THE MUSICAL SCALE AND MUSICAL INTONATION.
We have now considered as much of the phenomena of musical sounds
as may be considered to have a bearing upon the purpose of our
investigations. We may then devote some space to the matter of the
expression of musical ideas, and the intonation which has been devised
in order to reduce the mental products of composers to the limitations
of musical instruments. Music is expressed through the medium of a
scale of tones, all of which bear definite relations to each other as
to pitch. The “diatonic scale,” which is the foundation of musical
intonation, is composed of a series of eight tones which are named
after letters of the alphabet, the last tone having the same name as,
and being the octave to, the first. The frequencies of these tones
always bear the same ratios, one to another, whatever may be their
positions within the compass of any instrument. Now, considering the
frequency of the first tone to be unity, the frequencies of the others
are in the following proportions:
C D E F G A B C
1 9/8 5/4 4/3 3/2 5/3 15/8 2
If we now divide these proportionate numbers each by the other we have
the proportionate intervals that separate them. Doing this, we have the
following result:
C D E F G A B C
9/8 10/9 16/15 9/8 10/9 9/8 16/15
Now, it will be observed that we have in the above table three
different kinds of interval represented by the three ratios, 9/8,
10/9 and 16/15. The first of these is called the major tone and the
second the minor tone, while the third is known as the diatonic
semitone. Following out these ratios, we may obtain the frequencies of
any diatonic series. We shall choose the scale of which C 528 is the
key-note. Its frequencies are as follows:
C D E F G A B C
1 9/8 5/4 4/3 3/2 5/3 15/8 2/1
528 594 660 704 792 880 990 1056
Knowing as we do the ratios and frequencies already calculated, it
is obvious that we may similarly calculate the ratios and frequencies
for the diatonic scale, of which any given tone is the tonic or
key-note. Before doing this, however, it is well for us to remember
that the diatonic scale is not adequate to all the requirements of
music. Musicians have found it necessary to interpolate other sounds
in between those which form the diatonic progression. The reason for
this is that music, in order that it may have the greatest possible
freedom of expression, must be written in a larger number of keys,
and must contain more distinct sounds than the diatonic scale is able
to afford. For these and other cognate reasons the chromatic scale
was introduced. The addition of five chromatic semitones, obtained by
taking the difference between a minor tone and a diatonic semitone,
gives the chromatic scale thirteen semitones from key-note to octave.
Unfortunately, however, the same number of keys upon the pianoforte
cannot provide us with thirteen pure chromatic sounds in every key.
This may be demonstrated as follows: The ratio of a chromatic semitone
is 25/24. The sharp of C 528 is, therefore, 550. But in the diatonic
scale of D (the major second in scale of C), C sharp has a frequency
of 1113-3/4. The octave below this latter sound is the C sharp, which
is one chromatic semitone above C 528. We know the frequency of the
latter to be 550. The frequency of the octave below C sharp, 1113-3/4,
ought, therefore, to be 550. But we know that the octave below any
given note has a frequency that is one-half that of the given note.
Now, one-half of 1113-3/4 is 556-7/8. Therefore, we see that there is a
difference of 6-7/8 vibrations per second between the C sharp that is a
chromatic semitone above C 528 and the C sharp that is the octave below
the major seventh of the scale of D, and which ought to be the same
sound, as it is in the same position on the key-board as the former.
By carrying the same investigation further we are enabled to perceive
that sounds of the same name are not identical when played in different
keys, or, rather, that the same name does not imply that the sound so
denoted means the same thing when it is considered in its relation to
any tonic different to that to which it was first related. There is
another difficulty also that confronts us in the problem of playing
pure sounds upon the pianoforte; that instrument, as we know, does not
provide us with different keys for the sharp of one sound and the flat
of the sound next above it. There is a general belief that C sharp, for
example, and D flat are identical. But this is not so. The flat of D is
a chromatic semitone below that note, while the sharp of C is the same
interval above the latter. By referring to our former calculations it
will be seen that the chromatic semitone ratio is 25/24. The sharp of
C is, therefore, obtained by multiplying the frequency of C by 25/24,
and the flat of D is likewise evolved by an inverse process, namely, by
dividing the frequency of D by the same ratio. This is equivalent to
adding a chromatic semitone to C and subtracting the same from D. If we
take the notes C and D from the scale of C 528, we have the frequencies
of C and D as 528 and 594 respectively. Effecting the multiplication
and division as above we see that C sharp has a frequency of 550, while
that of D flat is 570-6/25. That is to say that these two notes differ
by no less than 20-6/25 vibrations per second.
