A NEW OR IMPROVED MODE OF AND APPARATUS FOR COMPARING MUSICAL KEYS, AND FOR EXAMINING AND EXPLAINING MUSICAL RELATIONS AND HARMONIES, CHORDS AND CONSONANCES
Year: 1869

page_text: PROVISIONAL SPECIFICATION left by Alfred Earnshaw, George
Edward Earnshaw, and Frederic Earnshaw at the Office of the
Commissioners of Patents, with their Petition, on the 8th Septem-
ber 1869.
We, ALFRED EARNSHAW and GEORGE EDWARD EARNSHAW, both of
Sheffield, in the County of York, Merchants' Clerks, and FREDERIC
EARNSHAW, of Harrington Street, in the County of Middlesex, Bankers'
Clerk, do hereby declare the nature of the said Invention for "A NEW
OR IMPROVED MODE OF AND APPARATUS FOR COMPARING MUSICAL KEYS, AND FOR
EXAMINING AND EXPLAINING MUSICAL RELATIONS AND HARMONIES, CHORDS AND
CONSONANCES," to be as follows:-
Our improved mode of comparing musical keys, examining and
explaining musical relations, and harmonies, chords, and consonances,
consists in the use and application for this purpose of graduated scales
marked with letters or signs instead of attaining the same result by
mathematical calculation only, as hitherto.
It has been long known to mathematicians that the musical interval
between any two musical notes of different pitch may be arithmetically
expressed by the difference of the logarithms of the two arithmetical

page_text: numbers which respectively denote the number of vibrations made in a
second of time by two stretched strings sounding those two notes.
This mathematical principle we have applied to the apparatus of our
Invention.
The apparatus consists of two circular discs, or it may be of two
parallel slides of suitable dimensions and material, which discs or slides
are marked or graduated on the well-known mathematical principle
of measure before referred to, viz., that the musical interval between
two notes is correctly measured arithmetically by the difference of the
logarithms of the numbers of vibrations per second of the strings sound-
ing those two notes. These graduations are severally denoted by figures,
letters, signs, and words representing the names by which musical notes
and intervals are known, so that by comparing the two discs (or as the
case may be the two slides) together the musical interval, for instance,
between any two musical notes marked upon said discs or slides may be
distinctly and correctly represented to the eye by the space between the
marks corresponding to the said musical notes, and similarly for the
other purposes mentioned,
We will first describe the arrangement and formation of the discs,
which are preferably of unequal diameters, and divided or graduated
near the circumferences in such manner that when the small disc is
applied concentrically on the larger one the two circles of graduations
will be exhibited, and may be compared the one with the other. The
circumferences of these discs are divided primarily into seven parts by
the marks or graduations above mentioned, each of which parts is again
subdivided by other marks into three portions, making twenty-one parts
in all. The seven principal graduations on the large disc and on one
side of the small disc are indicated by the letters A, B, C, D, E, F, G,
which represent the seven notes of which an octave is composed, so that
the space between one note and its octave occupies the whole of the
circumference of each disc, the longth of the principal spaces or portions
into which the circumference of each disc is divided thus being
dependent upon the diameter of the disc, or of the circle of graduations
marked thereon. The relative proportion of these spaces the one to the
other is however regulated by the mathematical principle of measure
before referred to, as any one of the spaces is proportional to the
difference of the logarithms of the number of vibrations per second of
the notes represented at each extremity of that space. For instance, the

