IMPROVEMENTS IN INSTRUMENTS OR APPARATUS FOR TEACHING AND TRANSPOSING MUSIC
Year: 1865

page_text: LETTERS PATENT to Richard Archibald Brooman, of 166, Fleet Street,
in the City of London, Patent Agent, for the Invention of "IMPROVEMENTS
IN INSTRUMENTS OR APPARATUS FOR TEACHING AND TRANSPOSING MUSIC."
- A communication from abroad by Ferdinand Bellour, of Toulon,
France.
Sealed the 6th July 1865, and dated the 12th January 1865.
PROVISIONAL SPECIFICATION left by the said Richard Archibald
Brooman, at the Office of the Commissioners of Patents, with his
Petition, on the 12th January 1865.
I, RICHARD ARCHIBALD BROOMAN, of 166, Fleet Street, in the City of
London, Patent Agent, do bereby declare the nature of the said Invention for
"IMPROVEMENTS IN INSTRUMENTS OR APPARATUS FOR TEACHING AND TRANSPOSING
Music," to be as follows:-
This Invention consists of a transposing gammometer so arranged that the
person using it need not be a musician; if he be of ordinary intelligence and
able to read he can with this apparatus solve the most arduous problems of
solfeggio with the greatest precision. The apparatus possesses the immense
advantage of rendering visible, and as it were tangible, the most abstract

page_text: notions of solfeggio. In using this gammometer a child can solve these
abstract questions instantaneously, and can easily remember them, because
the demonstration and application of the question are manipulated and placed
before him by the teacher in a similar manner to the explanation by means of
machines of chemical and philosophical problems.
Among the questions solved instantaneously by this gammometer I may
mention the following:-The formation of all the scales or gamuts, major, minor,
aud enharmonic; it shows precisely any interval by demi-tones; it distinguishes
the constitution of chords (accords) and the intervals which compose them,
and consequently their inversion; it gives tho relative minors of a tone, and
reciprocally it gives the transposition from one tone to another; it serves to
modulate, that is to say, to pass from one tone to another. By the application
of the gammometer to an instrument with either a fixed or a transposing key-
board the foregoing advantages are obtained instantaneously. It will thus be
seen that there only remains for the master to teach the pupil the musical
reading, the rhythm, and the intonation. I divide the tone into two equal
demi-tones, as in the key-boards of pianos and organs in which the same key
serves two uses.
The gammometer may be made either circular or rectangular. The con-
struction and action will be fully understood by the annexed view of a circular
gammometer and the following explanations :-
The largest blue circle shows precisely the number of demi-tones which
compose each interval. To ascertain, for example, the number of demi-tones
in the 7th interval, look for the word "seventh" in the second blue circle, and
the number 11 which is above gives the exact number of demi-tones comprised
in this interval.
The second blue circle indicates the number of sounds in relation to the
order (rang) they occupy in the gamut; the figures in this circle indicate the
number of sounds which form a gamut, the order they occupy, and the number
of tones or demi-tones comprised between one of these sounds and that which
immediately follows it.
The third blue circle gives the constitution of the chords and the intervals
which compose them, and consequently their inversion. The numbers written
under the indication of the chord designate the order in the gamut of the
sounds which constitute the chord; for this purpose it is sufficient to write
tbem by commencing from left to right by superposing them, say, for instance,
the chord of the 7th of the 4th kind indicated on "fa," we find the numbers
4, 6, 8, 10, the corresponding notes fa, la, do, mi, constitute the chord in
Question.

