PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Spectral and Mathematical Music Theory Analyses of the Ikoro Drum
Using Visualizations and Sonifications of Beat-Class Theory
Stephen G. ONWUBIKO1; Tracianne B. NEILSEN2; Andrea CALILHANNA3
1
2

University of Nigeria

Brigham Young University, United States
3

University of Sydney, Australia

ABSTRACTThe Ikoro, or Ekwe, is a wooden slit drum of the idiophone family. The Ikoro, which by virtue of its functions
and its predominance in the eastern parts of Nigeria with the Igbo tribe, plays an important role in the Igbo
musical culture and settings. The Ikoro produces a very deep rich bass sound, which can be heard from a
distance, and has a sustained reverberation important in rendering a background beat to music or as a soundof warning. This paper provides a multifaceted look at the features of Ikoro drum sounds. The historicalimportance and constructional process of the Ikoro are presented. A traditional frequency domain analysis isshown that indicates the sustained reverberation. In addition, the isomorphism between Ikoro music andmathematics are investigated by applying the narrative of mathematical music theory developed by Richard
Cohn (Yale). This work explores how the rhythm, meter and pitch of Ikoro music, can be represented throughusing various techniques such as visualizations and sonifications, by applying Cohn’s beat class set theory
with Andrew Milne’s (WSU) instruments of mathematical music theory, the cyclic graph XronoBeat, and
SkiHill apps. These analyses provide a deep understanding of the distinct sound of the Ikoro.
Keywords: African Drums, Ikoro, Musical theory

1. Introduction
For communication between humans and deity, humans and animals, and between people, the Igbo
people, in the South-east region of Nigeria, has historically relied on musical instruments. One specificdrum, that is central to Igbo music is the Ikoro or Ekwe, pictured in Figure 1. Ikoro drums are used insome Igbo areas of Anambra, Imo, Abia, Enugu and Ebonyi states of Nigeria. The Ikoro drums are amonumental symbol for the Igbo people and are normally positioned or mounted strategically for
optimal acoustic at the community’s public square (Ama) such as a market. An Ikoro drum is often
associated with a major deity or village government and is used to transmit coded and important
information to indigenes in the area, both before and after the advent of modern ways of disseminating
information.
Although the names Ikoro and Ekwe are used interchangeably—the name used generally varies
by tribe—each has distinctive characteristics. The Ikoro is made from the bore log of ufie (osisi obala),found in the Igbo region, also known as cam wood tree, while the Ikoro is made from mango tree/wood.
The different wood causes the Ikoro to have a deeper, bass sound, and the Ikoro to have a lighter sound.
The significant difference between the Ikoro and the Ekwe is the size: The Ikoro is typically larger thanthe Ekwe. In Nkwo market in Umunze, Orumba North Local Government Area of Anambra State,stands the biggest wooden slit drum in Igbo land, or perhaps, in the world (1): the Ikoro Obibiaku,
measuring about 9 feet in diameter and 8 feet long.

1stephen.onwubiko@gmail.com
tbn@byu.edu
3
a.calilhanna@gmail.com
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Figure 1 - a) An Ikoro being struck and b) and Ikoro.
The Ikoro holds a sacred place in Igbo culture. A community’s Ikoro is typically housed in a special
location. An example is shown in Figure 2 of where the ulo Ikoro is housed in Akoli-Imenyi of the
Bende local government area in Abia state Nigeria. An Oomu (palm front) on the door indicates that
the Ikoro is sacred and should be reverenced at all times. Traditionally, only men in general and malechildren are allowed to play the Ikoro in conjunction with specific ceremonies and festivals.
Although, the Ikoro and its music are generally representative of the culture of the Igbo tribe, theIkoro is mainly associated with the upper group in their stratified society. In most parts of Igbo land,
the Ikoro is typically employed in instrumental music associated with leaders and highly respected
people. For example, the Ikoro functions as one of the principal instruments of Ufie music in
traditional settings, limited to Ozo title taking, coronation of a king (Iwgwe), the Ofala festival,
funeral of a titled person, and the Iguaro or Iwaji festival. Nzewi (2) classified this type of music as
Event-Music with the subheading Egwu Ogalanya (Title music).

Figure 2 - The ulo Ikoro is housed in this building in Akoli-Imenyi of the Bende local government area in
Abia state Nigeria. An Oomu (palm front) on the door indicates that the Ikoro is sacred and should bereverenced at all times.

