In describing the form of Syren devised by Helmholtz
?, it was mentioned, that the lower revolving plate was pierced with four circles of 8, 10, 12, and 18 holes, and the upper with four circles of 9, 12, 15, and 16. If only the "8hole circle" on the lower and the "16hole" circle on the upper be opened, while the Syren is working, two sounds are produced, the
interval between which, the musician at once recognises as the
Octave. When the speed of
rotation is increased, both sounds rise in
pitch, but they always remain an
Octave apart. The same
interval is heard, if the circle of 9 and 18 holes be opened together. It follows from these experiments, that when two sounds are at the
interval of an
Octave, the
vibrational number of the higher one is exactly twice that of the lower. An
Octave, therefore, may be acoustically defined as the
interval between two sounds, the
vibrational number of the higher of which is twice that of the lower. Musically, it may be distinguished from all other intervals by the fact, that, if any particular
sound be taken, another
sound an
octave above this, another an
octave above this last, and so on, and all these be simultaneously produced, there is nothing in the resulting
sound unpleasant to the ear.
Since the
ratio of the
vibrational numbers of two sounds at the
interval of an
octave is as 2:1, it is easy to divide the whole range of musical sound into octaves. Taking the lowest sound to be produced by 16 vibrations per second, we have
1st Octave, from  16  to  32  vibrations per second.

2nd Octave, from  32  to  64  " "

3rd Octave, from  64  to  128  " "

4th Octave, from  128  to  256  " "

5th Octave, from  256  to  512  " "

6th Octave, from  512  to  1,024  " "

7th Octave, from  1,024  to  2,048  " "

8th Octave, from  2,048  to  4,096  " " 
Thus all the sounds used in
music are comprised within the
compass of about eight octaves.
Returning to the syren: if the 8 and 12 "hole circles" be opened together, we hear two sounds at an
interval of a
Fifth, and as in the case of the
octave, this is the fact, whatever the
velocity of
rotation. The same result is obtained on opening the 10 and 15, or the 12 and 18 circles. When, therefore, two sounds are at an
interval of a
Fifth, for every 8 vibrations of the lower
sound, there are 12 of the upper, or for every 10 of the lower there are 15 of the upper, or for every 12 of the lower there are 18 of the upper. But
8:12::2:3
10:15::2:3
12:18::2:3
[Read the above
proportion: 8 is to 12 as 2 is to 3, etc.]
Therefore two sounds are at the
interval of a
Fifth when their
vibration numbers are as 2 to 3: that is when 2 vibrations of the one are performed in exactly the same time as 3 vibrations of the other. This may be conveniently expressed by saying that the
vibration ratio or
vibration fraction of a
Fifth is 3 : 2 or
^{3}/
_{2}. Similarly the
vibration ratio of Octaves is 2 : 1 or
^{2}/
_{1}.
Again, on opening the circles of 8 and 10 holes, two sounds are heard at the
interval of a
Major Third. The same
interval is obtained with the 12 and 15 circles. Now 8 : 10 :: 4 : 5 and 12 : 15 :: 4 : 5. Therefore two sounds are at the
interval of a
Major Third, when their
vibration numbers are as 4 : 5; or more concisely, the
vibration ratio of a
Major Third is
^{5}/
_{4}.
With the results, thus experimentally obtained, it is easy to calculate the
vibrational numbers of all the other sounds of the musical
scale, when the vibration number
? of one is given. For example, let the vibration number
? of
d be 288, or shortly, let
d = 288; then the higher
Octave d' = 288 x 2 = 576. Also the
vibration ratio of a
Fifth =
^{3}/
_{2}; therefore the vibration number
? of
s is to that of
d, as 3 : 2; that is,
s =
^{3}/
_{2} x 288 = 432. Similarly the
interval {
^{m}/
_{d} is a
Major Third; but the
vibration ratio of a
Major Third we have found to be,
^{5}/
_{4}; therefore
m :
d : : 5 : 4, that is
m =
^{5}/
_{4} x 288 = 360. Again {
^{t}/
_{s} is a
Major Third; therefore
t =
^{5}/
_{4} x 432 = 540. Further, {
^{r'}/
_{s} is a
Fifth; therefore
r' =
^{3}/
_{2} x 432 = 648, and its lower octave
r =
^{648}/
_{2} = 324. It only remains to obtain the
vibrational numbers of
f and
l. Now {
^{d'}/
_{f} is a
Fifth, thus the vibrational number
? of
f is to that of
d' as 2 : 3; therefore
f =
^{3}/
_{2} x 576 = 384; and {
^{l}/
_{f} is a
Major Third, consequently
l = 384 x
^{5}/
_{4} = 480. Tabulating these results we have
d'  =  576

