Chapter M: T. M (2)

The effect of the remaining key-relationships involves contrasts between major and minor mode; but it is otherwise far less defined, since the primary tonic chord does not occupy a cardinal position in the second key. These key-relationships are most important from a minor tonic, as the change from minor to major is more vivid than the reverse change. The smoothest changes are those to "relative" minor, "relative" major (C to A minor; C minor to E[flat]); and mediant minor and submediant major (C to E minor; C minor to A[flat]). The change from major tonic to supertonic minor is extremely natural on a small scale, i.e. within the compass of a single melody, as may be seen in countless openings of classical sonatas. But on a large scale the identity of primary dominant with secondary subdominant confuses the harmonic perspective, and accordingly in classical music the supertonic minor appears neither in the second subjects of first movements nor as the key for middle movements.[6] But since the key-relationships of a minor tonic are at once more obscure harmonically and more vivid in contrast, we find that the converse key-relationship of the flat 7th, though somewhat bold and archaic in effect on a small scale, has once or twice been given organic function on a large scale in classical movements of exceptionally fantastic character, of which the three great examples are the ghostly slow movement of Beethoven's _D major Trio, Op. 70_, No. 1, the scherzo of his _Ninth Symphony_, and the finale of Brahms's _D minor Violin Sonata_ (where, however, the C major theme soon passes permanently into the more orthodox dominant minor).

Thus far we have the set of key-relationships universally recognized since the major and minor modes were established, a relationship based entirely on the place of the primary tonic chord in the second key. It only remains for us to protest against the orthodox description of the five related keys as being the "relative" minor or major and the dominant and subdominant with their "relative" minors or majors; a conception which expresses the fallacious assumption that keys which are related to the same key are related to one another, and which thereby implies that all keys are equally related and that classical composers were fools. It cannot be too strongly insisted that there is no foundation for key-relationship except through a tonic, and that it is through the tonic that the most distant keys have always been connected by every composer with a wide range of modulation, from Haydn to Brahms and (with due allowance for the conditions of his musical drama) Wagner.

b. _Indirect Relationships._--So strong is the identity of the tonic in major and minor mode that Haydn and Mozart had no scruple in annexing, with certain reservations, the key-relationships of either as an addition to those of the other. The smoothness of Mozart's style makes him prefer to annex the key-relationships of the tonic minor (e.g. C major to A[flat], the submediant of C minor), because the primary tonic note is in the second key, although its chord is transformed. His range of thought does not allow him to use these keys otherwise than episodically; but he certainly does not treat them as chaotically remote by confining them to rapid modulations in the development-portions of his movements. They occur characteristically as beautiful purple patches before or during his second subjects. Haydn, with his mastery of rational paradox, takes every opportunity, in his later works, of using all possible indirect key-relationships in the choice of key for slow movements and for the trios of minuets. By using them thus sectionally (i.e. so as not to involve the organic connecting links necessary for the complementary keys of second subjects) he gives himself a free hand; and he rather prefers those keys which are obtained by transforming the minor relationships of a major primary key (e.g. C to A major instead of A minor). These relationships are of great brilliance and also of some remoteness of effect, since the primary tonic note, as well as its chord, disappears entirely. Haydn also obtains extreme contrasts by changing both modes (e.g. C minor to A major, as in the _G minor Quartet, Op. 72_, No 6, where the slow movement is in E major), and indeed there is not one key-contrast known to Beethoven and Brahms which Haydn does not use with complete sense of its meaning, though his art admits it only as a surprise.

Beethoven rationalized every step in the whole possible range of key-relationship by such harmonic means as are described in the article BEETHOVEN. Haydn's favourite key-relationships he used for the complementary key in first movements; and he at once discovered that the use of the major mediant as complementary key to a major tonic implied at all events just as much suggestion of the submediant major in the recapitulation as would not keep the latter half of the movement for too long out of the tonic. The converse is not the case, and where Beethoven uses the submediant major as complementary key in a major first movement he does not subsequently introduce the still more remote and brilliant mediant in the recapitulation. The function of the complementary key is that of contrast and vividness, so that if the key is to be remote it is as well that it should be brilliant rather than sombre; and accordingly the easier key-relationships obtainable through transforming the tonic into minor do not appear as complementary keys until Beethoven's latest and most subtle works, as the _Quartet in B[flat], Op. 130_ (where we again note that the flat submediant of the exposition is temporarily answered by the flat mediant of the recapitulation).