page_text: The 4th blue circle gives the name of the sounds which form a gamut,
whether in the natural state or the accidental state; the large letters indicate
the natural and the small letters the accidental state. The full separating
bars indicate the notes which should be chosen, whatever be the accidental
before, to form the required gamut, or to transpose the piece before the operator.
For this purpose it is sufficient to take for the tonic name each of the names
in rotation which indicate the sounds of the natural gamut, say,-
do, re, mi, fa, sol, la, si,
re, mi, fa, sol, la, si, do, and so on.
The blue color in this transposing gammometer indicates the majors and the
red the minors.
Suppose it is required to know how. many flats there are in the gamut
of mi flat major, I introduce the small key suspended to the gammometer in
the opening left expressly for it at back, so that the point of this key exactly
coincides with the axis of the circumference of the gammometer; I turn it
round so that the line whether full or dotted which separates "re" sharp from
"mi" flat coincides perfectly with the triple full line of the fixed gammometer,
since in the gamut which it is desired to form mi flat the name of this sound
becomes tonic; then we find fa under the second full line, sol under the
third, la flat under the fourth, si flat under the fifth, do under the sixth, re
under the seventh, and under the eighth, which is the repetition of the first,
mi flat. Now, as the tonic sound mi is marked with a flat, in cominencing
the pupil will see that in this gamut there are three flats, si, mi, la. I proceed
in the same manner for all the major, minor, and enharmonic gamuts, taking
care in changing the mode to follow the proper color. To transpose with
this apparatus, suppose I have a piece in sol major, and it is desired to lower it
two tones or a major third, which gives the tone of mi flat, I form first the
gamut of mi flat, and I find, as above seen, three flats; I then move the dotted
line which separates re sharp from mi flat over the full line of sol in the fixed
gammometer, and every time there is in the piece of music a sol, mi flat is
read,—when a la, a fa,-and when a si, a sol, and the piece is transposed
instantaneously. To find the relative of a tone it is sufficient to place the
triple full lines in line, and we have all the relatives immediately. Thus we
see under do in the blue slide a la in the red slide, and la minor is the
relative of do major, and vice versâ ; all the relatives are thus placed before the
operator.
I now proceed to speak of modulations. To modulate being to pass from
one tone to another, the law of a modulation consists in the modifications which

page_text: the gamut of this tone is made to undergo to pass into another tone. Now
this transposing gammometer allowing of two gamuts being put in relation
instantaneously furnishes this law; thus I can immediately modulate to the
dominant or fifth, to the sub-dominant or fourth, or the minor homonymous
sound from its point of departure to the relative of its point of departure, to
the relative of the sound of the dominant, to the relative of the sound of the
sub-dominant. All this is obtained because I am able to place two gamuts in
relation, and I produce not only all these gamuts but also their major and
minor chords with their constitution. The preceding explanations of the
circular gammometer apply to the apparatus when made rectangular. In
teaching a class each pupil should hold a small apparatus in his hand
and follow on it the explanations given by the master on a largo apparatus.
The application of this transposing ganmoneter to instruments with a fixed
or a transposing key-board is, if possible, of still greater utility than that
before described, for not only in applying it to a key-board are all the advan-
tages herein-before cited obtained, but I am also enabled to transpose the
accompaniment of a piece, and that very rapidly. I may also add that it is
easy to make an accompaniment to any theme. To apply the gammometer
to instruments with a movable or transposing key-board I fix in relation to
the key-board a gammometer of two octaves constructed for the purpose
without slide, because in this case the transposing key-board replaces it. For
instruments with fixed key-boards I place in relation to the key-board a
gammometer of two octaves constructed for the purpose with slide.
For teaching the blind I construct the gammometer in relief. The move-
ment of the sliding parts may be produced by hand, by a crank handle, rack,
screw, or otherwise.
SPECIFICATION in pursuance of the conditions of the Letters Patent, filed
by the said Richard Archibald Brooman, in the Great Seal Patent Office,
on the 12th July 1865.
TO ALL TO WHOM THESE PRESENTS SHALL COME, I, RICHARD
ARCHIBALD BROOMAN, of 166, Fleet Street, in the City of London, Patent
Agent, send greeting.
WHEREAS Her most Excellent Majesty Queen Victoria, by Her Letters
Patent, bearing date the Twelfth day of January, in the year of our Lord One
thousand eight hundred and sixty-five, in the twenty-eighth year of Her reign,

