Contrary and Unconnected Keys. We have seen how the effect of modulation from major tonic to minor supertonic is, on a large scale, obscured by the identity of the primary dominant with the secondary subdominant, though the one chord is major and the other minor. The frequent attempts made by easy-going innovators to treat these key-contrasts on another footing than that of paradox, dramatic surprise or hesitation, only show a deficient sense of tonality, which must also mean an inability to see the intensely powerful effect of the true use of such modulations in classical music, an effect which is entirely independent of any ability to formulate a theory to explain it.
And this brings us to the only remaining subjects of importance in the science and art of harmony, namely, those of the tempered scale, enharmonic ambiguity and just intonation. Before proceeding we subjoin a table of all the key-relationships from major and minor tonics, representing the degrees by capital Roman figures when the second key is major and small figures 8 when minor. Thus I represents tonic major, iv represents subdominant minor, and so on.
A flat or a sharp after the figure indicates that the normal degree of the standard scale has been lowered or raised a semitone, even when in any particular pair of keys it would not be expressed by a flat or a sharp. But the direct artificial modulation is quite smooth, and rich rather than remote. Bach, whose range seldom exceeds direct key-relationships, is not afraid to drift from D minor to C minor, though nothing would induce him to go from D major to C major or minor.
Temperament and Enharmonic Changes. In practical performance the diminished 7th contains three minor 3rds and two imperfect 5ths such as that which is present in the dominant 7th , while the peculiarly dissonant interval from which the chord takes its name is very nearly the same as a major 6th. Now it can only be said that an intonation which makes nonsense of chords of which every classical composer from the time of Corelli has made excellent sense, is a very unjust intonation indeed; and to anybody who realizes the universal relation between art and nature it is obvious that the chord of the diminished 7th must owe its naturalness to its close approximation to the natural ratios of the minor scale, while it owes its artistic possibility to the extremely minute instinctive modification by which its dissonance becomes tolerable.
As a matter of fact, although we have shown here and in the article Music how artificial is the origin and nature of all but the very scantiest materials of the musical language, there is no art in which the element of practical compromise is so minute and so hard for any but trained scientific observation to perceive. If a painter could have a scale of light and shade as nearly approaching nature as the practical intonation of music approaches the acoustic facts it really involves, a visit to a picture gallery would be a severe strain on the strongest eyes, as Ruskin constantly points out.
Yet music is in this respect exactly on the same footing as other arts. It constitutes no exception to the universal law that artistic ideas must be realized, not in spite of, but by means of practical necessities. However independent the treatment of discords, they assert themselves in the long run as transient.
As much and no more, but that is no doubt a great deal. Yet the attempt to determine the point in modern harmony where just intonation should end and the tempered scale begin, is as vexatious as the attempt to define in etymology the point at which the literal meaning of a word gives places to a metaphorical meaning. And it is as unsound scientifically as the conviction of the typical circle-squarer that he is unravelling a mystery and measuring a quantity hitherto unknown. Just intonation is a reality in so far as it emphasizes the contrast between concord and discord; but when it forbids artistic interaction between harmony and melody it is a chimera.
It is sometimes said that Bach, by the example of his forty-eight preludes and fugues in all the major and minor keys, first fixed the modern scale.
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This is true practically, but not aesthetically. By writing a series of movements in every key of which the keynote was present in the normal organ and harpsichord manuals of his and later times, he enforced the system by which all facts of modern musical harmony are represented on keyed instruments by dividing the octave into twelve equal semitones, instead of tuning a few much-used keys as accurately as possible and sacrificing the euphony of all the rest. This system of equal temperament , with twelve equal semitones in the octave, obviously annihilates important distinctions, and in the most used keys it sours the concords and blunts the discords more than unequal temperament; but it is never harsh; and where it does not express harmonic subtleties the ear instinctively supplies the interpretation; as the observing faculty, indeed, always does wherever the resources of art indicate more than they express.
