Harmony Its Theory and Practice-prout

..-.- .^~f2™*" Un'»e«lty Library

MT

40.P96 1903

3

1924

021

757

285

CORNELL

UNIVERSITY

LIBRARY

Gift of

George

H.

Sabine

MUSIC

«

Cornell University

B

Library

The

original of

tliis

book

is in

tlie

Cornell University

Library.

There

are

no

known

copyright

restrictions

in

the

United States

on

the

use

of

the

text.

AUGENER'S

EDITION,

No. 918a

HARMONY

ITS

THEORY AND

PRACTICE

BY

EBENEZER

PROUT

THIRTIETH

EDITION

REVISED

AND

LARGELY

REWRITTEN

TWENTY-EIGHTH

IMPRESSION

AUGENER,

LTD.

LONDON

Boston,

Mass.:

THE

BOSTON MUSIC

CO.

Sole Agents for America

"Untwisting all the Chains that Tie

The

Hidden

Soul of

Harmony."

MILTON,

L'Allegro.

Stanbope]pres2 H.

GILSON

COMPANY

PREFACE

TO

THE

FIRST

EDITION.

So

large

a

number

of

works

on

Harmony

alreadyexists thatthe publication ofa

new

treatise

on

the subject

seems

to call for

explanation, ifnot forapology. "

_The

present

volume

isthe

out-come

of

many

years' experience inteachingthe theoryof music,

and

the author

hopes

that it containssufficientnoveltyboth in plan

and

in matterto plead ajustification for itsappearance.

Most

intelligentstudentsof

harmony

have

at times

been

per-plexed

by

their inability toreconcile passages they

have

foundin the

works

of the great masterswith the rulesgiven in the

text-books. If theyask the helpof theirteacher in their difficulty,

theyare probably told,

"Bach

is

wrong,"

or

"Beethoven

is

wrong,"

or, at best,

"This

is

a

licence."

No

doubt

examples

ofveryfreepart -writing

may

be found

in the

works

of

Bach

and

Beethoven, or

even

of

Haydn

and

Mozart

; several such are

noted

and

explained in the present work.

But

the principle

must

surely

be

wrong

which

placesthe rules of

an

early stage of musical

development above

the inspirationsof genius!

Haydn,

when

asked according to

what

rule

he

had

introduceda certain

harmony,

replied that '

'the rules

were

all hisveryobedient

hum-ble servants'

';

and

when

we

find that in our

own

time AVagner,

or

Brahms,

or

Dvorak

breaks

some

rule given in old text-books there is, to saythe least,

a

very strong presumption, not that the

composer

is

wrong,

but-that the ruleneeds modifying. In other words, practice

must

precede theory.

The

inspired

com-posergoes first,

and

invents

new

effects; it isthebusiness ofthe theorist not to cavil at every novelty, but to follow

modestly

behind,

and

make

his rules

conform

to the practice of the master. It is a significant fact that,

even

in the

most

recent

developments

ofthe art, nothing has yet

been

written

by any

composer

of

eminence

which

a

sound

theoreticalsystem

cannot

satisfactorily accountfor;

and

the objections

made

by

musicians

ofthe old school to thenovel

harmonic

progressionsof

Wagner

are little

more

than repetitionsof the severe criticisms

which

in the early years of the present century

were

launched

at the

works

ofBeethoven.

iv

Preface

to

the First

Edition.

It is

from

this pointof

view

that the present

volume

has

been

written.

The

rules herein given,

though

in

no

degree

in-consistentwith the theoretical system

expounded,

arefounded, not

upon

that, nor

on

any

other abstract system, but

upon

the actual practiceofthe greatmasters; so that

even

those musicians

who

may

differ

most

widely

from

the author's theoretical views

may

still

be

disposed to

admit

the forceof practical rules sup-ported

by

theauthoritypf Bach, Beethoven, or

Schumann.

The

system of theory

propounded

in the present

volume

is

founded

upon

the

dictum

of

Helmholtz, quoted

in

Chapter

II. ofthis

work

_(§ 42), that

"the

system of Scales,

Modes,

and

Harmonic

Tissues does not rest solely

upon

unalterable natural laws, butisat leastpartly also the resultofaesthetical principles,

which have

already changed,

and

will still further

change

with the progressive

development

of humanity.'

' While, therefore,

theauthorfollows

Day and

Ouseleyin takingthe

harmonic

series

as the basis ofhis calculations,

he

claims the right to

make

his

own

selection,

on

aesthetic grounds,

from

these harmonics,

and

touse only such of

them

as

appear

neeful toexplain the practice of the great masters.

Day's

derivation of the chords in

a key

from

the tonic, dominant,

and

supertonic is

adhered

to, but in otherrespectshissystem isextensivelymodified, itspurely

phys-icalbasisbeingentirely

abandoned.

Itwill

be

seen in

Chapter

II. _(§

₄₄₎

that

by

rejecting altogether the eleventh

and

thir-teenth notes of the

harmonic

series,

and

taking in their place other notes

produced

among

thesecondary harmonics, the chief objection

made

by

the

opponents

of all scientific derivation of

harmony

—

that

two

of the

most

important notes of the scale,

the fourth

and

the sixth, are

much

out of tune

—

^has

been

fully

met. In the

vexed

questionof the

minor

tonic chord,

Helm-holtz is followed to

a

considerable extent; but Ouseley's ex-planation of the

harmonic

originof the

minor

thirdisadopted.

Truth

is

many

sided;

and

no

writer

on

harmony

isjustified

in saying that his views are the onlycorrect ones,

and

thatall

others are wrong.

No

such claimis

made

forthe system herein

setforth; but it is

hoped

that itwill at least

be found

to

be

in^ telligible, perfectly consistent with itself,

and

sufficiently

com-prehensive toexplain the progressions of the

advanced

modern

school of composers.

It has

been

thought desirable to separate as far as possible the practical from the theoretical portions of this work.

The

latterare thereforeprintedin smaller type;

and

itwill

be found

advisable forbeginners,

who

may

take

up

this

work

without

any

previous

knowledge

of the subject, to

omit

at least Chapters II.

Preface

to

the First

Edition.

v

until

some

considerable progresshas

been

made

in the practical part of the volume.

The

exact point at

which

the studentwill

do

well toreturn tothe omitted portions will

depend

upon

his

progress

and

his general intelligence,

and

must be

left to the discretion ofthe teacher.

In the practical part of the

work

an

attempt has

been

made

to simplify

and

to codify the laws.

With

a view

of effecting these objects,

many

rules

now

obsolete,

and

contravened

by

the daily practice of

modern

writers,

have

been

altogether omitted,

and

others

have

been greatlymodified; while the lawsaffecting

the chords, especially the higherdiscords

—

the ninths, elevenths,

and

thirteenths

—

^have

been

classified,and, itishoped, materially simplified. It isof the utmost importance that students

who

wish tomaster the subject should

proceed

steadily

and

deliber-ately.

For

example, a proper understanding of the chords of the eleventh will

be

impossibleuntil thestudentisquite familiar with the chords of the ninth,

which

in their turn

must be

pre-ceded by

the chords of the seventh.