c. _Artificial Key-relationships._--Early in the history of the minor mode it was discovered that the lower tetrachord could be very effectively and naturally altered so as to resemble the upper (thus producing the scale C D[flat] E[natural] F, G A[flat] B[natural] C). This produces a flat supertonic (the chord of which is generally presented in its first inversion, and is known as the Neapolitan 6th, from its characteristic use in the works of the Neapolitan school which did so much to establish modern tonality) and its origin, as just described, often impels it to resolve on a major tonic chord. Consequently it exists in the minor mode as a phenomenon not much more artificial than the mode itself; and although the keys it thus connects are extremely remote, and the effect of their connexion very surprising, the connexion is none the less real, whether from a major or a minor tonic, and is a crucial test of a composer's sense of key-perspective. Thus Philipp Emanuel Bach in a spirit of mere caprice puts the charming little slow movement of his _D major Symphony_ into E[flat] and obliterates all real relationship by chaotic operatic connecting links. Haydn's greatest pianoforte sonata (which, being probably his last, is of course No. 1 in most editions) is in E[flat], and its slow movement is in F[natural] major (= F[flat]). That key had already appeared, with surprising effect, in the wanderings of the development of the first movement. No attempt is made to indicate its connexion with E[flat]; and the finale begins in E[flat], but its first bar is unharmonized and starts on the one note which most contradicts E[natural] and least prepares the mind for E[flat]. The immediate repetition of the opening phrase a step higher on the normal supertonic strikes the note which the opening had contradicted, and thus shows its function in the main key without in the least degree explaining away the paradoxical effect of the key of the slow movement. Brahms's _Violoncello Sonata Op. 99_, is in F; a prominent episode in the development of the first movement is in E[sharp] minor (= G[flat]), thus preparing the mind for the slow movement, which is in F[sharp] major (= G[flat]), with a central episode in F minor. The scherzo is in F minor, and begins on the dominant. Thus if we play its first chord immediately after the last chord of the slow movement we have exactly that extreme position of flat supertonic followed by dominant which is a favourite form of cadence in Wagner, who can even convey its meaning by its mere bass without any harmonies (_Walkure_, Act 3, Scene 2: "Was jetzt du bist, das sage dir selbst").

Converse harmonic relationships are, as we have seen, always weaker than their direct forms. And thus the relation of C major to B major or minor (as shown in the central episode of the slow movement just mentioned) is rare. Still more rare is the obtaining of indirect artificial relationships, of which the episode in the first movement just mentioned is an illustration in so far as it enhances the effect of the slow movement, but is inconclusive in so far as it is episodic. For with remote key-relationships everything depends upon whether they are used with what may be called cardinal function (like complementary keys) or not. Even a near key may occur in the course of wandering modulations without producing any effect of relationship at all, and this should always be borne in mind whenever we accumulate statistics from classical music.

d. _Contrary and Unconnected Keys._--There remain only two pairs of keys that classical music has not brought into connexion, a circumstance which has co-operated with the utter vagueness of orthodox theories on the subject to confirm the conventionally progressive critic in his conviction that all modulations are alike. 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. Now when the supertonic becomes major this difference no longer obviates the confusion, and modulation from C major to D major, though extremely easy, is of so bewildering effect that it is used by classical composers only in moments of intensely dramatic surprise, as, for example, in the recapitulation of the first subject of Beethoven's _Eroica Symphony_, and the last variation (or coda) of the slow movement of his _Trio in B[flat], Op. 97._ And in both cases the balance is restored by the converse (and equally if not more contradictory) modulation between major tonic and major flat 7th, though in the slow movement of the _B[flat] Trio_ the latter is represented only by its dominant chord which is "enharmonically" resolved into quite another key. 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.[7] There now remains only one pair of keys that have never been related, namely, those that (whether major or minor) are at the distance of a tritone 4th. In the first place they are unrelated because there is no means of putting any form of a tonic chord of F[sharp] into any form of the key of C, or vice versa; and in the second place because it is impossible to tell which of two precisely opposite keys the second key may be (e.g. we have no means of knowing that a direct modulation from C to F[sharp] is not from C to G[flat], which is exactly the same distance in the opposite direction). 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 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. Thus vi[flat] would, from the tonic of B[flat] major, express the position of the slow movement of Beethoven's _Sonata, Op. 106_, which is written in F[sharp] minor since G[flat] minor is beyond the practical limits of notation.

