An_Approach_to_Music-libre.pdf

An Approach to Music

Daniel Zachariah Franks

ii

KeArt Music Philosophy Limited Edition

c 2014 Daniel Zachariah Franks

This edition c MMXIV

All rights reserved. No part of this book may be reproduced,

stored in a retrieval system, or transmitted by any means,

electronic, mechanical, photocopying, recording. or otherwise,

without prior permission of the publisher.

To contact the author, to request copies of this

or other manuscripts, to inquire after lessons, or to point out a typo,

please contact [email protected].

Please note: This is an incomplete review copy.

To get the full picture please contact to purchase a copy.

This edition published by Ke Art

www.keart.co.za

mailto:[email protected]

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

I Harmony: Systems of Rational Relations 3

1 Consonance and Simplicity 5

1.1 Simple Number Ratios . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Relations by 3:2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Relations by 5:4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Relations by 6:5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Dissonance and Complexity 11

2.1 Divisions and Compounds of Intervals: Inverse proportions of pitchand vibrational length . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Addition / Multiplication,Subtraction / Division . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 Divisions 13

3.1 of an Octave into Fifths and Fourths . . . . . . . . . . . . . . . . . 13

3.2 of Fifths into Fourths and Tones . . . . . . . . . . . . . . . . . . . . 14

3.3 and into Ditones and Semiditones . . . . . . . . . . . . . . . . . . . 15

3.4 of Fourths into Ditones and Semitones . . . . . . . . . . . . . . . . 15

3.5 and into Semiditones and Tones . . . . . . . . . . . . . . . . . . . . 16

3.6 of Ditones into Semiditones and Semitones . . . . . . . . . . . . . . 16

3.7 and into Tones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.8 of Tones into Semitones . . . . . . . . . . . . . . . . . . . . . . . . 17

3.9 of Semitones, Dieses and Commas . . . . . . . . . . . . . . . . . . . 18

3.10 Further Divisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

iii

iv CONTENTS

II Time 21

4 Rhythmic and Metric Poetics 23

5 Rhythmics 25

6 Metrics 33

III Tetrachords 37

7 Tetrachords in general 41

8 Tetrachords in detail 43

9 The Varieties of the Ascending Species 45

IV Systema 49

10 Greater and Lesser Perfect Systems 51

11 Ametabolon: Tonality 53

12 Diatonic Systema 55

12.1 mixture of tense (syntonon) chromatic and tonic diatonic . . . . . . 55

12.2 mixture of soft (malakon) diatonic and tonic diatonic . . . . . . . . 55

12.3 tonic diatonic (unmixed) . . . . . . . . . . . . . . . . . . . . . . . . 55

12.4 mixture of tonic and ditonic diatonic . . . . . . . . . . . . . . . . . 55

12.5 mixture of tonic diatonic and tense diatonic . . . . . . . . . . . . . 55

13 Theory of Modality 57

13.1 Diatonic Systema . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

13.2 Chromatic Harmonic Systema . . . . . . . . . . . . . . . . . . . . . 57

V Modulation 59

14 Generally Tempered and Specifically Untempered Modulation 61

15 How the succession of dissimilar tetrachords may produce modu-lation 63

CONTENTS v

16 How natural melodic succession may produce modulation betweenthe genera within a single tetrachord or a succession of similartetrachords 65

17 How non-melodic adjustments may produce modulation within asingle tetrachord or a single genera 67

VI Mapping the Monochord 69

18 A Guide to the Practical Application of Tetrachord Theory forString Instruments 7118.1 Thesis: Layout of the Canon . . . . . . . . . . . . . . . . . . . . . . 7118.2 Dynamis: General Layout of Diatonic Tetrachords . . . . . . . . . . 7118.3 Just 5-limit Diatonic Intonations . . . . . . . . . . . . . . . . . . . 71

VII Merging Thesis and Dynamis: Key Systems (withspecific examples illustrating general concepts) 73

19 The nature of Enharmonic Key Systems 75

20 The nature of Chromatic and Diatonic Key Systems 77

vi CONTENTS

List of Figures

1.1 An extended Tetraktys . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Pythagorean Fifths ranging over six octaves. . . . . . . . . . . . . . 71.3 Pythagorean Fifths transposed to a single octave. . . . . . . . . . . 71.4 Ditone relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5 Septimal Ditone relations. . . . . . . . . . . . . . . . . . . . . . . . 81.6 Semiditone relations. . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.1 An octave divided into a fifth and a fourth. . . . . . . . . . . . . . . 133.2 A fifth divided into a fourth and an Epogdoic tone. . . . . . . . . . 143.3 ... an acute fourth and a sesquinonal tone. . . . . . . . . . . . . . . 143.4 ... a ditone and a semiditone. . . . . . . . . . . . . . . . . . . . . . 153.5 ... a Pythagorean ditone and a Pythagorean semiditone. . . . . . . 153.6 A fourth divided into ditones and semitones. . . . . . . . . . . . . . 153.7 ... into semiditones and tones. . . . . . . . . . . . . . . . . . . . . . 163.8 A ditone divided into semiditones and semitones. . . . . . . . . . . 163.9 ... into tones. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.10 Tones divided into semitones. . . . . . . . . . . . . . . . . . . . . . 173.11 Semitone divisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.12 Further divisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

vii

viii LIST OF FIGURES

LIST OF FIGURES 1

Introduction: The Philosophy of Musics Meaning

If I were not a physicist, I would probably be a musician. I oftenthink in music. I live my daydreams in music. I see my life in terms ofmusic. - Albert Einstein

History provides much cause for speculation. - Kirbuk Mas

In the West, tradition alleges to having been handed down from the classicalGreek scholars. The Greeks, for their part, especially in the later years of theirclassical civilisation, had only ever claimed to have preserved, and revised secondhand, a knowledge that had been passed down to them by history. The ancient sys-tems were ancient even then, and from them were moulded both practical notionsand more philosophic hypotheses. The ancients had the prescience to recognise thewealth of former achievement that lay behind them even then, its passage hiddenby the vastness of time.