The

learner's

motto must

be,

"

One

thingat

a

time,

and

that

done

thoroughly."

In preparingthe exercisesaspecial

endeavour

has

been

made

to render

them

interesting, as far as possible,

from

a

musical point of view.

With

thisobject theyare, with a fewexceptions, written in the

form

ofshort musical sentences, mostlyin four-bar

rhythm,

illustratingthe variousforms of cadence.

To

stim-ulate the pupil's imagination,

and

to encourage attempts at composition,

many

exercisesare inthe

form

of double chants or

hymn

tunes.

Each

bass can, ofcourse,

be harmonized

in several differentpositions;

and

thestudent's ingenuitywill

be

usefully

exercisedintryingtowrite asmelodious

an upper

part as possible forthese littlepieces.

Not

the least interesting

and

valuablefeatureof the

volume

will, it is believed,

be found

in the illustrativeexamples, con-siderably

more

than

300

in

number.

These have been

selected

chiefly,

though

not exclusively,

from

the

works

ofthe greatest masters,

from

Bach

and

Handel

down

tothepresentday. Earlier

examples

are not given, because

modern harmony

may

be

said tobeginwith

Bach

and

Handel.

While

it has

been

impossible without exceeding reasonable limits to illustrate all the points

mentioned,

it is

hoped

that at least

no

ruleof importance has

been

given without quoting

some

recognized author inits sup-port. It

may

at all events

be

positively said that,

had

want

of space not preventedtheir quotation,

examples might have been

found

to illustrate every rule laid

down

in the volume.

It

was

originallyintended to

have

included in the present

vi

Preface

to

the

First Edition.

The

volume

has, however,

extended

to so

much

larger

dimen-sions than

was

at first contemplated, that these chapters,

which

belong rather to practical composition than to

harmony

in its

strict sense,

have

been

reluctantlyomitted. It is intended to follow the present

work

by

atreatise

on

Composition, in

which

these

and

similarsubjects will

be

more

appropriately dealt with.

The

authordesires to

acknowledge

the valuable assistance

he

has received in the preparation ofhiswork, first

and

foremost

from

his son, LouisB. Prout, to

whom

he

isindebted fora very large

number

of the illustrative examples,

and

who

has also written

many

of theexercises. Valuable aid has also

been

re-ceived

from

the late

Rev.

Sir Frederick Ouseley, with

whom,

down

to the time of his

lamented

death, the author

was

in frequent correspondence

on

the subject of this work.

To

his friend, Dr. Charles

W.

Pearce, also, theauthor

must

express his thanks for

much

generous interest

and

many

most

useful sug-gestions, aswell as for his

kind

assistance in revising the proof-sheetsof thevolume. ,

It

would be

unreasonable to expect that the present

work

will

meet

with universal approval; butit

may

at least claimto appealto teachers

and

students as

an

honest attempt tosimplify ^hestudyof

harmony,

and

tobringit

down

to date.

vu

PREFACE

TO

THE

SIXTEENTH

EDITION.

It is

now

more

than twelve years since the firsteditionof

Harmony

: Its Theory

and

Practice

was

published;

and

the

great successwith

which

the

work

has

met

has

no

lesssurprised than gratified its author.

At

the

same

time

he must

saythat, after so

many

years' experience in teaching

from

it,

he

would

have

been

either hopelesslyignorantorincurably conceited

had

he

not

become

fully

aware

of its

numerous

defects

and

short-comings.

He

has felt that he could best

show

hisappreciation of its generous reception

by

themusical public

by

improving

it

as far as lay in his power.

For

some

years pastit has

been

his

intention to

do

this as soon^as the pressure of

work

allowed;

but

he

has thought it best to

complete

theseries of

which

this

forms the first

volume

before undertaking so serious a task as remodellingthis treatise.

Though

called

a

new

edition, it

would be

hardly too

much

todescribethe presentasa

new

book. Considerably

more

than half the text is either additional matter, or has

been

entirely rewritten.

A

short account of the modifications introduced is

necessary, that the reason for the

numerous

changes

made

may

be

understood.

First

and

foremost

among

these isthe virtual

abandonment

of the

harmonic

seriesasthe basis

on which

thesystemisfounded. Further investigation

and

thought

have

convinced

the author thatthe practical objections to the derivation of the higher

dis-cords

—

the ninths, elevenths,

and

thirteenths

—

from

thenatural

seriesof

upper

partials

were

far greater than

he

had

realized in

firstwriting the

volume.

That

the acoustical side of thesubject hasnevertheless

an

important bearing

on

harmony

he

stillholds

;

and

this matter is dealt with in

Appendix

B,

which

replaces

Chapter

II of previous editions.

But

the

modern

key,

whether

major

or minor, is so largely the resultof aesthetic, rather than ofscientific considerations that it is far better for the student that it should

be

dealt with

from

the former point of view. It

is obvious that this

change

has necessitated

an

entirely

new

treatment of the questionof the chromatic constituents of a

viii

Preface

to

the Sixteenth

Edition.

key.

The

plan

now

adopted

will,it isbelieved,

be found

much

simpler

and

easier, especiallyfor self-instruction, thanits

prede-cessor.

The

chromatic

element

is regarded as subordinate to the diatonic,

and

chromatic chordsareconsidered as

being

bor-rowed from

neighbouringkeys. This

view

itisbelieved,

was

first

propounded

by

the author's son, LouisB. Prout, in his

Harmonic

Analysis, to

which

little

work

theauthor

acknowledges

his

obli-gationsfor

many

valuablesuggestions.

The new

treatment ofthe subjecthas involved the

rearrange-ment

of agreatpartof thecontents ofthevolume.

The

whole

ofthe diatonic material of the key,

up

to

and

including the

chord

ofthe

dominant

thirteenth, isdealtwith before the chro-matic chordsareintroduced. This hasnecessitated the

remodel-ling,

and

in

some

cases the entire rewritingof the exercises.

To

avoid the inconvenience arising

from

the useof

two

books, theAdditionalExercises

have

been

incorporated in thevolume,

and

the exercises

on each

chapter

have been

graduated, asfaras

practicable, in the orderofdiiiSculty. This, however, does not applytothe

Chants

and

Hymn

Tunes,

which

ithas

been

thought advisable to place

by

themselvesatthe

end

ofeach chapter.

A

new

featureof the present edition is that,

from

thevery beginning, the

harmonizing

of simplemelodiesistaught simulta-neously with the

harmonizing

of figured basses. It is often

found

that pupils

who

can

do

the latterwith ease are hopelessly atsea

when

they attempt the former.

The

author's

own

experi-enceis that, ifthe

two

aretakentogether, the harmonization of

a melody

presents

no

verygreat difficulty

even

tobeginners of averageability,

and

that each

branch

of the studythrowslight

upon

the other.

Another most

important addition tothis

volume

is the full

analysisofthe

harmony

given throughout.

The

system adopted,

though

with considerable modification,is thatof E. F. Richter

;

theauthor has

extended

the idea of his predecessor

by making

a difference

between

' 'inversions ' '

and

' 'derivatives ' ' ofchords

(See § 252).