It thus becomes obvious that the expression of all the sounds within
the compass of an octave, in such a manner that absolutely correct
sounds in every key may be obtained, is a problem that calls for more
sounds than are provided by the pianoforte. As a correct understanding
of this most important subject is essential, a somewhat elaborate
treatment of it will be given here. The reader who takes the pains
to master the true inwardness of the problem of musical intonation
will have an insight into the matter which few pianoforte makers or
musicians possess.
“Just intonation” is the name given to that system whereby we are
enabled to command the expression of all the sounds that are required
to be heard within the compass of an octave in order that the degrees
of each and every possible scale may be correctly and exactly rendered.
It is not difficult to see that performers upon instruments which
do not have fixed tones should have no difficulty in adjusting the
intonation of every tone to correspond with the variations in pitch
required by the different positions in the scale that such tones may
occupy. Experiments have, in fact, been carried out with violinists
and it has been shown that artists upon this instrument do naturally
play the true diatonic and chromatic intervals when left to themselves
and when not forced to adjust their intonation to that of fixed tone
instruments.
In order to show with accuracy the total number of different sounds
that are required to produce “just intonation” in every possible key
the reader is invited to consider the following table, which shows the
smallest possible number of sounds that will give the true diatonic
intervals in twelve keys. The first note in each row is the key-note
and the last the octave thereto. The frequencies of those key-notes
that are not represented in the first scale (that of C) have been
calculated as follows:
The key-note to scale of B-flat is the perfect fourth to key-note of
scale F.
The key-note to scale of E-flat is the perfect fourth to key-note of
scale B-flat
The key-note to scale of F-sharp is the octave below major seventh of
scale G.
The key-note to scale of G-sharp is the octave below major seventh of
scale A.
The key-note to scale of C-sharp is the octave below major seventh of
scale D.
We therefore have the following results:
C D E F G A B C
528 594 660 704 792 880 990 1056
C-sharp D-sharp E-sharp F-sharp G-sharp A-sharp B-sharp C-sharp
556-7/8 626-3/64 696-3/32 742-1/2 835-5/16 928-1/8 1044-9/64 1113-3/4
D E F-sharp G A B C-sharp D
594 668-1/4 742-1/2 792 881 990 1113-3/4 1188
E-flat F G A-flat B-flat C D E-flat
625-4/9 703-45/72 781-34/36 833- 938-1/18 1042- 1172- 1250-8/9
25/27 11/27 59/72
E F-sharp G-sharp A B C-sharp D-sharp E
660 742-1/2 825 880 990 1100 1237-1/2 1320
F G A B-flat C D E F
704 792 880 938-2/3 1056 1173-1/3 1320 1408
F-sharp G-sharp A-sharp B C-sharp D-sharp E-sharp F-sharp
742-1/2 835-5/16 928-1/8 990 1113-3/4 1237-1/3 1392-3/15 1492
G A B C D E F-sharp G
704 792 880 938 1056 1173 1320 1408
G-sharp A-sharp B-sharp C-sharp D-sharp E-sharp Fx G-sharp
825 928-1/8 1031-3/4 1100 1237-1/2 1375 1546-1/8 1650
A B C-sharp D E F-sharp G-sharp A
880 990 1100 1173-1/3 1320 1466-2/3 1650 1760
B-flat C D E-flat F G A B-flat
938-2/3 1056 1173-1/6 1258-8/9 1408 1564-4/9 1760 1877-1/3
B C-sharp D-sharp E F-sharp G-sharp A-sharp B
990 1113-3/4 1237-1/2 1320 1485 1650 1856-1/4 1980
In order that the different sounds may more easily be separated,
they have been collated in linear progression, together with their
frequencies and the scales in which they or their octaves appear:
1. The sound C = 528 \ / C, F, G, B-flat.
2. " C = 521-11/54 | | E-flat
3. " C-sharp = 556-7/8 | A | D, B, F-sharp, C-sharp
4. " C-sharp = 550 | p | A, E, G-sharp
5. " D = 594 | p | C-G
6. " D = 586-2/3 | e | A, F, B-flat, E-flat
7. " D-sharp = 618-3/4 | a | E, B, F-sharp, G-sharp
8. " D-sharp = 626-31/64 | r | C-sharp.