t  =  540

l  =  480

s  =  432

f  =  384

m  =  360

r  =  324

d  =  288 
The vibrational numbers
? of the upper or lower octaves of these notes, are of course at once obtained by doubling or halving them.
It will be noticed that a
scale may be constructed on any vibration number? as a foundation. The only reason for selecting 288 was, to avoid
fractions of a
vibration and so simplify the calculations. As another example let us take
d = 200. Proceeding in the same way as before, but tabulating at once, for the sake of brevity, we get (underline added)
d'  =  200  x  2  =  400  (2)

t  =  ^{300}/_{1}  x  ^{5}/_{4}  =  375  (5)

l  =  266 ^{2}/_{3}  x  ^{5}/_{4}  =  333 1/3  (8)

s  =  ^{200}/_{1}  x  ^{3}/_{2}  =  300  (3)

f  =  ^{400}/_{1}  x  ^{2}/_{3}  =  266 ^{2}/_{3}  (7)

m  =  ^{200}/_{1}  x  ^{5}/_{4}  =  250  (4)

r  =  ^{300}/_{1}  x  ^{2}/_{3} x ^{1}/_{2}  =  225  (6)

d  =     =  200  (1) 
We may now adopt the reverse process, that is, from the
vibrational numbers, obtain the
vibration ratios. For example, using the first
scale, we find that the vibration number
? of
t is to that of
m as 540 : 360, that is (dividing each by 180, for the purpose of simplifying) as 3 : 2; or more concisely
{t 540 3
{ =  = —
{m 360 2
The
interval {
^{t}/
_{m} is therefore a
perfect fifth. Again
{
^{l}/
_{r} =
^{480}/
_{324}
Now the
vibration fraction of a
perfect Fifth =
^{3}/
_{2} =
^{480}/
_{320}, therefore {
^{l}/
_{r} is not a
perfect Fifth. We shall return to this matter further on, at present it will be sufficient to notice the fact. The student must take particular care not to subtract or add
vibrational numbers, in order to find the
interval between them; thus the difference between the
vibrational numbers of
t and
m in the second
scale is 375  250 = 125, but this does not express the
interval between them, viz., a
Fifth, but merely the difference between the
vibrational numbers of these particular sounds. To make this clearer, take the difference between the
vibrational numbers of
d and
s in the second table = 300  200 = 100, and between
d and
s in the first = 432  288 = 144. Here we have different results, although the
interval is the same. Take the
ratio, however, and we shall get the same in each case for
300 3 432 3
 =  and  = 
200 2 288 2
We shall now proceed to ascertain the
vibration ratios of the interrvals between the successive sound of the scale, using the first of the two scales given on the preceding page:
{d' 576 96 16
{ =  =  = 
{t 540 90 15
{d' 540 54 9
{ =  =  = 
{t 480 48 8
{d' 480 120 10
{ =  =  = 
{t 432 108 9
{d' 432 54 9
{ =  =  = 
{t 384 48 8
{d' 384 96 16
{ =  =  = 
{t 360 90 15
{d' 360 90 10
{ =  =  = 
{t 324 81 9
{d' 324 81 9
{ =  =  = 
{t 288 72 8
There are, therefore, three kinds of intervals between the consecutive sounds of the
scale, the
vibration ratios of which are
^{9}/
_{8},
^{10}/
_{9}, and
^{16}/
_{15}. The first of these intervals, which has been termed the Greater Step
? or Major Tone
?, occurs three times in the diatonic scale
?, viz.,
t s r
 – –
l f d
The next is the Smaller Step
? or Minor Tone
?, and is found twice, viz.,
l m
 –
s r
The last is the Sistonic Semitone
?, and also occurs twice, viz.,
d' f
 –
t m
We may now calculate the
vibration ratios of the remaining
intervals of the
scale. {
^{f}/
_{d} may be selected as the type of the
Fourth. Taking again the vibrational numbers
? of the first
scale, the
vibration ratio of this
interval is
384 96 8 4
 =  =  = 
288 72 6 3
This result may be verified on the Syren by opening the 12 and 9 or 16 and 12 circles.
Taking {
^{s}/
_{m} as an example of a
Minor Third, its
vibration ratio is
432 48 6
 =  = 
360 40 5
This can also be verified by the Syren with the 12 and 10 circles.
Again, the
vibration ratio of {
^{d'}/
_{m} a Minor Sixth
?, is
576 72 6
 =  = 
