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Intonation on trumpets

Article · July 1998

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Matthias A. Bertsch
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 Intonation on trumpets

Matthias Bertsch

Institut für Wiener Klangstil, University for Music and Performing Art in Vienna
A-1010 Vienna, Singerstrasse 26a, Austria e-mail:bertsch@magnet.at

Abstract: The purpose of this study was to determine the intonation properties of trumpets and to compare
empirical data of played trumpets with a.) theoretical tuning systems like equally tempered, Pythagorean
tuning or just intonation and b.) with the “objective intonation” which has been calculated by means of input
impedance measurements. Another aim of this study was to evaluate the size of inter- and intra-individual
variability in performances. Results show that there are great differences amongst players even playing the
same reference instrument. The arithmetic mean over all trials correlates best with the equally tempered sys-
tem. In the middle register, the calculated “objective intonation” matched played intonation even better.

INTRODUCTION

Tone generation on trumpets is influenced by many parameters. Variation of the played intonation are
caused by either the instrument, the player, or both. The intonation of the instrument is determined by the
mechanical dimensions of the instrument and the mouthpiece. The position of resonance frequencies, the so
called "objective intonation", can be calculated using the input impedance method. Statistical data taken
from 35 trumpets will be presented and compared with the pitch of notes blown by the player (so called
"subjective intonation"). The “subjective intonation” can vary for many reasons. It can be caused by the
physiological condition of the lips of a player or the increasing participation of higher harmonics in a
crescendo. The desired timbre of the sound can cause variations as well. These variations can be more than
50 cent with the same instrument as shown in a previous study (BERTSCH 1997). The main objective of
this study was to find out if players follow the tuning of the instrument or if rather they are trying to perform
one of the musical scale models as equally tempered, Pythagorean tuning and just intonation.

METHODS

MEASUREMENT OF THE "OBJECTIVE" INTONATION OF TRUMPETS : 36 trumpets in B-flat have been measured
using the "Brass Instrument Analysing System" BIAS, a Hard and Software system developed at the Institut
für Wiener Klangstil (IWK). [Widholm, 1995]. BIAS measures the input impedance of brass instruments.
Frequencies of impedance peaks are detected and set into relationship with the equally tempered scale. The
reference frequency for A4 is calculated in a way that the mean deviation of the impedance peaks 2-6 and 8
(which correspond to the natural tones Bb3, F4, Bb4, D5, F5, Bb5) is a minimum. Then the departure of all
notes from their ideal location is calculated taking all valve combinations of the instrument as well as the
refrence frequency - usually about 440 Hz …445 Hz, - into account. This calculation method (the so called
„without weighting“ method; WGT=0) assumes that the excitation signal is a sinusoidal signal. In reality

ISMA 98 - BERTSCH - page: 1

the excitation signal of a real player will instead have a sound spectrum containing many harmonics with
different amplitudes. Therefore not only the impedance peak at one frequency (the fundamental of the
particular note) has to be taken into consideration, but all multiple frequencies of the fundamental of the
virtually played note. The contribution of each partial of the excitation spectrum to the „over all“ imped-
ance of a particular note has to be weighted according to the relative amplitude of the excitation spectral
line. BIAS allows different weightings to simulate different dynamic conditions resp. sound spectra of the
excitation signal. Usually a “standard weighting” (WGT=2) is used, where the magnitude of the impedance
1 -1/(2
of higher partials is weighted by 1/x√x
(x being the index of the harmonic). This relationship has been
found in preliminary studies to correspond well with mezzoforte dynamic. The present study is another
approach to find an appropriate formula.

ANALYSIS OF PERFORMED INTONATION OF PLAYERS: trumpet players have been recorded in the anechoic
chamber of our institute playing several tasks in two sessions. The first trial was played on their own
instrument, while in the second trial, a reference trumpet in Bb (Romeo ADACI, Referenz 2001) together
with a reference mouthpiece (BRESLMAIR G1) had to be used. Only the rim of the mouthpiece could be
chosen by the player. The detailed set up was already described at ISMA 97 [Bertsch, 1997]
The subjects were 35 musicians having 18.5 years of experience in average. 20 of them were highly
trained professionals and members in established Vienna orchestras. The other 15 were advanced trumpet
students or amateurs with a wide range of experiences. 24 were playing a trumpet with rotary valves in the
first trial, 11 one with Perinét valves. The reference trumpet has Perinét valves, too.
The musicians were asked to play the given music as if they would perform on stage. No special instruc-
tions were given to concentrate on the intonation in order to receive realistic samples. Two scales in F major
have been analysed. Note that this corresponds to G major when the notation is for a trumpet in Bb (fig. 1).
Task F3-4 starts at F3 in the lower register of the instrument and ends in the middle register one octave
above. Task F4-5 covers the next octave from F4 to the beginning of the high register of the trumpet at F5.
One note, the F4 (written as g4) is played and analysed twice.