TABLES OF KEY-RELATIONSHIPS

_A. From Major Tonic_

I Direct Relationships ii iii IV V vi
----------------------|------------------------------|-----|------|----|-----|--
| | | | | |
| Indirect through both | | | | |
| i and the second key | | iv v |
|------------------------------|-----|-----------------|--
| | | |
| | | |
Indirect, through i \ Indirect through the | | |
III[flat] VI[flat] \ second key | III VI
--|-----------|----------\---------------------------|--------------------------
| | \ |
Doubly indirect through the \ |
former indirect keys \ |
iii[flat] vi[flat] \ |
-------------------------------\---------------------|--------------------------
\ |
\ |
Artificial, direct II[flat] \ | VII & vii
-------------\----------------|--------------------------
\ |
Artificial, indirect[8] ii[flat] \ |
-----------------\------------|--------------------------
\ |
\ |
\ |
\ |
\ |
-------------------------------------------------\---|--------------------------
\ |
\ | IV[sharp] & iv[sharp] =
Unrelated \| V[flat] & v[flat]
-----------------------------------------------------\--------------------------
|\
| \
| \
Contradictory II VII[flat] & vii[flat]
--------------------------------------------------------------------------------

_B. From Minor Tonic_[9]

i Direct Relationships III iv v VI VII
----------------------|------------------------------|-----|------|----|-----|--
| | | | | |
| Indirect through both | | | | |
| i and the second key | IV V | |
| -----------------------------|-----------------|-----|--
\ | | |
\ | | |
Indirect, through i \ Indirect through the | | |
iii[sharp] vi[sharp] \ second key iii vi |
--|-----------|------------\-------------------------------------------------|--
| | \ /
| | \ /
Doubly indirect \ /
III[sharp] VI[sharp] | /
------------------------------|----------------------------------------/--------
\ /
\ /
Artificial, direct \ II[flat] |
\ |
\-----------------------------|------------
Artificial, indirect[10] \ ii[flat] VII[sharp] | & vii[sharp]
\---------------------------|------------
\ |
\ |
\ |
\ |
\ |
------------------------------------------------\------------------|-------------
\ |
\ IV[sharp] & | = V[flat] &
Unrelated \ iv[sharp | v[flat]
----------------------------------------------------\--------------|--------------
\ |
\ |
\ |
Contradictory[11] ii II vii[flat]
----------------------------------------------------------------------------------

VI. _Temperament and Enharmonic Changes._--As the facts of artistic harmony increased in complexity and range, the purely acoustic principles which (as Helmholtz has shown) go so far to explain 16th-century aesthetics became more and more inadequate; and grave practical obstacles to euphonious tuning began to assert themselves. The scientific (or natural) ratios of the diatonic scale were not interfered with by art so long as no discords were "fundamental"; but when discords began to assume independence, one and the same note often became assignable on scientific grounds to two slightly different positions in pitch, or at all events to a position incompatible with even tolerable effect in performance. Thus, the chord of the diminished 7th is said to be intolerably harsh in "just intonation," that is to say, intonation based upon the exact ratios of a normal minor scale. 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. They resolve into permanent points of repose of which the basis is natural; but the transient phenomena float through the harmonic world adapting themselves, as best they can, to their environment, showing as much dependence upon the stable scheme of "just intonation" as a crowd of metaphors and abstractions in language shows a dependence upon the rules of the syllogism. 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. 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. 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--

and the chord of the augmented 6th, when accompanied by the fifth, may become a dominant 7th or vice versa, as in the passage already cited in the coda of the slow movement of Beethoven's _B[flat] Trio, Op. 97._ Such modulations are called _enharmonic_. We have seen that all the more complex musical phenomena involve distinctions enharmonic in the sense of intervals smaller than a semitone, as, for instance, whenever the progression D E in the scale of C, which is a minor tone, is identified with the progression of D E in the scale of D, which is a major tone (differing from the former as 8/9 from 9/10). But the special musical meaning of the word "enharmonic" is restricted to the difference between such pairs of sharps with flats or naturals as can be represented on a keyboard by the same note, this difference being the most impressive to the ear in "just intonation" and to the imagination in the tempered scale.

