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David Ackermann Fabian Brinkmann AND Stefan Weinzierl

# A Database with Directivities of Musical Instruments ††thanks:
Correspondence should be addressed to David Ackermann. E-mail:
<EMAIL_ADDRESS>
###### Abstract

We present a database of recordings and radiation patterns of individual notes
for 41 modern and historical musical instruments, measured with a 32-channel
spherical microphone array in anechoic conditions. In addition, directivities
averaged in one-third octave bands have been calculated for each instrument,
which are suitable for use in acoustic simulation and auralisation. The data
are provided in SOFA format. Spatial upsampling of the directivities was
performed based on spherical spline interpolation and converted to OpenDAFF
and GLL format for use in room acoustic and electro-acoustic simulation
software. For this purpose, a method is presented how these directivities can
be referenced to a specific microphone position in order to achieve a
physically correct auralisation without colouration. The data is available
under the CC BY-SA 4.0 licence.

##  1 INTRODUCTION

Studies of the sound radiation characteristics of the human voice date back to
the late 1930s [1], and studies of the directivity of musical instruments
began thirty years later (summarized in [2, Chapter 4]). While these early
measurements were often made with a single microphone moved around the source,
the radiation patterns of acoustic sound sources such as speakers, singers, or
musical instruments are today usually measured with the source at the center
of an enclosing microphone array in anechoic conditions.

Such a nearly full spherical array was used to analyze the directivity of 40
different human speakers, with measurements taken sequentially at 253
positions [3]. The radiation characteristics of eight opera singers [4] and
fifteen trained singers [5] were determined in the horizontal plane, with
measurements taken at nine and thirteen positions, respectively. High spatial
resolution was employed for the measurements of a professional male singer,
using an adjustable semi-circular microphone array with 24 receivers [6]. A
recent review of research on the sound radiation of singing voices is given by
Abe (2019) [7].

For the directivity of musical instruments, eight orchestral instruments were
measured using 64 microphones [8], while 22 instruments of a symphony
orchestra were measured using 22 microphones [9]. A recently generated
database for 14 instruments and a speaker contains radiation patterns measured
at 2522 positions on a sphere [10], but these data have only limited frequency
resolution in (third-)octave bands. The most comprehensive and publicly
available database was compiled for 41 modern and historic instruments,
measured with 32 microphones, and includes recordings of single notes within
the playable range of each instrument and directivities calculated from the
stationary parts of these notes [11, 12].

Based on these measurements, the aim of this work was to provide the
directivities of musical instruments in an open and standardised format,
including both the acoustic measurements, the results of the subsequent
processing, and important metadata, such as the exact position of the
microphone capsules, the tuning frequency of the instrument and the pitch for
which the directivity is valid. This should facilitate the exchange and use of
such data by the scientific and general acoustics community. To facilitate
this, we recently standardized the Spatially Oriented Format for Acoustics
(SOFA) convention FreeFieldDirectivityTF, as part of the AES69-2022 standard
[13, 14], but also provide the data in OpenDAFF [15] and Generic Loudspeaker
Library (GLL) [16] formats for use in room acoustic simulation software.

For each instrument, the database contains the single-tone recordings
(calibrated to an absolute sound pressure and equalized for the microphone
array transfer function), the extracted single-tone directivities, and the
one-third octave band-averaged directivities and corresponding finite impulse
responses (FIRs)111The data will be published after the review of this paper
with a DOI on the DepositOnce repository and can be accessed beforehand at
https://tubcloud.tu-berlin.de/s/8joeeK3fFingLgp.

##  2 METHODS

The directivities of 41 modern and historical musical instruments were
measured using a 32-channel full-spherical microphone array in an anechoic
chamber in an earlier study [11]. The following sections detail the recording
of the original database and highlightes the improved processing of the data
suggested in this work.

###  2.1 Data representation and format

The directivity of electro-acoustic sound sources, such as loudspeakers and
microphones, can be described relatively easily by using transfer functions in
the frequency domain or by finite impulse response (FIR) filters in the time
domain for each direction of radiation. In contrast, the directivity of
natural sound sources, such as human speakers, singers and musical
instruments, is more complex to describe, because the directivity of a musical
instrument depends not only on the frequency, but also on the note being
played, and sometimes on the fingering, so that the same note can have
multiple radiation patterns.

