7)/Vi
Ak> /;7
A STUDY OF DISSONANCE AND HIRIONIC TENSION IN THE
FUCrU.hS UF Tii LUDUS TONI.3IS BY .,-UL HIINDEIMITH
THLIS
Presented to the Graduate Council of the North
Texas State Teachers College in Partial
Fulf illment o, the Requirements
For the Degree of
LATERR OF TSIC
By
Otis Poe Harvey, Jr., B. K.
149325Ontario, California
August, 1947
149325
TMB LE QNTENTr%"O
LIST OF Th1BLES . . . 0 0 0 0 0 0
LIST OF ILLUSTbLTILNS . . 0 0 .0. .
ChapterI. ITR3ODUTKN ND P EURSA
Page. . . . . . . 0 iv
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IntroductionP-ro cedure
II. AALY5IS OF T ELV& FUGLS . . . . . . ..
Fuga Prima in CFuga Secunda in GFuga Tertia in FFuga uarta in AFuga auint a in EFuga Sexta in E,-FlatFuga )eptima in a-FlatFuga Octava in DFuga Nona in 3-FlatFuga Decima in D-FlatFuga Undecima in 3
IFuga Duodecifa in F-Sharp
III. SULvRY .D! C ONCLUSION . ... 0.. * 0 * 0 .
BIBLIOGRYHY
14
193
200
iii
0 0 . . . . . 0 0 ............
LIST OF Th13Lj
Table Page
1. Compilation of Tension Values, ,xtent of Usages,single and Total Absolute harmonic TensionValues .0.0. .0.0. . 0. . . 0. . 0. . 1.0 .0 .019
iv
LIST OF ILLUSTRATIONS
Fuga Prima
Fuga rimama
Fuga rrima
Fuga prima
Fuga Lrima
Luga Prima
Fuga Prima
Fuga rina
1uga rima
Fuga Prifaa
Fuga irima
(Measures 1-5) .
(Measures 6-9) .
(easures 10-13)
(Measures 14-17)
(Measures 18-21)
(Measures 22-25)
(Measures 26-29)
(Measures 30-33)
(Measures 34-37)
(Measures 38-41)
("vasures 42-46)('lsu, s47-46)
iage
.. . . . . . . 19
. .. . .. .20
. 21
.. . . . . . 22
23
.. . . . . . 24
. O . . . . . 23
27
50
9
13. Graph. Showing Averages of .bsolute HarmonicTension of All Measures of Fuga Prima
14. Fuga Secunda (Measures 1-5) . . . . . .
15. Fuga Secunda (Measures 6-11) . . . . . .
16. Fuga ecunda (Measures 12-17) . . . ...
17. Fuga 3ecunda (Measures 18-22) *-. . .
18. Fuga 3ecunda (Measures 23-27) . . . . . .
19. Fuga Secunda (Measures 28-32) . . . . . .
20. Fuga Jecunda (Measur-s 33-37) . . . .
21. Fuga Jecunda (Measures 38-41) . . . . . .
v
30
33
34
35
36
37
38
39
40
Figure
1.
2.
3.
4.
5.
6.
7.
8.
10.
11.
12
Figure Page
22. Fuga Secunda (Measures 42-46) . . . . . . . . . 41
23. Fuga Cecunda (Measures 47-51) . . . . . 42
24. Fuga Secunda (Measures 52-56) - - - - . 43
25. Fuga Secunda (Measures 57-61) . . . . . . 44
26. Fuga Secunda (Measures 62-65) - - - . . 45
27. Fuga 3ecunda (Measures 66-70). . ...... . 46
28. Fuga Secunda (Measures 71-75) ... .. . . 47
29. Graph Showing Avcrages of A bsolute HarmonicTension of All Measures of Fuga Secunda . . . 8
30. Fuga Tertia (Measures 1-6) . -.- -0 -. 51
31. Fuga Tertia (Mleasures 7-12) - - - - - - - 52
32. Fuga Tertia (Measures 13-17) . -. - - - - -. . 53
33. Fuga Tertia (IMeasures 18-22) . - - . . .- . . 54
34. Fuga Tertia (Measures 23-27) --...-...... 55
35.. Fuga Tertia measuress 28-32) - - - . - . . . . 56
36. Fuga Tertia (Measures 33-37) - - - . - . - . . 57
37. Fuga Tertia (Measures 38-42) * - -........ 58
38. Fuga Tertia (Measures 43-47) - - - - - - - . . 59
39. Fuga Tertia (Measures 48-53) . . . . . . . . . 60
40. Fuga Tertia (Measures 54-59) . . . . . . . . . 61
41. Graph Showing Averages of 1Absolute HarmonicTension of All Measures of Fuga Tertia . . . 62
42. Fuga arta (Measures 1-4) - - - . . . 65
43. Fuga Quarta (Measures 5-9) .. . . . . 66
44. Fuga ?,uarta (Measures 10-14) -. ....... 67
45. Fuga >arta (Measures 15-18) . . . . . 68
vi
-Figure
46. 'uga ,uarta (Measures 19-22)
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
6o.
Fuga CQuarta (Measurs 23-27)
iuga Turta (Measures 28-30)
Fuga Quarta (Measures 31-33)
Fuga uarta (Measures 34-36)
Fuga Quarta (Measures 37-40)
buga ,uarta (Measures 41-44)
Fuga Quarta (hleasures 45-48)
Fuga ;.uarta (Measures 49-52)
Fuga uarta (Measures 53-56)
Fuga .Qarta (Measures 57-60)
Fuga ,'uarta (Measures 61-64)
Fuga .uarta (Measures 65-68)
Fuga uarta (Measures 69-72)
Fuga Quarta (Measures 73-76)
61. Graph showing verages of bsolute HarmonicTension of All Measures of Fuga CQuarta . . . 84
62. Fuga Quinta (Measures 1-5) . . . . . . . . 86
63. Fuga 4uinta (Measures 6-11) . . . . . . 67
64. Fuga uinta (Measures 12-17) . . * . . . . 88
65. .uga Quinta (Measures 18-23) . . . . .* * . 69
66. Fuga Quint & (Measures 24-29) . * * . . . 90
67. Fuga 'uinta (Measures 30-35) * . * . . . . . 91
6$. Fuga uinta (Measures 36-41) * . . . . . . . 92
69. Fuga iQuinta (Measures 42-46) * - - - - * * . . 93
vii
i:age
. . . . . . . . . o9
- - - * - * . . . 70
. . . . 0 . 0 0 , 72- ---.. . 72
* 0 0 . * 0 - 0 0 73
. . . . . .. . . 74
. - - - 0 0 - . * 75
* 0 * . . . . . . 76
. . . . . . . . . 77
. . . . . 0 . . * 76
. . . . . * . . . 79
. . . . . . * . 80
*. - * - - . . . 618
* 0 0 . . - - - - 62
* 0 0 0 0 * * * 6
Figure Page
70. Fuga A.iuinta (Measures 47-52) . . . . . . . 94
71. Fuga Quinta (Measures 53-58) - - . . 95
72. Fuga '.uinta (Measures 59-64) . . . . . . 96
73. Fuga 6uinta (Measures 65-70) . . . . . . 97
74. Fuga 2Ruinta (Measures 71-75) . . . . . . 98
75. Fuga luinta (Measures 76-82) . . . . . . . 99
76, Graph Showing Averages of Absolute HarmonicTension of All Measures of Fuga :uinta .. 100