Not every progression of chords which is, so to speak, spelt enharmonically is an enharmonic modulation in itself. Thus a modulation from D flat to E major looks violently enharmonic on paper, as in the first movement of Beethoven's _Sonata, Op. 110_. 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. Thus, the passage just cited, put a semitone lower, becomes a perfectly diatonic modulation from C to E flat. But no transposition of the sixteen bars before the return of the main theme in the scherzo of Beethoven's _Sonata in E[flat], Op. 31_, No. 3, will get rid of the fact that the diminished 7th (G B[flat] D[flat] E[natural]), on the dominant of F minor, must have changed into G B[flat] D[flat] F[flat] (although Beethoven does not take the trouble to alter the spelling) before it could resolve, as it does, upon the dominant of A[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. Thus the whole series of keys now in practical use can be arranged in what is called the circle of fifths (C G D A E B F[sharp] [= G[flat]] D[flat] A[flat] B[flat] F C, from which series we now see the meaning of what was said in the discussion of key-relationships as to the ambiguity of the relationships between keys a tritone fourth apart). Now no human memory is capable of distinguishing the difference of pitch between the keys of C and B[sharp] after a wide series of modulations. The difference would be perceptible enough in immediate juxtaposition, but after some interval of time the memory will certainly accept two keys so near in pitch as identical, whether in "just intonation" or not. 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. For example, the keys of the movements of Brahms's _C Minor Symphony_ are C minor, E major, A[flat] major (= G[sharp]), and C (= B[sharp]). And the same circle occurs in the opposite direction in the first movement of his _Third Symphony_, where the first subject is in F, the transition passes directly to D[flat] and thence by exactly the same step to A (= B[flat][flat]). The exposition is repeated, which of course means that in "just intonation" the first subject would begin in G[flat][flat] and then pass through a transition in E[flat][flat][flat] to the second subject in C[flat][flat][flat]. As the development contains another spurious enharmonic modulation, and the recapitulation repeats in another position the first spurious enharmonic modulation of the exposition, it would follow that Brahms's movement began in F and ended in C sextuple-flat! 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. Brahms's movement begins and ends in F much more exactly than any wooden diagonal fits a wooden square.

The following series of musical illustrations show the genesis of
typical harmonic resources of classical and modern music.

[dagger]).]

).]

[dagger]).]

, while the second appoggiatura ([dagger]) is chromatic.]

at
* is really A[flat], and [Dagger] is no longer a note of resolution,
but a chromatic passing-note._

_Definitions._

(Intended to comprise the general conceptions set forth in the above
article.)

1. _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.

2. 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.

3. An _interval_ is the difference in pitch between two notes.

4. _Rhythm_ is the organization, in a musical scheme, of sounds in
respect of time.

5. _Melody_ is the organization, in a musical scheme, of rhythmic
notes in respect of pitch.

6. _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.

7. 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.

8. 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.

9. 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.

10. 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. 1 above);
the 5th, as between Nos. 2 and 3; and the 4th, as between Nos. 3 and
4.

11. All notes exactly one or more octaves apart are regarded as
harmonically identical.

12. 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.

13. A chord is inverted when its lowest note is not its root.

14. 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).

15. 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.

16. _Unessential discords_ are those that are treated purely as the
phenomena of transition, delay or ornament, in an otherwise concordant
harmony.

17. _Essential discords_ are those which are so treated that the mind
tends to regard them as definite chords possessing roots.

18. 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.)

19. This final note of a key is called its _tonic_.

20. The _major_ mode is that of keys in which the tonic triad and the
two other cardinal triads are major.

21. The _minor_ mode is that of keys in which the tonic triad and one
other cardinal triad are minor.

22. 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.

23. _Modulation_ is the passing from one key to another.

24. _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.

25. _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.

26. _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.
It probably accounts for many so-called "essential discords."

In the form of _drones_ the pedal is the only real harmonic device of
ancient and primitive music. The ancient Greeks sometimes 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.

27. The _fundamental bass_ of a harmonic passage is an imaginary bass
consisting of the roots of the chords.

28. 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. (D. F. T.)

FOOTNOTES:

[1] Musical intervals are reckoned numerically upwards along the
degrees of the diatonic scales (described below). Intervals greater
than an octave are called compound, and are referred to their simple
forms, e.g. the 12th is a compound 5th.

[2] It is at least probable that this is one of the several rather
obscure reasons for the peculiar instability of the 4th in modern
harmony, which is not yet satisfactorily explained.

