ee 
Nov. 24, 1870] 
MUSICAL INTERVALS 
i BRIEF summary of the remarkable papers on 
musical intervals, by MM. Cornu and Mercadier, 
published in the Comtes Rendus of February in last year, 
may perhaps interest those of the readers of NATURE 
who have not met with the original. 
The authors remark, in the first place, that two schemes 
of musical intervals haye been proposed, in which the 
ee of the number of vibrations in a given time are as 
ollows :— 
Octave. Fifth. Fourth, Major Minor Sixth. Seventh. 
Third, Third. 
3 4 3* 2° 33 a 
ee OS 
2 3 we 38 Za 27 
3 4 5 6 5 15 
ee 
2 3 4 5 3 8 
and the object of the paper is to examine the claims of 
each for adoption. 
On comparing these two systems, it may be observed 
that the Octave, Fifth, and Fourth are the same in both, 
and that the other intervals are connected by the following 
relations :— 
81 81 
ee, 
econ 2-05 3h SO 
A third view of the subject, according to which either 
of the two systems may be adopted indifferently, because 
they differ only by a “comma” (an interval represented 
by 81 : 80), may be at once dismissed, since, as a matter 
of fact, the ear is capable of appreciating intervals much 
smaller than the comma. 
As regards the scheme (2), it seems impossible not to 
admit that a major third is most harmonious when the 
resultant tone is exactly the double octave below the 
fundamental note, z.¢. when the interval is represented by 
the ration 5:4. Any deviation_from this proportion pro- 
duces unpleasant beats. This argument seems decisive 
in favour of (2). 
On the other hand, an ear which hears successively the 
notes emitted by an entire string, and by #ths of its length, 
will pronounce the third so formed to be too low. And, 
in fact, stringed instruments for concerted music are tuned 
by perfect fifths. Thus the intervals given by the violon- 
cello, alto, and violins, &c., in a concerto would be 
6,2, a, a, &. 
But this involves a third (c, e) defined by the ratio = 
for e must be the double octave below e”, and the interval 
(c, e”) is by hypothesis (3) ; hence 
waa b(y=3 
which is the ratio for a major third according to scheme (1). 
Thus, experiment apparently gives contradictory results, 
The authors of the paper then proceed to describe a 
series of experiments made with thevoice, violoncello, violin, 
organ-pipes, and monochord, all of which lead to conclu- 
sions reconciling this apparent contradiction, viz. :— 
That musical intervals do not belong to any single 
scheme, but that the ear is capable of distinguishing be- 
tween the intervals of the two schemes in question, and 
requires, 
(a) When notes are heard in succession, forming what 
is called melody, that the intervals should belong to a 
series of fifths, in accordance with scheme (1) ; 
(2) When notes are heard simultaneously, forming 
chords, or harmony, that the intervals should be adjusted 
according to scheme (1). 
The detaiis of the experiments, which are well worth 
study, would be too long to give in extenso; but the sub- 
NATURE 

a 
joined table will enable the reader to form a judgment of 
the results. 







| Major Third. Fifth. 
| 
Notes produced by | Harmony.| Melody. ||Harmony.| Melody, 
| | 
| | 
Voice aan a= 1.260 ~- 1.497 
Violoncello 1.25 1.266 1.499 1.508 
Violin : 1.249 1.264 1.504 1.504 
Organ pipes. 1.252 1.267 1.493 1.497 
Monochord ... — 1.271 — 1,500 
Mean of observation 1.251 1.266 1.499 I. 501 
Calculation .. «| $=1.250 | §}—=1.2656 || #}=1.500 1.500 




The direct experiments were made with these two inter- 
vals only ; but the same conclusions are shown to apply 
to the other intervals. 
The authors then proceed to inquire whether there is 
any reason for limiting the prime numbers which enter 
into the ratios of the harmonic intervals to those (1, 2, 3, 5) 
actually occurring in scheme (1). An answer to this 
question is found in the chord of the dominant seventh, 
usually defined as the common chord with the addition of 
a minor third (e. g. Do, Mi, Sol, Sib). The ratios of these 
intervals, according to scheme (2) are 
De 3 eS 6 
La ae : 
4 2 2 5 
and the simplest whole numbers near to these are 4: 5 : 
6: 7; and these, it is argued, are in fact an improvement 
on the former. For the ear, which alone can decide the 
question, will choose those notes which will form a chord 
devoid of beats, and whose difference tones do not intro- 
duce any notes foreign to the chord itself. Now 
— : — X — = 20: 25 : 30: 36 
7-6=1, 6-5 =1, 5-4=1 
=O eral 2) Ore Ar 2 
lie Aas 
So that from the chord 4: 5: 6: 7, we obtain the group 
of difference tones I, 2, 3, all of which belong to the 
natural series terminating with the chord itself. While 
from) the chord!) 20) 9) 25)29 3013" 360 — 4s) Sh) Oe tAae 
we at once derive an inharmonious difference tone. 
This a gréorz conclusion may be verified on the violin, 
by first tuning the two upper strings in unison ; then by 
shortening one of them so as to form a minor third 
(6:5) with a difference tone 1; and finally, shortening the 
other until the difference tone 1 is again heard. This 
will, of course, give an interval (7 : 6) perfectly agreeable 
when sounded simultaneously, but not so when sounded 
in succession. In the same way the ear might be called 
upon to decide whether the numbers 11, 13, &c., are or are 
not admissible in harmony. 
I trust that this very brief abstract may induce some 
of your readers to examine the paper itself, 
W. SPOTTISWOODE 

ON THE GREAT MOVEMENTS OF THE 
ATMOSPHERE * 
MY original paper was based on the mean monthly pressures 
calculated for 516 places, and on the mean monthly direc- 
tion of the wind calculated for 203 places over the whole surface 
of the earth. From these mean pressures the monthly isobars were 
drawn for every tenth of an inch, a pressure of thirty inches and 
upwards being represented on the charts by red-coloured isobars, 
and pressures of 29°9 inches and less by isobars coloured blue. 
Thus the distribution of the mass of the earth’s atmosphere from 
month to month was graphically represented, the red linesshowing 
* This paper, presented to the recent meejing of the “‘ British Associa- 
tion” at Liverpool, is a brief 7ésusé of a paper, and the discussion which 
followed, ‘‘On the Mean Pressure of the Atmosphere, and the Prevailing 
Winds over the Globe for the Months and for the Year,” originally read 
before the Royal Society of Edinburgh, and published in the Transactions 
of that Society in the beginning of December last, 