The dubiously progressive history of the school of theory we have attemptedto preserve and build upon must ultimately lead to revolution. As the wholecherished school of our modern theories and materials is cast into disrepute, mech-anisms must be forged to fulfill the holes in our theories. We cannot, nor shouldwe hope to, refashion the wheel, however (to abuse an analogy), we may strip theheavy burden of fashion that weighs upon our wheel and thus recover some wellconcealed utility and ease.

History is unfathomable in its extent. While we have been assured that we,the living, are the pinnacle of human progress, this is only true in a certain light.For all the power we hold, we have yet to put it to the use the ancients were ableto make of their primitive tools. Perhaps we are missing something essential,perhaps our approach is lacking.

Theories may be found wandering like ghosts in the graveyard of concensus,stillborn, fluctuating between the grave and the cradle. Interred in a new the-oretical system every hundred years or so, the great illusion of western culturalprogress, in music at least, has resulted merely from the fashionable timing of ex-humation, resuscitation and re-interment of material. To assist us, we have builtgreat libraries furnishing us with pigeonholes to guide our pedagogy and taste.Progress is perhaps not the word.

The equally tuned pianoforte has become the alter at which the modern theoryof practical music settles its disputes. The keyboard mechanism of the pianoforteis merely a sign of the general reduction of musical theory that has cleaved themodern theories from those of the ancients. From the string based theory of theancients to the modern keyboard methods of music, theory has sadly moved awayfrom the true mechanisms of string playing and the greater goal of imitating the

2 LIST OF FIGURES

harmonic purity of which a voice is capable. String instruments, including theclavichords that preceded the pianoforte, required a form of contact with stringsand exhibition of the nature of strings of which a modern piano and its equaltemperament is too often incapable.

If I seem to be overly hard on the pianoforte, it is born of well-intentionedconcern. The piano has after all a fine pedigree. Sadly, deeper understanding ofthe nature of harmony itself, as exhibited by the pianos strings themselves, iscompletely unnecessary to understand the pianos keyboard harmony. It is worthnoting that though the piano must take the bulk of the blame on this issue, it isnot alone.

The practices that have been hidden under the theory of the pianoforte mecha-nism are just as surely hidden by the modern equal temperament of fretted stringinstruments. The subtlety that we, as string players, are looking for is lost inthe abstruse nature of fret-harmony. In reducing the players relationship withthe string and its theory to the abstractions of piano keys and fret placement, weplace many things out of our sight that are right before us.

One can never hope to grasp that which one places out of sight. The truenature of harmony is eluded by the holes in our modern theories, which at presentcan furnish only analogies.

The truth we will seek is the theory with which to fully accompany our prac-tices.

(Excerpt from the authors manuscript A Theory of Strings.)

Part I

Harmony: Systems of RationalRelations

3

Chapter 1

Consonance and Simplicity

1.1 Simple Number Ratios

1:1 Unison

*

2:1 Octave Diapason

* *

3:2 Fifth Diapente

* * *

4:3 Fourth Diatessaron

* * * *

5:4 Third Ditone

* * * * *

6:5 Minor Third Semiditone

* * * * * *

Figure 1.1: The simple relations of whole numbers model the simple relations ofconsonances.

5

6 CHAPTER 1. CONSONANCE AND SIMPLICITY

The relations of these simple intervals provide an elegant system of harmonic rela-tions. By compounding their intervals, first combining like with like but graduallymixing them all together, we may obtain unchanging pictures of the relations un-derlying the changing harmonies.

Cycles of fifths reflects seven points, infilled with a further reflective circuit ofduality. This doubly reflective duality, diezeugnomon, runs deep throughout thesystem. Twelve points are related by fifths in their ranges by fourteen commas,a single diesis and a bouquet of semitones. Ditones furnish dieses and invert atextremities to reflect the greater diesis. This greater diesis, product of the commaand diesis, is in like wise the excess of three semiditones from their start. Theselayers are reflective and reflected, forming by their compaction a vast topography.The illusion of perspective produced by these reflections may be likened to thelight cast by a single flame in a room full of mirrors. As light only seems to recedeinto the mirrors infinite depth, so appearances only seem to recede in the well ofperception. All becoming something seeming other.

What follows is a map.

1.2. RELATIONS BY 3:2 7

16

5

12

5

9

5

27

20

81

80

243

160

729

320

2187

640

256

81

64

27

16

9

4

3

1

1

3

2

9

4

27

8

81

16

1280

729

320

243

80

81

40

27

20

9

10

3

5

1

15

2

45

4

Figure 1.2: Relations of Fifths and Fourths over six octaves. Left to right ascendingby 3

2. Central row depicting fifths from unity, rows above and below depict similar

relations a comma acute and grave respectively.