[3] The perfect concords are the octave, unison, 5th and 4th. Other
diatonic combinations, whether concords or discords, are called
imperfect.

[4] The submediant is so-called because if the subdominant is taken a
5th below the tonic, the submediant will come midway between it and
the tonic, as the mediant comes midway between tonic and dominant.

[5] See PLAIN SONG.

[6] Until Beethoven developed the resources for a wider scheme of
key-contrasts, the only keys for second subjects of sonata-movements
were the dominant (when the tonic was major) and the "relative" major
or dominant minor (when the tonic was minor). A wider range was
possible only in the irresponsible style of D. Scarlatti.

[7] Many theorists mistake the usual extreme emphasis on the dominant
chord of the dominant key, in preparation for second subjects, for a
modulation to the major supertonic, but this can deceive no one with
any sense of tonality. A good practical test is to see what becomes
of such passages when translated into the minor mode. Illusory
modulation to the flat 7th frequently occurs as a bold method of
throwing strong emphasis on to the subdominant at the outset of a
movement, as in Beethoven's _Sonata, Op. 31_, No. 1.

[8] Very rare, but the slow movement of Schubert's _C major String
Quintet_ demonstrates it magnificently.

[9] All the indirect relationships from a minor tonic are distinctly
strained and, except in the violently contrasted doubly indirect
keys, obscure as being themselves minor. But the direct artificial
modulation is quite smooth, and rich rather than remote. See
Beethoven's _C[sharp] minor Quartet._

[10] No classical example, though the clearer converse from a major
tonic occurs effectively.

[11] Not (with the exception of II) so violent as when from major
tonic. 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.

HARMOTOME, a mineral of the zeolite group, consisting of hydrous barium and aluminium silicate, H2BaAl2(SiO3)5 + 5H2O. 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.v.). Crystals are usually white and translucent, with a vitreous lustre. The hardness is 4(1/2), and the specific gravity 2.5.

The name harmotome (from [Greek: harmos], "a joint," and [Greek: temnein], "to cut") was given by R. J. Hauy in 1801, and has a crystallographic signification. Earlier names are cross-stone (Ger. _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. (L. J. S.)

HARMS, CLAUS (1778-1855), German divine, was born at Fahrstedt in Schleswig-Holstein on the 25th of May 1778, and in his youth worked in his father's mill. At the university of Kiel he repudiated the prevailing rationalism and under the influence of Schleiermacher became a fervent Evangelical preacher, first at Lunden (1806), and then at Kiel (1816). His trenchant style made him very popular, and he did great service for his cause especially in 1817, when, on the 300th anniversary of the Reformation, he published side by side with Luther's theses, ninety-five of his own, attacking reason as "the pope of our time" who "dismisses Christ from the altar and throws God's word from the pulpit." He also had some fame as a hymn-writer, and besides volumes of sermons published a good book on _Pastoraltheologie_ (1830). He resigned his pastorate on account of blindness in 1849, and died on the 1st of February 1855.

See _Autobiography_ (2nd ed., Kiel, 1852); M. Baumgarten, _Ein Denkmal
fur C. Harms_ (Brunswick, 1855).

HARNACK, ADOLF (1851- ), German theologian, was born on the 7th of May 1851 at Dorpat, in Russia, where his father, Theodosius Harnack (1817-1889), held a professorship of pastoral theology.

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 (1877-1878, 2 vols.). The liturgy of the Lutheran church of Russia has, since 1898, been based on his _Liturgische Formulare_ (1872).

