652 
If the order of the tints is recurrent, it is only using another 
word for the same fact to say that it is circular ; and it is possible 
so to arrange the colours of the spectrum in a circle that any 
two tints which are opposite to each other shall be complemen- 
taries. 
Grassmann’s results are purely theoretical, but they coincide 
in a great degree with the experimental determinations of 
Helmholtz, and of Clerk Maxwell. In such experiments the 
method is to mix two or more coloured lights by letting them 
fall on the same spot of white paper. Mixture of colouring 
stuffs will not give the same result. 
I now come to some remarks of my own on the theory of 
complementaries. 
If colours are so arranged on the circumference of a circle that 
every tint has its complementary opposite to it, as has been done 
by Newton and by Grassmann after him, any two tints which are 
180° apart are complementaries, and any two tints which are 
360° apart coincide. If then the theory of the octave is true, 
of two tints which are 360° apart, the number of vibrations in a 
second (or the frequency of vibrations) of one is twice that of 
the other. It might be expected that the ratio of the frequency 
of vibrations between any two tints which are 180° apart, would 
be the square root of this; or, in other words, that when the 
frequency of the vibrations of any colour is known, that of its 
complementary might be found by multiplying or dividing, as 
the case may be, by the square root of two. 
To put this in another form: If we so arrange the tints, from 
red to its octave, where purple turns red again, round the 360° 
of a circle, that any two tints separated by equal areas shall 
have their frequencies of vibration in equal ratios; then, as the 
frequencies of vibration of the two reds which are separated by 
360° stand to each other in the ratio of 2 to 1, the frequencies of 
vibration of any two tints which are separated by 180° will be to 
each other in the ratio of the square root of 2 to1. Nowif the 
theory of a chromatic octave be true, the pair of tints which are 
360° apart are exactly alike, and we might expect those which 
are 180° apart to be complementary to each other. 
But this is not the case. 
The ratios of the wave-lengths and of the frequencies of vibra- 
tion (which, of course, are in the ratios of the reciprocals of the 
wave-lengths), corresponding to various tints, have been deter- 
mined with great accuracy by Prof. Clerk Maxwell (Pilosophical 
Transactions, 1860), by means of an interference-spectrum. ‘The 
numbers in the following table, which are given on his authority, 
are the numbers of wave-lengths in the retardations ; each colour 
is written in the same line with its complementary. In the case 
of bluish-green, blue, and indigo, I take the middle one of three 
places in the same colour. 
Red 36°40 Bluish green 48°30 
Orange 39°80 Blue 51°80 
Yellow 41°40 Indigo 54°70 
If the frequency of vibration of the colours in the second 
column were to that of their complementaries in the first, in the 
ratio of the square root of 2 to1, the numbers would be— 
Bluish green 51°47 
Blue 56°28 
Indigo 58°54 
Thus the frequencies as observed were considerably less than 
as calculated from the hypothesis. The differences are all on the 
one side, and are much too great to be the result of any acciden- 
tal error. The complementary tints in the foregoing table are 
not precisely opposite, but approach each other by the green 
side of the circle ; and if from the portions of the circle from red 
to yellow, and from bluish green to indigo, any two tints are 
taken which stand exactly opposite, so that their frequencies of 
vibration are in the ratio of 1 and the square root of 2, their uni- 
son will not give pure white, but white with a blue tinge. 
But does this disprove the hypothesis that the true complemen- 
taries are those tints whereof the frequencies of vibration are in 
the ratio of 1 and the square root of 2? I think not. 
Complementaries are usually understood to be tints, which by 
combination form a colour sensibly identical with that of sun- 
shine. But is this correct? The solar spectrum is not pure, in 
consequence of the great number of absorption lines towards 
the violet end. That of the electric light, on the contrary, is 
free from absorption lines, and, in consequence of their absence, 
the electric light is sensibly bluer than that of the sun. If now 
the colour of the electric light, instead of that of sunshine, were 
taken as the true white, it appears probable that experiment 
INA ORE, 
[April 28, 1870 
would show the frequencies of vibration in any colour and its 
complementary to be in the ratio of 1 and the square root of 2. 
There are some remarks on this subject in the 2nd vol. of my 
work on ‘‘ Habit and Intelligence,” of which book you inserted 
a notice by Mr. Wallace on 15th Nov. and 2nd Dec., 1869, but 
it is more thoroughly thought out in this letter. 
JosEPH JOHN MuRPHY 
Old Forge, Dunmurry, Co. Antrim, April 16 
M. Rapa, in his ‘‘ Acoustique,” says :—‘* The disdain with 
which most musicians repel the invasion of their domain by the 
exact sciences is to a certain extent justified.” I venture to think 
it is very much justified, since little has been accomplished in aid 
of a technical theory of music by scientific men from Pythagoras 
down to Helmholtz. The highest service the mathematicians 
haye rendered was to assist in destroying the application of their 
own theories by establishing the universally received system of 
“equal temperament.” Now that the ‘‘effects” of colours are 
falling under the manipulation of mathematicians, could not the 
learned who are occupying your columns with the old discussion 
on ‘‘ ratios ” condescend to receive some warning from the history 
of ‘speculative music ” ? 
One of your correspondents asks for a ‘‘ white sound”! Seeing 
that a complementary colour completes the numerical value of 
the white rays more or less as the inversion of any musical in- 
terval completes the octave, is it unscientific to assume that the 
white ray must be the analogue of the monochord? Allow me 
to assume that it is so, and that white and black are complemen- 
taries, as M. Chevreul admits. Let me also assume that they are 
the two extremes of Jight and shade, including many gradations 
—many octaves—of intermediate shades of grey. 
Taking any normal gradation of light and shade, and calling 
it grey or white, as the generator, the primary colours and their 
complementaries correspond to the harmonics, thus :— 
yELLOW 
BLUE ry 
RED o)0 ee 
(GREY) Pirie ee. 
I 
(WHITE) 
4 2 
GREEN 6D 
ORANGE 2 
VIOLET 
The following series of figures 
Wane ey NL 
a i) SOG By & 3 gd) B mM f 
represent what musicians call a table of inversions in the octave. 
It must be understood that in the system of inversions of num- 
bers I employ, what is meant by a number and its inversion are 
the distinct notes in the scale the two numbers represent. For 
instance, I to 5 is C to G in ascending, and the inversion is 1 to 
4, C to ¥—always in ascending, and counting from the genera- 
tor No. 1. 
Hence the following table of abstract intervals and their in- 
versions, produced in regular order from the generator :— 
Unison Minor Major Minor Major 
Be _ sien 3rd 3rd 7th 7th 5th 
er one }Red. Indigo: sett Yellow. Temon, Red+Green (brown). 
A : : 5 : (tritone). 
= : Major Minor Major Minor Sharp 
I 4th 6th 6th and and 4th 
White 
Sane} Green. Chrome. Orange. Violet. Lavender. Green+Red (brown 
Musically speaking, the generator No. 1, as the root of a 
natural dissonant chord 1, 5, 3, 7b, becomes No. 5 of the scale, 
or dominant of the key, the tonic of which is four degrees higher. 
Consequently, if there be any analogy at all between sound and 
light, or between musical intervals and colours, the key-note of 
the spectrum would be gveex—the ray of medium refrangibility— 
and four degrees higher than the dominant or generator wAz/e. In 
modern views of harmony, I may remark it is not the concord or 
triad, but the dissonance, which is the basis of the technical theory. 
Collecting, then, the abstract intervals given above, and con- 