16

15

8

5

9

5

27

20

81

80

243

160

729

640

2187

1280

128

81

32

27

16

9

4

3

1

1

3

2

9

8

27

16

81

64

1280

729

320

243

160

81

40

27

10

9

5

3

5

4

15

8

45

32

Figure 1.3: Relations of Fifths and Fourths transposed to a single octave. Left toright ascending by 3

2. Central row depicting fifth relations from unity, rows above

and below depict similar relations a comma acute and grave respectively.

1.2 Relations by 3:2


512

375

128

75

4

3

5

3

25

24

125

96

128

125

32

25

8

5

1

1

5

4

25

16

125

96

192

125

48

25

6

5

3

2

15

8

75

64

375

256

128

45

16

9

10

9

25

18

125

72

144

125

36

25

9

5

9

8

45

32

225

128

Figure 1.4: Relations of Ditones transposed to a single octave. Left to rightascending by 5

4.


512

175

64

35

8

7

10

7

25

14

224

75

28

15

7

6

35

24

175

96

Figure 1.5: Relations of Ditones transposed to a single octave. Left to rightascending by 5

4.

1.4. RELATIONS BY 6:5 9

125

81

50

27

10

9

4

3

8

5

1296

625

125

108

25

18

5

3

1

1

6

5

36

25

216

125

648

625

125

72

25

24

5

4

3

2

9

5

27

25

162

125

25

16

15

8

9

8

27

20

81

50

Figure 1.6: Relations of semiditones transposed to a single octave. Left to rightascending by 6

5.


Chapter 2

Dissonance and Complexity

2.1 Divisions and Compounds of Intervals: In-

verse proportions of pitch and vibrational

length

The distance of an intervals vibrating length and the pitch distance of that inter-val, when reduced to numbered relations, are seen to be inversely proportionate.An interval of an octave may be generated by the process of halving a soundingstrings length. The pitch of the sound produced by the string is thereby doubled.

Length = 1 / Pitch

Pitch = 1 / Length

Pitch x Length = 1

An interval of 2:1 represents both doubling and halving, and must be consideredfrom both of these aspects. So too with all the other numbered relations.

11

12 CHAPTER 2. DISSONANCE AND COMPLEXITY

2.2 Addition / Multiplication,

Subtraction / Division

Addition and Multiplication are transformations whereby smaller intervals com-bine or grow to form larger ones. Subtraction and Division are transformationswhereby larger intervals dissemble or decrease to form smaller ones.

Addition of intervals takes place as an expression of multiplication.

Eg. 9:8 x 10:9 = (9x10) : (8x9) = 90:72 = 5:4

(Major Tone + minor tone = Ditone)

Subtraction of intervals is expressed by cross-multiplication,

or division.

Eg. 5:4 / 9:8 = (8x5) : (9x4) = 40:36 = 10:9

(Ditone / Major Tone = minor tone)

In like manner we derive the following relationships of intervals. Please note thefollowing comparisons are not sized to scale.

Chapter 3

Divisions

3.1 of an Octave into Fifths and Fourths

[------------------------2:1----------------------] Octave

Fourth [--------------4:3--------]-------------3:2-------- Fifth

Figure 3.1: The relation of doubling represented by the octave gives rise, by halv-ing, to the intervals of the fourth and fifth. These relations define the primerelations of 3 to 2 and, inversely, to 4.

13

14 CHAPTER 3. DIVISIONS

3.2 of Fifths into Fourths and Tones

[-------------3:2---------------] Fifth

Fourth [----------4:3----------]---9:8-- Greater Tone

(Epogdoic Tone)

Figure 3.2: The difference between a Fourth and a Fifth is equivalent to thatbetween the numbers 9 and 8, The Epogdoic Tone is naturally provided by systemscontaining overlapping fifths and fourths.

[-------------3:2---------------] Fifth

acute fourth [----------27:20---------]--10:9- lesser tone

(sesquinonal tone)

Figure 3.3: Raising a fourth by one comma (8180) reduces the difference between it

and the fifth to that between the numbers 10 and 9.

3.3. AND INTO DITONES AND SEMIDITONES 15

3.3 and into Ditones and Semiditones

[-------------3:2---------------] Fifth

semiditone [------6:5------]------5:4------- Ditone

Figure 3.4: Dividing the span of a fifth by half produces the two forms of 5-limitthirds

[-------------3:2---------------] Fifth

Pythagorean [-----32:27----]------81:64------ Pythagorean Ditone

semiditone

Figure 3.5: A comma may be redistributed according to the principles ofPythagorean temperament. This is particularly useful for treating systems con-taining parallel pure fifths.

3.4 of Fourths into Ditones and Semitones

[-----------4:3------------] Fourth

Ditone [-------5:4------]---16:15-- Diatonic Semitone

Pythagorean Ditone[-------81:64-----]-256:243- Pythagorean Limma

Augmented Ditone [--------32:25-----]--25:24- Minor Chromatic Semitone

Septimal Ditone [---------9:7-------]-28:27- Septimal 1/3rd Tone

Figure 3.6: Various distributions of semitones, showing how their combinationwith various ditones produces the span of a fourth.