The son pursued his studies at Dorpat (1869-1872) and at Leipzig, where he took his degree; and soon afterwards (1874) began lecturing as a _Privatdozent_. These lectures, which dealt with such special subjects as Gnosticism and the Apocalypse, attracted considerable attention, and in 1876 he was appointed professor extraordinarius. In the same year he began the publication, in conjunction with O. L. von Gebhardt and T. Zahn, of an edition of the works of the Apostolic Fathers, _Patrum apostolicorum opera_, a smaller edition of which appeared in 1877. Three years later he was called to Giessen as professor ordinarius of church history. There he collaborated with Oscar Leopold von Gebhardt in _Texte und Untersuchungen zur Geschithte der altchristlichen Litteratur_ (1882 sqq.), an irregular periodical, containing only essays in New Testament and patristic fields. In 1881 he published a work on monasticism, _Das Monchtum, seine Ideale und seine Geschichte_ (5th ed., 1900; English translation, 1901), and became joint-editor with Emil Schurer of the _Theologische Literaturzeitung_. In 1885 he published the first volume of his epoch-making work, _Lehrbuch der Dogmengeschichte_ (3rd ed. in three volumes, 1894-1898; English translation in seven volumes, 1894-1899). In this work Harnack traces the rise of dogma, by which he understands the authoritative doctrinal system of the 4th century and its development down to the Reformation. He considers that in its earliest origins Christian faith and the methods of Greek thought were so closely intermingled that much that is not essential to Christianity found its way into the resultant system. Therefore Protestants are not only free, but bound, to criticize it; indeed, for a Protestant Christian, dogma cannot be said to exist. An abridgment of this appeared in 1889 with the title _Grundriss der Dogmengeschichte_ (3rd ed., 1898). In 1886 Harnack was called to Marburg; and in 1888, in spite of violent opposition from the conservative section of the church authorities, to Berlin. In 1890 he became a member of the Academy of Sciences. At Berlin, somewhat against his will, he was drawn into a controversy on the Apostles' Creed, in which the party antagonisms within the Prussian Church had found expression. Harnack's view is that the creed contains both too much and too little to be a satisfactory test for candidates for ordination, and he would prefer a briefer symbol which could be rigorously exacted from all (cf. his _Das apostolische Glaubensbekenntnis. Ein geschichtlicher Bericht nebst einem Nachworte_, 1892; 27th ed., 1896). At Berlin Harnack continued his literary labours. In 1893 he published a history of early Christian literature down to Eusebius, _Geschichte der altchristl. Litteratur bis Eusebius_ (part 2 of vol. i., 1897); and in 1900 appeared his popular lectures, _Das Wesen des Christentums_ (5th ed., 1901; English translation, _What is Christianity?_ 1901; 3rd ed., 1904). One of his more recent historical works is _Die Mission und Ausbreitung des Christentums in den ersten drei Jahrhunderten_ (1902; English translation in two volumes, 1904-1905). It has been followed by some very interesting and important New Testament studies (_Beitrage zur Einleitung in das neue Testament_, 1906 sqq.; Engl. trans.: _Luke the Physician_, 1907; _The Sayings of Jesus_, 1908). Harnack, both as lecturer and writer, was one of the most prolific and most stimulating of modern critical scholars, and trained up in his "_Seminar_" a whole generation of teachers, who carried his ideas and methods throughout the whole of Germany and even beyond its borders. His distinctive characteristics are his claim for absolute freedom in the study of church history and the New Testament; his distrust of speculative theology, whether orthodox or liberal; his interest in practical Christianity as a religious life and not a system of theology. Some of his addresses on social matters have been published under the heading "Essays on the Social Gospel" (1907).

HARNESS (from O. Fr. _harneis_ or _harnois_; the ultimate origin is obscure; the Celtic origin which connects it with the Welsh _haiarn_, iron, has phonetic and other difficulties; the French is the origin of the Span. _arnes_, and Ger. _Harnisch_), probably, in origin, gear, tackle, equipment in general, but early applied particularly to the body armour of a soldier, including the trappings of the horse; now the general term for the gear of an animal used for draft purposes, traces, collar, bridle, girth, breeching, &c. It is usually not applied to the saddle or bridle of a riding animal. The word, in its original meaning of tackle or working apparatus, is still found in weaving, for the mechanism which shifts the warp-threads to form the "shed," and in bell-hanging, for the apparatus by which a large bell is hung. The _New English Dictionary_ quotes an early use of the word for the lines, rod and hooks of an angler (_Fysshing with an Angle_, c. 1450).