page_text: did, for Herself, Her heirs and successors, give and grant unto me, the said
Richard Archibald Brooman, Her special licence that I, the said Richard
Archibald Brooman, my executors, administrators, and assigns, or such. others
as I, the said Richard Archibald Brooman, my executors, administrators, and
assigns, should at any time agree with, and no others, from time to time and
at all times thereafter during the term therein expressed, should and lawfully
might make, use, exercise, and vend, within the United Kingdom of Great
Britain and Ireland, the Channel Islands, and Isle of Man, an Invention for
"IMPROVEMENTS IN INSTRUMENTS OR APPARATUS FOR TEACHING AND TRANSPOSING
MUSIC," (communicated to me from abroad by Ferdinand Bellour, of Toulon,
France,) upon the condition (amongst others) that I, the said Richard
Archibald Brooman, my executors or administrators, by an instrument in
writing under my, or their, or one of their hands and scals, should particularly
describe and ascertain the nature of the said Invention, and in what manner
the same was to be performed, and cause the same to be filed in the Great
Seal Patent Office within six calendar months next and immediately after the
date of the said Letters Patent.
NOW KNOW YE, that I, the said Richard Archibald Brooman, do hereby
declare the nature of the said Invention, and in what manner the same is
to be performed, to be particularly described and ascertained in and by the
following statement thereof, reference being had to the Drawings hereunto
annexed, that is to say:-
This Invention consists of a transposing gammometer by means of which
any person who can read is enabled to solve without difficulty the most difficult
problems of solfeggio with mathematical precision; this result, so favorable to
the art of music, is due to the fact that this instrument renders the most
abstract notions of solfeggio visible and palpable. The transposing gammo-
meter is applicable to instruments with a fixed or with a transposing key-
board; it may be independent of the key-board or connected thereto. It may
be placed on the face of a clock or watch, or be fitted in any article desired, or
on any convenient support, and made portable.
Figure 1 of the accompanying Drawings represents the gammometer
made of rectilinear form; Figures 2 and 3 represent it made circular; the
disc seen in Figure 3 must be supposed placed below the disc seen in Figure 2,
so as to turn on a central pivot in the form of a circular slide ;* Figure 4
shows the gammometer applied to a key-board.
I proceed to point out the problems which can be solved with the aid of
his transposing gammometer.
1

page_text: PROBLEMS SOLVED INSTANTANEOUSLY BY THE GAMMOMETER.
1. It gives the generic designation of each of the sounds forming a scale,
of the degree or order each of these sounds occupies, the name of these degrees,
and the interval which separates them.
2. It indicates the formation of all the major, minor, chromatic, and enbar-
monic scales, with their armures (armures).
3. The number of sharps or flats of a scale being given it serves to ascertain
the name of the scale.
4. It shows precisely any interval by tones and semi-tones, and determines
the name of this interval. (The necessity of simplifying has induced the
Inventor to divide the tone into two equal semi-tones, as on the key-board of
ก piano or organ, but he does not wish thereby to destroy the difference which
exists between the diatonic semi-tone and the chromatic semi-tone.)
5. It distinguishes the constitution of the direct or inverted chords (accords)
and the intervals which compose them.
6. It serves to ascertain all the chords which agree with each sound of a
scale, and reciprocally a chord being given it points out with which scales it
can agree.
7. It gives the relative minors of a major, and reciprocally.
8. It gives the transposition of a tone into another tone.
9. It distinguishes the modifications to be applied to a scale in order to
modulate (law of modulations).
10. It indicates the attractions of the different degrees of a scale to be
resolved for such and such degrees of the same scale rather than for others.
11. It designates on what degrees are the quaver, the semi-quaver, and tho
demi-semi-quaver of a scale.
12. By its application to any instrument, with either a fixed or a movable
key-board, any person who can read and write but does not understand
either the notes or the key-board can easily solve all the problems above
recited.
DESCRIPTION OF THE GAMMOMETER.
(See the three arrangements represented in the Drawings.).
The gammometer (measure of the scale) is, if it may be so called, a musical
meter, or better still, a calculating musical rule; that is to say, that a typical
scale being given, all the other scales can be formed after this type. It is by
the juxtaposition of two scales, one of which (the typical scale) is fixed, and