Trp.F) G-Dur
Bb Sequenzen task F3-4 task F4-5
#
Ó Œ ŒŒœ œ œ œ œ œ œ
œ œ
& œ œ ˙ ˙ œ œœœœ œœ
œ œ œ
F ˙ F
FIGURE 1 Musical context of task F3-4 and F4-5 played on trumpets in B-flat.

Duration and articulation was not defined more precisely. The dynamic should be mezzoforte. The grow-
ing tension of the ascending scales and the influence of the different dynamic range of the instrument in
lower and upper registers caused all players to perform a crescendo. In average, the F5 was played 13dB
louder than F3. i.e. an increase of 6 dB per octave. The inter individual variability was remarkably more
than 14 dB in each register which demonstrates a different interpretation of mezzoforte. Correlation be-
tween intonation and dynamic was part of the previous study.
Each single note played on a brass instrument varies in pitch, even without any vibrato. Most notes in a
ascending scale also ascend from their beginning to their end. To receive only one fundamental frequency
(f0) for each blown note, digital signal analysis has been applied. Fundamental frequencies have been

ISMA 98 - BERTSCH - page: 2

detected (fs: 44,1 resp. 48kHz; window Hz
f0 (C5)
lengths: 2048 Samples) when at least four
partials were fitting into a harmonic grid.
min freq. max freq.
Additionally the RMS was plotted. As rel- observed
50ms mean
evant frequency (in this study referred to frequency
as the oberseved mean frequency) the fre- RMS (C5)
dB
quency at the moment with maximum attack
amplitude (RMS) has been considered.
(Only in some cases a more constantly maximum
amplitude
played frequency was chosen.) The up-
per part of figure 2 shows the f0 played —» t
by one player as the note C5. Each data Figure 2. Detection of the played fundamental frequency.
point indicates the fundamental frequency as a result of a harmonic grid analysis. The first and last 50ms of
each tone, - one FFT window length -have been omitted. Absolute frequencies have been transformed into
relative intervals corresponding to an individual tuning. (A4 between 438 Hz and 445 Hz.) For the determi-
nation of A4, the minimum departure of all natural tones (no valve engaged) have been taken into consid-
eration. The arithmetic mean, 443 Hz (SD: 2 Hz), illustrates the actual custom to tune to a higher frequency
than the current international standard tuning frequency for western music, 440 Hz. Additionally the mini-
mum and maximum frequencies have been tracked to determine the variation of one note.

RESULTS

MEASURED INTONATION: BIAS measurements of 35 trumpets in B-flat have been made and the average
departure of the calculated intonation from the equally tempered system is shown in Fig.3. In the lower
register there is a great difference between a graph without weighting (WGT=0) - where notes are very flat,

55
50 MEAN WGT=0 MEAN WGT=2
45
40
35
30
25
20
15
10
5
0
-5
-10
-15
-20
-25
-30
-35
-40
-45
-50
-55
E3
F3
F#3
G3
Ab3
A3
Bb3
B3
C4
C#4
D4
Eb4
E4
F4
F#4
G4
G#4
A4
Bb4
B4
C5
C#5
D5
Eb5
E5
F5
F#5
G5
G#5
A5
Bb5
B5
C6
C#6
D6

Figure 3. Departure from equally tempered scale in cent (arithmetic mean of all BIAS measurements)

ISMA 98 - BERTSCH - page: 3

and one with standard weighting (WGT=2) - where notes in the 3rd octave are rather sharp. In the middle
and high register, variations between weightings are very small. For most values (except 3rd octave of WGT=0)
the analysis shows that if more than one valve is engaged t he effective length of the additional tube is too
short. The resulting tone is sharp. In fig. 4, the values (for WGT=2) for each valve combination which are used
in standard fingering are shown separately. This well known fact can on almost all instruments be corrected
manually using a trigger. Especially for B3 and C4 the use of a trigger is always recommended.
5 0 0 5 5 0
4 0 0 4 4 0
3 0
V. 0 0 3
V. 2 3 0
V. 1
2 0 0 2 2 0
1 0 0 1 1 0
0 0 0
-10 -10 -10
-20 -20 -20
-30 -30 -30
Bb3 F4 Bb4 D5 F5 Bb5 D6 A3 E4 A4 C#5 E5 A5 C#6 Ab3 Eb4 G#4 C5 Eb5 G#5 C6
5 0 0 5 5 0 5 0
4 0 0 4 4 0 4 0
3 0
V. 1+2 0 3 V. 2+3 3 0 3 0
2 0 0 2 2 0 2 0
1 0 0 1 1 0 1 0
0 0 0 0
-10 -10 -10 -10
-20 -20 -20 V. 1+3 -20 V. 1+2+3
-30 -30 -30 -30
G3 D4 G4 B4 G5 B5 F#3 C#4 F#4 F#5 F3 C4 E3 B3