To ensure maximum flexibility, the Freefield DirectivityTF convention
represents directivity data as complex transfer functions (TFs) at arbitrary
frequencies. This allows FIR filters of artificial sound sources, but also
multi-channel acoustic recordings of a musical instrument (see section 2.3.1)
to be stored as complex spectra with linear frequency resolution. In addition,
the directivity patterns of natural sound sources can be stored for arbitrary,
not necessarily equidistant frequencies such as the fundamental frequency and
the corresponding overtones (see section 2.3.2), or as one-third octave
frequency-band averaged data (see section 2.3.3). This representation is
mainly used in geometric acoustic simulation.

For a complete description of natural sound source directivities, information
about the instrument/singer and the way of playing is needed in addition to
the measurement setup. The SOFA standard allows these data to be stored as
metadata and uniquely assigned for easy handling and data exchange. An
overview of the available metadata can be found in section 3.

###  2.2 Measurement setup

The instruments were recorded with a fully spherical lightweight microphone
array in the anechoic chamber of the Technische Universität Berlin with a room
volume of approximately $1070$ m3 and a lower cut-off frequency of
$f_{\text{c}}=63$ Hz. 32 Sennheiser KE4-211-2 electret capsules of the
microphone array were located at the vertices of a pentakis dodecahedron with
a diameter of 2.1 m.

A height-adjustable chair was used to position the musicians with their
instruments so that the estimated acoustic centre of each instrument was as
close as possible to the centre of the microphone array. The musicians faced
the positive x-axis. The exact capsule positions of the array are given in
spherical coordinates, i.e., in azimuth ($\phi=0^{\circ}$ pointing in positive
x-direction, $\phi=90^{\circ}$ pointing in positive y-direction), colatitude
($\theta=0^{\circ}$ pointing in positive x-direction, $\theta=90^{\circ}$
pointing in positive z-direction) and distance (in metres) are included in the
metadata of the SOFA files (cf. section 3). The recordings were made with 24
bit resolution and a sampling frequency of $f_{s}=44.1$ kHz. The recordings
were calibrated – i.e., a digital amplitude of 1 corresponds to a pressure of
1 Pascal resp. a sound pressure level of $L_{p}=94$ dB – and compensated for
the frequency response of the microphone array and the capsules. The exact
measurement setup and a detailed description of the calibration procedure can
be found in [11].

###  2.3 Processing

The calibrated and equalised single-tone recordings in the dynamic range of
pianissimo (pp) and fortissimo (ff) provided as 32-channel WAV files, as
published in [12], form the basis for the processing described below.

####  2.3.1 Recordings

The real-valued single-note recordings $x_{q}[n]$ of even length $N$ are
available at discrete times $n\in\\{0,1,...,N-1\\}$ and for $Q=32$ channels of
the spherical microphone array with $q\in\\{1,2,...,Q\\}$. Because the
FreeFieldDirectivityTF convention requires frequency data, the recordings were
Fourier transformed

$X_{q}(k)=\sum_{n=0}^{N-1}x_{q}[n]\text{e}^{-\text{i}2\pi\frac{k}{N}n},$ (1)

with $\text{i}^{2}=-1$ being the imaginary unit, and saved in the SOFA files
as complex-valued single-sided spectra $X_{\text{S},q}(k)$ of length $N/2+1$

$X_{\text{S},q}(k)=\left\\{\begin{array}[]{l}X_{q}(k),\quad\quad\
\text{if}\>k=0\\\ 2\cdot X_{q}(k),\quad\text{if}\>0<k<\frac{N}{2}\\\
X_{q}(k),\quad\quad\ \text{if}\>k=\frac{N}{2}.\\\ \end{array}\right.$ (2)

Note that reconstructing the time domain recordings from the published data
thus requires the reconstruction of the both-sided spectrum of length $N$

$X_{q}(k)=\left\\{\begin{array}[]{l}X_{\text{S},q}(k),\quad\quad\quad\quad\
\text{if}\>k=0\\\ \frac{1}{2}\cdot X_{\text{S},q}(k),\quad\quad\ \ \
\>\text{if}\>0<k<\frac{N}{2}\\\ X_{\text{S},q}(k),\quad\quad\quad\quad\
\text{if}\>k=\frac{N}{2}\\\ \frac{1}{2}\cdot
X_{\text{S},q}^{\ast}(N-k),\quad\text{if}\>k>\frac{N}{2}.\end{array}\right.$
(3)

with $(\cdot)^{\ast}$ denoting the complex conjugate, before applying the
inverse Fourier transform (see [17] for details).

