Semper I

Explanation of the musical scale. The North American review. Volume 4,
Issue 10. November 1816

The pleasing effect of melody itself, however, is best explained, as
Dr. Franklin remarks in a letter to Lord Kaimes,* on the principle of
harmony; for although each sound be single, yet to be pleasing it must
harmonize with those preceding, the impression of which is retained in
the mind; musick being the effect of such risings or failings in
sound, as are agreeable to the ear, these are called by musicians
intervals; because a sort of space is left, or skipped over, in which
many others might be placed. Between the notes C and D, for instance,
on your piano, many other sounds might he placed, all higher than C
and not so high as D, of which a good ear would distinguish at least
twenty...
The semitones... which are obtained by dividing the five tones are
not, strictly speaking, so great as the two semitones major in the
natural scale; nor, indeed, could the whole tones be exactly divided;
nor would they give the concords precisely that we seek for if they
were ; for if we placed, for instance, a sharp to D just half ways
between D and E, it would not be a perfect minor 3d with C; and so
with the 6th and 7th minor to C. This could be demonstrated by a
calculation of the ratios of vibrations necessary to produce the
various concords, and it is an imperfection inherent in this and
probably every other scale that could be formed. It is this
imperfection that renders necessary what is called temperarnent, by
which we lessen a very little the semitones of the natural scale, and
enlarge those formed by dividing the tones; by which means all are
brought nearer to an equality; and the advantages of this are very
important, for by thus altering tbese and some of the other notes in
the natural scale, scarcely so as to be perceived, we are enabled to
command any concord from any note we please to start; and without
which, although some might be more rigidly exact, yet others would be
much less so. Of precisely the best mode of tempering instruments,
musicians are not agreed; it is generally admitted, however, that the
5ths and most other concords will bear a little alteration, but the
octaves none at all. The necessity of temperament may perhaps be
better conceived, however, from the following circumstance; if we tune
four 5ths upwards from C, and then from the highest point we arrive
at, tune downwards two octaves, it will fall upon E; but if this
tuning be rigidly exact, this E will not be an exact major 3d from C
as it ought, but will be perceptibly too high; and this also the
theory shows, for by calculation it would be about a tenth part of a
tone major too high, which is a difference easily perceived by a good
car. Now as the octaves must be exact, either the major 3ds must be
increased a little, and the 5ths decreased a little, or else the 5ths
must be lessened all this difference but as it can be divided among
the four fifths, making only a quarter part to each, this may be done
without any perceptible want of harmony in each step, and this is the
usual way. This alteration is temperament.

*In the Brit. Ency. this letter is quoted as being addressed to Dr.
Price.

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Hastings on musical taste. The North American review. Volume 15, Issue
37. October 1822

But if excellence of tone is necessary to produce the best effect, a
true intonation, `the art of singing or playing in tune,' is
indispensable for the production of any at all, save that of disgust.
Great faults in intonation cannot exist, where there is the slightest
ability to distinguish musical sounds, but those of smaller magnitude
are often to be noticed. Mr Hastings observes,`But though the ear,
that is misled by culture, learns to make its decisions with tolerable
uniformity; it yet instinctively revolts at the result of those
decisions, while it continues to persevere in them. An illustration of
this remark is often furnished among musicians of very considerable
attainments. Let one accuse them of an habitual error in intonation,
and they will deny and even retort the charge ; but let him
demonstrate to them the truth of the interval in question, by
variously combining it in harmony, (which is the only practical method
of demonstration) and they will at once be surprised and delighted at
the discovery. A note in melody which they had imagined to be
perfectly tuned, and which had yet, always, in all its harmonic
combinations, produced a disagreeable result to them, was now, by a
slight change in its pitch, found capable of producing the most
agreeable and harmonious effect.'
It would naturally be supposed that the imperfect chords are those
most liable to be mistuned, since the perfect ones cannot be altered
without producing an obvious dissonance. Such is the opinion of our
author. We should think, however, that the major seventh is oftener
tuned too low than too high, thus producing a minor instead of a major
third with the dominant as it ought.*
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A Phase of Medical Life. The Living age. Volume 50, Issue 641.
September 6, 1856