3.5 and into Semiditones and Tones

Semiditone [---------6:5------]-10:9- minor tone

Pythagorean semiditone [-------32:27-----]---9:8- Major Tone

Septimal Tone [-----8:7----]----7:6----- Septimal Semiditone

Figure 3.7: Commas may be redistributed in order to shade the various semiditonesand tones.

3.6 of Ditones into Semiditones and Semitones

[---------5:4------] Ditone

Semiditone [----6:5----]-25:24- Minor Chromatic Semitone

Pythagorean minor 3rd [---32:27--]--16:15- Diatonic Semitone

Figure 3.8: A comma may be redistributed according to the principles ofPythagorean temperament. This is particularly useful for treating systems con-taining parallel pure fifths

3.7 and into Tones

[---------5:4------] Ditone

lesser tone [--10:9---]---9:8--- Greater Tone

[--------81:64------] Pythagorean Ditone

Major Tone [---9:8----]---9:8--- Major Tone

[---------9:7--------] Septimal Ditone

Major Tone [---9:8----]----8:7--- Septimal Tone

Figure 3.9: Simple divisions of the various shades of ditones into various combina-tions of tones.

3.8. OF TONES INTO SEMITONES 17

3.8 of Tones into Semitones

--------------9:8-------------- Major Tone

Greater Chromatic ----135:128---[------16:15-----] Diatonic Semitone

Semitone

2/3rd Tone ------27:25------[----25:24----] Minor Chromatic

Semitone

Septimal 1/3rd Tone ----28:27----[-----15:14-------]

Major Septimal Semitone

--------------10:9------------ minor tone

Lesser Chromatic ----25:24----[------16:15-----]

Diatonic Semitone

Semitone

Pythagorean Limma ----256:243----[----135:128---] Greater Chromatic

Semitone

Figure 3.10: Simple divisions of tones into semitones.


3.9 of Semitones, Dieses and Commas

-------------27:25------------- 2/3rd Tone

Lesser Diesis ----128:125---[-----135:128----]

Syntonic Comma --81:80-[------16:15-----------] Diatonic Semitone

-------------16:15----------- Diatonic Semitone

Lesser Diesis --128:125--[------25:24------] Chromatic semitone

Syntonic Comma -81:80-[-------256:243-------]

Greater Diesis ---648:625---[----250:243----]

Diaschisma -20482025

-[--------135:128--------]

-----------256:243--------- Pythagorean Limma

Lesser Diesis --128:125--[----250:243----]

Diaschisma -20482025

-[--------25:24-------] Minor Chromatic

Semitone

-------135:128--------- Major Chromatic Semitone

Syntonic Comma -81:80-[----25:24------] Minor Chromatic Semitone

---------25:24-------- Minor Chromatic Semitone

Syntonic Comma -81:80-[---250:243---]

------648:625----- Greater Dieses

Syntonic Comma -81:80-[128:125--] Lesser Diesis

Figure 3.11: Some of the many varieties of semitones, commas and dieses.

3.10. FURTHER DIVISIONS 19

3.10 Further Divisions

-------------9:8------------- Major Tone

Syntonic Comma -81:80-[---------10:9--------] minor tone

-------------8:7---------------- Septimal Tone

Septimal Comma -64:63-[------------9:8---------] Major Tone

---------------7:6----------------- Septimal Semiditone

Septimal Diesis -49:48-[----------8:7-------------] Septimal Tone

Septimal 1/3 Tone -28:27--[----------9:8------------] Major Tone

Minor Septimal --21:20--[--------10:9------------] minor tone

Semitone

Figure 3.12: Some further varieties of divisions and combinations

Part II

Time

21

Chapter 4

Rhythmic and Metric Poetics

The number ratios used to express pitch relations in harmonic and melodic intervalsform relations that can be used to express musical durations in rhythmic relations.

1:1 equal ratio (dactylic genera)

2:1 duple ratio (iambic genera)

3:1 triple ratio (irrational form of dactylic genera)

3:2 hemiolic ratio (paeonic genera) 5-limit

4:3 epitritic ratio 7-limit

&c.

Metric and rhythmic composition may be analysed in these and other relationalunits, elaborating the affinities between musical and poetic composition. Theseratios represent a) the metric relation of long and short sounds; b) the metricarrangement of stresses and measures (the relation of successive sequences of up-beats [arses] and downbeats [theses] ) ; and c) further rhythmic subdivisions andadditions of a and b.

We may conceive the relations of these different characteristics by consideringhow rhythmic systems of relations are nested within the greater scope of a metricsystem.

Arsis (lit.: rising) = up-beat

Thesis (lit.: placing) = down-beat

short syllable / one unit = U

long syllable / two units = -

With reference to the placement of a figure we can only refer to points and ranges.Points construct the boundaries of figures, implying a limit which may or may not

23

24 CHAPTER 4. RHYTHMIC AND METRIC POETICS

be just that. If the boundaries of a figure are subject to a range of movement,perhaps even growth and transformation, then the very idea of using static pointsto convey motion becomes a complicated issue of interpretation, a field governed byhaughty and ill-informed performing opinion. Too many attempt at a systematisedsynthesis between number and music have become merely academic concern.