####  2.3.2 Single tone directivity data

To determine the directivity of the musical instruments, we used the
stationary part of the single note recordings. For all instruments producing
stationary parts, this was manually windowed by visual inspection, resulting
in durations between 200 and 2104 ms, with a median duration of 630 ms. For
the acoustic guitar and the harp, a quasi-stationary part was defined as the
part between the decay time and the release time as estimated with the Timbre
Toolbox [18]. For the transient timpani signals, the entire recording was
used, from the onset to the transition into the noise floor.

The directivities were estimated in two steps. First, the fundamental
frequency $f_{0}$ and the frequency of the overtones $f_{i}$ in Hz were
identified with $i\in\\{1,2,...,I\\}$ and $I$ being the highest identifiable
overtone. Secondly, the energy at these frequencies was estimated. Both steps
were based on the magnitude response $\left|X_{q}(k)\right|$ and ignored the
phase information because natural sound sources, unlike artificial sound
sources, do not have a stationary phase response, both for a given frequency
and direction of radiation [19]. Figure 1 shows the magnitude and phase
response for 0.5 seconds excerpt of a trumpet recording. Although the
amplitudes of the fundamental and harmonics remain constant throughout the
time window, the phase varies considerably, making it impossible to determine
it unambiguously.

This factor also complicates the use of natural sound sources for room
acoustic simulation without further modification, since interpolation of the
phase spectrum is highly susceptible to noise and can lead to errors,
especially at high frequencies [20]. We have therefore proposed the use of
absolute-valued directivities for interpolation [21]. This is in contrast to
the previous processing of the data, which used the complex valued spectrum
[11].

Figure 1: Extract from the spectrogram for the signal of a trumpet playing
note A4 ($f_{0}=440$ Hz) at fortissimo (microphone 4), with amplitude in dB
(top) and phase in degree (bottom).

An estimate of the fundamental frequency $f_{0}$ was made by identifying the
frequency with the highest amplitude within a window of $\pm 100$ cent
bandwidth around the frequency corresponding to the tuning pitch indicated by
the musicians. This was usually 442 or 443 Hz for modern instruments, 430 Hz
for instruments of the Classical period, and 415 Hz for instruments of the
Baroque period. This frequency was obtained for all 32 microphone recordings,
and the most frequently occurring frequency over all 32 extracted values was
chosen as $f_{0}$.

In the next step, the frequencies of the partials were estimated by placing a
search window of $\pm 10$ cents around each harmonic frequency corresponding
to $f_{0}$ and identifying the most frequently occurring, highest amplitude
within this window, again considering all 32 microphone recordings. The search
window takes into account the fact that, for physical reasons, the partials do
not always lie exactly at the harmonic multiples of the fundamental frequency,
and that the fundamental was not always held exactly constant over the
duration of the stationary part. If the identified frequency deviated from the
harmonic frequencies by more then five cents, the procedure was stopped, and
the signal energy was considered to be below the noise floor. A visual
inspection of the detected signal components confirmed that all relevant and
clearly identifiable partial tones had been found by this procedure.

At the frequencies $f_{i}$ that could be estimated, the directivity was
calculated using the power spectral density (PSD)

$S_{xx,q}(k)=\frac{1}{f_{s}N}|X_{q}(k)|^{2}\,$ (4)

which is a measure of the power of each frequency component in a signal with
the implicit unit Pa2/Hz. We have used the Welch method to improve the
robustness of the PSD against noise by

dividing the signal into eight segments of equal length with 50 % overlap,

applying a Hanning window function to each segment to reduce the effect of
spectral leakage caused by the finite length of the segments,

computing the periodogram of each segment, which is an estimate of the PSD of
that segment, and

averaging the periodograms across the segments to obtain an estimate of the
overall PSD of the signal. To obtain an estimate of the power at each
frequency, each estimate of the PSD was scaled by the equivalent noise
bandwidth of the window (in Hz).

[capbesideposition=right,top,capbesidewidth=4cm]figure[]

Figure 2: Normalized FFT (grey) and scaled PSD (black) of the signal of the
modern oboe playing note C4 (262 Hz) at fortissimo, recorded with microphone
4. The asterisks indicate the detected power values and frequencies $f_{0}$
and $f_{i}$ that do not necessarily coincide with the discrete bins of the
PSD.