Now it frequently happens that discords or artificial chords are not merely obscure in their intonation, whether ideally or practically, but as produced in practice they are capable of two sharply distinct interpretations.
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And it is possible for music to take advantage of this and to approach a chord in one significance and quit it with another. Where this happens in just intonation in so far as that represents a real musical conception such chords will, so to speak, quiver from one meaning into the other. And even in the tempered scale the ear will interpret the change of meaning as involving a minute difference of intonation. The chord of the diminished 7th has in this way four different meanings—.
Such modulations are called enharmonic. Not every progression of chords which is, so to speak, spelt enharmonically is an enharmonic modulation in itself. But E major with four sharps is merely the most convenient way of expressing F flat, a key which would need six flats and a double flat.
The reality of an enharmonic modulation can be easily tested by transporting the passage a semitone.
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Thus, the passage just cited, put a semitone lower, becomes a perfectly diatonic modulation from C to E flat. But though there is thus a distinction between real and apparent enharmonic modulations, it frequently happens that a series of modulations perfectly diatonic in themselves returns to the original key by a process which can only be called an enharmonic circle.
And hence the enharmonic circle of fifths is a conception of musical harmony by which infinity is at once rationalized and avoided, just as some modern mathematicians are trying to rationalize the infinity of space by a non-Euclidian space so curved in the fourth dimension as to return upon itself. A similar enharmonic circle progressing in major 3rds is of frequent occurrence and of very rich effect. So much, then, for the application of bad metaphysics and circle-squaring mathematics to the art of music.
Neither in mathematics nor in art is an approximation to be confused with an imperfection. The following series of musical illustrations show the genesis of typical harmonic resources of classical and modern music. Musical sounds , or notes , are sensations produced by regular periodical vibrations in the air, sufficiently rapid to coalesce in a single continuous sensation, and not too rapid for the mechanism of the human ear to respond.
The pitch of a note is the sensation corresponding to the degree of rapidity of its vibrations; being low or grave where these are slow, and high or acute where they are rapid. Melody is the organization, in a musical scheme, of rhythmic notes in respect of pitch. Harmony is the organization, in a musical scheme, of simultaneous combinations of notes on principles whereby their acoustic properties interact with laws of rhythm and melody.
The harmonic series is an infinite series of notes produced by the subdivision of a vibrating body or column of air into aliquot parts, such notes being generally inaudible except in the form of the timbre which their presence in various proportions imparts to the fundamental note produced by the whole vibrating body or air-column.
A concord is a combination which, both by its acoustic smoothness and by its logical origin and purpose in a musical scheme, can form a point of repose. A discord is a combination in which both its logical origin in a musical scheme and its acoustic roughness show that it cannot form a point of repose.
The perfect concords and perfect intervals are those comprised within the first four members of the harmonic series, namely, the octave, as between numbers 1 and 2 of the series see Ex. The root of a chord is that note from which the whole or the most important parts of the chord appear if distributed in the right octaves as members of the harmonic series. The major triad is a concord containing three different notes which octaves being disregarded are identical with the first, third and fifth members of the harmonic series the second and fourth members being negligible as octaves.
The minor triad is a concord containing the same intervals as the major triad in a different order; in consequence it is artificial, as one of its notes is not derivable from the harmonic series. Unessential discords are those that are treated purely as the phenomena of transition, delay or ornament, in an otherwise concordant harmony. Essential discords are those which are so treated that the mind tends to regard them as definite chords possessing roots.
A key is an harmonic system in which there is never any doubt as to which note or triad shall be the final note of music in that system, nor of the relations between that note or chord and the other notes or chords. In this sense the church modes are either not keys or else they are subtle mixtures of keys. The major mode is that of keys in which the tonic triad and the two other cardinal triads are major. The minor mode is that of keys in which the tonic triad and one other cardinal triad are minor. A diatonic scale is a series of the notes essential to one major or minor key, arranged in order of pitch and repeating itself in other octaves on reaching the limit of an octave.