The reader is welcome to skip this chapter, however, be it at his peril. Thefact remains, unless points are correlated wih their transformations, all attemptsat describing musical motion will fall staggering upon our blinkers.

The following is a summation from the fragments of the ancients, notably theworks of Aristoxenus and those who followed him.

Chapter 5

Rhythmics

Primary Durations

Incomposite / Simple

A single indivisible syllable / note / interval / movement.

Composite / Multiple

Duple, triple, quadruple, &c.

Rhythmical (maintaining simple ordered ratio)

Non-rhythmical (disordered, maintaining no simple ratio)

Quasi-rhythmical (possessing aspects of both order and disorder,

eg. Accelerating abbreviated rhythms,

& Decelerating expanded rhythms.)

The Genera of Feet (from primary durations)

7 differences between feet:

i) magnitude (of primary units)

ii) genus

iii) composition (division into simple or compound durations)

iv) rational or irrational (irrational = irregular / dotted rhythms)

v) types of division

vi) arrangement of divisions

vii) in respect of antithesis

25

26 CHAPTER 5. RHYTHMICS

The Genera of Rhythmic Units

1:1 equal ............................................ from 2 to 16 units

2:1 duple ............................................ from 3 to 18 units

3:2 hemiolic ........................................ from 5 to 25 units

4:3 epitritic ....................................... from 7 to 14 units

5:4 epipentic ....................................... from 9 to 27 units

(& c. becoming more irrational as the ratios become more complex)

Composite made from 2 or more genera

Coupled - combinations of 2 simple and dissimilar feet

Periodic combination of more than 2 feet

Incomposite using only one genus of feet

Mixed analysed sometimes into durations and sometimes into rhythms

eg. The dactylic and iambic nature of 6 units

27

Dactylic Rhythms

6 incomposite rhythms

i) simple prokleumatic ( t : a )

short thesis, short arsis U : U

ii) double prokleumatic ( t : a )

thesis of 2 short units, arsis of 2 short units U U : U U

iii) anapaest a maiore ( t : a )

long thesis, arsis of 2 short units - : U U

iv) anapaest a minore ( a : t )

arsis of 2 shorts, long thesis U U : -

v) simple spondee ( t : a )

long thesis, long arsis - : -

vi) greater spondee ( t : a )

thesis of 2 long units, arsis of 2 long units - - : - -


2 coupled rhythms

i) ionic a maiore ( t : a : t : a )

simple spondee followed by simple prokleumatic - : - : U : U

ii) ionic a minore ( t : a : t : a )

simple spondee followed by simple prokleumatic U : U : - : -

Iambic Rhythms

4 simple rhythms

I) Iambus ( a : t )

thesis half arsis, arsis double thesis U : -

or - : --

II) Trochee ( t : a )

double thesis, single arsis - : U

or -- : -

III) Orthian ( t : a )

arsis of 4 units, thesis of 8 units UUUU : ----

IV) Semantic ( t : a )

thesis of 8 units, arsis of 4 units ---- : UUUU

29

2 coupled composite rhythms

The Two Bachii

I) Iambus, Trochee ( a : t : a : t )

U : - : - : U

II) Trochee Iambus ( t : a : t : a )

- : U : U : -

12 Periodic Rhythms

Four of One Iambus and Three Trochees

I) Trochee ab Iambo ( a : t : t : a : t : a : t : a )

iambus in first place U : - : - : U : - : U : - : U

II) Trochee a Bacchio ( t : a : a : t : t : a : t : a )

iambus in second place - : U : U : - : - : U : - : U

III) Bacchius a Trochaeo ( t : a : t : a : a : t : t : a )

iambus in third place - : U : - : U : U : - : - : U

IV) Epitritic Iambus ( t : a : t : a : t : a : a : t )

iambus in last place - : U : - : U : - : U : U : -


Four of One Trochee and Three Iambi

I) Iambus a Trochaeo ( t : a : a : t : a : t : a : t )

trochee in first place - : U : U : - : U : - : U : -

II) Iambus a Bacchio ( a : t : t : a : a : t : a : t )

trochee in second place U : - : - : U : U : - : U : -

III) Bacchius ab Iambo ( a : t : a : t : t : a : a : t )

trochee in third place U : - : U : - : - : U : U : -

IV) Epitritic Trochee ( a : t : a : t : a : t : t : a )

trochee in last place U : - : U : - : U : - : - : U

Four of Two Trochees and Two Iambi

I) Simple Bacchius ab Iambo ( a : t : a : t : t : a : t : a )

iambi first U : - : U : - : - : U : - : U

II) Simple Bacchius a Trochaeo ( t : a : t : a : a : t : a : t )

trochees first - : U : - : U : U : - : U : -

III) Intermediate Iambus ( t : a : a : t : a : t : t : a )

iambi in centre - : U : U : - : U : - : - : U

IV) Intermediate Trochaeo ( a : t : t : a : t : a : a : t )

trochees in centre U : - : - : U : - : U : U : -

31

Paeonic Rhythms

2 incomposite feet

Paion Diaguios ( t : a )

- : U -

Paion Epibatos ( t : a : t : a )