Though

the

method

looksatfirstsight

complex,

it isinpractice extremelysimple, ifsystematically pursued

from

the

beginning

;

and

the insightintothe

harmonic

structure ofa

composition

which

isobtained

by

its

means

will

be

found

by

the earnest student invaluable.

For

hisguidance,

and more

partic-ularlyfor teachers

who

may

use this book, special attention is

called tothe

new

Key

totheExercises. In this, every

chord

in each exercise is analyzed

on

the system here taught.

Though

the

Key

shouldin

no

case

be

usedasa

"

cram,"

itwill

be most

useful toconsult itafterthe exercises

have been worked,

and

the analyses written

beneath

them.

Preface

to

the

Sixteenth

Edition.

ix

Of

the

two

Appendices, the

second

has

been

already

spoken

of.

The

firstcontainsanecessarily brief

and

incomplete account of the Ecclesiastical

Modes, a

subjectof

which most

students

know

little or nothing, but a slight acquaintance

with which

will

be found

ofgreat use inaiding their

comprehension

of

much

of themusic of the seventeenth

and

eighteenthcenturies.

The

sketch

—

forit professes to

be

nothing

more

—

has

been compiled

from

many

sources, chief

among

which

should

be

named

Marx's

Composition

and

Charles Child Spencer's

Brief Account

ofthe

Church Modes.

The

author

would acknowledge

his obligations to

many who

have

kindly assisted

him

with advice

and

suggestions in the preparationof this

new

edition.

From

his son

he

hasreceived

much

valuable help;

and

he

also desires especially to

thank

his

friends Dr. A. _J. Greenish

and

Mr.

R.

Orlando

Morgan

for

many

useful practicalhints.

He

isalsoindebted to several

kind

friends for theirhelp in readingthe proofs

—

a

more

thanusually laborioustask,especially asregardsthe

Key,

owing

tothemulti

plicityof detail.

It is hoped, not without

some

confidence, that the present edition will

be found

not only

more

complete, but far simpler

for teaching purposes than the

work

in its earlier

form

; if it

smooths

over the difficulties in the path of the student, the author will feel himself well repaid for the year's hard

work

spentinitspreparation. ^

NOTE.

In consequenceof the rearrangementof thesubjectmatterinthe present

edition, both Chapters and Sections have been renumbered.

As

there are manyreferencesto

Harmony

in the followingvolumesof theseries,atableis

here given of the Chapters and Sections referredto,with the corresponding

numbersinthenewedition. Asthe othervolumesof theseriesarereprinted,

the necessaryalterationsof referenceswill be madeinthetext. Intwocases

(^§423, 434), theywill not befound, because the passages in question have beenreplacedintheneweditionbyothers.

XI

TABLE

OF CONTENTS.

INiB.

—

Thenumbersreferin every instancetothe sections,nottothepages ^

CHAPTER

I.

—

Introduction page I

Amountofknowledge presupposed, i

—

A

musical sound; pitch, 2

—

Melody

andharmonydefined, 3

—

Interval defined,4

—

Semitone, 5

—

Enharmonic

interval, ₅ (note)

—

Diatonic and chromatic semitones, 6

—

A

Tone 7

—

Scales, 8

—

Diatonic scales, g

—

Thechromatic scale, 10

—

Names

of the degreesofthediatonicscale,11-13

—

Consonance anddissonancedefined,

14-16

—

TheResolution of a Dissonance, 16

—

Discords, 17

—

How

intervals are reckoned, 18

—

Compound

intervals,ig

—

Different kinds ofintervals,

20

—

Perfect and major intervals, 21

—

Minor intervals, 22

—

Augmented and diminished intervals, 23, 24

—

Inversion ofintervals, 25,26

—

Inver-sionofcompound intervals,27

—

Consonantanddissonantintervals,28

—

Perfectandimperfectconsonances, 2g

—

Tableofintervals, 30.

CHAPTER

II.—Key, or Tonality page 13

DefinitionofJCey,31-33

—

Developmentofkey fromitstonic,

34—

Thediatonic

and chromatic elementsin a key, 35

—

A

Chord: majorand minor

com-mon

chords,36

—

Majorand minorkeys,37

—

The "primarynotes " of a key,38

—

Thediatonicmaterialof amajor key, 3g,40

—

How

tomarkthe

roots,41

—

The diatonic materialofa minorkey,42

—

Thedifference

be-tween major and minor keys,43

—

Positionof the semitones,44

—

Other

keys than C,45

—

Tetrachords,46

—

Keyswith sharps,47-49

—

Keyswith

flats,50,51

—

How

to find the tonicofany majorkey,52

—

Tableof

key-signatures; enharmonic keys, 53

—

How

to find the signature of a key

containingmorethanseven sharps orflats,

54-S^-CHAPTER

III.

—

The

General Laws

of Part-Writing page 22

A

Partdefined,57

—

Rules of melodic progression; "conjunct" and "dis-junct motion," 58

—

Diminished intervals,

how

treated, 59

—

Augmented intervals,60

—

Largeintervals,61

—

The leap ofa seventh withone

inter-mediate note, 62.

—

Leaping to an accented note, 63

—

Harmonic pro-gression; similar,oblique, and contrarymotion,64

—

Four-partharmony

;

namesof the voices,65

—

Rulesofpart-writing,66

—

Consecutive unisons

andoctaves,67-70^—Consecutivefifths,71—73

—

Hiddenoctavesandfifths,

74—77

—

Consecutivefourths,78

—

Consecutive seconds, sevenths,andninths, 79,80

—

Theprogressionfromasecondtoa unison, 81

—

Approachingand

leavingtheunisonbysimilarmotion,82-84

—

Overlappingandcrossingof

parts,85

—

An

unprepareddiscordbestapproached bycontrarymotion,86

—

Thechiefdifficultiesofharmonyfoundintheearlierstages, 87.

CHAPTER

IV.

—

The

Diatonjc Triadsof

the

Major

Key

page 34

A

Triad, 88

—

^The diatonic triads of a major key, 8g

—

The Diminished

Triad, go

—

"Doubling"a note, 91

—

Compassof the voices, 92

—

Close

and extended position,93

—

The best positionofharmony, 94-96. The best notes to double,97-99

—

Treatmentof the leading note, 100,101

—

Omission ofone note of a chord, 102

—

Three rules for part-writing,

xii

Contents.

define the key,

112—

The connection of the primary triads with one

another,

113—

The positionof the first chord,

114—

Exercises with bass given,

115—

Harmonizingamelodywith primarytriadsonly; the import-ance of hearing themusic mentally,116

—

The Cadence: Authentic and

Plagal Cadences, 117

—

Thepositionof the cadence,

118—

Thechoiceof chords,

119—

A

melody harmonized, 120-126

—

The stiiifeffect of this

harmony: the reason, 127

—

Melodiesgivenforharmonization, 128.

CHAPTER

V.

—

The

Diatonic Triads of

the Major

Key

(Continued).