77. Fuga Sexta (Measures 1-5) . . . . . . . 102
78. Fuga Sexta (Measures 6-11) . . . . . . . 103
79. Fuga 6exta (Measures 12-17) . . . . . . 104
80. Fuga Sexta (Measures 18-22) -- --- . . . . . . 105
81. Fuga Sexta (Measures 23-27) . . . . . . 106
82. Fuga Sexta (Measures 28-32) . . . . . .
83. Fuga 3exta (Measures 33-38) .. .... . .... 108
84. Fuga Sexta (Measures 39-44) . . . . . . 109
85. Fuja ae.ta (Measures 45-50) . . . . . . 110
86. Graph Showing Averages of Absolute HarmonicTension of Lll Measures of Fuga 6exta . . , 111
87. Fuga Septima (Measures 1-5) . . . . . . . . . 113
88.6 Fuga Septiiia (Measures 6-9) . . . . . . , . . 114
89. Fuga 3eptima (Measures 10-13) . . . . . . . . 115
90. Fuga Septima (Measures 14-17) . . . . . . . . 116
91. Fuga &epti::a (Measures 18-21) . . . . . . . . 117
92. Fuga Septia (Measures 22-25) . . . . . . . . 118
viii
Figure aage
93. Fuga $eptima (Measures 26-29) . . . . . . . . 119
94. Fuga Septima (Measures 30-33) . . . . . . . . 120
95. Fuga ;eptima (Pleasures 34-37) . . . . . . . . 121
96. Fuga eptilila (Measures 38-41) . . . . . . . . 122
97. Fugu eptitma (Measures 42-45) .. ... . .. 123
98. Fuga Septima (Measures 46-49) . . . . . . . . 124
99. Fuga Septima (Measures 50-54) . . . . . . . . 125
100. Graph Showing averages of .bsolute HarmonicTension of ,ll Measures of Fuga 2eptima . . 126
101. Fuga Octava (Measures 1-3) - .. . . . . . . 128
102. iuga Uctava (Measures 4-6) . . . . . . . . . 129
103. 1uga Octava (Measures 7-9) . . * . - . . - . 130
104. Fuga Octava (,esures 10-12) - - - - - . - . 131
105. Fuga Octava (Measures 13-15) . . . - . - . . 132
106. I-uga Octava (Measurres 16-18) - - - . . . . . 133
107. Fuga QCtLva (Measures 19-22) - - - - . . . . 134
108. Fuga Octava (Measures 23-26) - - . . . . . . 135
109. Graph Showing Averages of Absolute HarmonicTension of All Measures of Fuga Octava - . 136
110. Fuga Nona (Measures 1-5) - . . . . 138
111. 2uga Nona (Measures 6-10) - - - - - - - - * . 139
112. Fuga Iona (Ieasures 11-15) . . ...... . 140
113. Fuga Nona (Measures 16-19) . . . . . . . -141
114. fuga ona (Measures 20-24) . . . . . . . . . 142
115. 2uga aLona (Measures 25-29) - - - - - - - - . 143
ix
Figure Page
116. Fuga Nona (Measures 30-34) . . . . . . . . , 144
117. Fuga Nona (Measures 35-39) . . . . . . . . . 145
118. uga Nona (Measures 40-44) . . . . . . . . 146
119. Fuga 1ona (Measures 45-4.9) . . . . . . . . . 147
120. Fuga Mona (Measures 50-54) . . . . . . . . . 148
121. Fuga Mona (Measures 55-59) . . . . . . . . . 149
122. Fuga Mona (Measures 60-64) - . . - - - . . . 150
123. Fuga Mona (Ldeasures 65-69) . . . . . . . . . 151
124. Fuga iona (Measures 70-73 . . . . . . . . . 152
125. FWa Iona (Measures 74-77) - . - - - - -# . 153
126. Fuga Mona (Measures 78-82) - - - - . . . 154
127. Graph Showing Xvorages of Xbsolute HarmonicTension of All Measures of Fuga Nona . . . 155
128. Fuga Decima (Measures 1-4) - - - . . . - . . 157
129. Fuga Decima (Measures 5-7) - - - - -0 - . . 158
130. Fuga Decima (Measures 8-9) - - - - - - . . . 159
131. Fuga Decifia (Measures 10-12) . . . . . . . . 160
132. Fuga Decima (Measures 13-15) . . . . . . . . 161
133. Fuga Decima (Measures 16-8) . . . . . . . . 162
134. Fuga Deci ma (Measures 19-22) . . . . . . . . 163
135. Fuga Decima (Measures 23-25) . . . . . . . . 164
136. Fuga Decima (Measures 26-27) , . . . . . . 165
137. Fuga DeciLma (Measures 28-29) . . . . . . . . 166
138. Fuga Decirma (Measures 30-32) . . . . . . . . 167
x
Figure age
139. Fuga Decima (Measures 33-36) . . . . . . . . 168
140. Graph Shoiing .verages of ,bsolute HarionicTension o f ll Measures of Fuga Decira . . 169
141. Fuga Undecima (Leasures 1-4) . . . . . . . . 171
142. uga Undeciiiia (Leasures 5-8) . . . . . . . . 172
143. Fuga Undecia (L.asures 9-12) . . . . . . . 173
144. Fuga Undecima (Measures 13-1) . . . . . . . 174
145. uga Unochima (L&asures 17-19) * . . . . . . 175
146. Fua Undecia (Measures 20-23) . . . . . . . 176
147. Grraph Showing average of bsolute I rm-onicTension of .11 hesures oIf IuLsa Undecica . 177
148. Fuga Duodecima (Leasures 1-3 . . . . . 179
149. uga Duodecinia (Measures 4-6) . . . . 180
150. Pua Duodeciia (ieeasures 7-9).... ... . 131
151. uj.auodoc ia (Me-surs 10-12) . . .. . 132
152. Fuga DuodecL 1la (itjasures 13-15) . . . . . . 183
133. cuga Duodeci:a measuress 16-18) . . . . . 134
154. Fu"a Duodecica (,,asures 19-21) . . . . . 135
155. uga uOdeci,--a (7e ues22- 2 4)
156. uga Duodeciafe (Teasures 25-27) . . . . . . 187
157. Fuga Duooi 1a (I easures 23-30) . . . . . . 183
158. Fu;a Duodecila (Measures 31-33) . . . . . . 189
159. Fuga Duode.cina (Measures *4-37) . . . . . . 190
xi
160. ra..v lhown -vraes of .bsolute-armonic Tension oi l L Ia1suresof 2u.,.Duodecima .. . . . . . ... .191
lol. GrLI1h Zoin, tr Total vlranls ofabsolutee L rronic Tension of l1Twelve Fugues . . . . . . . . . . . . . . 192
xii
giur e rag
CHJAPTL I
INTROJUCTIOTN .ZD PRU2DURE
Introduction
Although works of the composers of the twentieth
century are continually being studied, it is quite evident
that studies of various aspects of ,.iusic of the modern
period do not nearly ecual the exainations made of composi-
tions written before 1900. It should, therefore, be a dvan-
tageous to any person int-rested in the stylistic features
of .. odern compositions that more an, "'Oi information be ob-
tained through bet> broad and technic i analyses of any and
l
as ect13 of music of the present
period.