Finally, the power was determined by simple peak picking in the scaled PSD
around the frequencies $f_{i}$ of the tone’s harmonics and converted to the
sound pressure value $p_{q}(f_{i})$ by taking the square root. Figure 2
illustrates the process for one note played by the modern oboe.

####  2.3.3 One-third octave band averaging

Geometric acoustics applications typically require a single directivity
pattern for a sound source, which is usually provided in one-third octave band
resolution.

This was calculated by energetic averaging of the partials of all individual
notes of an instrument falling into the one-third octave frequency bands
according to IEC 61260-1 [22], for the $M=30$ centre frequencies from 25 Hz to
20 kHz. All partials of the $J$ individual notes $p_{q}(f_{i,j})$ from section
2.3.2 were used for the one-third octave band representation. The data were
averaged for each of the $Q=32$ receivers of the spherical microphone array
individually, by calculating the averaged amplitude as

$\bar{p}_{q,m}=\sqrt{\frac{1}{L}\sum_{i=0}^{L-1}p_{q,i}^{2}},$ (5)

where $L$ indicates the number of partials identified in one one-third octave
band. Figure 3 illustrates the averaging procedure over all partials for the
modern oboe in one-third octave bands from 400 Hz to 2500 Hz.

Figure 3: Estimated sound pressure levels of all partials of the modern oboe
plotted against frequency (grey) in the one-third octave bands from 400 Hz to
2500 Hz, recorded with microphone no. 13. As an example, the fundamental and
the first five overtones for the note A4 ($f_{0}=442$ Hz) are highlighted in
black. The dashed vertical lines mark the boundaries of the third-octave
bands. The horizontal solid black lines indicate the energetically averaged
sound pressures $\bar{p}_{q,m}$ according to equation 5, shown in dB.

At this point, the data still contains the direction-dependent frequency
response of each instrument. If the directivities are used for auralizations
in which a simulated (binaural) room impulse response is convolved with an
anechoic recording of an instrument, it should be noted that the anechoic
recording also contains the frequency response of the instrument in the
direction of the recording microphone. To avoid an unnatural coloration during
auralizations, the frequency response has to be removed from the third-octave
averaged directivities. In theory, this normalization should also consider the
position of the microphone from which the anechoic recording of the instrument
was made, i.e., this direction should be normalized to 0 dB. If this position,
however, is unknown or unstable due to movements of the instrument relative to
the microphone, it may be most robust to equalize the directivity so that
equal energy is radiated across all directions within each band, i.e., to

$\bar{p}_{\text{diff},q,m}=\frac{\bar{p}_{q,m}}{\sqrt{\sum_{q=1}^{Q}\bar{p}_{q,m}^{2}\cdot
w^{\prime}_{q}}},$ (6)

where $w^{\prime}_{q}$ are the normalized area weights of the measurement grid
with $\sum w^{\prime}_{q}=1$. This representation of the radiation patterns is
called diffuse equalization in the following and will be discussed in more
detail in section 4.

The final step was to calibrate the directivity to the sound power of the real
instruments, as the previous diffuse equalisation (cf. equation 6) had lost
the absolute sound power reference. This was done by averaging the sound
pressure level over the effective one-third octave bands for each microphone.
Third-octave bands with no sound radiation were set to zero.

The average sound pressure level of the diffuse equalized directivity per
microphone was then calculated as

$L_{\text{P},\text{3rd},q}=10\lg\left(\frac{\frac{1}{M}\sum\limits_{m\in
M}\bar{p}^{2}_{\text{diff},q,m}}{p^{2}_{0}}\right),$ (7)

where $M$ indicates the number of effective one-third octave bands and
$p_{0}=2\times 10^{-5}$ Pa. Its average over the surface of the spherical
envelope is given by

$\bar{L}_{\text{P},\text{3rd}}=10\lg\left(\frac{1}{Q}\sum\limits_{q=1}^{Q}10^{0.1L_{\text{P},\text{3rd},q}}\right).$
(8)

The sound pressure of the reference, i.e. the corresponding instruments, was
calculated from the calibrated recordings as follows

$L_{\text{P},\text{ref},q}=10\lg\left(\frac{\frac{1}{NJ}\sum\limits_{n=1}^{N}\sum\limits_{j=1}^{J}x^{2}_{q,j}[n]}{p^{2}_{0}}\right),$
(9)

where $N$ is the number of samples of the stationary part, $J$ is the number
of single notes and averaged over all microphones to