Sequences f^g' 49

The" secondarytriads," 129

—

Which notes todouble, 130

—

Theintermixing of primaryandsecondarytriads, 131

—

Three-partharmonywithonepart

doubled, 132

—

Sequence defined,133

—

Tonalsequence,134

—

Lengthof

pattern,_13S

—

Intervalofimitation,136

—

Irregularsequence;?-ifa/sequence,

137

—

Licenses permittedin repetitionsofa sequence,

138—

Changingthe

positionof chordsina sequence, 139, 140

—

Directionsforworking,141

—

Exerciseswithtrebleandbassgiven,142

—

Ditto with bass only given, I43

—

Harmonizingmelodies: choice of chords,144

—

How

tobegin andend,

145

—

Root progressions: the leapof a fourth,146

—

Theleap of athird,

147

—

Progressions by step, 148, I49

—

General principles, 150-152

—

A

melodyharmonized, 153-158

—

Melodiesforharmonization, 159.

CHAPTER

VI.

—

The

Inversionsof

the

Triads of

a

Major

Key

page61 Inversionin general,160

—

Inversionof achorddefined; thenumberof

possi-ble inversions,161

—

Thefirstinversion,162

—

FiguredBass, 163

—

Figuring of thefirst inversion, 164

—

How

to show theposition of chords

when

markingthe roots,165

—

Whichnotestodouble, 166-168

—

Treatmentof a seriesoffirst inversions,169-171

—

The second inversion,172

—

Its

in-complete effect, 173

—

How

to find the rootofa chord fromthe figured

bass, 174, 175

—

Secondinversionsofprimarytriads, 176

—

The "cadential

5" of the tonic chord, 177, 178

—

Itsposition inthebar,179

—

^Thesame chordused non-cadentially,180

—

Thesecondinversionof thesubdominant

chord, 181

—

^Progressionof thevoices ina cadential_J,182.

—

The

second

inversionof the dominant chord, 183

—

Second inversions of secondary

triads, 184-187

—

Rules for approaching a second inverson, 188

—

Rules

for quitting a second inversion,

189—

The best note to double, igo

—

Special exercises for the treatment ofsecond inversions, 191

—

Exercises

on a figured bass, 192

—

Enlarged meaningof theterm "progression of

roots," 193

—

^Additional rules for root progression, 194-I99

—

A

melody

harmonized,200-203

—

Melodiesgivenforharmonization,204

—

The com-positionoffour-barphrases, 205.

CHAPTER

VII.

—

The

Minor Key:

Its Diatonic Triads

and

their

Inversions fage 80

The

Harmonic Minor Scale,206

—

Older forms ofscale: theAeolian,207

—

The Dorianscale,208

—

Melodicminorscale,209

—

Theleading-note ofa

minor keyneverwritten inthesignature,210

—

Relativemajorand minor

keys,211

—

Tonic majorand minorkeys,212

—

Thediatonictriadsof the

minor key, 213

—

The augmented triad on the mediant, 214

—

The

diminished triad on the supertonic, 215

—

The "Dorian sixth": its employment in the subdominant chord, 217, 2i8

—

A

modem

example

Contents.

xiii

progression between dominant and submediant in the minorkey,221

—

The doubling of the submediant, when to be avoided, 222

—

First

in-versions:

how

to avoid the augmented second,223

—

Chords containing the minor seventh of the Icey, 224-226

—

How

to indicatethe rootsof these chords, 227

—

Second inversions in the minor Icey, 228

—

The

Tiercede Ficardie,229

—

Exercises, 230.

CHAPTER

Vin.

—

The

Chord

of

the

Dominant

Seventh page 94

Effect of tonic and dominant harmony, 231

—

The chord of the dominant: seventh: a "fundamental discord " defined, 232

—

The seventh a

disso-nance, requiring resolution, 233

—

Treatmentof the third andseventh of the chord, 234

—

Omission of the fifth, 235

—

Resolution on the tonic

chord,236

—

Ornamentalresolutionsof the seventh, 237-239

—

Resolution

on

^t

firstinversion of the tonic chord,240

—

Reasonof theimportance of the chord of the dominant seventh,241

—

Resolutionon the submedi-ant chord, 242

—

Resolution on the subdominant chord, 243

—

The

in-versionsof thechord ofthedominantseventh,244

—

Theirfiguring, 245-247

—

The first inversion: its various resolutions, 248-250

—

Omissionof the generator in this inversion, 251

—

Distinctionbetweeninversionsand

derivatives,252

—

How

to mark the rootsofderivatives,253

—

Examples

of the first inversion anditsderivative,254-256

—

Thesecondinversion

:

its most usual resolution,257

—

Itsresolutiononih^firstinversionof the

tonic chord, 258, 259

—

Other resolutions, 260

—

The derivative of the

second inversion,261

—

Examples by Handel, 262

—

The third inversion

:

its usual resolution,263

—

Resolutiononthesecondinversionof the

sub-mediantchord: theonlysatisfactoryposition,264

—

The derivativeof the

thurd inversion,265

—

Changes in the positionof a chordof the seventh before resolving, 266

—

Treatment of the dominant seventh in a full

cadence,267

—

Exceptionalresolutions,268

—

Exercises,269.

CHAPTER

IX.

—

Key

Relationship

—

Modulation

to nearly related

Keys

—

FalseRelation page 115

Modulation and Transition defined, 270

—

Key

Relationship, 271

—

Related majorkeys,272

—

Unrelated keys,273

—

Chords

common

tonearlyrelated

keys,274

—

Tableof nearlyrelatedkeys, 275,276

—

How

theirkey signa-tures differ, 277

—

How

to effect modulation,278

—

Figuring the bass in

modulations, 279

—

Immediate and gradual modulation, 280

—

How

to

regardambiguous chords, 281

—

Changeof the primarynotes,282

—

The^

modulation from a major keytoits supertonic minor,283

—

Modulations from minor keys, 284—Examples of modulations analyzed, 286-291

—

General rules formodulatingtonearlyrelatedkeys,292

—

Modulation by meansofirregular resolutionsof thedominantseventh,293

—

Thechoice of modulations,294

—

TransitionalDominants, 295

—

False Relation,296

—

Exercises, 297

—

Harmonization of Chorals: the Cadences, 298

—

The

Inverted Cadence,299

—

The Interrupted Cadence, 300

—

Positionof the cadences, 301

—

Implied modulations,302

—

Chorals to harmonize, 303.

CHAPTER

X.

—

Unessential Discords(I)

—

AuxiliaryNotes, Passing

Notes,

and

Anticipations page 131

Notes unessential to the harmony, 304

—

Auxiliatynotes defined,305

—

How

takenandleft,306

—

When

a toneandwhena semitonefromtheharmony note,307

—

Chromaticauxiliary notes,308

—

Auxiliary notes inmorethan

onepart,309

—

PassingChords, 3 10

—

Examplesreferredto,312

—

XIV

Contents.

notes,313

—

Singlechangingnotes,314

—

Auxiliary notes takenbyleap,315

—

Passingnotes,31 6

—

Dittoinaminorkey,317

—

Two

successivepassing

notes,318

—

Chromaticpassingnotes,319

—

Passing notes in severalparts

bycontrarymotion,320

—

Passing notes quitted byleap,

321—

Examples referred to,322

—

Auxiliary notescannot make falserelation,323,324

—

Anticipations, 325, 326

—

Summaryofrules for employmentof auxiliary

and passingnotes,327

—

Exceptional treatment ofauxiliary notes,

328—

Theirintroduction,329

—

A

choralharmonizedwithauxiliary notes,330

—

Notesof amelodytreated as accentedauxiliaryandpassingnotes,331

—

Directionsforwork,332.