This study considers only one as ipect of .iusic of the
twentieth century--that of dissonance. Through an analysis
of harmonic tension in the tv, elve fugues of the Ludus Tonalis
by Paul Hindemith, a two-fold significance is notable. irst,
consideration of the broad tendencies of modernism is neces-
sary. second, ,ith hindemith as a chosen representative of
certain aspects of the twentieth century style, the study
attempts to show more specifically certain characteristics of
the composer himself in the use and treatment of dissonance,
one of the more technical features of style in modern music.
Paul Hindemith was born in Hanau, Germany, on November 16,
1
2
1895. Running away from home in his eleventh year because
of his fathers objection to a -musical career, Hindemith
earned his living playing the violin in dance orchestras,
motion picture houses, and cafes. tt the same time he entered
the Hoch Conservatory in Frankfurt. In 1915 he became concert
master of the orchestra of the Frankfurt Opera and later the
conductor. He first achieved fame as a composer when his
works in modern counterpoint were played on tours between
1929 and 1932 by the Iamous Amar String quartet which he
founded.1
"His style as a composer can be said to have become
crystalized with a concerto for piano and twelve solo instru-
ments (Karaermusik) introduced in 1925 at a festival of modern
music in Venice.'2
<iith the successful production of his first opera,
Cardillac, one year later the attention of the music world
was focused on him. Then came the jazz opera, PNeues vom
, and later the well known Mathis der Maler. He has
written'hiany works for chamber music groups and orchestra and
a great variety of smaller pieces for pianola, radio, and
talking pictures which have been grouped by critics under the
classification of Gebrauchsmusik (workaday music)--a term in-
ventea ror lindemith."'3
lDavid Ewen, The Book of Modern Cm rs, p. 301.
2Ibid., p. 302.
3Ibid., p. 302.
3
As a composition teacher in the Berlin Hochschule for
ten years, beginning in 1927, "he became a storm center in
Nazi Germany when the Kulturkammer condemned his music as
antagonistic to the spirit of the "new Germany," and decreed
that it did not meet the specifications of a true aryan
composer."4
In 1935 hindemith was commissioned by the Turkish govern-
ment to reorganize its musical life. In tile spring of 1937
on an invitation fram the Elizabeth Sprague oolidge Founda-
tion, he visited the United States for the first time, tiring
the country as violist in performance of his works. Since
then he has moved to this country and is now a member of the
music faculty of Yale University. 5
From the article, "The Composer Speaks," Paul Hindemith
himself says:
The basis of all worth-while compositions must beof course, inspiration and worth-Vhile musical ideas;after that comes technique. There seems to be an im-pression that there is today too much technique. Itis my impression that there is not nearly enough tech-nique. One cannot learn how to be a composer, in themodern sense, by a few years of harmony, counterpointand theory in a conservatory. It requires years ofdaily intimacy with all kinds of music, not merely theprocess of playing it, or hearing it, but that of in-vestigation 6 and studying this music as a great naturalphenomenon.
4 Ibid p. 302.
5Ibid p. 302.
bIbid., .303.
4
Hindemith has developed a system of tonal laws that
includes all harmonic phenomena of even the most modern
music, but which derives from tonal foundations. Through the
most complicated physical calculations he is determined to
bring all problems of composition into the system. This means
a revolution in harmonic theory, especially the tearing down
of Riemann's doctrine of function. F1 or many years hindemith
has been using this system of tonal logic in his teaching of
composition. 7 In his Craft of Musical COMposition, Book I,
Theoretical Part, Hindemith describes the laws governing
tonal material, regardless of any style or technique. Iis
"system" is simply meant to supply an analytical background
for all types of music.
In order to consider Hindemith's general style, or any
technical aspect of his general style, it is necessary to
consider the modern period, and more specifically modernism.
ItModernism in riusic may be broadly defined as that in which
there is manifest some aspect of musical style or form that
departs in some significant respect from common practices af
the preceding period. "1
The development of modernism in music may be classified
into three periods. The first period is the last two decades
711. H. Stuckensch.idt, "Hindemith Today," Modern Music,XIV (January, 1937), p. 67.
6Hugh M. Miller, in Outline-Hisr of Music, p. 171.
5
of the nineteenth century, which showsdefinite signs of de-
parture from general nineteenth century practices. The
second period, roughly defined as beginning wvith the turn of
the century and continuing to the opening of i.orld 'ar I, is
one of open revolt against German romanticism and is marked
by radical experimentation. The present period, from the
close of world :ar I until the close of Yorld var II seems to
be one of assimilation of new principles. although much ex-
perimentation still exists and romanticism is not completely
abandoned, there is clear evidence of a prevailing objectivity
in music of the present period. This is manifest in the
general restraint of emotional content, in the simplifica-
tion of materials and structures, and in the greater atten-
tion to musical craftsmanship. I classical spirit called
"neoclassicism" prevails. 9
iNeoclassicism is a definite return to a classical point
of view. omantic subjectivity is disc.a.rded in favor of a
modern objectivity. It consists in general of a simplifica-
tion of material, form, and medium. iore specifically, it
is represented by a recognition of eighteenth century ideals
in the use of counterpoint anid formal clarity, but clothed
in tx.entieth century harmonic idiom, key schemes, orchestra-
tion, and melodic style. One of the principal leaders in the
9lbid., pp. 171-172.
6
neoclassical trends is Paul findemith who is an exponent of
modern counterpoint.
One of the most significant developments of the t;ventieth
century is the revival of interest in polyphonic writing, an
attribute of the neoclassical tendencies of the present.
baroque contrapuntal forms are again employed (fugue, canon,cantus firmus), but with tie linear freedom allowed by modern
harmonic concepts. Greater attention is ,iven to the melodic
contour of lines, the linear aspect, than to harmonic effects. 1 1
hindemiith's early works show him to have combineddefinite romantic tendencies xJith an extraordinarymelodic gift and a virtuoso command of technique.Convinced of the value of modern harmonic develop-ments, he has arrived at his own harmonic idiom by aprocess of melodic development. His problem as toreach a melodic style th--t would Uictate atonalharmony. The process of evolution led hindemith tothe development of dissonant counterpoint, a termwhich 1 has become used quite frequently in connectionwith contemporary music. Although he allows the wordatonal to be used in a bibliography of his works, hismusic does not confom to the strict rules of atonalitysuch as used by Schonber. he retains the basic,even conventional chord progressions of key tonality,but disguises them aith an often highly dissonantsuperstructure.12
In brief,
The style of Hindemith is essentially contra-puntal, harmonically dissonant as a result of linearindependence, often melodically angular with dissonantskips, essentially tonal in that it begins and ends ina key, but tonality is often obscured and remote from
10Ibid., p. 173.
llIbid., p. 182.
Iheodore M. Yinney, A History of music, p. 581.
7
the starting point, and neoclassical in its economyof material and clarity of texture.13
aul Rosenfield, in his article, "Lindemith describes
hinaemith as the most fruitful of contemporary German com-
posers, and one of the most conspicuous representatives of
vhat is perhaps the most sharply defined. :overent in contem-
porary music--thlat of neoclassicism--an evolution'l4
The Ludus Tonalis, subtitled Studies in Couterpoint,
Tonal i and iano i as written in 1943.It consists of twelve fugues, one in each key of the
chromatic scale, connected by interludes and -receded and
followed by a iraeludium and a P:ostludium. The Jostludium
is the Iraeludiumi written backiard;s and upside doxwn, thus
the name Ludus Tonalis, a Latin tern neaninc "musical ;ames."