page_text: are obtained. In the apparatus for instruments with a fixed key-board the
typical scale is movable, and the other is fixed and corresponds to the key-
board itself. The full black lines or marks designate the name of the sound
which it is necessary to choose to form such and such a scale, or the key on
which the finger must be placed. As in music there are two modes, tho
major and the minor, the fixed part (movable for instruments with a fixed
key-board) carries two scales. The solutions of the major will be found in the
blue part, and those of the minor in the red part.
METHOD OF USING THE TRANSPOSING GAMNOMETER, WHETHER CIRCULAR OR
RECTILINEAR.
1. It gives the generic designation of each of the sounds forming a scale, of the
degree or order each of these sounds occupies, the name of these degrees, and the
interval which separates them.-There will be observed in the Drawings DO
marked in large characters above the numeral 1 surmounted by the word
TONIC; consequently the first sound of a scale is called "tonic," the fourth
"fourth," and so on. The numeral indicates the order or degree which these
sounds occupy in the scale. In the circular gammometer and in the red part,
that is to say, the minor, the names of the different degrees of the scale are
marked in the last front circle, and in the rectilinear gammometer with two
octaves the generic names of the sounds are marked in each of the divisions
to the right of the numeral. It will be observed that the interval comprised
between the full black lines or marks carrying the names DO, RE, is double
the interval comprised between the full black lines or marks carrying tho
names MI, FA, for example. The interval which is equal to half of the first
is called "semi-tone," and the first which is double is called "tone." The
name and the kind of the interval which separates each of the sounds of the
scale are thus known.
2. The gammometer indicates the formation of all the major, minor, chromatic,
and enharmonic scales, with their armures.-The gammometers of circular form
and rectilinear form (containing two octaves) are specially intended for
persons who wish to learn solfeggio and the first notions of harmony, and the
rectilinear gammometer with key-board (and containing only one octave)
allows of the application to every instrument with a key-board of the means
before recited. To form a scale with the gammometer applied to a piano, and
to know which is the armure (armure) there will be at the clef, I proceed as
follows:-I place the gammometer (which may be folded) in the small space
behind the keys of the piano so that FA on the key-board of the gammo-
meter exactly coincides with the white finger of the same name on the real

page_text: key-board; the black and white keys act as a guide. The gammometer
proper is the small movable slide which falls back upon the piano. The fine
lines traversing the gammometer perpendicularly indicate the separation of the
keys of the piano, and the full black marks the keys on which it is necessary
to place the finger in order to form the desired scale or chord; by the triple
line at the left sight is not lost of the name of the sound, the point of depar-
ture. Thus, if the thick black line of the triple line be placed on the line of
separation between SI, DO, of the gammometer key-board, it will be seen
that DO coincides with DO, RE with RE, and so on; but it will not be the
same if the gammometer be made to slide so that the triple line coincides
with the line which separates do # or reb from RE. Suppose it is desired to
compose and execute this scale (RE major). For this purpose the finger
must be placed on the keys indicated by the full black marks, and if we follow
in the blue part we make the scale of RE major; after the solution, if it be
desired to render this scale minor, the red part must be followed without
changing the position of the gammometer, and we will then have the scale
of RE minor. We see first that the finger must be placed on the key
named RE, since it is divided by full black marks which have the name DO
on the gammometer; MI is divided by RE, but on FA there is a dotted line,
consequently the scale of RE major is not identical to the scale of DO major
marked on the gammometer. Let us see then on which key the full black
mark named MI will come. It comes on a black key on which is marked
fa ; therefore we have already a sharp before the FA. To continue, SOL
coincides with FA, LA with SOL, SI with LA, but the same obstacle is
met with at DO. Let us see then on which key the full black mark will
come; on a black key, which is marked do #. There are therefore in the
scale of RE major two sharps, fa and do. It will be the same for all the
other scales. For the rectilinear gammometer with two octaves it is only
necessary to push the slide from right to left, and for the circular gammometer
I introduce the key A (shown attached thereto) in the small opening bebind,
and I turn the slide from right to left, or vice versa, and the same solutions are
obtained. The gammometer with a key-board allows of studying at the same
time the composition of two scales of the same name, one of which is major
and the other minor, for example, SI major and SI minor; when the major
scale has been composed it is only necessary to apply the names of the sounds
of this scale to the full black marks of the red part, and the scale of SI
minor is obtained, and so on, The circular gammometer and the rectilinear
with two octaves even allow of studying at the same time a major scale
and the minor homonymous scale, as well as the homonymous scales, one of