CHAPTER

XI.

—

Unessential Discords(II)

—

Suspensions page 145

The differentkinds ofunessential discords, 333

—

Suspension defined,₃₃₄

—

Which notes can be suspended,336

—

Suspensions notto bemarked in indicating roots,337

—

Their preparation,338

—

Their position inthebar,

339

—

Length ofpreparation, 340

—

When

it

may

be sounded with the note ofits resolution,341

—

Incorrect progressions,342

—

Alwaysresolved

on the chord over which it is suspended, 343

—

Practical limitations to

suspension, 344, 345

—

How

to

know

a suspension fromthefiguring,346,

347

—

The inversions of suspensions, 348-350

—

Ornamental resolutions,

3SI> 352

—

Examples from the great masters, 353-356

—

Suspensions

re-solving upwards, 357-363

—

Double suspensions, 364, 365

—

Suspensions ofcomplete chords,366

—

Directionsforwork,367.

CHAPTER

XII.

—

The Chord

of

the

Dominant Ninth

page 161

The chord of the dominant ninth different in major and minor keys,368

—

Whichnotetoomit,370

—

Figuring,371

—

Thechordresolvingonits

own

root; theninthproceedingtotheroot,372,373

—

Theninthproceedingto

the third, 374, 375

—

Irregularresolutions on fifth or seventh of chord,

376

—

Position of the major ninth, _377, 378

—

The chordresolved on a

different root, ₃₇₉

—

Treatmentof the fifth, 380

—

The inversions of the

chordrare,381

—

Derivatives of thedominant ninth;

how

figured,382

—

Inversions of thederivatives,383

—

The LeadingandDiminishedSevenths,

384

—

Resolution of these derivatives, 385

—

Examples from the great

masters,386-389

—

Changingthe positionof the chord beforeits resolu-tion,390

—

Furtherderivatives; the supertonic chords ofmajor and minor

keys, 391, 392

—

Exercises, 393.

CHAPTER

XIII.

—

The

Chord

of

the Dominant

Eleventh... page 172 Thechordof the eleventh,394

—

Thedifferencebetweenthe eleventhandthe

fourth,395

—

Which notes mostly omitted,396

—

Figuring,397

—

Gradual

resolutionof the higher discords,398

—

Resolution,399

—

Inversions,400

—

First inversion, 401

—

Second inversion,402

—

More than one analysis

sometimes possible, 403

—

The other inversions, 404

—

Derivative of the

firstinversion (rare),

405—

Dittoof thesecondinversion,406

—

Its resolu-tion ona dorhinant chord, ₄₀₇

—

Ditto on a tonic chord,

408—

How

to distinguish between chords that are identical inappearance,

409—

Other

derivatives,

410—

^-^^ ChordoftheAdded Sixth,

411—

Resolved on a

dominantchord,

412—

Resolved onatonicchord,413

—

Usedin approach-ing a cadence,

414—

Theremainingderivatives,_415,416

—

The

figuringof thesechords;

how

totelltheir real nature,

417—

The_{subdominant chord}

Contents.

xt

CHAPTER

XIV.

—

The

Chord

of

the Dominant Thirteenth

page185

Thethirteenth completes the series of dominant chords, 419

—

It differs in

major and minor keys, 420

—

Contains every note of the diatonic scale,

421

—

The thirteenth a consonance with the root, 422

—

The inversions,

423

—

The lower notes of the chord,

how

treated,424

—

Resolutions,₄₂₅

—

^The thirteenth and the ninth,426

—

Forms in use,427

—

Root,third, andthirteenth ;

how

to distinguishfromthemediantchord, 428

—

Which

notesto double; figuring,429

—

Resolutionon dominant seventh,430

—

Dittoonthetonicchord,431

—

Thefirstinversion,432

—

Thelastinversion,

433

—

Hoot,third, seventh, andthirteenth,434

—

Resolutionon the

domi-nant seventh,435

—

Alternative explanation,436

—

Resolutiononthetonic

chord,437

—

Root, third, fifth,

and

thirteeenth,438

—

Otherforms of the chord, 439

—

Derivatives; the seventh on the subdominant, 440

—

Ditto resolvedonatonicchord, 441

—

The triad on the submediant,442

—

The

chord of the thirteenth in its complete form, 443

—

Secondary Discords,

444

—

Secondary sevenths;

how

they differfrom fundamental sevenths,

445

—

Theirharmonicorigin tobedisregarded, 446

—

Rulesfortheir

treat-ment,447

—

Examples by Handel, 448

—

A

series of secondary sevenths,

449

—

Secondaryninths,450

—

Exercises, 451.

CHAPTER

XV.

—

Chromatic

Triads

—

The

Chromatic

Scale page198

Chromaticnotes in akey,452

—

Chromaticchordsdefined,₄₅₃

—

Illustrations,

454,455

—

Chromatic chords are borrowed chords,

456—

The keys from which theyare borrowed,457

—

Thechromatic scale,458

—

How

formed fromthediatonic; itsharmonicform,459

—

Themelodic chromaticscale,

460,461

—

Why

only nearlyrelated keys are used for borrowing from,

462

—

In a minorkeyonly the neighbouringminorkeys borrowed from,

463

—

Three methodsof averting the modulations suggested bychromatic

notes,464-466

—

How

tomark the rootsof chromatic chords,467

—

The

chromatic

common

chords of a minorkey,468

—

Theionic majorchord,

469

—

Seldom used except as a passing chord,470

—

The chord on the

flattenedsupertonic,471

—

Itsfirst inversion,the "Neapolitansixth,"₄₇₂

—

Its second inversion,473

—

The major chord on the supertonic,474

—

Treatment of the third of the chord, 475

—

Examples analyzed, 476

—

Rarerchromatic chords, 477,478

—

Listofchromatic

common

chordsina major key,479

—

Which chords are restricted in theirprogression,480

—

Examplesofchromaticchordsinamajorkey,481

—

Modulation bymeans

of chromatic chords, 482.

CHAPTER

XVI.

—

Chromatic

Chords

of

the Seventh

page 215

The real nature of chromatic chords, 483

—

The chromatic sevenths are the

dominantsevenths of nearlyrelatedkeys, borrowed,484

—

The Supertonic Seventh: its chromatic notes,485

—

Treatmentofthe third,486

—

Treat-mentof the seventh,487

—

The inversionsof the chord,

488—

Examples

ofits employment, 489

—

The seventh rising tothefifthof the dominant

chord,490

—

Theseventhleaping,491

—

^Aseldomusedprogression,492

—

Examplesof the inversions, _493, 494

—

Derivativesof the chord, 495

—

The Tonic Seventh: itsresolution,496

—

Progressionof the third,497

—

Progression of the seventh,498

—

The inversionsof the chord,499

—

Ex-amplesof the tonic seventh in root position,500

—

Ditto in thefirstand secondinversions,501

—

Dittointhelastinversion,502

—

Thederivatives;

how

to indicate the roots, 503

—

Examples, 504, 505

—

Modulation by means of chromatic sevenths, 506

—

Sequences ofmodulating sevenths,

507-xvi

Contents.