HARO, CLAMEUR DE, the ancient Norman custom of "crying for justice," still surviving in the Channel Islands. The wronged party must on his knees and before witnesses cry: "Haro! Haro! Haro! a l'aide, mon prince, on me fait tort." This appeal has to be respected, and the alleged trespass or tort must cease till the matter has been thrashed out in the courts. The "cry" thus acts as an interim injunction, and no inhabitant of the Channel Islands would think of resisting it. The custom is undoubtedly very ancient, dating from times when there were no courts and no justice except such as was meted out by princes personally. The popular derivation for the name is that which explains "Haro" as an abbreviation of "Ha! Rollo," a direct appeal to Rollo, first duke of Normandy. It is far more probable that haro is simply an exclamation to call attention (O.H.G. _hero_, _hara_, "here"!). Indeed it is clear that the "cry for justice" was in no sense an institution of Rollo, but was a method of appeal recognized in many countries. It is said to be identical with the "Legatro of the Bavarians and the Thuringians," and the first mention of it in France is to be found in the "Grand coutumier de Normandie." A similar custom, only observed in criminal charges, was recognized by the Saxon laws under the name of "Clamor Violentiae." Thus there is reason to think that William the Conqueror on his arrival in England found the "cry" fully established as far as criminal matters were concerned. Later the "cry" was made applicable to civil wrongs, and, when the administration of justice became systematized, disappeared altogether in criminal cases. It naturally tended to become obsolete as the administration of justice became systematized, but it was long retained in north-western France in cases of disputed possession, and was not actually repealed until the close of the 18th century. A survival of the English form of haro is possibly to be found in the "Ara," a cry at fairs when "settling time" arrived.

HAROLD I. (d. 1040), surnamed Harefoot, the illegitimate son of Canute, king of England, and Aelfgifu of Northampton. On the death of his father in 1035, he claimed the crown of England in opposition to Canute's legitimate son, Hardicanute. His claims were supported by Leofric, earl of Mercia, and the north; those of Hardicanute by his mother, Queen Emma, Godwine, earl of the West-Saxons and the south. Eventually Harold was temporarily elected regent, pending a final settlement on Hardicanute's return from Denmark. Hardicanute, however, tarried, and meanwhile Harold's party increased rapidly. In 1037 he was definitely elected king, and banished Emma from the kingdom. The only events of his brief reign are ineffectual inroads of the Welsh and Scots. Hardicanute was preparing to invade England in support of his claims when Harold died at Oxford on the 10th of March 1040.

HAROLD II. (c. 1022-1066), king of the English, the second son of Earl Godwine, was born about 1022. While still very young (before 1045) he was appointed to the earldom of the East-Angles. He shared his father's outlawry and banishment in 1051; but while Godwine went to Flanders, Harold with his brother Leofwine took refuge in Ireland. In 1052 Harold and Leofwine returned. Having plundered in the west of England, they joined their father, and were with him at the assembly which decreed the restoration of the whole family. Harold was now restored to his earldom of the East-Angles, and on his father's death in 1053 he succeeded him in the greater earldom of the West-Saxons. He was now the chief man in the kingdom, and when the older earls Leofric and Siward died his power increased yet more, and the latter part of Edward's reign was virtually the reign of Harold. In 1055 he drove back the Welsh, who had burned Hereford. In 1063 came the great Welsh war, in which Harold, with the help of his brother Tostig, crushed the power of Gruffyd, who was killed by his own people. But in spite of his power and his prowess, Harold was the minister of the king rather than his personal favourite. This latter position rather belonged to Tostig, who on the death of Siward in 1055 received the earldom of Northumberland. Here, however, his harshness soon provoked enmity, and in 1065 the Northumbrians revolted against him, choosing Morkere in his place. Harold acted as mediator between the king and the insurgents, and at length agreed to the choice of Morkere, and the banishment of his brother. At the beginning of 1066 Edward died, with his last breath recommending Harold as his successor. He was accordingly elected at once and crowned. The men of Northumberland at first refused to acknowledge him, but Harold won them over. The rest of his brief reign was taken up with preparations against the attacks which threatened him on both sides at once. William challenged the crown, alleging both a bequest of Edward in bis favour and a personal engagement which Harold had contracted towards him--probably in 1064; and prepared for the invasion of England. Meanwhile Tostig was trying all means to bring about his own restoration. He first attacked the Isle of Wight, then Lindesey, but was compelled to take shelter in Scotland. From May to September the king kept the coast with a great force by sea and land, but at last provisions failed and the land army was dispersed. Harold then came to London, ready to meet whichever enemy came first. By this time Tostig had engaged Harold Hardrada of Norway to invade England. Together they sailed up the Humber, defeated Edwin and Morkere, and received the submission of York. Harold hurried northwards; and on the 25th of September he came on the Northmen at Stamford Bridge and won a complete victory, in which Tostig and Harold Hardrada were slain. But two days later William landed at Pevensey. Harold marched southward as fast as possible. He gathered his army in London from all southern and eastern England, but Edwin and Morkere kept back the forces of the north. The king then marched into Sussex and engaged the Normans on the hill of Senlac near Battle (see Hastings). After a fight which lasted from morning till evening, the Normans had the victory, and Harold and his two brothers lay dead on the field (14th of October 1066).