page_text: which is relative of the first; take, for example, DO major. We can study DO
and DO minor, as before explained, plus LA minor and LA major; now LA
major and LA minor are two homonymous scales, and LA minor is the
relative minor of DO major. So again RE minor, RE major, and FA major,
FA minor, &c.
3. The number of sharps or flats of a scale being given, the gammometer serves
to ascertain the name of the scale.-This scale may be major or minor. The
following is how to act for the major scale with sharps; there will be read on
the blue part
0 2 4 6
with sharps DO RE MI fa#
1 3 5 7
SOL LA SI do #
dob reb mib FA
7 5 3 1
sol b lab sib DO with flats
6 4 2 0
Although this formula is not indicated on the gammometer, it is very clear
and easily recollected. The scale will be divided into two parts, each com-
posed of four sounds, each of which is separated by an interval of a tone or
two semi-tones, which gives the above table.
For major scales with sharps the reading will commence at the left, and on
each of the first four sounds the even numerals commencing with 0 will be
applied. Thus it will be seen that in DO major there is 0 sharp; in RE
major, two; in MI major, four; and in fa #major, six. For the four remain-
ing sounds the uneven numerals will be applied; thus in SOL major there is
one sharp; LA major, three; SI major, five; and do major, seven. For
the major scales with flats the same rule is followed by reading from right to
5 left; the even numerals are applied to the four first, and the uneven to the
four last. Thus in the four first la b major will have four flats, and in the
four last reb major will have five flats.
We need
0 2 4 6
with sharps LA SI do# re #
1 3
5 7
MI fa sol #la #
mib FA SOL LA with flats
20
lab sib DO
RE
7 5 3
1
6
4

page_text: the desired solution, for having determined the major scale which answers
the question, we have only to take the relative minor written below on the
gammometer, and this minor has the same armure. In fact, if we read
RE major in the blue part we immediately find SI minor below in the red
part; now RE major has two sharps as well as SI minor its relative, and so on.
4. The gammometer shows precisely any interval by tones and semi-tones, and
determines the name of this interval.-Suppose the question be, How many
semi-tones are their in a tenth ? In the rectilinear gammometer with two
octaves the answer is written, for there will be noticed below the generic
.names of the sounds small numerals between two lines; now below large
numerals 10 (tenth) we find 16, that is the number of semi-tones comprised
in this interval. But in the circular gammometer and the rectilinear applied
to the piano there is only one octave; consequently to find the required solu-
tion it is only necessary to observe that in commencing from left to right the
same DO which bore the number 1 becomes number 8, and therefore RE 9,
MI 10, &c. Now, since from DO TONIC to DO OCTAVE there are
twelve semi-tones, we say DO 12, and the semi-tone designated by the
numeral 1 becomes 13, and so on. In the gammometer applied to the piano
the semi-tones are indicated in the white space separating the blue and red
parts, and in the circular gammometer they are between the two smallest
fixed circles blue and red. Now to find the generic and specific name, the
number of tones and semi-tones which compose an interval, proceed as
follows: Suppose it be required to find the name of the interval from FA
to reb for example. It is necessary to make some brief explanations which
will enable this solution to be found. The firsts (tonic), fourths, fifths, and
octaves cannot be minors, they are "just or "perfect." The first "just"
is also called unison; these intervals are similar and identical in the two
modes. All the other intervals commencing from DO in the blue part (major)
with the natural notes are majors, and taken in the red part (minor) com-
mencing from LA with the natural notes (without accidentals) are minors.
It is necessary to accept the SECOND, which is major in the two modes.
The minor second is an interval of a semi-tone, as MI FA. It will be seen
that a minor interval has a semi-tone less than the major interval of the same
The augmented intervals have a semi-tone more than the just or
perfect or major intervals, and the diminished intervals a semi-tone less than
the just or perfect or minor intervals. These explanations having been given,
the generic and specific name of the interval indicated by any two notes can
be determined with the aid of the gammometer. Whatever be the aforesaid
interval from FA to reb the gammometer is made to slide, so that FA may
name.
"