- : - : -- : -

Species of Rhythm generated by mixtures of genera

2 Dochmii

First Dochmii Iambus, Paion Diaguios

Second Dochmii Iambus, a Dactyl, a Paeon

Prosodiacs

of 3 elements Pyrrhic, Iambus, Trochee

of 4 elements Iambus added to the above

of 2 coupled feet Bacchius, Ionic Majore

2 Irrational Choreii

Iambic ( a : t )

dactylic in rhythm but iambic in respect of parts - : U U

Trochaic ( a : t )

dactylic in rhythm but trochaic in respect of parts U U : -


6 Other Mixed Rhythms

Cretic Trochee thesis, Trochee arsis

Iambic Dactyl Iambus thesis, Iambus arsis

Bacchaic ( a Trochaeo ) Dactyl Trochee thesis, Iambus arsis

Bacchaic ( ab Iambo ) Dactyl Iambus thesis, Trochee arsis

Choreic ( Iambic species ) Dactyl Iambus arsis, Iambic thesis

Choreic ( Trochaic species ) Dactyl Trochee arsis, Trochee thesis

Chapter 6

Metrics

Syllables

vowels

long

short

ambiguous (dichrona)

semi-vowels

double

liquids

sigma

mutes

simple

aspirated

intermediate

Systema of Feet (from syllables)

Combinations [systema] of 2 Syllables produce 4 Kinds of Feet

i) Both short Pyrrhic / Pariambic

ii) Both long Spondee

iii) Short first, long second Trochee

iv) Long first, short second Iambus

33

34 CHAPTER 6. METRICS


i) Three shorts Choreios

ii) Three longs Molossus

iii) One long, two shorts Dactyl, Amphibrach, & Anapaest

iv) One short, two long Bacchius, Amphimakros & Palimbacchius


i) Four shorts Prokleusmatic

ii) Four longs Dispondee

iii) Two shorts, two longs Ionic a Minore

iv) Two longs, two shorts Ionic a Majore

v) Long, short, short, long Choriamb

vi) Short, long, long, short Antispastos

vii) Long, short, long, short Ditrochee

viii) Short, long, short, long Diambus

ix) One long, others short 4 Paeons

1. long in first place First Paeon

2. long in second place Second Paeon

3. long in third place Third Paeon

4. long in fourth place Fourth Paeon

x) One short, others long 4 Epitrites

(named after the same manner as the Paeons)

35

Disyllables and trisyllables combine to produce 32 pentasyllabic feet.

Trisyllables and trisyllables combine to produce 64 hexasyllabic feet.

36 CHAPTER 6. METRICS

Part III

Tetrachords

37

39

The most striking characteristic of Greek music, especially in its ear-lier periods, is the multiplicity and delicacy of the intervals into whichthe scale was divided. A sort of frame-work was formed by the divisionof the octave into tetrachords, completed by the so-called disjunctivetone within the tetrachord the reign of diversity was unchecked. Notonly were there recognised divisions containing intervals of a fourth, athird, and even three-eights of a tone, but we gather from several thingssaid by Aristoxenus that the number of possible divisions was regardedas theoretically unlimited. Thus he tells us that there was a constanttendency to flatten the moveable notes of the Chromatic genus, andthus diminish the small intervals, for the sake of sweetness or in or-der to obtain a plaintive tone; - that the [third sound] of a tetrachordmay in theory be any note between the Enharmonic and the Diatonic;- and that the magnitude of the smaller intervals and division of thetetrachord generally belongs to the indefinite or indeterminate elementin music. (Monro, 1894:110-111)

Chapter 7

Tetrachords in general

The Genera of the Tetrachords:

Diatonic fourth of 2 tones and a semitone,

and a peculiarly diatonic raised fourth of 3 tones

Chromatic fourth of 2 semitones and a semiditone

Enharmonic fourth of 2 quartertone dieses and a ditone

The General Species of the Genera

Diatonic 3 species

semi

* tone * tone * tone *

1.

semi

* tone * tone * tone *

semi

2 . * tone * tone * tone *

3. * tone * tone * tone *

(raised fourth)

41

42 CHAPTER 7. TETRACHORDS IN GENERAL

Chromatic 2 species

semi semi

1. * tone * tone * semiditone *

semi semi

* semiditone * tone * tone *

semi semi

2. * tone * semiditone * tone *

Enharmonic 1 species

pyknon

1. * * * ditone *1

4tones

(dieses)

pyknon

* ditone * * *1

4tones

(dieses)

Pyknon (Lit. : Compression. A term used by the ancients todescribe the melodic phenomenon inherent in intervals smaller thanthe semitones. An all inclusive term describing the tension createdby intervals that closely approach consonance yet may be made tomomentarily defy the inevitable gravity towards consonance.)

Chapter 8

Tetrachords in detail

Hypate = index finger(literal translation: the superior. Explained by its strength and defin-ing position)

Parhypate = middle finger(lit.: the furthest superior. Explained by its length and dominancein fingering)

Lichanos = ring finger(lit.: the licking one. Explained by its suitability in ornamentation)

Mese = little finger(lit.: the middle. Explained by the fourth, whose span divides theoctave span by half.)

Proslambonomenos = the note immediately above or below therange of the tetrachord in each position. (lit.: the adjoined)

These terms help to describe the ranges of the melodic successionsthat are possible within the consonant interval of a fourth, in particularrelation to the fingered fourth of a string.

