Dept. for Speech, Music and Hearing
Quarterly Progress and
Status Report
Statistical computer
measurements of the
tone-scale in played music
Fransson, F. and Sundberg, J. and Tjernlund,
P.
journal:
volume:
number:
year:
pages:
STL-QPSR
11
2-3
1970
041-045
http://www.speech.kth.se/qpsr
STL-QPSR 2-3/1970
IV.
A.
MUSICAL ACOUSTICS
STATISTICAL COMPUTER MEASUREMENTS
O F THE TONE-SCALE IN PLAYED IvIUSIC
F. F r a n s s o n , J . Sundberg, and P I Tjernlund
An e s s e n t i a l a s p e c t of played m u s i c i s the tone s c a l e , i. e. the s e r i e s of
fundamental f r e q u e n c i e s u s e d in playing.
In a n e a r l i e r r e p o r t a method f o r
m e a s u r i n g t h i s s e t of fundamentals was d e s c r i b e d ( '). The method o p e r a t e s
with a h a r d w a r e fundamental frequency m e a s u r i n g equipment connected to a
computer.
The computer m e a s u r e s the p e r i o d t i m e and p r o c e s s e s the data
statistically,
The r e s u l t s a r e p r e s e n t e d in the f o r m of fundamental f r e -
quency his tog ran is^
In t h i s p a p e r a study will be r e p o r t e d of the tone s c a l e
u s e d by p r o f e s s i o n a l m u s i c i a n s playing melodies on different instruments.
Problem
Some y e a r s ago, one of the author d i s c o v e r e d that the s e r i e s of fundam e n t a l f r e q u e n c i e s blown on a flute i s strongly dependent on the eventual
p r e s e n c e of a tonal interpretation of the notated m u s i c .
Thus, if a flutist
plays a f t e r notes, he adopts one s c a l e , and if he plays a f t e r s u c c e s s i v e dictation of single notes, he adopts a different s c a l e , that is r a t h e r closely r e l a t e d
to the r e s o n a n c e f r e q u e n c i e s of the flute(2), s e e P i g . IV-A- 1.
s i m i l a r r e s u l t s have been r e p o r t e d ( 3 ) .
L a t e r on,
This fact r a i s e s the question, which
a r e the f a c t o r s that d e t e r m i n e the s c a l e of a played m e l o d y ? How do the
s c a l e s of different m u s i c i a n s playing different m e l o d i e s and i n s t r u m e n t s
compare ?
Material
T h r e e p r o f e s s i o n a l f i r s t r a t e m e m b e r s of a symphony o r c h e s t r a w e r e
u s e d a s subjects: a flutist, a n oboist, and a violinist.
They a l l played the
s a m e melody on t h e i r own i n s t r u m e n t s and, a s f a r a s the wood-wind ins t r u m e n t a l i s t s w e r e concerned, a l s o on a n i n s t r u m e n t that was unknown to
them.
The melody was replayed and t a p e - r e c o r d e d s e v e r a l t i m e s , and the
subjects s e l e c t e d t h r e e v e r s i o n s f o r analysis.
b a r s f r o m a solo cadence of a flute concerto.
octaves.
The melody was a couple of
It h a s a n a m b i t u s of 2 1/2
Finally, the subjects a l s o improvized on t h e i r own i n s t r u m e n t s o r
played another melody that they choosed t h e m s e l v e s .
STL-QPSR 2-3/1970
Analysis
The tape - r e c o r d i n g s of the s e melodie s w e r e analyzed with r e s p e c t to
the fundamental frequency distribution.
E x p e r i m e n t s showed that the i n t e r -
v a l s between the s c a l e tones can be d e t e r m i n e d with a n a c c u r a c y of f 5 cents,
w h e r e a s the absolute frequency values of the toncs could be m e a s u r e d only
within
+
17 c e n t s ( 1 $).
Due to the l a r g e a m b i t u s of the melodies and the
s p e c t r u m p r o p e r t i e s of thc toncs, the cutoff frequency of the variable lowp a s s f i l t e r of the fun2amental frequency m c a s u r i n g equipment had to be
v a r i e d synchronously with the fundamental frequency of the melodies.
Results
In F i g . IV-A-2, a s in the following f i g u r e s , the m e a s u r e d f u n d x n e n t z l
f r e q u e n c i e s a r e c o m p a r e d with those of the equal t e m p e r e d s c a l e and the
deviations a r e e x p r e s s e d in the ccnt-unit.