Figure 4. Intonation error in cent for each valve combination (BIAS WGT=2)
Besides this characteristic, which is connected to the valve combination, it is remarkable that all notes
blown using valve 1 are sharp (except C6). Furthermore, notes blown at the 6th resonance frequency tend to
be very sharp (Eb5: +17 cent; F5: +10 cent)

PLAYED INTONATION: Recordings of 35 trumpet players have been analysed regarding of their intonation
performance within two F major scales. In Fig. 5 a huge inter-individual variation of the observed mean
frequencies can be seen. The overall distance between maximum and minimum is about 30-40 cent, and in
the lowest register even more. Astonishing is the variety of played intonation on the reference trumpet.
Compared to the equally tempered system, the arithmetic mean of all played intonations fluctuates be-
tween sharp and flat. (See table 1) Remarkable is the note A3, which is more than 16 cent flat and the notes
C4, G4 and F5 which are played more than 10 cent higher than the equally tempered scale. The associated

60
Figure 5. The actually
50
played intonation of all
40 trumpeters (departure
30 from the equally tem-
20
pered scale in cent). The
black curve indicates the
10
arithmetic mean over all
0
sessions. The individual
-10
values are shown as black
-20 narrow bars for values
-30 played with their own in-
-40 dividual instrument and
-50
grey bars for the session
played on the reference
-60
trumpet.
F3

G3

A3

Bb3

C4

D4

E4

F4

F4

G4

A4

Bb4

C5

D5

E5

F5

ISMA 98 - BERTSCH - page: 4

standard deviation (SD) for tones in the lower register is Table 1. arithmetic mean, in cent of departure
about 15 cent, in the middle and upper register from the equally tempered intonation over both
approxamately 7 cent. The SD for trials on the reference trials (I+R) together with standard deviation and
trumpet (R) is only slightly higher (1 cent) than for the tri- for each trial seperatly.
========================================
als played with individual instruments (I). I+R I+R R I
Intonation differences between I and R exist for some MEAN SD MEAN MEAN
F3 -7,1 17,3 -2,9 -11,7
particular tones, which are caused by the type of the trum- G3 -9,2 15,8 -10,6 -8,3
pet. On instruments with Perinét valves (like R), the A4 A3 -16,9 12,4 -18,4 -16,0
Bb3 -8,4 10,4 -10,2 -7,2
and Bb4 are about 6 cent sharper, C5 and D5 are about 7 C4 15,0 13,0 16,0 13,5
cent more flat as on instruments with rotary valves (like D4 6,9 10,5 5,6 8,4
E4 -6,1 8,8 -8,2 -3,7
most of I).
F4 -3,1 7,2 -3,6 -2,1
The fact that the second F4 is in average 7 cent higher F4 3,7 7,2 3,0 4,5
than the first shows the importance of musical context. In G4 12,0 8,3 13,5 10,4
A4 -0,6 7,0 2,2 -3,1
this case the second F4 is played right after the G4, which Bb4 1,6 6,8 4,9 -1,8
is very sharp. C5 -0,8 7,1 -2,2 0,5
D5 -4,7 6,3 -6,3 -3,0
If the intonation of both tasks is compared, (see fig. 6) E5 4,5 8,6 6,6 2,6
almost no common trend concerning interval relationship F5 10,9 8,0 12,2 9,7