360 45 5
and this, too, may be confirmed on the Syren, with the 16 and 10 circle.
The
vibration ratio of {
^{l}/
_{d'} a
Major Sixth, is
480 60 5
 =  = 
288 36 3
which may be confirmed with the 15 and 9 circles.
The
vibration ratio of the
Major Seventh {
^{t}/
_{d} is
540 136 15
 =  = 
288 72 8
and this can be verified with the 15 and 8 circles.
The
vibration ratio of the
Minor Seventh {
^{f}/
_{s1} is
384 96 16
 =  = 
216 54 9
capable of verification with the 16 and 9 circles.
The
vibration fraction of the Diminished Fifth
? {
^{f}/
_{tl} is
384 64
 = 
270 45
and that of the Tritone
?, or Pluperfect Fourth
? {
^{t}/
_{f} is
540 90 45
 =  = 
384 64 32
In order to find the
vibration ratio of the sum of two intervals, the
vibration ratios of which are given, it is only necessary to multiply them together as if they were vulgar
fractions, thus, given
{
^{s}/
_{m} =
^{6}/
_{5}, and {
^{m}/
_{d} =
^{5}/
_{4} ; to find {
^{s}/
_{d} :—
{
^{s}/
_{d} =
^{6}/
_{5} x
^{5}/
_{4} =
^{6}/
_{4} =
^{3}/
_{2} ;
which we already know to be the case. The reason of the process may be seen from the following considerations. From {
^{s}/
_{m} =
^{6}/
_{5}, and {
^{m}/
_{d} =
^{5}/
_{4} we know that,
for every 6 vibrations of s, there are 5 of m;
and every 5 vibrations of m, there are 4 of d ;
Therefore for every 6 vibrations of s, there are 4 of d;
that is for every 3 vibrations of s, there are 2 of d.
Again, in order to find the
vibration ratio of the difference of two
intervals, the
vibration ratios of which are given, the greater of these must be divided by the less, just as if they were vulgar
fractions. For example, given
{
^{d'}/
_{d} =
^{2}/
_{1}, and {
^{m}/
_{d} =
^{5}/
_{4}, find {
^{d'}/
_{m}:
{
^{d'}/
_{m} =
^{2}/
_{1} ÷
^{5}/
_{4} =
^{2}/
_{1} x
^{4}/
_{5} =
^{8}/
_{5}
The reason for the rule will be seen from the following considerations. From the given
vibration ratios we know that,
for every 2 vibrations of
d', there is 1 of
d;
that is for every 8 vibrations of
d', there are 4 of
d;
and for every 4 vibrations of
d, there are 5 of m;
therefore for every 8 vibrations of
d', there are 5 of
m.
We shall apply this rule, to find the
vibration ratios of a few other intervals. The Greater Chromatic Semitone
? is the difference between the Greater Step
? and the Diatonic Semitone
?. {
^{fe}/
_{e} is an example of the Greater Chromatic Semitone, being the difference between {
^{s}/
_{f} a Greater Step, and {
^{s}/
_{fe} a Diatonic Semitone. Now {
^{s}/
_{f} =
^{9}/
_{8}, and {
^{s}/
_{fe} =
^{16} /
_{15} (for it is the same interval as {
^{d'}/
_{t}); therefore
{
^{fe}/
_{f} =
^{9}/
_{8} ÷
^{16} /
_{15} =
^{9}/
_{8} x
^{15}/
_{15} =
^{135}/
_{128}.
The Lesser Chromatic Semitone
? is the difference between the Smaller Step
? and the Diatonic Semitone
?; {
^{se}/
_{s}, for example, which is the difference between {
^{1}/
_{s} and {
^{1}/
_{se}. Now {
^{1}/
_{s} =
^{10}/
_{9} and {
^{1}/
_{se} = {
^{d'}/
_{t} =
^{16}/
_{15}; therefore
{
^{se}/
_{s} =
^{10}/
_{9} ÷
^{16}/
_{15} =
^{10}/
_{9} x
^{15}/
_{16} =
^{25}/
_{24}.
This is also the difference between a
Major and a
Minor Third, for
^{5}/
_{4} ÷
^{6}/
_{5} =
^{5}/
_{4} x
^{5}/
_{6} =
^{25}/
_{24}
The
interval between the Greater
? and Lesser Chromatic Semitones
? will be
^{135}/
_{128} ÷
^{25}/
_{24} =
^{135}/
_{128} x
^{24}/
_{25} =
^{81}/
_{80};
which is usually termed the Comma
? or Komma
?.
Referring to the third table of
vibrational numbers in this chapter, we have 1 = 480, and
r = 324; therefore
{
^{1}/
_{r} =
^{480}/
_{324} =
^{40}/
_{27};
and thus, as noticed above, it is not a
Perfect Fifth. To form a
Perfect Fifth with 1, a note
r` would be required, such that
{
^{1}/
_{r`} =
^{3}/
_{2}.
It is easy to find the vibration number
? of this note if that of 1 be given, thus:
{^{1}/_{r'}   =  ^{3}/_{2},