2. Physical Characteristics and Construction of the Ikoro DrumThe basic form of an Ikoro drum is a big slit drum from a cylindrical tree trunk. Except for its huge
size and the monumental importance attached to it, the Ikoro has little or no difference from other slit
drum types in Igbo land. The Ikoro drums with extra projections and body decorations, in theirsimplest forms represent visual imagery of a male figure lying on his back. The abstract cylindricalmass of the drum forms the main body. It is on this cylindrical mass that a cavity is opened to produce
the Ikoro sound. The mouth of this cavity is in the form of two rectangular shapes joined by a narrowslit, as shown in Figure 1. Some areas around Umuahia (like Ohuhu) have examples of such Ikoro

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drums lacking extra projections or relieved designs on their bodies. This type, however, does notreally help us much in understanding the Ikoro because it is in the minority.

3. Results and Analyses
The sounds of the Ikoro drum are examined to quantify the difference between the forward andbackward tones using a spectral analysis and to identify the rhythm, meter, and pitch using musicalbeat-theory. The audio was extracted from a video of the Ikoro drum played for 12 seconds at a tempo
of approximately q= 120 beats per minute. The Ikoro was struck in the forward-backward pattern fivetimes followed by the backward-forward pattern five times. The audio, shown in Figure 3a, wassaved as a WAV file, which scaled the amplitudes of the recording between +/- 1, thus all amplitudes
and levels herein are relative to this scaling. The prominent resonances sound like C3 (Low) and E3(High).
3.1 Spectral Analysis
The acoustical differences between the sounds from the forward and back striking locations can beseen in both the waveforms and the spectra. The waveforms and spectrograms for the entire
recording are shown in Figure 3. The division between striking location is annotated in the waveformin Figure 3a. The corresponding spectrogram is shown in Figure 3b, which shows how the
spectrum—frequency content—varies with time. The spectrograms show clearly the difference inhigher frequency content between the forward and back sounds, Figure 3c provides a zoomed inversion that is more useful for seeing the lower frequency differences. Differences at low frequencies
correspond to the perceived change in pitch, while the difference over the rest of the frequencies
accounts for the differences in timbre or tone color. To look more closely at the difference between theforward and back striking locations, zoomed in versions are shown for two note pairs. The
waveforms and spectra from one of the forward-back sets are displayed in Figure 4.
The waveforms and spectrograms from the Ikoro drum recording show differences in the sounds
due to striking location. The waveforms in part a) of Figures 5-6 show that the sound amplitudes forthe forward striking location are longer and show a more gradual decrease than the back location.
The waveform amplitudes for the back location has an initial decay followed by a slight increase inlevel before the final decay. The spectrograms in parts b) and c) of Figures 5-6 show these same
temporal characteristics and also show differences in the frequency content of the two sounds. Thereis a definite shift in low frequency content between the two: the forward striking location produces asound with large levels at lower frequencies than the backward striking location. This changes causesa shift in the perceived pitch between the two notes with striking in the forward location sounding
lower than striking in the back. In addition, the high frequency content of the two sounds are different.
The forward location produces a tone with more energy in the 1000-2000 Hz band, but the back tone
has more energy above 2000 Hz.
The difference can be seen more easily when average spectra are compared, as in Figure 5. The
spectra shown are the average of the spectra for each tone’s most steady section from the five times thetone is struck first (in Figure 3). The comparison of average spectra in Figure 5 shows the main spectral
differences between the forward (blue) and back (red) locations. The lowest peak in the spectrum istypically related to perceived pitch. (Note, all the frequencies presented herein are approximate due tothe 3 Hz resolution in the spectrum.) For the forward case, the lowest frequency peak is at 140 Hz andis larger than any other spectral peak by at least 10 dB. For the back case, the lowest peak is at 158 Hz,
but the largest peak is at 205 Hz. (For comparison, here are the frequencies associated with notes ona piano, C3 = 131 Hz, D3 = 147 Hz, E3 = 165 Hz, F3 = 175 Hz, and G3 = 196 Hz, and A3 = 220 Hz.).
The peak frequency for the forward case of 140 Hz lies between C3 (131 Hz) and D3 (147 Hz), while
the lowest peak frequency for the back location of 158 Hz lies between D3 and E3 (165 Hz). The 18 Hz
difference between these lowest frequency peaks corresponds approximately in this octave to theinterval of a major third. The 205 Hz frequency of the highest peak in the average spectrum indicatesthat the sound for the back location has a strong frequency component between G3 (196 Hz) and A3(220 Hz), which likely contributes to the perception that the back sound is a fifth above the forward
sound.