CHAPTER

XVII

—

Chromatic

Chords

of

the

Ninth—

False Nota-tion

—

Enharmonic Modulation

J>''ge 231 The chromatic ninths,508

—

Their various forms,509

—

;The dominant minor

ninth in a major key, 510

—

Its derivatives, 511, 512

—

The dominant major ninth in the minor key,when possible, 513

—

Resolutions of the chromatic dominantninth,514

—

The supertonic ninth,515

—

Its resolu-tion upon itsown root,

516—

Resolutionuponadifferent root,517

—

Pro-gression of the ninth, 518

—

The supertonic minor ninth taken in the

major key; False Notation,519

—

Law

oftheSharpest Note,520-522

—

How

to detect false notation,523

—

When

falsenotationistobemetwith,

524

—

Examples ofderivatives of the supertonic ninth, 525-527

—

The

tonicninth: itsresolution,528

—

Progression of the ninth, 529

—

Double

false notation, 530

—

Resolution of the chordupon a supertonic discord,

531

—

Examplesofderivativesof thetonic minor ninth,532-535

—

Ditto

' _ofmajorninth, 536

—

A

seriesof diminished sevenths,537

—

Rarer

deriva-tives, 538

—

Modulation by means of the diminished seventh, 539

—

Ex-treme keys; enharmonic change of notation, _540, 54'

—

Enharmonic

modulation, 542, 543

—

Can be effected between any two keys by the

diminished seventh, 544

—

Example from Bach's Chromatic F'antasia

analyzed,545

—

Exampleby Beethoven, 546

—

Cautiontothe student, 547.

CHAPTER

XVIII.

—

Chromatic

Chords

of

the Eleventh and

Thir-teenth page 250

These chords comparativelyrare,548

—

The chromaticchordof thedominant

eleventhinamajorkey, 549,550

—

The tonic eleventh, 551

—

The super-tonic eleventh,552

—

The

chromaticdominantminorthirteenth,₅₅₃

—

Its

resolution: false notation, 554

—

Examples analyzed, 555, 556

—

Tonic and supertonic thirteenth, 557

—

Derivatives of chromatic thirteenths

:

rules fortheir identification, 558

—

Necessity of ascertaining the key,559

—

Thederivatives: I. Third, ninth,

and

thirteenth,560

—

Chordsthatare

only partially chromatic, 561

—

False notation: "False Triads," 562

—

Examples, 563-565

—

II. Third,fifth, ninth,

and

thirteenth; a "False Tetrad,"566

—

ExamplebyMozart,567

—

III. Third, seventh ninth,

and

thirteenth,568

—

IV. Fiflh,seventh, ninth,

and

thirteenth,569

—

V. Third,

fifth, seventh,ninth

and

thirteenth, 570

—

VI.Fifth,ninth,eleventh,and thirteenth,571

—

VII. Seventh, ninth,eleventh,andthirteenth used chro-matically, 572-575

—

Enharmonic modulation by means of chromatic chords of the thirteenth, 576-579

—

The difficulty of these chords; im-portanceofcareful analysis,580.

CHAPTER

XIX

—

The

Chord

of

the

Augmented

Sixth page 269 The chord of the Augmented Sixth,581

—

Derived from twotonics,582

—

On

which degrees of the scale found, 583

—

It hasa "doublegenerator,"

584

—

Resolutions of the intervalof the augmented sixth, 585

—

Peculiar

constructionof the chord,586

—

Itsusual forms,587

—

Itsfiguring,588

—

' Distinctivenamesof the three forms,589

—

How

the chordsareindicated

in analysis,590

—

Theinversions,591

—

Thechord onthesixth degree of thescalethecommonest, 592

—

TheItalian Sixth; itsresolutions, ₅₉₃

—

Itsinversions, 594

—

Examples,595

—

TheFrenchSixth, 596

—

Its resolu-tions,597

—

Itsinversions,598

—

Examples, 599-601

—

The

German

Sixth;

itsresolutions, 602

—

Itsinversions,603

—

Examples, 604-609

—

Rareforms of the chordof the augmented sixth,610

—

Modulation bymeansof the chord, 611

—

Enharmonicmodulation onlypossiblewiththeGermansixth.

Contents.

xvii

CHAPTER

XX—

Pedals page 289

A

/Vi/a/ defined, 620

—

Which rotes can be usedasPedals, 621,622

—

Treat-mentof the harmonywhen the Pedal is notanote of the chord,623

—

How

to mark the analysis of a " Pedal point," 624.

—

Examplesof

dominant pedals, 625

—

A

tonic pedal; introduction ofadditional parts,

626

—

An

" Inverted Pedal,"627

—

Pedalaboveandbelow,628

—

A

pedal

in a middle voice,629

—

A

pedal point generally ends with a chord of

which thepedal note forms a part; exceptions, 630

—

Modulation on a pedal, 631

—

Examples, 632-634

—

Ornamentationof apedal note,635

—

A

double pedal, 636.

CHAPTER

XXI

—

Harmony

in

Fewer and

More

than

Four

Parts page 300

Harmony

continuously in four parts rare, 637

—

Threepart harmony, 638

—

Characteristic notesto be retained, 639

—

Broken chords, 640

—

Position

of chordsin three-partharmony,641

—

Thecadence,642

—

Motionof the separateparts, 643

—

Examples, 644

—

Two-partharmony,645

—

Examples

referredto,646

—

Examplesgiven,647-650

—

Harmonyin morethanfour parts, 651

—

Greater freedomofpart-writing allowed, 652

—

Which notes

to double, 653-^Five-part harmony, 654, 655

—

Six-part harmony, 656,

657

—

Seven-part harmony, 658

—

Eight-part harmony, 659-661

—

Direc-tions forwork,662

—

Conclusion, 663,

APPENDIX

A

—

The

Ecclesiastical

Modes

page 311 Difference between the ancient modes and modern keys, 665

—

How

the

modes were formed,666

—

The Authentic Modes, 667

—

The Dominant, 668

—

The Plagal Modes, 669

—

How

they differed from the Authentic

Modes, 670

—

Tableof Plagal Modes, 671

—

_{Transposition of}the modes, 672

—

Major and minor modes, 673

—

Characteristicnotesunalterable,674

—

Modulation, 675

—

Alteration of non-characteristic notes, (>"](>—The Dorian Mode, 677,678

—

The Phrygian Mode, 679-681

—

The Lydian Mode,682,683

—

The Mixolydian Mode, 684,685

—

The Aeolian Mode, 686-688

—

The Ionian Mode, 689

—

Difficulties of the subject, 690

—

Natureofitsinterest,691.