9. " E-flat = 625-4/9 | s | B-flat
10. " E = 660 | | C, G, A, E, B-flat
11. " E = 668-1/4 | i | D
12. " E-sharp = 696-3/32 | n | F-sharp, C-sharp
13. " E-sharp = 687-1/2 | | G-sharp.
14. " F = 704 | t | C, F, B-flat
15. " F-sharp = 742-1/2 > h < G, D, E, B, F-sharp, C-sharp
16. " F-sharp = 753-1/3 | e | A
17. " G = 792 | | C, D, F, G
18. " G = 782-4/13 | s | B-flat
19. " G = 781-34/36 | c | E-flat
20. " Fx = 773-6/16 | a | G-sharp
21. " G-sharp = 825 | l | A, E, B, G-sharp
22. " G-sharp = 835-5/16 | e | F-sharp, C-sharp
23. " A-flat = 833-25/27 | s | E-flat
24. " A = 880 | | C, E, F, A
25. " A = 881 | o | D
26. " A = 891 | f | G
27. " A-sharp = 928-1/8 | | B, F-sharp, C-sharp, G-sharp
28. " B-flat = 938-2/3 | | F, B-flat, E-flat
29. " B = 990 | | C, G, D, A, E, B, F-sharp
30. " B-sharp = 1031-1/4 | | G-sharp
31. " B-sharp = 1044-8/64 / \ C-sharp
Thus we see that thirty-one different sounds are required to give the
true diatonic intervals in only twelve keys. But it is not necessary to
remind the reader that there are more keys than these used in music.
We have, in fact, not yet considered the keys of A flat, D flat and
G flat. The frequencies of the keynotes of these scales have been
calculated as follows:
A-flat is the perfect fourth to E-flat, which as calculated above =
625 therefore A-flat = 833-25/27.
D-flat is the perfect fourth to A-flat, which as calculated above =
833-25/27 therefore D-flat = 555-154/162.
G-flat is the perfect fourth to D-flat, which as calculated above =
555-154/162 therefore G-flat = 741-130/486.
We are therefore able to construct these following additional scales:
A-flat B-flat C D-flat E-flat F G A-flat
833- 938- 1042- 1111- 1250- 1389- 1563- 1666-
25/27 26/316 73/236 73/81 8/9 21/52 132/216 50/54
D-flat E-flat F G-flat A-flat B-flat C D-flat
555- 624- 694- 741- 833- 926- 1042- 1111-
154/162 640/1290 365/648 130/486 150/162 284/486 528/1296 146/152
G-flat A-flat B-flat C-flat D-flat E-flat F G-flat
741- 823- 926- 988- 1111- 1285- 1389- 1482-
130/486 3600/3888 1136/1944 520/1458 308/456 650/1458 1408/3088 260/486
By examining the last table the reader will perceive that we have
obtained fourteen new sounds. They are shown graphically in this manner:
In the scale of A-flat the new sounds are B-flat, C, D-flat, F and G.
In the scale of D-flat the new sounds are E-flat, F, G-flat, and A-flat.
In the scale of G-flat the new sounds are A-flat, C-flat, D-flat,
E-flat and F.
None of these sounds had been obtained in the scales given before and,
consequently, we have to consider that there are fourteen more sounds
to be added to the thirty-one that we have already found.
The above calculations would suffice to provide us with the diatonic
intervals in all the keys that are used in music. Harmony demands,
however, certain other intervals. These are minor thirds, minor
sevenths, dominant sevenths and minor sixths. Accordingly, if we desire
to probe the matter of just intonation to its depths, we must calculate
the sounds that are required to make up these intervals in such scales
as are now without them. Examining the tables already prepared, we find
that there are wanting the following members:
Minor thirds to the key-notes of the scales C, D, E-flat, F, G,
B-flat, A-flat, D-flat, G-flat.