I had laid my hand on the door, to exchange my dormitory for the
breakfast-room, when I was arrested by the clear notes of a bugle,
sounded, and repeated with marked emphasis; then other wind
instruments swelled in the air, till I was thrilled by the outburst of
a full band in the sweeping chords of delicious harmony which open
Mendelssohn's "Wedding March." I hasted on deck, and found that the
melody, which yet rang in the surrounding air, proceeded from the
artillery band then playing in the Royal Arsenal, close under which
the Unite' convict hospital-ship lay in moorings. Each morning was
that pleasure renewed; and every time that the full band struck up the
"Wedding March" did my thoughts fly to Willington. I longed for Mary,
that, hearing the just intonation of the band, she might catch
something of the exultant spirit of the original.
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The Living age.Volume 54, Issue 689. August 8, 1857

Theory and Practice of Just Intonation: with a View to the Abolition
of Temperament. As Illustrated on the New Enharmonic Organ. Effingham
Wilson.
It is an attempt of no small magnitude to construct an organ capaple
of the same perfect correctness in all its keys and intervals as the
violin or the human voice. The experiment has been tried in the
enharmonic organ, and we have here a description of the manner in
which it has been done. The different lengths of string necessary for
the various intervals in use having been carefully calculated, new
tones are introduced where the ordinary tones do not supply those
intervals with accuracy. Thus we have two major seconds and two minor
sevenths, respectively designated grave and acute, and distinguished
from each other on the instrument by white and black keys. This system
is followed out as far as it is practicable, and introduces, of
course, a very complicated mechanism and great additional difficulties
to the performer. But this is a small price to pay for perfect purity
of tone. In the construction of common pianofortes and organs the
object is, not to obtain perfect correctness, but to conceal
inevitable incorrectness with the utmost art. The inaccuracies are
carefully extended over the whole range of the octave, in order that
they may not be harshly perceptible in any one note. This process is
termed tempering. It is the best method which until now has been found
of overcoming a mechanical difficulty, but there is no doubt that it
is but a poor compromise with the requirements of art. With a view to
abolish "temperament" our author has devised the enharmonic organ, and
if it offer no more serious practical difficulties than appear in the
description, he will have rendered an invaluable service to the cause
of music. His instrument contains three boards. The distinctions
between the notes are made, as usual by black and white keys, but the
colors are differently disposed. An ingenious device is introduced for
the use of the blind, the black keys being all serrated at the edge,
whereas the white ones are smooth.
The treatise concludes with an appendix tracing the identity of design
with the enharmonic of the ancients.
-Economist.
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Modern Musical Instruments. Scientific American. New Series, Volume
20, Issue 8. Feb 20, 1869

A wide field of discovery and invention still remains unexplored in
the department of musical instruments. It is well known that musicians
make a distinction between those musical instruments which permit of
perfect intonation and those which are "tempered" or modified in their
intervals. The latter include organs, pianos, and melodeons. Such
instruments are not, however, to be regarded as seriously defective on
this account, although it would be desirable to so improve them that
perfect intonation could be obtained. Indeed this has been attempted
and with considerable success too, but for some reason the improved
instruments have not enjoyed large popularity.
...they [bugle instruments] are deficient in what musicians call
portamento in singing, the gliding by insensible gradations from one
tone to another; one of the most charming musical effects when
delicately performed with the voice, violin, or trombone. This defect
is common to all keyed instruments, if we except the flutes, upon
which gliding from one tone to another can be partially attained by a
skillful performer...
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Musical Intervals. Scientific American. New Series, Volume 20, Issue
22. May 29, 1869