43

44 CHAPTER 8. TETRACHORDS IN DETAIL

Ascending Form

[-----------------Tetrachord Hypaton--------------------------------]

Hypate

Mese

Parhypate may span the range from the least enharmonic diesis tothe greatest tone, a range of the largest semitone. Lichanos may spanthe range from the pyknon to the greatest semiditone, a range of a tone.

Descending form

[-----------------Tetrachord Hypaton--------------------------------]

Hypate >>>>>>

Chapter 9

The Varieties of theAscending Species

1:1 4:3

* *

(formed by varying the notes between)

hypate (and) mese

36:35 28:27

Enharmonic Genus * * * 5:4 *

[---16:15--]

36:35 septimal 1/4 tone


16:15 greater diatonic semitone / limma

5:4 maj 3rd / ditone

45

46 CHAPTER 9. THE VARIETIES OF THE ASCENDING SPECIES

25:24 128125

* * * 5:4 *

[---16:15--]

25:24 lesser chromatic semitone

128

125lesser diesis

128

125

250

243

* * * 81:64 *

[-256:243-]

250

243diesis-grave pythagorean semitone

81:64 pythagorean major 3rd / acute ditone

27:26 26:25

* * * 100:81 *

[---27:25---]

27:26 lesser 1/3 tone (13-limit ) / tridecimal comma

26:25 greater 1/3 tone (13-limit )

27:25 2/3 tone / large limma

100:81 comma-grave maj 3rd / comma-grave ditone

47

Chromatic Genus

Tense * 16:15 * 15:14 * 7:6 *

[--------8:7------]

15:14 lesser septimal semitone

7:6 septimal min 3rd

Tonic * 16:15 * 25:24 * 6:5 *

[-----10:9------]

25:24 lesser chromatic semitone

10:9 lesser tone

6:5 semiditone

* 27:25 * 25:24 * 32:27 *

[-------9:8-------]

27:25 2/3 tone / large limma

9:8 greater tone

32:27 pythagorean minor 3rd

Hemiolic * 24:23 * 23:22 * 11:9 *

[------12:11------]

24:23 lesser 3/8 tone

23:22 greater 3/8 tone

11:9 even ditone (11-limit)

12:11 even semitone (11-limit) / 6/8 tone

Soft *27:26 *26:25 * 100:81 *

[----27:25----]

100:81 grave maj 3rd

48 CHAPTER 9. THE VARIETIES OF THE ASCENDING SPECIES

Diatonic Genus

Soft * 21:20 * 10:9 * 8:7 *

[----------7:6--------]

21:20 septimal semitone

8:7 septimal tone

Tonic * 28:27* 8:7 * 9:8 *

[----------32:27---------]


32:27 pythagorean min 3rd

Ditonic * 256:243 * 9:8 * 9:8 *

[----------32:27--------]

256:243 pythagorean limma

Tense * 16:15 * 9:8 * 10:9 *

[-----------6:5----------]

16:15 greater diatonic semitone / limma

Even * 12:11 * 11:10 * 10:9 *

[-----------6:5----------]

12:11 11-limit semitone

11:10 11-limit tone

There are many additional varieties, though these will suffice fornow.

Part IV

Systema

49

Chapter 10

Greater and Lesser PerfectSystems

Systema spanning an octave or more are formed from tetrachords com-bined by conjunction and disjunction

The Greater Perfect System

Systema spanning two octaves formed by four tetrachords

Tetrachord Hyperbolean 3:1---------------4:1

conjunction 3:1

Tetrachord Diezeugnemon 9:4---------------3:1

disjunction (2:1----9:4)

Terachord Meson 3:2---------------2:1

conjunction 3:2

Tetrachord Hypaton 9:8---------------3:2


Proslambonomenos 1:1

51

52 CHAPTER 10. GREATER AND LESSER PERFECT SYSTEMS

The Lesser Perfect System

Systema spanning a twelfth formed by three tetrachords


Tetrachord Synnemennon 2:1---------------8:3

conjunction 2:1

Tetrachord Mese 3:2---------------2:1

conjunction 3:2

Tetrachord Hypaton 9:8---------------3:2



Chapter 11

Ametabolon: Tonality

The Ancient Diatonic Octave Genera

mixture of tense (syntonon) chromatic and tonic diatonic

[--------tense chromatic---------]disjunction[-------tonic diatonic-----------]

[-------7:6--------]

* 16:15 * 15:14 * 7:6 * 9:8 * 28:27* 8:7 * 9:8 *

1:1 16:15 8:7 4:3 3:2 14:9 16:9 2:1

[--------tonic diatonic----------]disjunction[---------tense chromatic---------]

[-------32:27-------] [------8:7-----]

* 9:8 * 8:7 * 28:27 * 9:8 * 7:6 * 15:14 * 16:15 *

1:1 9:8 9:7 4:3 3:2 7:4 15:8 2:1

15:14 lesser septimal semitone

7:6 septimal min 3rd


8:7 septimal tone

9:8 greater tone

9:7 septimal ditone

7:4 septimal flat seventh

15:8 harmonic major seventh

53

54 CHAPTER 11. AMETABOLON: TONALITY

mixture of soft (malakon) diatonic and tonic diatonic

[--------soft diatonic----------]disjunction[--------tonic diatonic----------]

* 21:20 * 10:9 * 8:7 * 9:8 * 28:27* 8:7 * 9:8 *

1:1 21:20 7:6 4:3 3:2 14:9 16:9 2:1

[--------tonic diatonic---------]disjunction[---------soft diatonic----------]

* 9:8 * 8:7 * 28:27 * 9:8 * 8:7 * 10:9 * 21:20 *

1:1 9:8 9:7 4:3 3:2 12:7 40:21 2:1

21:20 septimal semitone

10:9 lesser tone

9:8 greater tone

8:7 septimal tone


Please note: This is an incomplete review copy. This is where itends. To get the full picture please contact the author to purchase acopy.