In his revievi of the Ludus Tonalis in Uhicago in February,
1944, critic Cecil Smith says:
In its scope and rane:ent it mdy thus becompared to LEch's ell Tempered Clavichord. anthe purely t0chni 1CA level, bo has~kTh0ance ocont Ptuntal skill probably unequaled since .ch andas an investiGation of the p0 >ibilities of the Aanoas a virtuoso instrument, the vork is staggering, anddeserves a special iono5raph. This aspect of 1iinde-Lith'isp.oIers probably surrises rio one. But the depthe ofexpressiveness and the v, riety of -oods are ,ulitiesor which s one, unaware of tho continual ripening ofHindemith's musical persoiality in the last decade,,iay have been unprepared. 5
1 3"'iller, op. cit., p. 193.
+ wen, .2. Acit., . 305.1 5 Cecil Smith, "Composers to Chica, "
1.I (June, 1944), 243.
8
.another reviewer, Hans Rosenwald, says:
Jor those interested in a closer correlation ofpiano playing: and study of mod4Lrn music it offersmany problems as well as surprises. And it would lotbe a work conceived in the real craftsLaInship-spiritof the neopolyphonist if it failed to surprise you,too, with its Lany contrapwLtal tricks ana devices.It is "lard to decide whether the telve piano fuguesin t .elve keys which cover the gamut of emotions andtneir dancelike interludes are a more airect parallelto Each's iell-Tempered keyboard, which was goarotherhere, or to that master's Art of the Fugue. There isa great deal of sensitive feeling-in both the freeand strict forms, and there is a great deal of intri-cacy which you will have to conquer gradually. andthat is clear, there are few masters today who writetheir fugues with the ease o: hindeiith, so that instudying the work the old Reger words came to my mind:"Others compose fugues; I can only think in fugues."10
Thus it may be seen that this work represents the modern
Paul Hindemith well, clearly refle cting neopolyphony in theemployment of- one of the Baroque contrapuntal forms--the
fugue--although in a modern setting, thus creating modern
counterpoint because of the linear 1 7 and vertical dissonance
and modern rhythmic patterns. It typifies the neoclassical
hindemith in that it is a contrapuntal composition, materially
economical, formally clear, and structurally objective.
Procedure
The purpose of this study is to discover any notable
fact or characteristic in the use of dissonance by Paul
hans hosen-vald, speakingg ofus ic, " Jusic News .E&N(Lecetmber, 1943), 11.
17The term linear emphasizes the horizontal aspect ofcontrapuntal dissonance, as opposed to the harmonic (vertical)aspect.
9
Hindemith which may or may not be typical of the composer's
present style of composition. The means of the study is the
Ludus Tonalis.
To pursue such a task the investigator read several
books and parts of other books and magazines to gain a broad
perspective of the present period and to see how the composer
fits into the whole picture. Also a considerable number of
articles on P1-aul hindemith were read to determine the promi-
nent characteristics of his style in both his earlier writings
and particularly his more recent ones. ll available informa-
tion on the composition itself was collected to determine its
typicalness in the present period, and more specifically its
representativeaess of the modern Hindemith style. The writer
was slightly handicapped by the lack of specific information
due to the newness of the work. Ludus Tonalis is a representative
work of one of the foremost exponents of . odern counterpoint.
To limit the problem of analysis, the investigator chose only
the twelve fugues of the Ludus Tonalis for comprehensive study
of dissonance and harmonic tension. The limitation of voices
(in the fugures there are only three voices used) made more
practical the chosen method for measuring harmonic tension.
The writer has applied 'n Objective Method for the tudy of
Harmonic Tension which was devised by Hugh h. Miller in 1946.18
18A paper read before the annual meeting of the 4rmericanMusicalogical Society, 9rinceton University, December, 1946,to be &ublished in the Society bulletin No. 12.
10
This method assigns two properties to harmonic tension.
One is termed absolute" and the other "relative." The
latter involves such -- 'ctors as duration, accent, root ten-
sion, bass tension, altered tones, and possibly other rae-
tors. In this study only the property of "absolute" tension
is considered. Absolute tension is determined by the dis-
sonance value of component intervals in a combination of
tones. Dissonance may be defined as "a term used to describe
the effect of certain tonal combinations which represent the
element of irregularity and disturbance.1 9 Dissonant har-
monies and melodies are strongly characteristic of music of
the present period, of which Paul Hindemith'ss works, and par-
ticularly the Ludus Tonalis, arerepresentative. The method
for study is based upon the arbitrary assignment of propor-
tionate numerical values to all intervals. This necessitates
a classification of all intervals. The perfect fourth, fifth,
and octave are described as "perfect" consonants with no
tension value. Major and minor thirds and sixths are des-
cribed as "imperfect" consonants, also with no tension value.
The augmernted fourth and dimixshed fifth are described as
mild dissonance -with a tension value of one degree. The
augmented triad is arbitrarily ao signed a tension value of
one degree, also being described as mildly dissonant. The
19 illi pel, Harvard Dictionary of Music, p.18o.
1L
major second and ninth, and the minor seventh are asgned a
value of two deg-:rees tension and described as ioderate dis-
sonance. Tle interVals ef the minor second an4. ninth, and
major seventh are as-igned a tension value of three degr:es,
and thus being the greatest in tension value, they are de-
scribed as strong or sharp dissonances. Thfe erect fourth
is arbitrarily assigned a tension value of one degrEie when
it occurs between the lowest eouing voice and any other
voice. Le addition o a perfect octave to a dissonant inter-
val do;s not change the value of tension. Tension is lost
v-en a distance of :more than four octaves ,earates the t o
intrvals. 1he tension via.e of any co inatiLon of tones is
the saLer s its en1Grse nic e uivalent. an irresLar spelling
of t ,o or -- ore tones does not change if the enhiL rrnic equiva-
lent c onformos to one of the -previ ously stated interval classi-
ficLtions.
11e inve tIgator has .,urE all intervals in the t-elve
fugues of thc Ludus Tonalis, clas-ifyin 1 then: according to
this system of arbitrarily determined numerical values, and
assirninr a tension. velue of from one to three degrees to
every possible interval. ith the combination of three tones
the Lighest possible tension value is eight decrees; i.e.,
three plus three lus to, since any greater dissonance vauLd
be impo, sible according to the method employed. In the
t:eIvc futues, none hevin. -ore than three voices, the greatest
single absolute tension value is eight degrees. The numilerical
value of aC0 solute tension of co-ai1ponent intervals in all the
12
possible combinations of tones in the twelve fugues has been
plotted on a series of graphs which are drawn directly under
every line in each fugue. Each line of every fugue is listecd
by a Figure number and may be found in the List of Illustra-
tions. The graph represents degrees ranging from zero de-
grees (complete consonance) to eight degrees of tension, the
highest found in this study. The factor of duration cannot
be avoided when considering absolute tension, although it is
actually a specific factor of relative tension. The horizon-
tal line shows the duration of the tension value and conso-
nance and the vertical line shows the increase and decrease
of absolute tension, registering the exact degree of absolute
harmonic tension of every tonal combination throughout each
fugue. Preceding every graphed fugue analysis is a brief
description of the analysis pointing out any notable features
of dissonance and dealing specifically with the presence and
absence, frequency and amount of increase and decrease of
absolute harmonic tension. Also accompanying these short
reports are graphs registering the averages of total absolute
harmonic tension of all the measures in each fugue. The last
graph registers the total averages of absolute harmonic ten-
sion in all twelve fugue. Tile contour of this line shows
the tension relationship of each fugue to the other as to
its order in the Ludus Tonalis.