$\bar{L}_{\text{P},\text{ref}}=10\lg\left(\frac{1}{Q}\sum\limits_{q=1}^{Q}10^{0.1L_{\text{P},\text{ref},q}}\right).$
(10)

Finally, the calibration of the diffuse equalized directivity from equation 6
is given by

$\bar{p}_{\text{cal},q,m}=\bar{p}_{\text{diff},q,m}\cdot
10^{\frac{\bar{L}_{\text{P},\text{ref}}-\bar{L}_{\text{P},\text{3rd}}}{20}}.$
(11)

The one-third octave band averaged directivity of each instrument is
calculated for the dynamic fortissimo (ff) and provided in the
FreeFieldDirectivityTF SOFA convention.

####  2.3.4 Interpolation

Several applications that rely on the directivity of sound sources, such as
room acoustic simulations, require the use of continuous or high resolution
data. Consequently, the measurement data must be spatially resampled
(interpolated) to match the required sampling grid.

By sampling the actual sound pressure function $f(\theta,\phi)$ with a $Q$
channel spherical microphone array, the samples $p_{q}=f(\theta_{q},\phi_{q})$
are given at the positions $(\theta_{q},\phi_{q})$ of the respective
microphones for $q\in\\{1,2,...,Q\\}$. The general mathematical formula for
interpolation can therefore be expressed as

$\hat{f}(\theta_{r},\phi_{r})=\sum\limits_{q=1}^{Q}f(\theta_{q},\phi_{q})\cdot
L_{q}(\theta_{r},\phi_{r}),$ (12)

where $\hat{f}(\theta_{r},\phi_{r})=\hat{p}_{r}$ is the estimated sound
pressure at the $R$ points $(\theta_{r},\phi_{r})$ of the interpolation grid
for $r\in\\{1,2,...,R\\}$ and $L_{q}(\theta_{r},\phi_{r})$ being the
interpolation function derived from the known sound pressure $p_{q}$ at the
position $(\theta_{q},\phi_{q})$. The specific choice of the interpolation
function depends on the interpolation method being used.

There is a plethora of techniques for interpolating real-valued scattered data
that make different assumptions about the distribution of the discrete set of
known data points [23]. For musical instruments, the thin-plate pseudo-spline
method [24, 25] of order 1 has been found to be a good method, producing lower
interpolation errors than spherical harmonics (SH) interpolation [26, 27, 28]
or three-dimensional Vector-Based Amplitude Panning (VBAP [18]) when applied
to sparsely sampled directivity measurements and evaluated against the
directivity of different musical instruments measured at high resolution as a
reference [21].

We chose an equiangular grid with an angular resolution of $5^{\circ}$ in
azimuth and colatitude as the target for the interpolation, resulting in
$R=2522$ sound pressure values $\hat{p}_{r,m}$ for each of the $m\in M$ one-
third octave bands. The closed form spherical spline interpolation for order 1
was realized with AKtools using the function AKsphSplineInterp() [29] based on
the directivity patterns according to equation 11.

(a)

(b)

Figure 4: One-third octave band-averaged radiation pattern of the modern
bassoon at 400 Hz, with original measurement resolution (32 points, a) and
spatial interpolation to 2522 points, using spherical splines of 1st order
(b). Balloon representation with radius corresponding to the sound pressure
level in dB

.

####  2.3.5 3rd octave smoothed FIRs

To enable musical instruments to be used as sound sources for simulating room
acoustic and electroacoustic environments with software that uses FIR filters
to represent directivity, $R=2522$ FIR filters were calculated using the above
mentioned grid. For this purpose, a one-third octave band spectrum according
to IEC 61260-1 [22] with a frequency resolution of 1 Hz was generated from the
estimated sound pressure values $\hat{p}_{r,m}$ (gray line in figure 5(a)).
The one-sided spectrum with odd N was then smoothed with a one-third octave
filter (black line in figure 5(a)). After transformation to a two-sided
spectrum according to (equation 3) the spectrum was transformed into the time
domain using IFFT and the phase was made minimum phase using the AKTools
function AKphaseManipulation(). Finally, the FIR filter was reduced to 8192
samples according to AES56-2008 [30] as shown in figure 5(b).

(a) Magnitude spectrum

(b) Energy Time Curve (ETC)

Figure 5: Third-octave spectrum (gray) and smoothed third-octave spectrum
(black) for the instrument bassoon modern (a) for the radiation direction
$(\theta=0;\phi=-10)$ and the corresponding energy-time curve (b).

