Dept. for Speech, Music and Hearing
Quarterly Progress and
Status Report
Fundamental frequency
adjustment in barbershop
singing
Hagerman, B. and Sundberg, J.
journal:
volume:
number:
year:
pages:
STL-QPSR
21
1
1980
028-042
http://www.speech.kth.se/qpsr
I n both q u a r t e t s the e x e r c i s e s consisted of chord -sequences,
i .e. cadences, which are sunq on a CV-syl lablt-.
I-icwvc?r, thc c-a-
dences d i f f e r e d regarding L o t h chords and s y l l ~ ~ b 1 c . s .Quartet A
r e t a i n e d b a s i c a l l y the same chord frequencies '11.1 the t h e while
quartet B r a i s e d t h e frequencies by a half tone s t e p b e b e e n each
rendering.
Quartet A repeatedly c h a n g a the s y l l a b l e while quartet
B f i r s t used [ma] i n 1 1 versions of their cadence and then [m 1 i n
7 versions.
The cadences are given i n Ekample 1.
QUARTET B
QUARTET A
Example 1.
Cadences sung by t h e two q u a r t e t s . The symbols below
t h e chords i n d i c a t e their harmonic function.
The recordings *re
made i n an ordinary room.
Each of the
singers had a contact microphone (acceleramcter) glued t o the skin
of the trachea s m c e n t i r w t e r s below t h e thyroid c a r t i l a g e .
T k
s i g n a l s picked up by these four contact micro~~horles
wre recortlcd
on a four t r a c k tape recorder.
This recording was used f o r meas-
uring fundarrental fretjucncy i n the s u b s c p n t a n a l y s i s .
'1%
signal
fram an ordinary microphont? was recorcl~don a second tape recmrder.
A fundamental frequency measurement c q u i p n t described else-
where was used (see Askenfelt, 1979)
.
Basically it measures the
p e r i o d i c i t y by means of a hardware fundamental frequency d e t e c t o r
which a p p l i e s a double peak picking s t r a t e g y .
The r c s u l t l n g spare-
wave s i g n a l is fed t o t h e computer which cc>nwrts it i n t o frequency
data. P l o t s of the fundamntal frequency versus tirrre as *l.l a s
fundamental frequency histograms may be obtained. When the operat o r has marked the nude frequency of each histogram peak by means
iI
PROBABILITY ( '10, ARBITRARY SCALE)
INTERVAL MEAN IN [ma]
~ i g .II-A-2.
- CHORDS
(CENT)
intaw11 mragas of 11 m m i i e r d
of chords using the syllable [ma] and 7 renderings of
the same chords using the syllable [m3
The chords
*re selected fran the cadence sung by quartet B.
~orrelaticnbe-
.
STANDARD DEVIATION (CENT)
averages and standard deviations of these minor seconds are l i s t e d
in Table 11-A-111.
The average is s l i g h t l y greater than the Pytha-
gorean value of 90 cents and much n a r r w r than the 112 cents of
just intonation.
cents.
The standard deviations a u n t t o 5.7 t o 7.7
This is not much greater than the smallest standard deviaThus, these singers seem
tions observed for h m n i c intervals.
t o arrive a t a very high degree of accuracy even i n melodic intervals, i . e . when the standard is merely stored a s an internal reference.
Table 11-A-111.
Interval average and standard deviations ( i n
cent) f o r lead singing the tones constituting
the interval i n succession.
Quartet
Chords
Mean
SD
From the above it seems reasonable t o canclude t h a t the interv a l averages represent reliable information and t h a t even small d i f ferences betwen interval averages may be significant.
I t is then
interesting t o compare the interval averages with interval s i z e s
prescribed by the Pythagorean and the just scale.
As
f a r as the
seventh is concerned, a t h i r d version is relevant, namely the int e r v a l betwen the fourth and the seventh p a r t i a l of a d n i c
spectrum.