From these near pictures showing the value of absolute
harmonic tension one should be able to derive certain facts
13
regarding the use and treatment of dissonance. It is the
hope of the investigator that discoveries of any innovations
and certain characteristics by t.iis study of harmonic tension
may lead the reader to a clearer insight into the composer's
style.
CHkPT:R II
ANLY5IS OF T .JL>J FUGU 15
Fuga Prima in C
The following graphs, represented by Figures 1-13, show
the rise ana :all of absolute harmonic tension in 1Fuga crima
in U. In this first fugue, as in all but one fugue, there
is no tension at the very beginning. It is registered as
zero degrees on the graph, Figure 1. &his is obviously typi-
cal of at least the first measure in all fugues except one,
aue to tie fact that there is only one voice present--the'
fugal subject. There is no tension until the entry of the
second voice, in measure four of Figure 1, and not then until
the third count of the measure, after this point there is a
consistent rise and tall of tension, but not more than four
degrees at one time until measure nineteen o Figure 5, Vhiere
there is a sudden drop to zero degrees from a point of five
degrees. The first tension high point is reached on measure
twenty-three of Figure 6, ana after this point there i s a
notable increase in both the amount and extent of tension
change. another high point of eight degrees tension is
noticed on measure twenty-seven of Figure 7. It is immediately
followed by a sudden drop of seven degrees, thus supporting
14
15
the fact regarding an increased tension change. Beginning
with Figure 8 we see a more moderate rate of change, and also
of lesser amount. The horizontal graph line which registers
the duration of tension and consonance is less frequently
interrupted by the vertical line which shows the amount of
increase and decrease. This moderate tension change prevails
until measure thirty-six of Figure 9, where there is a sudden
rise from complete consonance, registered as zero degrees, to
another high point of eight degrees. This high value is left
by a sudden decrease of five degrees to a point of three de-
grees. The horizontal line still registers a fairly moderate
duration of tension, or a rather restraineUc amount of ten-
sion change, even though the vertical line shows a high point
of eight degrees at least three more times, once each in
Figures 10, 11, and 12. In Figure 11 there is evidence ofa greatly decreased tension average, probably due to the re-
peated use of complete consonance. This consonance, however,
is interrupted by several sudden increases in tension (in-
cluding one of eight degrees in measure forty-four of Figure
11), but these, in themselves, are quite brief and therefore
do not alter the general decrease in total harmonic tension.
slightly more frequent tension chLnzje is noticed after
measure forty-four of Figure 11L, althougli the amount of change
does not exceed three degrees except in measure forty-seven
of Pigura 12 where another sudden increase to a high point of
eight degrees is noticed. This last high point, itself of
16
small duration, is followed by a tension value of three de-
grees, showing a sudden decrease of five degrees. The general
tension average from measures forty-eight through fifty-one
is approximately the same, although there is a continual and
consistent return to less tension at fairly regular intervals,
and finally a return to complete consonance on the final
measure.
Fuga 2irima, as will be seen in others, begins and ends
with consonance, Although there may bo argument as to the
actual establishment of the key of C at the beginning, there
can be no doubt that it is the composers intention to return
to the proposed key at the end. Thus it is that in this
fugue--and later it shall be noted in the others--there is a
constant contrast betxueen dissonance and consonance, tension
and repose. Of all the tonal combinations many may be seen
to contain dissonance. The contrast in the various degrees
of dissonance in these tonal combinations is ouite evident,
and their dissonant quality is made more prominent by the more
contrasting consonances, interspersed throughout the fugue.,
This contrast of dissonance and consonance, and degrees of
dissonance is a notable fact which is characteristic of all
the fugues, as the study will show later.
From a broader perspective seen in Figure 13 which
registers the averages of absolute harmonic tension of all
measures of Fuga Prima on a graph may be noticed a curve from
low to high and return. The general rise to a high point
17
about in the middle of the fugue is noticed, followed by a
fairly gradual return to complete consonance. ThisE general
contour is frequently i at.rrupted by changes from low to
high, but does not alter the fact that there is a gradual
increase in harmonic tension from the beginning which, when
climaxed about half-way in the fugue, is brought to a gradual
return to complete consonance by the final measure.
0C12
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31
Fuga 3ecunda in G
In this fugue, as in the precedinG one, there maay be
seen the typical rise and fall of harmronic tension, regis-
tered on graphs hich are listed as figures 14-2 and accom-
panied by a graph shoving the General contour o: tension
change, derived from the average O 1, r onic tenision in all
the adesures listed as Figure 29. This fugue is slightly
less dissonant than Fuga nPrina, w -hich -ay be explained by
the fact that there is a Llore frequent return to comLplete
consonance. . though the high point of absolute tension is
eight degrees, it is reached only one tie, whereas in the
preceding{ fugue it as attuinecd six tiraes. The frequency of
tension change is about the same as buq;a irixa, although the
average amount is slightly less. ',nother notable fact is
that the frequency of tensional change itself is aore notably
contrasted in this fugue. There is evidence through the
graphs that certain sections register consistently low fre-
quency of change, then are suddenly contrasted by a consis-
tently high frequency change. There are longer sections of
either a consistently high or consistently low frequency of
tension change than in thie preceding fugue. There is, gener-
ally, a longer duration of tension values and consonances in
this fugue than in Fuga Prinia, although only slight in re-
gard to dissonance, notably longer durations of complete
consonance.
32
Prom a broader point of view, noted by reference to the
graph listed as figure 29, there riay be seen a slight differ-
ence in the general contour of the line registering the in-
crease and decrease of tension. althoughh typical in i ts be-
ginning with complete consonance, there is a fairly prominent
increase in total absolute tension vhich, vhen reaching a
high point rather quickly, -uhich is gradually increased
after a brief decline to another nigh point), begins a per-
sistent return to consonance, but it turns again rather
sharply and continues in an upward direction to the Iiighest
point of tension average jus t before the end and drops sud-
denly to much less tension at the close, whereas Fuga Prima
not only began but ended with an avera e of complete conso-
nance in its first and final measureS, Fuga Secunda, although
returning on its last note to consonance, does not close with
a totally consonant final measure. Therefore, the contour of
the graph line register ing tension in this fugue does not
conform to t he general cur ve of the line registering tensionin the first fugue. This is perhaps m ore notable when we
consider the fact that Fuga Qecunda is totally less dis-
sonant.
33
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49
Fuga Tertia in F
Typical is thi s fuguet s beginning xith complete conso-
nance, but more notable is its employment of consonance
totally through the sixth measure, shown on graph in Figure
30. This is obviously due to the long subject of six measures.
Tension interrupts consonance on the seventh measure, 1 figure
31, although the value is slight, and continues thus for at
least six more mecures. Tuga Tertia is totally less dissonanttitan the preceding fugues, althugLh it doEs not have s great
a frequency of return to complete consonance as does Tuga
Secunda. The igiihest absolute tension value i6 only Live de-
grees, this being reached only nine times. however, theaverage of the mature uith g-,atest tension is 4.2, relatively
more than the high measure averages of the preceding fugues.