Moreover, in a l l c a s e s the devia-
tions w e r e balanced with r e s p e c t to t h i s scale.
This m e a n s that the influence
on the r e s u l t s of a n o v e r a l l mistuning of the i n s t r u m e n t a s well a s of f a u l t s
in the absolute frequency measurements was eliminated.
Fig. IV-A-2 shows, in the d e s c r i b e d way, the s c a l e of the flute under
v a r i o u s conditions.
It i s s e e n that the scale i s not v e r y dependent neither
of the melody, nor of the instrument: a l l c u r v e s l i e r a t h e r close together,
and the distance between them i s in m o s t c a s e s s m a l l e r than some 15 cents.
Fig. IV-A-3 shows the corresponding c u r v e s f o r the oboist.
In this c a s e ,
the d i s t a n c e s between the m e a s u r i n g points f o r a given tone ?-re s t i l l s m a l l e r
than f o r the flute.
Thus we m a y conclude, that a l s o in the c a s e of the oboe,
the melody and the type of i n s t r u m e n t do not contribute v e r y m u c h to the
determination of the scale.
Fig. IV-A-4 shows the s c a l e s u s e d by the violinist.
t h i s c a s e the played s c a l e i s r a t h e r stable.
Note that a l s o in
This is astonishing in view of
the f a c t that the violin i m p o s e s no r e s t r i c t i o n s a t a l l on the p l a y e r a s r e g a r d s
the choice of fundamental frcquencies.
The r n c a s u r e m e n t s on the melody
choosen by the violinist failed to give a complete s e r i e s of fundamentals.
Howevcr, on thc toncs that the m e l o d i e s have in common the frequency
valucs l i e quitc close to e a c h other.
CENT
40
.
-
f?
I \
I
-
Fig. IV-A-2.
.
Deviations f r o m the equal tempered scale in flute playing
m e a n values obtained f r o m t h r e e recordings of a solo cadence played
on the m u s i c i a n ' s own flute.
m e a n values obtained f r o m t h r e e recordings of the s a m e solo
cadence played on a flute unknown to the flutist.
V values obtained when the flutist improvized on h i s own flute.
I
\
\
/
P
CENT
40
Fig. IV-A-3.
Deviations from the equal t e m p e r e d scale in oboe playing.
'
m e a n values obtained r r o m t h r e e versions of a solo cadence played
on the m u s i c i a n ' s own flute.
m e a n values obtained f r o m t h r e e versions of the s a m e cadence
O played on an oboe unknown to the player.
v
values obtained when the oboist improvized on h i s own instrument.
CENT
40
F i g . IV-A-4.
Deviations f r o m t h e equal t e m p e r e d s c a l e in violin playing.
m e a n v a l u e s obtained f r o m t h r e e r e c o r d i n g s of a solo cadence.
0
v a l u e s obtained when the violinist played the melody "Schon Rosemarin".
44.
STL-QPSR 2-3/1970
contexts in the melody.
Such a n influence might p e r h a p s be quite s t r o n g and
i t would a p p e a r m a i n l y a s a broadening of the p e a k s in the fundamcntal f r e quency histogram.
The substantial differences in the values of the tone F4
f o r the violin do indeed point towards such a dcpendencc.
However, t h i s
question f a l l s outsidc thc scope of the p r e s e n t investigation.
Our m c a s u r c m e n t s have shown that the m c a n s c a l e s of a soloplaying
flutist, oboist, and violinist a r e quite s i m i l a r .
equal t e m p e r e d s c a l e , but a s t r e t c h e d scale.
They a l l employ not the
What a r e the possible r e a s o n s
f o r this s t r e t c h ?
F r o m the a c o u s t i c s of the wood-wind i n s t r u m e n t s we know that thc r e s -
.
onance f r c q u c n c i c s of those i n s t r u m e n t s m a k c a s t r e t c h e d s c a l e expectable( 8 )
On the other hand, we know that the p l a y c r cen play according to the equal
t e m p e r e d s c a l e , if he wants, by adjusting h i s blowing technique(9).
It i s
noteworthy that a l s o thc violinist in o u r e x p e r i m e n t s s e e m s to p r e f e r a
slightly s t r e t c h e d scale.