page_text: space from A to D is proportional to the difference of the logarithms
of the number of vibrations corresponding to note A. and note D. Thus
the seven spaces representing an octave. into which each disc is divided
are of three different sizes. The spaces between AB, OD, FG, are
the largest, and are, equal to one another (the differences between the
logarithms of the numbers of vibrations corresponding to those notes
being the greatest and equal), and any one of the intervals between
these notes is called a "major tone." The next in size are DE, GA,
which are also equal for the same reason, and represent the "minor
tones." The smallest are BC, EF, also equal, and each interval is
called a "limma."
As before mentioned, each of these seven spaces on the large disc and
on one side of the small disc is subdivided into three parts by other
graduations distinguished from the principal graduations and from one
another by the signs or marks usually employed to designate sharps
and flats. The positions of these subdivisions or secondary graduations
are ascertained and regulated similarly to those of the principal gra-
duations or naturals, that is, by aid of the mathematical principle of
measure, as before described. The other face of the small disc, although
not subdivided for sharps and flats, is however provided with the seven
principal graduations or naturals, which are marked Do, Re, Mi, Fa, Sol,
La, Si, and also figured 1, 2, 3, 4, 5, 6, 7, respectively, and represent the
diatonic scale, the face first described representing the chromatic scale.
The diatonic face is moreover provided with marks indicating the
places of the minor third, minor sixth, and minor seventh, which fall
respectively between the graduations marked Re and Mi, Sol and La,
La and Si. The figures which are also annexed, to the graduations on
the diatonic scale refer to the principles of harmony, as these figures
correspond with the figures used by composers to denote chords in
the bass.
In forming the discs of ivory or other similar substance we prefer to
recess the small disc in the larger one so as to turn freely therein by
hollowing out the face of the latter, so that the surfaces of the two discs.
will be flush and the comparison of the two scales thereby facilitated.
The apparatus when straight instead of circular in form consists of a
straight bar or rule, having a longitudinal groove along its face from end
to end, in which is placed a sliding piece capable of being moved along
the groove to any position therein. The sliding piece is provided with

page_text: graduations representing one octave, and marked similarly to the small
disc to which it corresponds in use. The graduations are arranged along
either edge of the sliding piece, and the two edges are marked with
letters, words, figures, and signs to represent the chromatic and diatonic
scales respectively, as do the two faces of the small disc. The face of 5
the bar or rule along either edge of the groove is graduated and marked
in a similar manner to the larger of the two discs before described (to
which the rule also corresponds), except that the rule is of such
dimensions as to comprise two octaves in its length. The reason for
1
this is obvious, as one octave on the sliding piece must always face a
whole octave on the part in which its slides, or vice versa, therefore
requiring two octaves to be marked either on the sliding piece or on the
rule along which it is moved; whereas in the case of a circular disc
the octave repeats itself by reason of the circle returning into itself.
The length of the spaces and the positions of the graduations on the 1
slides are regulated and obtained in a precisely similar manner, to that
before described with reference to the discs.
We will now proceed to describe the mode of adjusting the apparatus
(which is the same whether discs or slides are employed) in order to
ascertain the results required.
As an example, we will suppose it is desired to compare different
keys, say, the key of A with that of G. For this purpose the small disc
is placed upon the larger diso with the face upwards, that is, graduated
and marked with letters and signs similarly to the larger disc. Now
the scale of key A. is,―
•
ABCDEFG
and opposite to these marks on the larger disc we find on the smaller disc
GABCDEF#;
the results thereby obtained are, 1st, we see how to transpose a tune
from the key of A to that of G, and conversely from the key of G to that
of A; 2ndly, it shows that there is only one sharp in the key of G to

page_text: As another example suppose it is required to ascertain the musical
interval between note A and note D. The smaller disc is placed as
before on the larger disc, but with the numbered face upwards, and the
figure 1 thereon is made to coincide with A on the larger disc; then D
on the larger disc will be found to be slightly beyond the figure 4 on the
smaller disc, thereby showing that the musical interval between notes A.
and D is rather larger than a perfect fourth, or, musically expressed, that
D is a sharp 4th to A. The apparātus is adjusted in a similar manner in
order to effect the various other objects mentioned.
As musical scales differ from each other, being varied according to the
system of tuning the musical instrument employed, each system would
require its peculiar set of discs or slides calculated to suit the system,
said discs or slides being graduated on the principle before referred to,
so that any two discs or any two slides which represent the same or
different musical scales may be adjusted in such manner that the
graduations and necessary marks upon both of the discs (or slides) may
be seen at the same time, and their agreements and disagreements
observed and estimated.