The fr2eiuency of tension chan e is not great, and the amount
rarely e:xceeds three degres. Thre is At Leust one notable
chan, e in the fre uency of tension change, and thc extent of
the clianges, this beginning around measure foty-ninc in
Figure 39 and continuing to the last measure, Pigure 40.
ere may be seen an increase in the chan&:e from great to
small dissonance, or the change from dissonance to consonance.
Wso, the amount of the tension change is greater. There is
a greater and more 1reuent contrAst in dissonant tonl combi-nations, and in dissonant and consonant combinations. at
the end there is a r tuin to thie tonic tone which was in this
fuSue well established ;.t th beginning.
50
The general contour of tension averages seen on graph
in Figure 41, shows a con:s tant increase of tension to a high
point about half-bay through the fugue which, vhen beginning
its decline, turns rather sharply to reach a lighter point
near the fugue s end before dropping suddenly to much less
tension. ihis is quit e sisiilar in this respect to uga
Jecunda. It, like the preceding uga uecun a, does not con-
Iormi to the symmizretrical curve ozc Fuga Prima.
51
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FUga cuarta in A
Typical in its beginning, with complete consonance it,like Fuga -erima, returns to a final measure of L totally
consonant average. The total tension average of Fug'a "uartais less than any of the three preceding futures. Its high
point of absolute tension is eight degrees, this being found
four times. Of notable interest is the formal structure
which has particular influence upon the broad picture of
tension. Divided into three sections, the middle one of con-
trasting style and new subject material and the latter a re-turn to the first, this fugue allows for the use of consider-
able consonance which, when pictured on a grapb. in Figure 61,
has a definite effect on the general contour of the lineregistering tensional averages. Actually, the middle sectionwhich is in itself a different fugue when pictured w ith theentire fugue, allows for considerable contrast in the lineregistering tension. ith the ending of the first section
on complete consonance, the statement of the new subject pre-vents the use of dissonance until the second voice is entered.This may be seen on the graph listed as Figure 44. Notable
is the almost extreme duration of dissonance values of
Figures 45 and 46, which in themselves are not great. Thisis contrasted to a greater frequency of tension change, be-ginning with Figure 47 and continuing fairly consistently tothe close of the fugue. lore notable, perhaps, is the increase
64
in the amount of these tension changes, many of which includea complete return to consonance.
The broad picture of tension rises and declines, Figure61, shows at least three high points. After the first highpoint, seen about one-third of the way through the fugue, isthe rapid decline to a prolonged use of consonance. This isfollowed by a persistent rise, at least one more decline andrise before decreasing to the final measure of complete con-soiiance.
65
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Fuga uinta in 1
Hith a fugue subject of seven meL;esures, this fugue
registers no tension until measure eight, Figure 62, and then
only slightly. persistent return to complete consonance
characterizes this, one of the lesser dissonant fugues. The
frequency of tension change is relatively small, although the
extent of the change is probably not below the average of
the preceadiig fugues. Yith a tension high of eight degrees,
vhich is reached only oc, this fugue acquires its moderate
tension average by prolonged tension values. The factor of
durationis quite prominent, particularly in regard to con-
sonance. There is a greater degree of prolonged consonance
than prolonged dissonance.
Beginning and ending in the key, Fuga -,uinta is charac-
teristic in its return to complete consonance which, in this
case, allows for a totally consonant average for the final
measure. The broad picture of tension averages, seen inFigure 7u, shows a persistent rise, followed by a slight de-
cline and return to a high point near the fugue's close, be-
fore the usual drop to complete consonance. The general
contour is interrupted by frequent drops in harmonic tension.
86
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101
iuga extra in .- lat
This fugue employs a tension high of only five degrees,
and. this is reached only three tliaes. Its irnoderate tension
value is pr obably due to the prolongation o G its individual
tension values. The frequency of tension change is rather
slight, and the extent of the change is usually sail, rarely
ever exceeding four degrees. also of interest is the pro-
longed use of consonance wahichi about equals the duration of
prolonged tension values.
F1or a broao perspective Uigure 86 sho.s continual in-
crease in harL.ronic tension viich, when V liaed a"out aL- ay
through the fugue, goes into decline to the ena o the fugue,
thus conforming to the curve" of the first fugue. TLis
general cont our is typically interrupted by t pension average
changes. notable is the final chord in second inversion
-ich, conforming to the applied method of determining ten-
sion values, receives a tension value of one degree, thus
preventing a complete return to total conisonance, a practice
used so consistently as to be considered a characteristic.
102
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112
Fuga Septima in -Flat
This fugue has the same appro imate harmonic t nsiIn
average as Fug> Auinta, that of 1.lu .hiC is less t'han the
av-ra: of all fugues. It is also 1i:e Ue fifth rugue in
that it has a tension high point of it de ee - iC is
ataine only one time. The usual b ininy, und nding bith
corleto consonance is in evi dnce in 2u. eptima. The
duration of tension values is slightly loss 1nIOe than in
-reViOs fugues.
Thus it may be said that their is pe-hapsat le ast a slight increase in the fre quency of tension change .
1otJDl, o1ev-r, is the lach of any persistently high value:,
althouji near tie beginning and the end thir is some evi dence
of a prolonged moderate tension; i.e., Igures 9, 94, 97,
anJ 98. There is a frequent return to complete consonance
and the duration of consonance is quite variable.
Figure 100 shoTs the curve of harmonic tension averages
of all the measures. This fugue conforms to the general
curve seen in ecrl-ir examples. There is noticed the typical
beginning >.-ith complete consonance, the climatic rise of ten-
sion, and gradual decline to total consonance.
113
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4
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127
Fuga Octuva in D
Typical in its beginning and ending ith complete con-
sonance, this fugue is quite representative of the characteris-
tically frequent contrast of consonance and dissonance. This
high frequency of change, both in tension values (the contrast
of various degrees of dissonance) and in contrast between dis-
sonance and consonance, is the most notable point of interest.
Consequently the duration of tension values is quite small,
and although the duration of consonances is slightly more, it
too is relatively small. Of notable contrast is the amount
of change in both values of tension and in the return to
complete consonance. Therefore, the general picture repre-
sented by the graphs listed as Figures 101-108 is one of
freuent shifts from low to high, the extent of these in-
creases and decreases being notably greater than usual. This
is particularly interesting when noting that the tension high
'oint is only Live degrees. This point of five degrees, how-
ever, is reached eighteen times, thus adding proof to the
fact of a greater frequency of change.
The usual curve from consonance to greatest tension and
return is evidenced by the graph of measure averages, Figure
109.
128
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Fuga iiona in B-Flat
Like Fuga Octava, this fugue is notable because of its
highi frequency of chu.nge. Lore specifically, it deals ith
a ChLnge to complete consonance Tuore frequently than the oe-
vious fugue. The tension high point is also five degrees in
this fugue, this being reached, as in uga ctavs, eighteen
tiaes. :Tiereas tile preceding fugue had a high frequency of
change in tension vlues, ti C rugue has or. ohanes from
dissonance to complete consonance. Also of interest is the
prolonged use of consonance, best exemplifiea in figures lb,
118, l22, and 123. The amount of change is vriubl.