page_text: coincide with DO, and if reb be sought on the slide, if the gammonieter be
circular or rectilinear with two octaves, or on the gammometer key-board if
the gammometer be applied to a piano, it will be seen that reb falls on the
full line of SIXTII. FA minor is therefore a minor sixth.
5. The gummometer distinguishes the constitution of the direct or inverted
chords (accords) and the intervals which compose them.-Chords are generally
composed of superposed thirds, except some, the name of which indicates the
exceptional constitution, say the chords of the sixth and of the fourth and sixth.
In fact, these chords are only inversions. Now the gammometer indicates the
different chords which can be placed on such and such a sound of a scale
whatever be the point of departure. Thus there will be observed, for example,
on the fifth of a major or minor scale "chord of seventh of dominant, 5, 7, 9, 11."
If we look for the names of these different sounds we find-5, SOL; 7, SI;
9, RE; 11, FA.
We write these figures and the names of these sounds, going from left to
5
. SOL
and we see that from SOL to SI there is an interval of third, from SOL
to RE an interval of fifth, and from SOL to FA an interval of seventh. It
is evident that by changing the name of the fundamental sound of this chord,
the name of the sounds composing it will change also, but their order or rank
will remain invariable. This explanation applies to the circular gammometer
and to the rectilinear gammometer with two octaves. With regard to the
gammometer applied to the piano it is only necessary to place the fingers on
the keys indicated by the full black marks, and we obtain and execute at the
same time the chords which are indicated at the head of the major and minor.
With regard to the chord of fourth and sixth augmented in the minor
mode, we find after the figures 11 the sign + which signifies plus, that is to
say the sixth plus or semi-tone, since it is augmented. Now in the division
between the eleventh and the twelfth we find below the dotted line the same
sign. It is there that the solution of this chord is found.
6. The gammometer serves to ascertain all the chords which agree with each
sound of a scale, and reciprocally a chord being given it points out with which
scales it can agree.-I will divide the solving of this question into two parts.
Firstly, it serves to ascertain all the chords which agree with each sound of
a scale." Take, for example, mib major. 1 place nib in relation with DO

page_text: of the blue part, and the fundamentals of the different chords are the degrees
corresponding to the different degrees of the typical scale of DO, above which
we read the name and the constitution of these chords. We are easily assured
that the second, the third, and the sixth can be the fundamental notes of per-
fect minor chords, for on putting the note LA of the red part successively in
relation with these three notes, we sce that the three notes LA, DO, MI of
the perfect minor chord correspond to three notes of the scale of mib major.
We see that the fourth and the fifth are the fundamental notes of perfect
major chords, since these notes being put in relation with DO, the fundamental
sound of the perfect major chord, the other notes MI, SOL of this chord
correspond to notes of mib major. We also see that the tonic mib is the
fundamental note of a chord of seventh major. The chords of sixth augmented,
and of fourth and sixth augmented, are also sometimes employed on the fourth
degree of a major scale; the chord of seventh of dominant is employed on the
dominant of a minor scale as on the dominant of a major scale; the chord of
fifth diminished is employed more particularly on the second of the minor
mode; the chord of seventh of the fourth kind or seventh major is employed
on the sixth degree of a minor scale; the chords of ninth are not indicated on
the gammometer, because the study thereof is correlative to that of the chord
of seventh of dominant of two modes, and because their employment is not
frequent. Secondly, "and reciprocally a chord being given it points out with
which scales it can agree.
To determine the converse, it is necessary pre-
viously to learn the constitution of the different chords of a scale by taking
the sound DO as a fundamental base; this will be studied as indicated in
division 4 of this Specification, and with the aid of the movable part of the
gammometer by placing DO successively in relation with the tonic or funda-
mental sound of each different chord, and by reading the composition of the
chord sought for on the movable part with DO for tonic. With the apparatus
for the piano it is the tonic of each different chord which is moved above DO
of the fixed part; take, for example, the chord of fourth and sixth augmented
FA, LA, SI (natural), re #. The slide is moved so that FA coincides with
DO on the gammometer key-board; in relation with the chord FA, LA, SI,
re, we read DO, MI, fa, la #. This therefore is the composition of the
chord of fourth and sixth augmented by taking DO for the fundamental sound,
and so for all the other chords, say DO, MI, SOL, perfect major chord; DO,
inib, SOL, perfect minor chord; DO, mib, SOL, sib, chord of seventh of
the second kind, and so on. Suppose SOL, si V, RE be the given chord;
by putting DO of the movable part above SOL of the fixed part we see that
the chord corresponds to DO, mib, SOL; therefore it is a perfect minor
"