HARP (Fr. _harpe_; Ger. _Harfe_; Ital. _arpa_), a member of the class of stringed instruments of which the strings are twanged or vibrated by the fingers. The harp is an instrument of beautiful proportions, approximating to a triangular form, the strings diminishing in length as they ascend in pitch. The mechanism is concealed within the different parts of which the instrument is composed, (1) the pedestal or pedal-box, on which rest (2) the vertical pillar, and (3) the inclined convex body in which the soundboard is fixed, (4) the curved neck, with (5) the comb concealing the mechanism for stopping the strings, supported by the pillar and the body.

(1) The _pedestal_ or _pedal-box_ forms the base of the harp and
contains seven pedals both in single and double action harps, the
difference being that in the single action the pedals are only capable
of raising the strings one semitone by means of a drop into a notch,
whereas with the double action the pedals, after a first drop, can by
a further drop into a second and lower notch shorten the string a
second semitone, whereby each string is made to serve in turn for
flat, natural and sharp. The harp is normally in the key of C flat
major, and each of the seven pedals acts upon one of the notes of this
diatonic scale throughout the compass. The choice of this method of
tuning was imposed by the construction of the harp with double action.
The pedals remain in the notches until released by the foot, when the
pedal returns to its normal position through the action of a spiral
spring, which may be seen under each of the pedals by turning the harp
up.

(2) The _vertical pillar_ is a kind of tunnel in which are placed the
seven rods worked by the pedals, which set in motion the mechanism
situated in the neck of the instrument. Although the pillar apparently
rests on the pedestal, it is really supported by a brass shoulder
firmly screwed to the beam which forms the lowest part of the body, a
connexion which remains undisturbed when the pedal box and its cover
are removed.

(3) The _body_ or _sound-chest_ of the harp is in shape like the
longitudinal section of a cone. It was formerly composed of staves
joined together as in the lute and mandoline. Erard was the first to
make it in two pieces of wood, generally sycamore, with the addition
of a flat soundboard of Swiss pine. The body is strengthened on the
inside, in order to resist the tension of the strings, by means of
ribs; there are five soundholes in the back, which in the older models
were furnished with swell shutters opened at will by the swell pedal,
the fourth from the left worked by the left foot. As the increase of
sound obtained by means of the swell was infinitesimal, the device has
now been discarded. The harp is strung by knotting the end of the
string and passing it through its hole in the centre of the
soundboard, where it is kept in position by means of a grooved peg
which grips the string.

(4) The _neck_ consists of a curved piece of wood resting on the body
at the treble end of the instrument and joining the pillar at the bass
end. In the neck are set the tuning pins round which are wound the
strings.

(5) The _comb_ is the name given to two brass plates or covers which
fit over both sides of the neck, concealing part of the mechanism for
shortening the strings and raising their pitch a semitone when
actuated by the pedals. On the front plate of the comb, to the left of
the player, is a row of brass bridges against which the strings rest
below the tuning pins, and which determine the vibrating length of the
string reckoned from the peg in the soundboard. Below the bridges are
two rows of brass disks, known as forks, connected by steel levers;
each disk is equipped with two studs for grasping the string and
shortening it. The mechanism is ingenious. When a pedal is depressed
to the first notch, the corresponding lower disk turns a little way on
a mandrel keeping the studs clear of the string. The upper disk, set
in motion by the steel levers connecting the disks, revolves
simultaneously till the string is caught by the two studs which thus
form a new bridge, shortening the vibrating length of the string by
just the length necessary to raise the pitch a semitone. If the same
pedal be depressed to the second notch, another movement causes the
lower disk to revolve again till the string is a second time seized
and shortened, the upper disk remaining stationary. The hidden
mechanism meanwhile has gone through a series of movements; the pedal
is really a lever set upon a spring, and when depressed it draws down
the connecting rod in the pillar which sets in motion chains governing
the mandrels of the disks.

The harp usually has forty-six strings, of gut in the middle and upper
registers, and of covered steel wire in the bass; the C strings are
red and the F strings blue. The compass thus has a range of 6(1/2)
octaves from [music notes]. The double stave is used as for the
pianoforte. The single action harp used to be tuned to the key of
E[flat] major.