The broad picture, represented by F7igure 127, showVs the
typical curve, Uith the high point o: tension average coining
abouI to-thirds through the fugue. There is a return to
consonance, although the final chord, Igure 120, oeing in
second inversion, has a tensions vulue of one degree.
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156
Fuga De cima in D)-Flat
Probably the most notable fact about this fugue isthat it is the least dissonant of all the fugues. egistering
a total tension average of sligh-tly more than .83, Fuga Decimaemploys not only fewer high tension values, but a persistentuse of complete consonance, the duration of which is quitenotable. Because of this fact the frequency of change fromconsonance to dissonance and shifts from various degrees ofdissonance is comparatively small. The tension high pointof eight degrees is reached only one time, in measure twoof Figure 133. The extent of the changes in various degrees
o0 tension and the changes from any tension value to c ompleteconsonance is variable.
Figure 140 shows a curve in tension averages, thesebeing frequently interrupted by drops in degrees of tension,or comiDlete returns to consonance. The high point of tension,according to measure averages, is about half-way through thefugue, this being followed by a notable drop to & prolongeduse of consonance.
157
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170
Fuca Undecima in B
This, the shortest of the fugues, is actually written
in the form of a canon. although it begins with consonance
on the first note, the second count of the first measure,
Figure 141, registers a tension value of three degrees.
Like 2uga Decima, its single high point o- tension is eight
degrees, also reached only one time. however, the total
harmonic tension value is more than the tenth fugue although
it is below the average of all twelve. There is a rather
constant shift in degrees of tension and in the change from
dissonance to consonance. although the frequency of change
is not great, it is rather steady. The extent of the change
is variable.
Notale in the broad picture of the measure avecages of
harmonic tension, as seen on graph in Figure 147, is the
triple curve. Cftr the first rise there is an Litediate
decline to complete consonance, a sharp increase to the ten-
sion high point of averag6es, another decline and rise before
the final drop vuhich is rather abrupt. Although not registering
total consonance in the first measure, there is here the now
characteristic return to complete consonance for hIe final
measure.
17l
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178
Fuga Duodecima in F-Sharp
This is the most dissonant of all the fugues. Regis-
tering a tension value of slightly more than 1.77, Fuga
Duodecima is at least .3 above the averagE of all the fugues.
The single high point of tension is eight degrees and ore
thLan any of the other fugues, it is reuclied seven times.
Although there is in some places, such as measure one of
Figure 140, measure three of Figure 152, and measure three
of Figure 153, a prolonged use of consonance, there is also
a prolonged use of dissonance, in evidence by the duration
of tension values throughout the fugue. Thire is a moderate
frequency of chuxnge, both in degrees of dissonance and in
consonance and dissonance. The extent of these changes is
slightly higher than the average, this being explaine& by
the more frequent use of higher tension values.
TIe contour of measure averages of tension, seen on
graph in Figure 161, shoxvs a prominent increase from complete
consonance to the highest point of tension average, a return
to consonance, another sharp incline, then a grad ual decline
to the typical consonant ending, now so characteristic in
these fugues.
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ummary
In the foregoing analyses the writer ha tried to show
accurately and specifically the presence and absence, fre-
quency and extent of use of dissonance, one of the prominent
aspects, not only of the Ludus Tonlis and other works of
Paul 1findemith, but of the modern period. By the application
of an objective method for io asuring harmonic tension the
writer has considered every tonal combination for its deter-
mined amount of absolute harmonic tension. In this investi-
getion there has been no regard for the tonal relationships
of diaLferent combinations, a factor in the aplied method
termed as "relative" tension. ach individual interval or
t onal combination as assi ae d a nu ri3c-al value of from one
to eight degrees, de ending u on the amount of absolute har-
monic tension, and this valuevas registered on a graph by a
vertical line. The horizontal line registered the duration
of the tension or the consonance, the latter being indicated
on the lowest line of the graph and labeled "zero" degrees.
These graphs accompany every line of each fugue and show the
exact amount of tension and duration of the tension, or cona-
plete lack of tension, for every tone or tonal combination.
193
194
Preceding each series of graphs for every fugue is a brief
report, a description of the analysis pointing out any
notable feature or characteristic in the use or lack of use
of dissonance. These deal specifically with the presence and
absence, a.ount of increase and decrease of absolute tension,
and are intended to bring to the reader's attention certain
facts he might otherwise miss merely on observing the graphs.
The graphs which follow each series of graphs showing the
tension value of each fugue are intended to show the total
averages of absolute harmonic tension in each measure of the
fugue. These averages, being plotted on these graphs and
connected by straight lines, show the continual rise and fall
of tension, thus permitting a broader picture of the tension
relationships in each fugue. Figure 161 is a graph showing
the total averages of absolute harmonic tension In all twelve
fugues. The contour of this line shows the tension relation-
ship of each fugue to the other.
Preceding the analyses is a literary presentation of the
subject and its significance. The advantages of Laking such
a study are pointed out, whether considered from a broad
point of view or from a more technical standpoint. The need
for additional information on the technical aspects of music
of the present period is in evidence; therefore, it has been
the hope of the writer, not only to locate and present these
aspects (in the consideration of only one factor which is
195
characteristic of the period--that of dissonance), but also
to correlate these features with the composer and in turn
place him into the broad picture of the modern period.
Conclusion
From the findings of this study, limited to the extent
of considering only one factor of harmonic tension, that of
the absolute tension value of each tonal combination, it is
perhaps of greatest interest to note that the total amount
of tension used in this respect is comparatively small. The
total tension average barely exceeds 1.39. -hen considering
that the music "sounds" dissonant and of course is, this fact
is of notable interest. although there is evidence of con-
siderable dissonance in some tonal combinations, for the most
part each tonal combination contains only a relatively small
amount of dissonance. Of course, the limited use of three
voices ex:cludes the possibilities of a tension value much
greater than eight degrees, the highest found in the fugues.
These dissonant combinations are frequently approached and
left by complete consonant combinations. This fact lessens
the general average value of tension, but probably does not
decrease the tension value when considering the relation of
one tonal combination to the other. Therefore, these fugues
are dissonant, partially because of the single dissonance
value of each tonal combination, but more because of the
relationship of one tone or tonal combination to another.
196
Thus it is that linear dissonance and not vertical dissonance
is of most prominence. Linear dissonance is a characteristic for
which Paul Hindemith has for several years been noted.
The frequency of tension change, the numLer of times one
dissonant tonal combination changes to another tonal combina-
tion of greater or lesser dissonance, and the frequency of
change from dissonance to consonance is quite variable. There
are probably more shifts in degrees of tension, or contrasts
in various degrees of dissonance than in the contrasts of
dissonance and consonance. Both the contrasts in various
degr es of dissonance and in dissonance and consonance are
fairly equalized, and their frequency changes have definite
bearing on the total feeling of dissonance. The greater the
contrast the greater the tension ill seem. The extent of
these changes is of course ,uite variable, ranging in changes
from one to eight degrees. The change of about three degrees
is most prominent, although more abrupt changes are notable.
The duration of consonance slightly exceeds that of dissonance.
Although more dissonance is found than consonance, this is
true because of the values of these dissonant combinations and
not necessarily because of their prolon:_ed duration. Whereas,
in the case of consonance, although at tises of short dura-
tion, the duration is longer in comparison.