page_text: chord belonging consequently to the major scales of which SOL is the second,
or the third, or the sixth, like FA, mib, sib, major, or belonging to the minor
scales of which SOL is the tonic or the fourth, like SOL, RE, &c.
7. The gammometer gives the relative minors of a major, and reciprocally.
These are found immediately on the gammometer; thus DO major in
the blue part has for its minor relative LA in the red part below; do #
major in the blue has for its minor relative la # in the red, and vice versa.
8. The gammometer gives the transposition of a tone into another tone.-
Suppose we have a piece of music in RE major, and we wish to transpose it
a third higher, which gives FA major, we begin by composing the scale of
FA major, in which we find a flat before the SI which determines the armure
(armure) of the clef. For the gammometer applied to the piano the slide is
moved to make RE coincide with FA on the gammometer key-board, and
whenever we find a RE in the piece of music we read FA, MI, SOL; fa #,
LA, &c.; it will be the same for the chords. For the rectilinear gam-
mometer with two octaves and the circular gammometer the slide is placed
so that the black line having the name FA on the slide coincides with the
black line having the name RE on the fixed part, and the solution is
obtained.
9. The gammometer distinguishes the modifications to be applied to a scale in
order to modulate (law of modulations).-To modulate being to pass from one
tone to another, the law of a modulation consists in the modifications which
the scale of the tone to be quitted should undergo to pass into that of the
new tone. Now, as this gammometer allows of any two scales being placed
in relation to each other it supplies this solution instantaneously; thus we
can modulate to the fifth or dominant, to the fourth or sub-dominant, to the
minor homonymous sound from its point of departure to the minor relative
of its point of departure, to the minor relative of the sound of the sub-
dominant, and to the minor relative of the sound of the dominant. Take, for
example, DO as the fundamental sound; we can modulate with SOL, domi-
nant; FA, sub-dominant; DO, minor homonym of the fundamental sound;
LA, minor relative of the fundamental sound; MI, minor relative of the
sound of the dominant; and by this fact alone we can put any two scales in
relation, and we can not only obtain these scales with their armures, but also
the constitution of their major chords, minor chords, &c. Thus by putting
the scale of DO and that of SOL in relation, we see that to modulate from
DO major into SOL major, the fourth, FA, of the first scale is sharpened.
10. The gammometer indicates the attractions of the different degrees of a scale

page_text: to be resolved for such and such degrees of the same scale rather than for others.―
There will be noticed two kinds of arrows on the gammometer; one kind
forms part of the black mark connecting one sound to another, and the other
kind replaces the black mark. The following is the meaning-In the major,
that is to say in the blue part, there will be seen between the third and the
fourth, the seventh and the octave, two arrows replacing the black mark; they
indicate the natural attractions of the seventh for the octave, and of the fourth
for the third. In the red part the attractions indicated between the fifth and
the sixth, the seventh and the octave, are the most imperious. The other arrows
forming part of the black mark which connects the sounds indicate less im-
perious attractions. The point of the arrow indicates that such a sound is
attracted, and is resolved on such a sound rather than on another. There-
fore, a chord being given, we know and can designate what will be the names
of the sounds which compose the chord on which the normal resolution should
be made.
11. The gammometer designates on what degrees are the quaver, the semi-
quaver, and the demi-semi-quaver of a scale.-To find the principal harmonic
quavers, we see on the gammometer that on "tonic" in the full black mark
or line there is a black oval mark which indicates the perfect absolute quaver;
on the fifth, a black parallelogram which designates the semi-quaver; and on
the third, half of the parallelogram to show the demi-semi-quaver.
12. By the application of the gammometer to any instrument with either a
fixed or a movable key-board, any person who can read and write but does not
understand either the notes or the key-board can easily solve all the problems above
recited. The application of the gammometer to any instrument with a key-
board is one of the most important of which it is capable, for not only can
the advantages above enumerated be obtained, which are very considerable
for teaching music in schools, as all the rules of solfeggio and the first notions
of harmony are contained in a small apparatus, but also any person of ordinary
intelligence can on a piano or an organ, without knowing either the key-board
or music, establish all kinds of scales, produce harmony, transpose without
error or hesitation logically and mathematically. The gammometer key-
board allows of a complete study of the piano with or even without the
instrument.
For teaching the blind I construct the gammometer in relief. The move-
ment of the slide may be produced by hand, by a crank handle, rack, screw,
or otherwise.
And having now described the nature of the said Invention, and in what

page_text: manner the same is to be performed, I declare that I claim constructing
instruments or apparatus for teaching and transposing music, substantially as
herein-before described and represented in the accompanying Drawings.
In witness whereof, I, the said Richard Archibald Brooman, have
hereunto set my hand and seal, this Eleventh day of July, One
thousand eight hundred and sixty-five.
R. A. BROOMAN. (L.S.)