APPENDIX

B—

The

Harmonic

Series page 322

Practical use of the study, 692

—

How

harmonics are produced, 693,694

—

Pitch and vibration,695

—

"Upper partials,"696

—

The HarmonicSeries

fromC,697

—

Ratios ofintervals,698

—

Compound tones, 700

—

How

far

modern scales are derived from nature, 701

—

The useof the harmonic

series in determiningkey-relationship, 702,703

—

Thescientific

explana-tionof consonance and dissonance,704

—

Booksrecommendedforstudy,

HARMONY:

ITS

THEORY

AND

PRACTICE.

CHAPTER

I.

INTRODUCTION.

1.

A

certain

amount

ofelementary

knowledge

of

music

will

be

necessaryto the student before

beginning

the studyof the present work. It will

be

assumed

that

he

is acquainted with the

names

ofthe notes,the

meanings

ofthe variousmusical signs (accidentals, etc.), the relativetimevaluesof notes of different lengths,

and

such other matters as are treatedof in ordinary text-books

on

the

Elements

of Music.

2.

A

musical

sound

is

produced

by

the periodicvibrationof the air, that is tosay, its

motion

ata uniformrate.

When

the

air

moving

at

a uniform

rate

comes

in contactwith thenerves ofhearing, there isproduced, provided the

motion

issufficiently

rapid,

what

is calleda musical sound, or note.

The

pitchof

a

sound

(that is, its being

what

is called

a

high or

a

/SaTnote),

depends

upon

the rapidityof thevibration. (See

Appendix

B.

)

3. Ifsounds of different pitch are heard

one

afteranother,

we

get

what

iscalled

Melody

;* if sounds of differentpitch are heard together,

we

get

Harmony

. It is the laws of

harmony

that

we

shallexplaininthis

book

;but it*will

be

seenas

we

pro-ceed that the question of

melody

is often so closely

connected

withthatof

harmony,

that it is impossibletotreatof

one

with-out also

paying

some

attention to the other.

4. If

two

differentnotesaresounded, either insuccessionor together, itis clear that

one

of the

two must be

the higher,

and

theother the lower.

The

difference in pitch

between

the

two

sounds is called the Interval

between

them. This difference

may

be

so small as to De nardly recognizable

by

the ear; or it

may

be

as great as

between

the lowest

and

highest notes of a *This isonlya very general definition; differenceof pitch aloneis not

sufficient to make a good melody; but for the present purpose theideasof

melodyasasuccessionof sounds andofharmonyas a combination of sounds

will suffice. Fora more complete definitionof melody,see Musical Form,

2

Harmony.

[Chap.i.

largeorgan, or anything

between

the two.

An

infinite

number

of intervals is possible; but in

music

we

make

aselection, the

Jiatureof

which

will

be

explained later.

For

the present

we

^re

merely

defining the

meaning

of the

word "

Interval." 5.

The

smallest interval usedin

music

iscalled

a

Semitone

.*

We

may

definea Semitone, asthe distancebetween

any

onenote,

and

thenearestnotetoit, above orbelow,

on any

instrument

which

hasonlytwelve sounds in the octave.

For

example,

on

the piano, the nearest noteto

C

is

B

on

the

one

side (below),

and

C|f (or D*')

on

the otherside (above).

From

B

to C,

and

from

C

to

C

jf(or

D

b) are therefore

both

semitones.

Similarly

from

F^

to

Y\,

and

from

F|f to

G

will

be

semitones; but

from

G

to

A

will

not

be

a semitone; for

A

is not thenearestnoteto

G

;

G

jf (or

Ab),

comes between

them.

6.

There

are

two

kindsof semitone. If

we

look atthe

two

here given,

one

above,

and

the other

below

C

J],

itwill

be

seen that thereis

a

difference

between them.

C

and

B

are

on two

different placesof the staff;

one

is

on a

line

and

theother

on

a space; but

C

and

C

\are

both

on

the

same

place inthestaff; but the latternote has

an

accidental beforeit.

A

semitone of

which

the

two

notes are

on

different degreesof the

staff iscalleda.diatonic

semitone

; the

word

'

'diatonic

'

'

means

'

'throughthe tones, ordegrees ofthe scale.

'

'

A

second

mean-ing

which

is attached to the

word

will

be

explained later.

When

the

two

notesof the

semitone

are

on

the

same

degree of the staff,

and

one

ofthe

two

is altered

by

an

accidental {e.g.,

C

to _Cjf) the semitqjie is called chromatic.^ a

word

literally

meaning "

coloured."

This

useof the

word

will

be

further ex-plainedlater.

7.

The word "

semitone "

means

halfa tone.

A

Tone

is

an

interval, the

two

notesof

which

are on 'adjacentdegreesof the

staff,

and

which

contains

two

semitones.

But

if

we

take

two

diatonicsemitones

one above

another.

*Inonesense this statement isnotstrictlyaccurate, asthe"enharmonic

diesis"{j..e.,thevery smallintervalbetween twonotes represented bythesame soundon the piano,suchas

F#

andGb, orClJand B(),issometimes usedin

modulation. For ordinary purposes, however, the statement in the textis correct.

Chap.I.]

_{Introduction.}

the resulting intervalwill

be from

B

to

D

|7 ;

which

isnot

a

tone as the

two

notes are not

on

thenext degrees of thestaff to

one

another.

And

if

we

take

two

chromatic semitones,

Ex.3.

it is equally clear that theywillnot

make

a tone; for

now

the

resulting notes

C

[7

and

C

%are

both

on

the

same

degree of the

staff.

We

see therefore that ofthe

two

semitones

which

make

a tone, one

must

be diatonic

and

theother chromatic. Itmatters not

which

of the

two

is thelower.*

..-^'^8.

A

Scale

is

a

successionof notes arranged according to feome regular plan.

Many

different kinds of scales

have

been

used at various times

and

in different.partsof the

world

; in

modern

European

music

only

two

are

employed,

which

are called the diatonic

and

the chromaticscale.

9.

The word "

diatonic'

' has

been

already explainedin§ 6

as

meaning

'

'through the degrees.

'

'

A

diatonic scale is

a

suc-cessionof notesin

which

thereis

one

note, neither

more

norless,

on

each degree ofthestaff

—

thatis to say,

on

eachline

and

space.

The way

in

which

the diatonic scales areconstructedwill

be

ex-explained later {see

Chapter

II) ; at present

we

simply give the

forms of them.

There

are

two

varietiesof the diatonic scale,

known

as the

major

(or greater)

and minor

(orless) scalefrom thenatureof the interval

between

the first

and

third notesof thescale.

Major

Scale. % £z.4.

P

Minor

Scale. Ex,B.

i

-&S =2^

Other

forms

of

the

minor

scale frequently to

be

met

with

will

be

explained later. It will

be

seen that each of these scalescontainsonly seven different notes. This is because the eighth note, or

Octave

_(ZaA«,

"octavus"

=

eighth), is

a

repetitionof the first note ata different pitch;

and from

this

note theseries

recommences.