This indicates that the m u s i c a l e a r f o r some un-
known r e a s o n demands a s t r e t c h e d scale.
Another i n t e r e s t i n g f a c t i s that a l s o the piano i s tuned in a s t r e t c h c d
s c a l e ( 10) It h a s been shown that this probably i s a n a c o u s t i c a l consequence
.
This
of the inharmonicity of the p a r t i a l s of the piano tone spectrum(").
(12)
s t r e t c h belongs to the c r i t e r i a of a good piano
Thus, a l s o in the c a s e
.
of the piano, the m u s i c a l e a r s e e m s to demand a s t r e t c h c d scale.
T h e r e i s a difference bctwecn the octave p e r c e i v e d a s p u r e and the
octave corresponding to the frequency r a t i o i:2(I3).
a bigger i n t e r v a l than the m a t h e m a t i c a l octave.
p e r i m e n t s with sine waves.
Tho subjective octave i s
This h a s been shown by ex-
However, in view of the r e s u l t s p r c s e n t e d h e r c ,
i t s e e m s likely that a l s o when complex tones a r c concerned, the m u s i c a l
e a r wants s t r e t c h e d octaves and s t r c t c h c d i n t e r v a l s of o t h e r s i z e s a s well.
P r e s e n t l y cxpe rilnents a r e c a r r i e d out in o r d e r to shed s o m e light o v e r this
problem.
Refercnce s:
(1) Sundberg, J. and Tjernlund, P. : "Computer M e a s u r e m e n t s of the
Tone Scale in P e r f o r m e d IvIusic by Means of Frcquency
Histograms", STL-QPSR 2-3/1969,
p. 33.
(2) F r a n s s o n , F. : " h ~ e a s u r c m e n t sin F l u t e s f r o m Different P e r i o d s " ,
STL-QPSR 4/1963, p. 12.
r e f s . continue on next page
CENT
40
Fig. IV-A-5.
Mean deviations f r o m the equal t e m p e r e d s c a l e in
0
flute playing.
V oboe playing.
violin playing.
CENT
40
Fig. IV-A-6.
Mean deviations f r o m the equal t e m p e r e d s c a l e i n flute-, oboe-, and violin playing. The solid line
shows a model s c a l e consisting of p u r e tone s t e p s ( 2 0 4 c e n t s ) and s m a l l s e m i tone s t e p s ( 9 7 . 5 cents).
STL-QPSR 2-3/1970
Coltman, J. 7 7 . : "Acoustics of the F l u t e t t , P h y s i c s Today 2 1: 1I
(1968), p. 27.
Sundberg, J. : "The ' Scale' of Musical I n s t r u m e n t s t t , Svensk Tids k r i f t f o r Musikforskning 49
-(1967), p. 126.
Greene, P. C. : "Violin Intonation", J. Acoust. Soc. Am.
9 (1937),
p. 43.
Nickerson, J. F. : "Intonation of Solo and E n s e m b l e P e r f o r m a n c e of
the Same Melody", J.Acoust. Soc.Am. 2f (1949), p. 593.
L o t t e r m o s e r , TI. and Meyer, Fr. -J. : "Frequcnzmessungen a n gcsungcnen Alclcordentt, Acustica 10 (1960)) p. 18 1.
Bcnade, A. H. : "On the Mathematical Theory of Tloodwind F i n g c r
Holes", J.Acoust.Soc.Am. 32:12 ( 1 9 6 0 ) ~p. 1591.
Meycr , J. : Akustik d e r Holzblasinstrumente in E i n z e l d a r stellungen
( ~ r a n k f u r a.
t M., 1966).
Schuclc, 0. H. and Young, R. T I . : "Observation on thc Vibrations of
P i a n o Strings", J. licoust. Soc.Am. 15:1 (1943), p. 1.
Young, R. T I . : "Inharmonicity of P l a i n Wire P i a n o Strings", J.
Acoust.Soc.Am. 24:3 (1952), p. 267.
Martin, D. T I . and V a r d , TT. D. : "Subjective Evaluation of Musical
33:5
Scale T e m p e r a m e n t in Pianos", J. Acoust. Soc. Am. (1961), p. 5G2.
VJard, 17. D. : Itsubjective Musical Pitch", J. Acoust. Soc. Am. 26:3
(1954), p. 369.