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
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
[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
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)
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
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
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:
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.,
The next is the Smaller Step or Minor Tone, and is found twice, viz.,
The last is the Sistonic Semitone, and also occurs twice, viz.,
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
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
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
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
which may be confirmed with the 15 and 9 circles.
The vibration ratio of the Major Seventh {^{t}/_{d} is
and this can be verified with the 15 and 8 circles.
The vibration ratio of the Minor Seventh {^{f}/_{s1} is
capable of verification with the 16 and 9 circles.
The vibration fraction of the Diminished Fifth {^{f}/_{tl} is
and that of the Tritone, or Pluperfect Fourth {^{t}/_{f} is
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} :â€”
{^{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,
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} Ã· ^{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} Ã· ^{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} Ã· ^{16}/_{15} = ^{10}/_{9} x ^{15}/_{16} = ^{25}/_{24}.
This is also the difference between a Major and a Minor Third, for
^{5}/_{4} Ã· ^{6}/_{5} = ^{5}/_{4} x ^{5}/_{6} = ^{25}/_{24}
The interval between the Greater and Lesser Chromatic Semitones will be
^{135}/_{128} Ã· ^{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:
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
but {^{f}/_{r`} is a Minor Third, for its vibration ratio is
The interval between r and r` is the comma, its vibration ratio being evidently
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
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:
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}.
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
Â 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
Â {tÂ Â Â 540Â Â Â 3
Â {Â Â =  = 
Â {mÂ Â Â 360Â Â Â 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
300Â Â Â Â 3Â Â Â Â Â Â 432Â Â Â Â 3
Â =Â Â andÂ Â =Â 
200Â Â Â Â 2Â Â Â Â Â Â 288Â Â Â Â 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'Â Â Â 576Â Â Â 96Â Â Â 16
{Â Â =Â  =  = 
{tÂ Â Â Â 540Â Â Â 90Â Â Â 15
{d'Â Â Â 540Â Â Â 54Â Â Â 9
{Â Â =Â  =  = 
{tÂ Â Â Â 480Â Â Â 48Â Â Â 8
{d'Â Â Â 480Â Â Â 120Â Â Â 10
{Â Â =Â  =  = 
{tÂ Â Â Â 432Â Â Â 108Â Â Â Â 9
{d'Â Â Â 432Â Â Â 54Â Â Â 9
{Â Â =Â  =  = 
{tÂ Â Â Â 384Â Â Â 48Â Â Â 8
{d'Â Â Â 384Â Â Â 96Â Â Â 16
{Â Â =Â  =  = 
{tÂ Â Â Â 360Â Â Â 90Â Â Â 15
{d'Â Â Â 360Â Â Â 90Â Â Â 10
{Â Â =Â  =  = 
{tÂ Â Â Â 324Â Â Â 81Â Â Â Â 9
{d'Â Â Â 324Â Â Â 81Â Â Â 9
{Â Â =Â  =  = 
{tÂ Â Â Â 288Â Â Â 72Â Â Â 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.,
tÂ Â sÂ Â r
Â Â â€“Â Â â€“
lÂ Â fÂ Â d
The next is the Smaller Step or Minor Tone, and is found twice, viz.,
lÂ Â m
Â Â â€“
sÂ Â r
The last is the Sistonic Semitone, and also occurs twice, viz.,
d'Â f
Â Â â€“
tÂ Â 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
384Â Â Â 96Â Â Â 8Â Â Â 4
=Â Â =Â Â =Â 
288Â Â Â 72Â Â Â 6Â Â Â 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
432Â Â Â 48Â Â Â 6
=Â Â =Â 
360Â Â Â 40Â Â Â 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
576Â Â Â 72Â Â Â 6
=Â Â =Â 
360Â Â Â 45Â Â Â 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
480Â Â Â 60Â Â Â 5
=Â Â =Â 
288Â Â Â 36Â Â Â 3
which may be confirmed with the 15 and 9 circles.
The vibration ratio of the Major Seventh {^{t}/_{d} is
540Â Â Â 136Â Â Â 15
==Â 
288Â Â Â Â 72Â Â Â Â 8
and this can be verified with the 15 and 8 circles.
The vibration ratio of the Minor Seventh {^{f}/_{s1} is
384Â Â Â 96Â Â Â 16
=Â =Â 
216Â Â Â 54Â Â Â Â 9
capable of verification with the 16 and 9 circles.
The vibration fraction of the Diminished Fifth {^{f}/_{tl} is
384Â Â Â 64
=Â 
270Â Â Â 45
and that of the Tritone, or Pluperfect Fourth {^{t}/_{f} is
540Â Â Â Â 90Â Â Â 45
==Â 
384Â Â Â Â 64Â Â Â 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} :â€”
{^{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 5Â vibrations ofÂ m, there are 4 of d ;
Therefore for every 6 vibrations of s, there are 4 of d;
that isÂ 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} Ã· ^{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} Ã· ^{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} Ã· ^{16}/_{15} = ^{10}/_{9} x ^{15}/_{16} = ^{25}/_{24}.
This is also the difference between a Major and a Minor Third, for
^{5}/_{4} Ã· ^{6}/_{5} = ^{5}/_{4} x ^{5}/_{6} = ^{25}/_{24}
The interval between the Greater and Lesser Chromatic Semitones will be
^{135}/_{128} Ã· ^{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
384Â Â Â Â Â 96Â Â Â Â Â 32
Â =Â Â =Â 
324Â Â Â Â Â 81Â Â Â Â Â 27
but {^{f}/_{r`} is a Minor Third, for its vibration ratio is
384Â Â Â Â Â 48Â Â Â Â Â Â 6
Â =Â Â =Â 
320Â Â Â Â Â 40Â Â Â Â Â Â 5
The interval between r and r` is the comma, its vibration ratio being evidently
324Â Â Â Â Â 81
Â =Â 
320Â Â Â Â Â 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
{GÂ Â Â Â Â Â Â {s
{E,Â Â orÂ Â {m, for example, â€“ are as
{CÂ Â Â Â Â Â Â {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}.