For use in the software EASE 222www.afmg.eu/en (accessed May 2, 2023), this
data was stored in the proprietary GLL format.

##  3 DATABASE

The database contains the calibrated single-note recordings, the single-note
directivities, and the frequency-averaged directivity patterns in SOFA format
FreeFieldDirectivityTF convention under a Creative Commons share alike licence
(CC-BY-SA 4.0). In addition, high spatial resolution interpolated radiation
patterns averaged over one-third octave bands, are provided in openDaff format
and as FIR filters in GLL format. All data are freely accessible 333The data
will be published after peer review with a DOI on the TU Berlin data
repository and can be accessed beforehand at https://tubcloud.tu-
berlin.de/s/8joeeK3fFingLgp. A list of the available musical instruments can
be found at table 1.

The SOFA files can be read using a variety of APIs444cf.
www.sofaconventions.org/mediawiki/index.php/
Software_and_APIs (accessed May 2, 2023). They contain the recorded signals
and extracted directivities as complex transfer functions (TFs) together with
metadata describing the data in detail. This includes the name of the
instrument in the entry GLOBAL_SourceName, the name of the musician in
GLOBAL_Musician, the manufacturer of the instrument in
GLOBAL_SourceManufacturer and a verbal description of the position of the
instrument during the measurement in SourceView_Reference. The arrangement of
the capsules of the microphone array is described in ReceiverPosition.

In the SOFA data of the recordings, the respective note and the dynamic level
are indicated in GLOBAL_Description, e.g. note = A4; dynamic = ff, the MIDI
number in MidiNote, e.g. 69 for A4, and the frequency for A4 corresponds to
the tuning frequency in the entry for SourceTuningFrequency. In the SOFA data
of the original recordings, SteadyPart indicates the range of the manually
determined stationary portion in samples. A detailed description of the
structure of the database can be found in the accompanying documentation.

###  3.1 SOFA recordings

The single note recordings are available as a one-sided complex TF for each
instrument and note. The data can be converted into a two-sided spectrum
according to equation 3 and converted to the time domain by means of an IFFT.
The naming of the data follows the scheme SourceName_dynamic_midi
Note_recordings.sofa, and the recordings are stored in the Data.Real and
Data.Imag fields. The fields have the dimension MxRxN, where M (measurement)
is always 1, R (receiver) is the number of capsules of the microphone array,
with R = 32, and N indicates the length of the TF. The field N contains the
frequencies of the bins of the TF in Hz. Note that the calibrated 32-channel
WAV recordings on which this data set is based are still freely accessible
[12].

###  3.2 SOFA single note

There is also a separate SOFA file for each instrument and note for the
single-note directivity data; the naming of the file corresponds to the scheme
SourceName_midiNote_singleTones.sofa. The purely real sound pressure levels
are stored as complex transfer functions in the field Data.Real with the
dimension MxRxN, where M is always 1, R = 32, and N refers to the number of
the $I$ extracted partials. The Data.Imag field with the dimension MxRxN is
included in the dataset for consistency reasons, but contains only zeros.The
field N indicates the frequencies of the $I$ partials in Hz.

###  3.3 SOFA One-third octave band

For the one-third octave band-averaged directivities, there is one SOFA file
for each instrument with the naming scheme SourceName_3rdOctave.sofa. The
averaged and calibrated sound pressures from equation 11 are stored in
Data.Real with the dimension MxRxN, where M is always 1, R = 32, and N=30
refers to the nominal centre frequencies from 25 Hz to 20 kHz according to IEC
61260-1:2014 [22]. The data field Data.Imag has been filled with zeros. The
field N indicates the centre frequencies of the one-third-octave bands in Hz.
In this case, the entries SourceTuningFrequency, SteadyPart and MidiNote are
not included in this data representation.