that is,  ^{480}/_{r}  =  ^{3}/_{2};

therefore  ^{r`}/_{480}  =  ^{2}/_{3},

r` =  ^{2}/_{3} x ^{480}/_{1}  =  320. 
This
note has been termed
rah or grave
r, and may be conveniently written
r`. Similarly {
^{f}/
_{r} is nopt a true
Minor Third, for its
vibration ratio is
384 96 32
 =  = 
324 81 27
but {
^{f}/
_{r`} is a
Minor Third, for its
vibration ratio is
384 48 6
 =  = 
320 40 5
The interval between r and r` is the comma, its
vibration ratio being evidently
324 81
 = 
320 80
Summary.
The sound used in
Music lie within the compass of about eight
Octaves.
The
vibration ratio or
vibration fraction of an
interval, is the
ratio of the
vibrational numbers of the two sounds forming that
interval. [This
ratio is always
proportional.]
The
vibration ratio of the principal musical intervals have been exactly verified by Helmholtz
?'s modification of the Double Syren.
It may be shown, by means of this instrument, that the
vibrational numbers of the three tones of a Major Triad
?, in its normal position
{G {s
{E, or {m, for example, – are as
{C {d
4 : 5 : 6.
Starting from this experimental foundation, the
vibrational numbers of all the tones of the modern scale can readily be calculated on any basis; and from these results, the
vibration ratio of any
interval used in modern
music may be obtained.
Vibration ratios must never be added or subtacted.
To find the
vibration ratio of the sum of two or more intervals, multiply their
vibration ratios together.
To find the
vibration ratio of the difference of two intervals, divide the
vibration ratio of the greater
interval by that of the smaller.
The
vibration ratios of the principal intervals of the modern musical scale are as follows:
Komma  ^{81}/_{80}

Lesser Chromatic Semitone  ^{25}/_{24}

GreaterChromatic Semitone  ^{135}/_{128}

Diatonic Semitone  ^{16}/_{15}

Smaller Step or Minor Tone  ^{10}/_{9}

Greater Step or Major Tone  ^{9}/_{8}

Minor Third  ^{6}/_{5}

Major Third  ^{5}/_{4}

Fourth  ^{4}/_{3}

Tritone  ^{45}/_{32}

Diminished Fifth  ^{64}/_{45}

Fifth  ^{3}/_{2}

Minor Sixth  ^{8}/_{5}

Major Sixth  ^{5}/_{3}

Minor Seventh  ^{16}/_{9}

Major Seventh  ^{15}/_{8}

Octave  ^{2}/_{1} 
To find the
vibration ratio of any of the above intervals increased by an
Octave, multiply by
^{2}/
_{1}; thus the
vibration ratio of a Major Tenth
? is
^{5}/
_{4} X
^{2}/
_{1} =
^{10}/
_{4} =
^{5}/
_{2}.