* The two semitones composinga tone are notof exactly the samesize.

A

diatonic semitone is larger than a chromatic; neither semitoneistherefore

exactly

^a^the

tone; but as the difference isofnopractical importancein

harmony,the student need not regard it. Itisonlymentioned here forthe

4

HARMOiTY.

rChap.I.

10.

A

chromaticscale isa scale consisting entirely

of

semi-tones,

and

it is called chromatic because

some

of itsnotes

re-quire accidentals (flatsor sharps) before

them

(§ 6).

Ez.6.

As

will

be

explained later, the chromatic scale is frequently-written in

a

different

way

from

that here given; but,

however

written, itequallyconsists of semitones.

11.

The

different degrees of the diatonic scale

_(§9)

are

known

by

different names, with

which

it is necessary that the student should

be

perfectly familiar, as they are of constant occurrence.

The

first noteof the scaleis calledthe

Tonic,

or

Key-note.

Thisisthe note

which

gives its

name

tothe scale

and

key.

The

scales in

_§

_9, for instance, are the scalesof

C

major

and

C

minor,

and

it will

be

seen that they

both

begin withthe note C.

The

term

"

tonic

"

isusedin

harmony

much

more

frequentlythan

"key-note."

The

most

important note in

a key

after the tonicisthe fifthnote of thescale.

For

this

reason it is called the

Dominant,

or ruling note of the key.

The

fourthnote of the scale lies atthe

same

distance below the tonic that thefifthnotelies aboveit. Thiswill

be

seenat

once

by

beginning

at the top of the scale

and

descending. This fourthnote (the next in

importance

tothe

dominant),

is there-fore called the

Subdominant,

or lower

dominant.

We

have

now

got appropriate

names

forthe three chief notesin thekey.

12.

About

midway

between

tonic

and

dominant

lies the third note ofthe scale.

We

shall see presently that in the

major

scaleitisrathernearer tothe

dominant,

and

inthe

minor

rather nearer to the tonic; but, roughly speaking, it is in the

middle between

the two. It is therefore called the

Mediant,

that is, the

middle

note.

The

sixth degree of the scale lies

midway

between

the tonic

and

subdominant,

just as the third

lies

between

tonic

and

dominant.

We

therefore call this sixth note the

Submediant,

or lower mediant.

Some

writers

on

harmony

call this note the

"

Superdominant,

"

or note

above

the

dominant

; but the

name

Submediant

is

much

more

usual,

and

in every

way

preferable.

The

second

note of the scaleis

called the

Supertonic,

i.e., thenote

above

the tonic;

and

the seventh note ofthescale, which, it will

be

seenlater, has

a

very strong

tendency

tolead up, or rise tothe tonic,is

on

that ac-countcalled the

Leading

Note.

Itissometimes,

though

rarely,

called the

"Subtonic," from

itsposition as thenext note

below

the tonic.

Chap.I.]

Intr od

uction.

13.

Having shown

the origin

and meaning

of these different

names,

we

will

now

tabulate them.

First

Degree

of the Scale

=

Second

Third

Fourth

Fifth Sixth

Seventh

Tonic

(Key-note). Supertonic. Mediant.

Subdominant.

Dominant.

Submediant (Superdominant).

Leading Note

(Subtonic).

14. Before proceedingto treatofthe

names

and

classification

of Intervals, itwill

be

needful to define

and

explain

two

terms

which

we

shallveryfrequently

have

touse inspeaking of them.

These

are the terms

Consonance and

Dissonance.

15.

A

consonantinterval, or

Consonance,

isa

combination

of

two

sounds,

which by

itselfproduces a

more

orless

complete

and

satisfactoryeffect,i.e.,

which

does notnecessarily require to

be

followed

by

some

other combination.

For

example, if the studentwill strike

on

the piano

any

of the following pairs of notes, pausing

between

each.

Ex.7.

m

-»-he

will find thateach is

more

orless satisfactory.

A

consonant

chord

is a

chord

of

which

allthenotes

make

consonant intervals with

one

another.

Ex.8.

Harmony.

[Chap.I.

Everyone

willfeel the incomplete effectof these combinations,

and

that they require to

be

followed

by

something'else to

be

satisfactory. .Let us try.

We

will putafter

each of

these

dis-sonances

a

consonance,

and

itwill

be

at

once

feltthat the

com-pleteness

which was

before

wanting

has

now

been

obtained.

Ez,10.

The

consonance

which

follows the dissonance is called the

Resolution

of the dissonance.

The

laws according to

which

dissonancesareresolvedwill

be

learned later.

17.

A

dissonant chord, or

Discord,*

is

a chord

which

con-tains atleast

one

dissonance

among

the intervals

made

between

the various notes.

Like

a dissonant interval,

a

dissonant

chord

has

by

itself

an incomplete

effect.

Let

the student play the following dissonant chords,

and

he

willfeel this.

Ex.11.

Now,

as beforewith the dissonant intervals, letusput after

each

chord,

a consonant chord

for its resolution.

The

satisfac-toryeffect is feltatonce. In general, it

may

be

said that

con-sonance

is

a

position of rest,

and

dissonance

a

position of

un-rest.

123456

1^

^

te

^

Ez.l2.

te

=J:"^" J=. si-\ra _bs

18. Intervals are always

reckoned

upwards, unless the con-trary

be

expressly stated.

Thus "

the thirdof

C

"

always

means

the third

above

C

; if the third

below

is intended, it

must be

so described.

The number

of

an

interval is always

computed

according tothe

number

of degrees of the scalethat itcontains, including

both

thenotes

forming

the interval.

Thus

from

C

to

E

is calledathird, because itcontainsthreedegreesof thescale,

C,

D,

E. Beginnersareapt to getconfused

on

this point,

and

*Theterm "Discord " isalsosometimesappliedmerelytothe dissonant noteitself.

Chap.I.]

Introduction.

7

to think of

D

asthe first note

above

C,

and

E

as the second.

But

thenote

C

isitself

counted

asthe first note of the interval.

Similarly,

from

G

to"

D

is a fifth,

from

F

to

D

a

sixth;

and

so

on

in all other cases.

The

same

reckoning, but inthe reverse direction, applies to the intervals below.

Thus

A

is the third

below

C,

D

isthefourth

below G,

etc. Letthestudent

examine

the

major

scaleof

C

in

_§

_9,

and he

willfindwithinthe

compass

of the octave there given

two

7ths, three 6ths, four Sths, five

4ths, six 3rds,

and

seven 2nds. Itwill

be a

useful- exercisefor

him

todiscover

them

forhimself.

19.

An

interval larger than

an

octaveiscalled

a

f^w/^«/Kif

interval.

Thus

the interval i

^

*^

'

is

compounded

of the octave,

C

toC,

and

the third,

C

toE.

(The

octave isprinted here as

a

small note.) Obviously, in addition to the thirdat

the top, the interval containstheseven notes ofthelower octave

from

C

to B.

The

upper

C

is already

counted

as partof the

third.

Thus

the

number

of

a

compound

interval is always ₇

more

than that of the simple interval to

which

the octave is added. Therefore,