Table 1: List of modern (m) and historical (h) instruments whose individual directivities are available in the database in the indicated pitch range for the playing dynamics pianissimo (pp) and fortissimo (ff) and additionally averaged in one-third octave bands. Instrument | Era | pp | ff
---|---|---|---
Violin | m | G3–G6 | G3–G6
Violin | h | G3–C7 | G3–C7
Viola | m | C3–F7 | C3–F7
Viola | h | C3–C6 | C3–C6
Cello | m | C2–G6 | C2–G6
Cello | h | C2–D5 | C2–D5
Double bass | m | E1–E5 | E1–D#5
Double bass | h | E1–E3 | E1–E3
Acoustic guitar | m | C2–B5 | E2–B5
Double action harp | m | C1–F4 | C1–F4
Oboe | m | A#3–A6 | A#3–F#6
Classic oboe | h | C4–D#6 | C4–D#6
Romantic oboe | h | B3–C6 | D4–G5
English horn | m | E3–G#5 | E3–G#5
Clarinet | m | D3–A#6 | D3–A#6
Clarinet | h | D3–F6 | D3–F6
Bass clarinet | m | A#1–D#5 | A#1–D#5
Bassoon | m | A#1–E5 | A#1–E5
Classic bassoon | h | A#1–C5 | A#1–C5
Baroque bassoon | h | A1–F#4 | A1–F#4
Contrabassoon | m | A#0–D#3 | A#0–F3
Dulcian | h | C2–G4 | C2–G4
Alto saxophone | m | C#2–C#5 | C#2–C#5
Tenor saxophone | m | B1–A4 | B1–A4
Flute | m | B3–A#6 | B3–D7
Transverse flute | h | D4–F#6 | D4–A#6
Keyed flute | h | C#4–A6 | C#4–A6
Trumpet | m | F#3–F6 | F#3–F6
Natural trumpet | h | D3–D6 | D3–D6
French horn | m | D2–F#5 | D2–G5
Natural horn | h | A1–B4 | A1–B4
Basset horn | h | F2–A#5 | F2–A#5
Tenor trombone | m | G1–F5 | G1–F5
Alto trombone | h | D2–D#5 | C#2–D#5
Bass trombone | m | B1–B4 | C1–B4
Tenor trombone | h | E1–D5 | E1–D5
Tuba | m | G#0–C5 | F0–E5
Timpani | m | D2 | D2
Pedal timpani | m | D2 | D2
Soprano | – | G3–G#5 | G3–G#5

###  3.4 OpenDAFF One-third octave band

The open source format openDAFF can be read with several APIs555cf.
www.github.com/svn2github/opendaff (accessed May 2, 2023). The naming of the
data follows the scheme SourceName.daff. The directional patterns are stored
with a spatial resolution of $5^{\circ}$ (azimuth and colatitude), i.e. each
file contains one-third octave magnitude spectra at 2522 points. These data
can be used directly in the acoustic simulation environment RAVEN [31]. For
evaluating the directivity, both of the individual notes and of the frequency-
averaged directivity patterns, in arbitrary spatial resolution, we provide a
Matlab script as part of the database (cf. section 3.6).

###  3.5 GLL One-third octave band FIRs

The proprietary GLL format allows the integration of complex sound sources
into the acoustic simulation environment EASE. The directional patterns
averaged over a one-third-octave for all 21 modern musical instruments and for
a soprano singer were stored as FIR filters with 8192 taps, and with a spatial
resolution of $5^{\circ}$. For the exact naming scheme, please refer to the
documentation of the dataset666The data will be published after the review of
this paper with a DOI on the DepositOnce repository and can be accessed
beforehand at https://tubcloud.tu-berlin.de/s/8joeeK3fFingLgp.

The data can be converted to UNF (used by Ulysses777www.ifbsoft.de (accessed
May 2, 2023)) and the CLF/CIF format (used by ODEON 888www.odeon.dk (accessed
May 2, 2023) and CATT-Acoustic999www.catt.se (accessed May 2, 2023)) using the
proprietary SpeakerLab101010www.afmg.eu/en/ease-speakerlab (accessed May 2,
2023) software from AFMG.

###  3.6 Tools

Part of the database is the Directivity_demo.m Matlab script that makes it
possible to read the recordings from the SOFA data, display their spectrum
graphically and transform them into the time domain by IFFT. This data can
then be saved as a WAV file and played back with common media players.

The script also allows for the three-dimensional display of single-note and
frequency-averaged directivity in the form of balloon plots based on the SOFA
data provided. Finally, the data can be evaluated at any sampling quadrature
using spherical spline interpolation. Prerequisite for all processing steps is
the installation of AKTools111111cf.
www.tu.berlin/ak/forschung/publikationen/open-research-tools/aktools (accessed
May 2, 2023) and the SOFA API for Matlab (SOFAToolbox121212cf.
www.github.com/sofacoustics/SOFAtoolbox (accessed May 02, 2023)) contained
therein.