Minor sixths to the key-notes of the scale C, E-flat, B-flat, A-flat,
G-flat, and D-flat.
Dominant sevenths to the key-notes of the scales E-flat, F and B-flat.
Minor sevenths to the key-notes of the scales A-flat, D-flat, and
G-flat.
We shall have no difficulty in calculating the frequencies of the
required notes by the same processes that we have followed heretofore.
Minor Minor Dominant Minor
Key-notes-- thirds-- sixths-- sevenths-- sevenths--
6/5 Ratio 8/5 Ratio 16/9 Ratio 9/5 Ratio
C E-flat A-flat
528 633-3/5 841-4/5
D F
594 712-4/5
E-flat G-flat C-flat D-flat
625-4/9 750-4/55 1000-32/45 1111-80/81
F A-flat E-flat
704 844-4/5 1251-5/9
G B-flat
792 950-2/5
B-flat D-flat G-flat A-flat
938-2/3 1125-11/15 1501-13/15 1668-20/27
A-flat C-flat F-flat G-flat
833-25/27 1000-106/135 667-38/276 741-64/243
D-flat F-flat B double flat C-flat
555-146/152 667-66/810 889-358/810 988-359/810
G-flat B double flat E double flat F-flat
741-124/486 889-1330/2430 593-10/2400 658-2908/4374
The result of these calculations may now be collated and summarized.
We find that there are no less than sixty-six separate sounds required
for the production of the necessary intervals in all the possible
scales. These sounds are thus classified:
Different sounds in twelve diatonic scales 31
Sounds wanting to complete the diatonic scales of A-flat,
D-flat, G-flat 14
Minor thirds wanting in scales of C, E-flat, F, G, B-flat 6
Minor sixths wanting in scales of C, E-flat, and B-flat 3
Dominant sevenths wanting in scales of E-flat, F and B-flat 3
Minor thirds wanting in scales of A-flat, D-flat and G-flat 3
Minor sixths wanting in scales of A-flat, D-flat and G-flat 3
Minor sevenths wanting in scales of A-flat, D-flat and G-flat 3
--
Total number of sounds in an octave 66
Now the obvious conclusion to be drawn from this analysis is that
the true sounds of the just musical scales are very different from
any that we hear upon the pianoforte. Indeed, we may properly carry
the reasoning a step further. If the expression of all the degrees
of the true musical scales requires this formidable array of sounds,
then surely, the sounds that are produced upon the piano are not all
of the required true sounds, but are totally unlike any of them. For
it is evident that if the sixty-six true sounds within the compass
of an octave have to be reduced to the thirteen that are found upon
the pianoforte, the process of compression to which the former must
be subjected will force the latter into the position of so many
compromises. In fact, with the exception of the standard tone from
which all calculations and all tuning must start, and its octaves,
there is no tone upon the piano, as it is now tuned, which is identical
with any sound of the justly tuned scale. The process to which we have
alluded, and which is necessary to secure to the piano and all other
instruments with fixed tones the ability to perform music in all keys
which are desired for the proper expression of the composers’ ideas,
is called temperament. Upon the skill and cunning with which this
compromise with natural laws is effected depends the whole beauty of,
and the whole of our pleasure in, music as we are accustomed to hear
it. It would be vain to pretend that tempered intonation is preferable
to that which is pure and just, but it is equally vain and foolish to
decry the accepted system of temperament until the mechanical skill of
manufacturers of musical instruments and the taste of performers have
risen to the point of appreciating the beauties of pure intonation and
of devising mechanical means of attaining it. Until that time arrives
we must fain be content to accept what we have and make the best of
it. There have, of course, been attempts to provide instruments that
could be used to give the pure intervals in every key, but they have
been invariably failures. Most of them have been forced to depend
upon tempered intonation to a certain extent, while others have been
mechanically impossible.
In any case we must remember that the pianoforte, as at present
constructed and played, depends entirely upon an equally tempered
intonation. So strongly has the pianoforte entrenched itself in popular
favor, indeed, that music and tempered intonation have become, to most
people, exactly synonymous. It is proper that we should be able to draw
true distinctions, however, as the practical work of piano building
ought to be largely guided by the considerations induced from the
necessity and fact of temperament.