The present musical scale, to which all modern musical instruments are
attuned, has been made the subject of study by eminent scientific men,
among whom Helmholz may be said to be the most prominent. Tyndall, in
his lectures on sound, touches very lightly upon this topic. He
defines a musical sound to be one which "is produced by sonorous
shocks which follow each other at regular intervals with a sufficient
rapidity of succession." The octave of any tone is produced by double
the number of vibrations which produce that tone. The division of the
interval of the octave into intervals including five tones and two
semitones makes the modern diatonic scale. If the whole of this scale
be divided into semitones, we have the chromatic scale of twelve
semitones. The discussion of this subject has lately been quite
prominent. Several papers have been read upon it before the French
Academy. M M. A. Cornu and E. Mercadier have expressed the opinion
that a single musical scale will not satisfy all conditions. They
affirm that the intervals in a scale of melody are not precisely the
same as in a scale of harmony. They remark that sounds that are
pleasing in succession as melodies, are not necessarily pleasing when
superposed as harmonies, and we may even be astonished that the
intervals, hitherto considered the most perfect, ~ the octave, the
fifth, and fourth, do not satisfy both conditions. The ear detects
faulty intonation in melody much more readily than in harmony, unless
the volume of tone be subdued. Musical composers avail themselves of
imperfect chords in passages where large volume of sound is employed,
and powerful organs cover up discords that would be intolerable in
instruments of less power. The subject is beset with many
difficulties. The instruments, which have been constructed with a view
to remedy the defects of those which require what is called
temperament in tuning, have never become popular. They have required
too complex mechanism, and new systems of notation and fingering.
We believe that the maxim, "let well enough alone," may aptly apply to
those who are engaged in the discussion of this subject Are not the
instruments we now possess sufficiently accurate in their intonation
to satisfy the refined ear? We think they are, and that in this
respect they had better be let alone. There is little doubt that
instruments may be devised that would add to the resources of the
orchestra, and that there is still room for improvement in the action
of such instruments as require a keyboard as well as in other
respects. There is also room for improvement in the mechanism of brass
instruments, especially those known as valve instruments; but we think
an attempt to reach any further refinement of intonation unnecessary
and impracticable.
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White, Richard Grant. Richard Wagner. The Galaxy. Volume 17, Issue 6.
June 1874

...[A] square melody..., like most airs or "tunes," consists of four
strains of equal length, and which, starting upon one harmony, the
tonic, passes into a second, the dominant, and often into a third, the
sub-dominant, and returns at its close to the harmony of the tonic
from which it started; thus corresponding to the lines of the four
sides of a square. Nearly all the great melodies that have been
written-all that have lived in the ear and heart of the world-are
constructed upon this model. Formality and symmetry are of the very
essence of melody-as essential to it as the scientific division of the
octave into the diatonic scale, without which the simplest melody and
harmony are absolutely impossible. Hence we may be sure that the
Hebrews and the Greeks, with all their rhapsodies about music, did not
really know what music-that which we call music-was.
___

Church, Prof. J. A. Musical Duodenes, Scientific Miscellany, The
Galaxy. Volume 19, Issue 3. March 1875

AN interesting paper on the correct tuning of keyed instruments was
read before the British Royal Society by Mr. A. J. Ellis. The ordinary
keyed instruments, such as the organ and pianoforte, are limited to
twelve sounds in the octave, while a fall series, to produce just
intonation, would consist of seventy-eight sounds. He exhibited a
chart of these sounds distinguished by the ordinary musical signs,
with the addition of certain marks to denote the pitch of the notes.
The great number of these notes is troublesome in practice, and by
admitting a slight error, which is not sufficient to offend the ear,
the number has been much reduced. In carrying the system into practice
the twelve notes of the ordinary keyboard are retained, and it is
necessary to make choice of certain sets of twelve notes to be used
when playing in corresponding keyes. These sets Mr. Ellis calls
duodenes, and the object of his paper was to show how these sets were
to be selected. He also played some harmonical passages, first on a
harmonium of the ordinary kind, next on another, with absolutely just
intonation, and finally on a newly-constructed harmonium, tuned on
Handel's plan of the old organ temperament, but which had other notes
added, enabling music to be played on in in all keys. These additional
notes were brought into use by draw stops. In discussing the paper,
Dr. Pole pointed out the reason why unaccompanied vocal music by good
performers is thought delightful. It is because the singers, when
uncontrolled by falsely-tuned instruments, are guided by their correct
ears, and produce absolutely true harmony. This separation of vocal
from instrumental music is comparatively infrequent, and to lovers of
music it has the charm of an agreeable novelty. Such performances are
rapturously applauded, and it is becoming the fashion to introduce
unaccompanied vocal solos into oratorio music. The only other
opportunity of hearing true harmony is when a stringed quartet is
played by fine players. While upon the subject of music, we will
mention that an enthusiastic student of University College, Oxford,
Mr. Hamilton, left his university course unfinished for the purpose of
perfecting the AEolian harp. His success has been great; for by
employing a reed in addition to the string, he imitates the notes of
organ pipes, securing all the volume and sweetness of the pipe in
addition to the sympathy and blending qualities peculiar to strings.
He also exhibited (before the Physical Society) a string which could
not be put out of tune, and a new pianoforte string of surprising
purity and volume.

Clark, this is metatuning! Music stuff (unless it's purely
a discussion of ET music) is off-topic here.

-Carl