Chapter 12

Diatonic Systema

12.1 mixture of tense (syntonon) chromatic

and tonic diatonic

12.2 mixture of soft (malakon) diatonic

and tonic diatonic

12.3 tonic diatonic (unmixed)

12.4 mixture of tonic and ditonic diatonic

12.5 mixture of tonic diatonic and tense

diatonic

55

56 CHAPTER 12. DIATONIC SYSTEMA

Chapter 13

Theory of Modality

13.1 Diatonic Systema

13.2 Chromatic Harmonic Systema

57

58 CHAPTER 13. THEORY OF MODALITY

Part V

Modulation

59

Chapter 14

Generally Tempered andSpecifically UntemperedModulation

61

62CHAPTER 14. GENERALLY TEMPERED AND SPECIFICALLY UNTEMPEREDMODULA

Chapter 15

How the succession ofdissimilar tetrachords mayproduce modulation

63

64CHAPTER 15. HOWTHE SUCCESSION OF DISSIMILAR TETRACHORDSMAY PRODUCE

Chapter 16

How natural melodicsuccession may producemodulation between thegenera within a singletetrachord or a succession ofsimilar tetrachords

65

66CHAPTER 16. HOWNATURALMELODIC SUCCESSIONMAY PRODUCEMODULATIO

Chapter 17

How non-melodicadjustments may producemodulation within a singletetrachord or a single genera

67

68CHAPTER 17. HOWNON-MELODIC ADJUSTMENTSMAY PRODUCEMODULATIONWITHIN

Part VI

Mapping the Monochord

69

Chapter 18

A Guide to the PracticalApplication of TetrachordTheory for StringInstruments

18.1 Thesis: Layout of the Canon

18.2 Dynamis: General Layout of Diatonic

Tetrachords

18.3 Just 5-limit Diatonic Intonations

71

72CHAPTER 18. A GUIDE TO THE PRACTICAL APPLICATION OF TETRACHORD THEOR

Part VII

Merging Thesis andDynamis: Key Systems(with specific examples

illustrating general concepts)

73

Chapter 19

The nature of EnharmonicKey Systems

75

76 CHAPTER 19. THE NATURE OF ENHARMONIC KEY SYSTEMS

Chapter 20

The nature of Chromatic andDiatonic Key Systems

77

78CHAPTER 20. THE NATUREOF CHROMATIC ANDDIATONIC KEY SYSTEMS

79

Barker (1989)

80CHAPTER 20. THE NATUREOF CHROMATIC ANDDIATONIC KEY SYSTEMS

Bibliography

Barker, A. 1989. Greek Musical Writings, Volume II: Harmonic andAcoustic Theory. Cambridge: Cambridge University Press. 79

Monro, D.B. 1894. The modes of ancient Greek music. ClarendonPress. 39

81

IntroductionI Harmony: Systems of Rational RelationsConsonance and SimplicitySimple Number RatiosRelations by 3:2Relations by 5:4Relations by 6:5

Dissonance and ComplexityDivisions and Compounds of Intervals: Inverse proportions of pitch and vibrational lengthAddition / Multiplication, Subtraction / Division

Divisionsof an Octave into Fifths and Fourthsof Fifths into Fourths and Tonesand into Ditones and Semiditonesof Fourths into Ditones and Semitonesand into Semiditones and Tonesof Ditones into Semiditones and Semitonesand into Tonesof Tones into Semitonesof Semitones, Dieses and CommasFurther Divisions

II TimeRhythmic and Metric PoeticsRhythmicsMetrics

III TetrachordsTetrachords in generalTetrachords in detailThe Varieties of the Ascending Species

IV SystemaGreater and Lesser Perfect SystemsAmetabolon: TonalityDiatonic Systemamixture of tense (syntonon) chromatic and tonic diatonicmixture of soft (malakon) diatonic and tonic diatonictonic diatonic (unmixed)mixture of tonic and ditonic diatonicmixture of tonic diatonic and tense diatonic

Theory of ModalityDiatonic SystemaChromatic Harmonic Systema

V ModulationGenerally Tempered and Specifically Untempered ModulationHow the succession of dissimilar tetrachords may produce modulationHow natural melodic succession may produce modulation between the genera within a single tetrachord or a succession of similar tetrachordsHow non-melodic adjustments may produce modulation within a single tetrachord or a single genera

VI Mapping the MonochordA Guide to the Practical Application of Tetrachord Theory for String InstrumentsThesis: Layout of the CanonDynamis: General Layout of Diatonic Tetrachords'Just' 5-limit Diatonic Intonations

VII Merging Thesis and Dynamis: 'Key' Systems (with specific examples illustrating general concepts)The nature of Enharmonic 'Key' SystemsThe nature of Chromatic and Diatonic 'Key' Systems

Documents

An_Approach_to_Music-libre.pdf