Chromatic notes and chords are those which do not belong to the diatonic scale of the passage in which they occur, but which are not so used as to cause modulation. Enharmonic intervals are minute intervals which never occur in music as directly measured quantities, though they exist as differences between approximately equal ordinary intervals, diatonic or chromatic.
In an enharmonic modulation, two chords differing by an enharmonic quantity are treated as identical. Pedal or organ point is the sustaining of a single note in the bass or, in the case of an inverted pedal , in an upper part while the harmonies move independently. Unless the harmonies are sometimes foreign to the sustained note, it does not constitute a pedal. In modern music pedals take place on either the tonic or the dominant, other pedal-notes being rare and of complex meaning.
Double pedals of tonic and dominant, with tonic below are not unusual. The device is capable of very free treatment, and has produced many very bold and rich harmonic effects in music since the earlier works of Beethoven.
In the form of drones the pedal is the only real harmonic device of ancient and primitive music. The ancient Greeks sometimes 10 used a reiterated instrumental note as an accompaniment above the melody. These primitive devices, though harmonic in the true modern sense of the word, are out of the line of harmonic development, and did not help it in any definite way. The fundamental bass of a harmonic passage is an imaginary bass consisting of the roots of the chords.
A figured bass , or continuo , is the bass of a composition supplied with numerals indicating the chords to be filled in by the accompanist. Thorough-bass Ger. Generalbass is the art of interpreting such figures. Intervals greater than an octave are called compound, and are referred to their simple forms, e.
Other diatonic combinations, whether concords or discords, are called imperfect. A wider range was possible only in the irresponsible style of D. A good practical test is to see what becomes of such passages when translated into the minor mode. Usually a small amount of potassium is present replacing part of the barium. The system of crystallization is monoclinic; only complex twinned crystals are known.
A common and characteristic form of twinned crystal, such as is represented in the figure, consists of four intercrossing individuals twinned together according to two twin-laws; the compound group resembles a tetragonal crystal with prism and pyramid, but may be distinguished from this by the grooves along the edges of the pseudo-prism. The faces of the crystals are marked by characteristic striations, as indicated in the figure.
Twinned crystals of exactly the same kind are also frequent in phillipsite q. Crystals are usually white and translucent, with a vitreous lustre. Earlier names are cross-stone Ger.
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Kreuzstein , ercinite, andreasbergolite and andreolite, the two last being derived from the locality, Andreasberg in the Harz. Morvenite from Morven in Argyllshire is the name given to small transparent crystals formerly referred to as phillipsite. Like other zeolites, harmotome occurs with calcite in the amygdaloidal cavities of volcanic rocks, for example, in the dolerites of Dumbartonshire, and as fine crystals in the agate-lined cavities in the melaphyre of Oberstein in Germany.
It also occurs in gneiss, and sometimes in metalliferous veins. At Andreasberg in the Harz it is found in the lead and silver veins; and at Strontian in Argyllshire in lead veins, associated with brewsterite a strontium and barium zeolite , barytes and calcite. At the university of Kiel he repudiated the prevailing rationalism and under the influence of Schleiermacher became a fervent Evangelical preacher, first at Lunden , and then at Kiel He resigned his pastorate on account of blindness in , and died on the 1st of February See Autobiography 2nd ed.
Harms Brunswick, Theodosius Harnack was a staunch Lutheran and a prolific writer on theological subjects; his chief field of work was practical theology, and his important book on that subject, summing up his long experience and teaching, appeared at Erlangen , 2 vols. The liturgy of the Lutheran church of Russia has, since , been based on his Liturgische Formulare The son pursued his studies at Dorpat and at Leipzig, where he took his degree; and soon afterwards began lecturing as a Privatdozent.
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These lectures, which dealt with such special subjects as Gnosticism and the Apocalypse, attracted considerable attention, and in he was appointed professor extraordinarius. In the same year he began the publication, in conjunction with O. Zahn, of an edition of the works of the Apostolic Fathers, Patrum apostolicorum opera , a smaller edition of which appeared in Three years later he was called to Giessen as professor ordinarius of church history.