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 "8-hole circle" on the lower and the "16-hole" 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

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.