Table 11-A-IV
lists a l l interval averages i n the material
including standard deviations, confidence intervals, and deviations
from just and Pythagorean intonation.
can be made.
Many interesting observations
The just and Pythagorean values, which agree f o r the
three simplest intervals (octave, f i f t h , and f o u r t h ) , f a l l within
the confidence intervals of the corresponding averages for these
simplest intervals with one exception.
Eventhough mst of these
intervals wre sung s a w cents n a r r m r than the ideal according t o
Pythagorean and just intonation, it seems justified t o assume that',
these intervals wre sung in accordance with these theoretical values.
A l l major t h i r d s show confidence intervals excluding the just version
and, with one single exception, a l s o the Pythagorean.
38.
STL-QPSR 1/1980
The chords in the material analyzed can be divided i n t o three
types :
(1)
tonic chords
(2)
major t r i a d s including seventh
(3)
minor t r i a d s w i t h o r without seventh
Disregarding f o r the m m n t the f a c t that the lead seems t o serve
a s the c m n reference f o r the singers, w can ccanpute the intervals
r e l a t i v e t o the root of each chord.
In t h a t way, we may compare in-
tonation within and between each of the three types of chords mentioned.
This has been done in Table 11-A-V
.
The agreemnt k t w n the tvm
quartets is amazingly high in the case of the tonic chord.
The octave
is pure, the f i f t h is 4 o r 5 cents wider than pure and the t h i r d is
4 and 5 cents narrower than Pythagorean.
This means t h a t the t h i r d of
the major t o n i c is 18 and 17 cents wider than a pure third!
I t seems
t h a t the pure t h i r d is replaced by the qrthagorean t h i r d minus a small
correction i n the tonic chord.
With respect to the major t r i a d s with
a seventh, there is a very high degree of agre-t
third.
in the case of the
Interestingly it is tuned t o a value which is 5 t o 11 cents
wider than the value of just intonation.
The f i f t h shows less con-
sistency but is smaller than the f i f t h of the tonic chords i n a l l cases.
The seventh of the subdominant seventh chord has a confidence interval
excluding a l l other versions of seventh.
This suggests that a seventh
is performed narrowr when the chord has the function of a dminant
than when it has another hanmnic function. The seventh of the daninantal t r i a d s exceeds the interval betwen p a r t i a l s 4 and 7 of a harmonic spectrum by 8 to 16 cents, and a l l the confidence intervals
exclude t h i s hanmnic interval. The two minor chords are dissimilar
with respect to the c m n intervals: both confidence intervals exclude the average of the other. Howver, the major t h i r d between
the t h i r d and the f i f t h of these t r i a d s agree within one cent: 394
and 393 cents. This is very close to the values of the major t h i r d s
of the major t r i a d s with seventh. W e may speculate that the major
I
t h i r d has a s o r t of key function in Barbershop intonation.
Discussion
!
There are t m observations made above that require saw comrrents.
One is the magnitudes of the standard deviations, and the other is the
puzzling discrepancies betwen a l l k n m interval s i z e s and those perf o m d by our Barbershop singers.
T a b l e 11-A-V.
Intervals (in c e n t ) relative t o the root of the chords. Symbols as in Table 11-A-I.
A
T
(luartet
Chord
Wan
.
Conf int
.
Mean
. .
C o n f int
Mean
. .
Conf int
Minor t r i a d
Major triad w i t h seventh
Major triad
B
A
B
B
T
D7
D7
Mod
A
B
T7
S7
A
Sp7
A
Tp
STL-QPSR 1/1980
40.
W e found that the standard deviation of the lead, in performing
a melodic interval of a minor second, was 5.7 t o 7.7 cents. The
three smallest standard deviations f o r chord intervals *re found t o
be between 4.3, 5.1, and 5.2 cents ( c f . Table 11-A-I).
These num-
bers can be canpared with the difference l i m n f o r frequency.