can be found. This makes an underlying general “theoretical system” unlikely. Remarkable is that both
keynote octaves are larger than 1200 cent. Moreover, 7 of 9 occurring
20
15 octaves are stretched. In order to compare the played intonation with
10 theoretical tuning systems like equally tempered, Pythagorean and just
5
0
intonation, the departures have been calculated and plotted in Fig. 7. In
-5 general, trumpet performance most closely conforms with equally tem-
-10 task F3-4
-15 task F4-5
pered intonation. Departures from each model are much greater than
-20 differential threshold (about 3 cent).
F G A Bb C D E F
Additionally, the played intonation of certain groups of players se-
Figure 6. upper / lower octave (∆ ET)
lected from all players has been statistically analysed. As a result, little
significant difference was found between professional and student players, between younger and older
players, or between male and female musicians. Diversity was found to exist only on notes with extreme
deviation. For example professionals played the C4 and G4 five cent less sharp than amateurs and students.
Especially the C4 was played very sharp by players with less experience.
The examination of the played tones reveals 25 [ø] played ∆ ET ∆ PT ∆ JI
20
further a great difference between minimum and 15
maximum for each note within the scale. 26 cent 10
5
in average among all players for the trials on their 0
own instrument, and 28 cent for trials with the -5
-10
reference trumpet. In the lower register the aver- -15
age variation of some notes was even more than -20
-25
50 cent. Of course, during the ascending scale, F3 G3 A3 Bb3 C4 D4 E4 F4 F4 G4 A4 Bb4 C5 D5 E5 F5

most slurs have been upwards. Figure 7. Departures from theoretical intonations in cent. Or-
dinate corresponds to played intonation on indiv. instruments

ISMA 98 - BERTSCH - page: 5

 40
MEASURED VERSUS PLAYED INTONATION: 35 [ø] played ∆ BIAS WGT=0 ∆ BIAS WGT=2
30
A comparison of the arithmetic mean of the variety 25
20
of subjective intonations of an instrument as played by 15
10
a player and the “objetive intonation” as measured us- 5
0
ing BIAS shows a good matching in the middle regis- -5
-10
ter. In Fig. 8 a disagreement can be only found in the -15
-20
lower register, where weighting influences the result to -25
-30

a great extent. The standard weighted intonation meas- -35-40

F3 G3 A3 Bb3 C4 D4 E4 F4 F4 G4 A4 Bb4 C5 D5 E5 F5
urement approaches the played intonation more closely
which can be taken as an indication that an improve- Figure 8. Departures of BIAS measurements from the
ment of the weighting algorithm could be a good way played intonation in cent. Ordinate corresponds to played
intonation on indiv. instruments.
to further improve correlation between played and
measured intonation.
SUMMARY
15 MAD task F3-4
13,4 MAD task F4-5
11,8
The main results of the present study are summarized in
10,3
10 8,9
9,7 table 2 in which the arithmetic mean of the MAD values
7,5 and the associated standard deviations are tabulated. Visual
6,1
5,3 inspection of Figure 9 shows: a.) that trumpet performance
5 4,1 4,4
in regard to theroretical tunings most closely conforms to
equally tempered intonation and b.) that the standard
0 weighted “objective intonation” model matches played in-
∆ BIAS ∆ BIAS ∆ ET ∆ PT ∆ JI
WGT=0 WGT=2
tonation especially well in middle register. Therefore, it can
F 3-4 13,44 11,84 8,87 10,33 9,65 be concluded that trumpeters follow the tuning asserted by
SD 16,13 12,21 4,73 7,11 7,10 the instrument rather than trying to match a theoretical scale.
For the player, a perfect „objective intonation“ that
F4-5 5,31 4,11 4,45 6,15 7,47
matches his „intended intonation“ could free him to con-
SD 4,24 4,44 3,62 3,86 5,18
centrate on other aspects. Since technical tools are able to
Table 2. Mean absolute difference (MAD) in optimise the „objective intonation“ of brass instruments
cent between observed played intonation on (Anglmayer and Kausel 1998), the question arises which
players own instruments and BIAS measure-
ideal intonation a musician expects from his instrument.
ments without (WGT=0) and with standard
Usually only extreme departures are considered as a real
weighting (WGT=2). Further MAD and theo-
retical musical systems: equal tuning (ET), Py- problem, because much energy is needed for correction.
thagorean tuning (PT) and just intonation (JI).
Figure 9. shows MAD for each task seperately. ACKNOWLEDGMENTS

I am grateful to all trumpet players who have participated in this study.

Bertsch, M., “Variabilities in trumpet sounds,” Proceedings of the ISMA`97, Edinburgh, UK. pp. 401-406, 1997.
Widholm, G., Brass Wind Instrument Quality Measured and Evaluated by a new Computer System,
“Proceedings of the 15th ICA, Trondheim, Norway. pp. 517-520, 1995.
Anglmayer,P., Kausel W., Recent research on brass instruments can be found on the IWK homepage: http://iwk.mhsw.ac.at

ISMA 98 - BERTSCH - page: 6

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