##  4 DISCUSSION

The present dataset contains recordings and radiation patterns of the
individual notes of 41 modern and historical musical instruments, measured
with a 32-channel microphone array in anechoic conditions. The recordings and
directivities are provided in standardised SOFA format in convention
FreeFieldDirectivityTF. From these data, averaged directivities have been
calculated for each instrument, which are suitable for use in acoustical
simulation and auralisation. In addition, spatially high-resolution
directivities in OpenDAFF and GLL formats have been generated, allowing direct
use in software such as RAVEN and EASE.

The absolute quality of the interpolation methods used for this spatial
upsampling obviously depends on the characteristics of the sound source, such
as its acoustically effective size, the modal patterns of its sound radiating
parts, and the resulting complexity of the radiation pattern. For acoustically
small sources, such as a trumpet and trombone bell or a violin, a fairly
accurate interpolation can be expected even based on a measurement at 32
points. For extended sources with more complex radiation patterns, however, a
sparse sampling grid may lead to increasingly poor estimates of the far field
directivity [21]. If other types of interpolation prove superior in the
future, such as recently investigated, physically informed interpolation
methods using the Euler equations as constraints [32], the interpolations
applied may be revised in the future.

For the physically correct auralisation of musical instruments in virtual
acoustics, frequency-averaged directivities are a compromise that has to be
made for technical reasons. Current simulation applications do not allow a
straightforward exchange of single tone representations of such directivities
within a simulation run. Due to the frequency averaging of the input data, a
tonal colouration of the simulation result may occur [33]. The magnitude of
this effect and its influence on instrumental and room acoustic perception
will have to be determined in a subsequent study using the data presented.

When an anechoic recording of an instrument and its directivity is used to
auralize a virtual acoustic environment, it is essential to normalize the
directivity of the source to a suitable reference. In theory, this would be
the position of the microphone used to make the anechoic recording. Since the
recorded signal already contains the directional timbre characteristic of the
instrument in that direction, it should not be altered by the applied
directivity, as this would result in unwanted colouration of the simulation.
This means that the directivity should be normalized, so that only a frequency
response relative to this reference direction will be obtained. This will be
achieved by calculating

$\hat{p}_{\text{pt},r,m}=\frac{\hat{p}_{r,m}}{\hat{p}_{\text{mic},m}},$ (13)

where $\hat{p}_{\text{mic},m}$ is the interpolated and frequency-averaged
sound pressure of the $M$ one-third octave band in the reference direction.

In a real recording situation, however, the instrument being played by a
musician is a moving sound source. This can result in an angular displacement
of the instrument which can easily reach up to $47^{\circ}$ when played in a
standing position and up to $36^{\circ}$ when played in a sitting position
[34]. To compensate for the movement of the instrument when referencing, we
consider it useful to equalise the directivity not to a point, but to the
average of a larger spherical surface area $A$. The distribution of
orientations over this surface can be taken into account by using a weighting
function, i.e., by calculating

$\hat{p}_{\text{area},r,m}=\frac{\hat{p}_{r,m}}{\sqrt{\sum\limits_{r_{A}=1}^{R_{A}}\hat{p}_{r_{A},m}^{2}\cdot
w^{\prime}_{r_{A}}\cdot g^{\prime}_{r_{A}}}},$ (14)

where $w^{\prime}_{r_{A}}$ and $g^{\prime}_{r_{A}}$ are the normalized area
and the two-dimensional function weights, respectively, with

$\sum\limits_{r_{A}=1}^{R_{A}}w^{\prime}_{r_{A}}\cdot g^{\prime}_{r_{A}}=1,$
(15)

and $r_{A}\in\\{1,2,...,R_{A}\\}$ are the $R_{A}$ grid points of the area $A$
over which the mean is to be calculated.

If no information about the position of the microphone during the recording is
documented, a diffuse equalised directivity (cf. equation 6) can be used to
minimise the sound colouration on average. For this reason, the directivity
averaged over one-third octave bands has been provided in this representation
for uniformed immediate use.

The extent to which the different referencing methods of the recording
position affect the perceived sound event and the room acoustic parameters
will be clarified in a later investigation on the basis of these data.

##  5 ACKNOWLEDGMENT

We would like to thank Stefan Feistel and Silke Bögelein (AFMG) for their
support in producing the GLL datasets.

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