Iinother feature may be considered characteristic, not
only of the composer in the fugues of the Ludus Tonalis, but
197
other works as well. This is the establishment of a key,
complete departure from that key, and a final return to
tonic. In reference to tonality Hindemith began each fugue
in an intended key, starting with complete consonance, then
through the many contrapuntal devices at his cCoMmand departed
to a great extent from the original key feeling, but always
he brought the fugue to a close in the key or related key,
and always to complete or near-complete consonance. In the
sections where there is the furthest departure from the
originally intended key feeling are found the greatest over-
all tension values. These ranges in tension values, or in-
creases and decreases of dissonance, are usually seen to
form a curve. (See Figures 13, 29, 41, 1, 7', 66, 100,
109, 127, 140, 147, and 160). The high point of the curve
comes somewhere neur the middle of the fugue, so that the
fugues usually begin with a rise in total tension, reach a
climax, and return to consonance.
In these contrapuntal v;orks, then, the following con-
clusions may be drux;n:
1. The fugues are harmonically dissonant as a result
of linear dissonance, although the variable absolute or
vertical dissonance values of tonal combinations add to the
total feeling of dissonance.
2. The tonal combinations are essentially dissonant,
but not to great extent.
3. The frequent and extensive use of consonance serves
198
as a contrastin6 element which in itself creates a feeling
of dissonance.
4. The contrasts in various degrees of dissonance are
notable,
5. Te independence of the voice lines, and not a pre-
conceived harmonic arrangement of tonal conibinations, is
more largely responsible for the dissonance, a technical
aspect not only of this work and other works by "aul
iindemith, but of nearly all composers of the present period.
199
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BIBLIOGRk HY
Books
Apel, "illi, Harvard Dictionary of Music, Cambridge, Lass.,Harvard University Printing Ofrice, 1945.
Eschman, Karl, Chanin Forms in Modern Music, Boston, Aass.,E. C. chirmer Music Co., 1945.
Eien, David, The Book of Modern Composers, New York, AlfredK Knoph, 1942.
Ewen, David, Composers of Todoy, New Yor, 11. -. "ison Co.,1936.
Finney, Theodore y., oHistory of Music, New York, Harcodrt,Brace & Co., 1935.
Fraser, ndrew, ssas on E0i ,, rU.L~res, 190. gic, LondonI Oxford 'University.Press,.193.
Ilindemith, -aul, Craft o Musical Compoosition, Book I,Theoretical c-frt, Ne YorkT-,Tssociated Music ublisiers,Inc., London, 3. 3chott & Co., 1937.
Hindemith, Pau, Llementary TaiinnforMusicians, ew York,associatedd Musiculishers, Inc.1946.
Iiindeith, Paul, Traditional Earmony, New York, associatedMusic x1ublishers, Inc., 1944.
Miller, Hugh M., n Outline-History of Music, New York,Barnes & Noble, Inc., 1947.
Pannain, Guido, Modern Composers, New York, E. . Dutton &Co., Inc., 93
Articles
Blitzstein, Marc, "1in und Zuruck in Philadelphia," ModernMusic, V (May, 1928), 34.36.
Llitzstein, Marc, "Tame Season in New York," Modern Music,IX ( January, 1932), 79.-85.
2UU
201
3litzstein, Marc, premieress and Experinents," Modern usic,IM (March, 1932), 121-127.
Blom,, Iric, "The Twentieth Slump," Modern Music, A (larch,1933), 123-129.
Carter, Elliott, "Sprint season of Mindenith and iZericans,"modern Music, LVI (Lay, 1939), 249-254.
Chanler, Theodore, "New York, 1934," Modern Music, -I (March,1934), 142-147.
Citkovitz, Israel, "Spring Finale in e York," Modern Music,X (May, 1933), 197-199.
Citko-itz, Israel, "y-r:phonic and Chber music, " modernMusic , zI (May, 1935)), 187-192.
Oopland, -aron, "4ljyinJ 2afe at Lurich, 4Modern usic, IV(November, 1926), 28-31.
Copland, zaaron, "Laden-aden, 1927)," Modern Music, V (Lovem-ber, 1927), 31-34.
Copland, Aaron, "Music Lince 1920," Modern Music, V (March,1928), 16-20.
Copla d, .aron, "Contemporaries at Oxford, 1931," Modernmusic, iZ (November, 1931), 17-23.
Copland, aron, "Stravinsky and inderaith," Modern Music,IX (January, 1932), 85-88.
Einstein, Afred, "9au1 Mindemith," Modern Music, III (M'arch,1926), 21-26.
Felber, irwin, "Step Children of Music," Modern Music, IV(Nove, ber, 1926), 31-33.
Frankenstein, 1fred V., "lindemrith and Bach at Chicugo,1Mo dern Music, VIII (Novem ber, 1930), 35-37.
Gutman, Hans, "Young Germany, 1930," Modern Music, VII(February, 1930), 3-10.
Heiden, Ber hard, "H indemith' s System, New Lpproach,"Modern Music, XIX (January, 1942), 102-107.
202
Hindemith, PauL, "Method of iusic Theory," Musical uartrly,.,= (January-October, 1944), 20-28.
Jacobi, Frederick, "Liege, 1930," Modern Music, VIII(November, 1930), 10-20.
1endel, irthur, "First Fruits of the Season," Modern Music,VI (January, 1929), 30-32.
Muser, Frani 3., "A Recent lork of faul Hindemith,"" MusicalQ(arterly,X (January-October, 1944), 29-36.
Reich, Jilli, "Paul Hindemith," Musical uarterly, NXII(January-October, 1931), 486-496.
Rosenwald, Hans, "Speaking of usic," Music News, IXXV(January-December, 1943), 10-11.
Saminsky, Lazare, "Jurg enatch," Modern Music , VII(December, 1929), 34-39.
Sanborn, Pitts, "The 1925-1926 Season," Modern Music, III(May, 1926), 3-9.
Schillenger, Joseph, "Electricity, "i Musical Liberator,"Modern Music, VIII (March, 1931), 26-31.
Schlee, Aifred, "The Germans in Paris," Modern MusiC, A
(November, 1932), 42-46.
Sessions, Roger, "Hindemith's Mathis der Maler," ModernLUusic, .II (November, 1934), 13-17.
Sessions, Roger, (Recent 3ooks) "hindemith on Theory,"Modern Music, XV (November, 1937), 57-63.
Smith, Cecil M., "Hindemith and the Coolidge Festival,"Modern Music, "IV (May, 1937), 208-215.
Smith, Cecil M., "Breaking Trails in Chic, go," Modern Music,XXI (June, 1944), 107-108.
Smith, Cecil 1,., "Composers to Chicago," Modern Music, DXI(June, 1944), 243-246.
Straus, Henrietta, "The Fruits of Victory," Modern Music, V(March, 1928), 26-28.
Stuckenschmidt, K. K., "Hindemith Today," Modern Music, XIV(January, 1937), 67-75.
203
le isnann, Adolph, rUnebp ected German Developments , " oerniusic, III (March, 1926), 36-38.
feiimani, .olph, "G er.-an 's Latest apsras Iodern Lusic,(<say, 1927), 25-26.
Reports
huiller, THu J., An Objective method f= the Study ofHarmonic Teni, ev York, c. Kirer, Inc.,~~946.