If
musically m11 trained subjects repeatedly adjust the frequency of
a response tone to pitch agr-nt
1
1
with a preceding standard tone,
the difference l i m n may be a s l o w a s 6 cents, which incidentally is
very close t o the standard deviations f o r the m l o d i c minor seconds.
When musically trained subjects adjusted tm synthesized vibrato
v o w l s t o different intervals, and thus had no beats t o use a s a
c r i t e r i o n , the average intervals, and the average standard deviations shown in Table 11-A-VI
wrk)
.
were observed (Agren, unpublished t h e s i s
With the major second a s a possible exception the standard
deviation averages a l l f a l l within the standard deviations observed
f o r the melodic version of the minor second a s performed by the
leads ( c f . Table 11-A-111)
.
Thus, the lmst standard deviations i n
our Barbershop quartets are of the sarne order of magnitude a s those
obtained i n psychoacoustic pitch matching and interval matching exp e r i m n t s with musically w e l l trained subjects.
F r m t h i s # conclude t h a t the fundamental frequency control can be trained t o a
very high degree of s k i l l in singers.
The accuracy with which a
subject adjusts the pitch of a synthetic tone by tmmmg a knob is
about the s m a s the accuracy with which a singer can reproduce an
interval.
Table 11-A-VI.
Averaged intervals and standard deviation (in cent)
from 8 subjects showing the lowst standard deviat i o n of a grmp of 17 musically t r ~ i n e dsubjects who matched prescribed intervals betmen tm
simultaneously sounding synthetic sung vibrato
-1s.
(According to Agren, unpublished t h e s i s
mrk )
.
l 'I
SD
Interval
Octave
Fifth
Major t h i r d
Major second
Average
1203
705
400
20 3
&an
Min
Max
6.2
5.6
5.7
8.7
3.6
4 .O
4.4
5.6
8.8
6.8
9.4
13.7
1
I
I
Conclusions
The accuracy with which the fundamental frequencies are chosen
in Barbershop singing is extremely high and does not seem t o depend
on the v-1
t o any great extent. The lead serves a s the reference
t o which the other singers adjust their fundamental frequencies so
a s t o produce the desired chords.
shop theorists.
This is in a g r e m n t with Barber-
The nmber of camon p a r t i a l s (within a given fre-
quency range) between t~ tones constituting an interval tends t o aff e c t the d i f f i c u l t y with which the intervals is tuned.
Thus, it
seems e a s i e r t o tune simple intervals, which share many p a r t i a l s than
t o tune intervals with few c m n p a r t i a l s .
Most intervals in Barbershop singing deviate systematically from
the corresponding values according t o just and Fythagorean intonation.
The major t h i r d of the major t r i a d having the harmonic function of a
dominant may be interpreted a s a stretched version of a pure third,
while i n the tonic chord it can be regarded a s a flattened version
of a Pythagorean third.
The major t h i r d contained in a minor t r i a d
shows the saw width a s the major t h i r d of major t r i a d s including
seventh.
These deviations from just intonation do not give rise t o
beats. The reason for t h i s muld be the f i n i t e degree of periodicity
of the tones produced by the singers. *
References
(1979) : "Automatic notation of played music",
Farter Actis ?!usicaeU, pp. 109-120.
ASKENFELT, A.
LARSSOtJ, B.
(1977):
"Pitch tracking of music signals", STL-QPSR
1/197'7, pp. 1-8.
Society f o r Preservation and Encouragmnt of Barbershop Quartet
Singing i n America (Kenosha, W I , USA): Contest and Judging
Handbook.
W e are indebted t o the quartets "Happiness Emporium" and "St&j w e t " f o r t h e i r expedient cooperation.
This m r k was supported by
the FRN , NFR , and HSFR.
*
T h i s paper was presented a t the m e t i n g of the Acoustical Societ y of the Nordic countries i n
Finland, June 1980.