation of sound) by v^1 (Vib. of the air), and of
  4. s^2 (Sensation of light) by v^2 (Vib. of the Aether)".

The Universal Frame
By J.D. North
QB 29. N67

(Note: Summations:

  1. b^1 (Vib. sounding body) causes v^1 (Vib. in air).
  2. b^2 (Vib. luminous body) causes
  3. v^2 (Vib. in Aether).
  4. v^1 (Vib. in air) causes s^1 (Sensation of sound).
  5. v^2 (Vib. in Aether) causes (Sensation of light)).

Copyright. Robert Grace. 1999.

67.49 Harmonics in Strings and Membranes, Bells and Rods 

"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.

Impossible Correspondence Index

Copyright. Robert Grace. 1999