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a)

b)

c)

Figure 3 a) Waveform and b) spectrogram of the first recording showing a series of strikes on the Ikoro drum.
The first five pairs have the forward tone first followed by the back tone. The second five have back andthen forward. Zoomed in version of the spectrogram in c). These amplitudes and levels are relative levels
after the sound from videos was converted to a WAV file, which scales waveform amplitudes between +/- 1.
3.2 Rhythm, meter and pitch
The rhythm, meter and pitch of the Ikoro drum sounds are analysed using Cohn’s mathematicalmusic theory (3) the first theory of meter to focus on the measurable qualities of both music andmathematics in the psychoacoustic experience of listening to music. To do this, we introduce
visualizations and sonifications of mathematical music theory (beat-class theory) to analyse the musicof the Ikoro drum to determine the meter and rhythm and, in conjunction with spectral analysis, we
determine the mathematical properties of pitch internalized by the listener. Then, the spectral analysis
also examines in detail quantifiable elements, such as, volume, frequency, and timbre.
3.2.1 Background
Cohn’s beat-class theory represents modern meter theory instead of, for instance, propagating
notation-based understandings of meter and rhythm, which propagate inaccurate quantifications ofmeter. Traditional, notational-based meter theory doesn’t recognize all metric levels and only twopulses are recognized by the meter signature. Cohn’s modern meter theory, on the other hand,recognizes all of the pulses the listener hears because it is based on the understanding that meter is “aninclusionally related set of distinct, notionally isochronous time-point sets” and it acknowledges
research which indicates that musical meter is located in the listener as an embodied psychoacoustic
or “mind and body” response to the stimulus of sound (4). In a sense, Cohn’s theory provides auniversal theory of music because it is based on mathematics, which can be applied to the embodied
psychoacoustic experience of music from around the world. Cohn’s beat-class theory ( 5 , 6 ) isvisualized through the instruments of mathematical music theory: the ski-hill graph (7) and the cyclic

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graph. Computerized versions of these—the SkiHill app (8) and XronoBeat (9)—enable listeners torecord the quantifiable mathematical properties they experience subjectively when listening to music.

a)

b)

c)

Figure 4 a) Waveform and b) spectrogram from a forward then back strike on the Ikoro drum. Zoomed inversion of the spectrogram in b).

a)b)
Figure 5 Comparison of the average spectra for forward and back striking.

The ski-hill graph, which is a two-dimensional matrix designed by Cohn to represent hemiolas, isarguably the most compact and efficient instrument of music theory for the listener to quantify meter(10). The nodes of the ski-hill graph are empty to represent that meter is understood as soundexperienced rather than as the notation. The listener has the option to choose to map the quantified
results of their psychoacoustical embodiment of sound through mapping the pulses they hear that form

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meter as empty nodes, traditional notation, fractions, polygons, and a choice of sonifications (seeFigure 3). Unlike linear-notation, by mapping adjacent pulses in a relation of inclusion in a ratio of
2:1 duple meter to the left direction and adjacent pulses in a relation of inclusion in a ratio of 3:1 triple
meter to the right direction, the two different metric pathways or subdivisions provide a useful
distinction visually and through sonifications (in the computerized version). Minimal meters areformed in any music where there are duple and triple minimal meters such as those heard in the Ikoro
music of the following analysis. Meter is represented as ordered set notation where minimal duple
meter is represented as <2>, minimal triple meter as <3> and deep meter as, for example <232>.
The cyclic graph and its computerized version XronoBeat (9) also measures the mathematicalproperties of meter but in a circular graph which, unlike the ski-hill graph, also has the capacity torepresent rhythm and pitch with integers, polygons and sonifications. Rhythm is defined here as
timepoints selected from the metric hierarchy and pitch is defined as frequency. Meter, rhythm andpitch are all presented as sets on cyclic graphs in visualisations of beat-class theory as integers,
polygons, and sonifications. In this way rhythm, meter and pitch are quantized through the listener’s
own reporting of these elements as sets using beat-class and pitch-class theory and spectral analysis invisualizations and sonifications.
3.2.2 Application
The results of using the SkiHill app (8) and the XronoBeat (9) on the Ikoro drum sounds are now
shown. According to the SkiHill graphs, shown in Figures 1a)-c), the Ikoro drum music exhibits themeter <232> with adjacent pulses in a relation of inclusion in a ratio of 2:1, and minimal triple meterto the right pathways, adjacent pulses in a relation of inclusion in a ratio of 3:1. Figure 1a) represents
the meter with traditional staff notation (and sonifications). Figure 1b) represents a hearing of themeter through fractions and polygons which represent a C12 cycle through divisions, periodicity, andsonifications. The results from XronoBeat (9) are shown in Figure 1c) are cyclic graphs, where
beat-class theory is represented in sets as integers, polygons, and sonifications representing C12{0-11}.

a)b)c)Figure 6 - Representations of the deep meter <232> a)-b) ski-hill graphs; c) polygons; d) cyclic graph withintegers and polygons.
The selected nodes and their edges in Figure 3 form a metric pathway to represent adjacent pulses
in a relation of inclusion in sets (a set refers to pairs of pulses where one of the notes is either two or
three times faster or slower than the other) of minimal duple meter in a ratio of 2:1 sloping to the left
direction, such as,orand minimal triple meter in a ratio of 3:1 sloping to the right

directionor. Unlike pulse stacks in a linear-vertical orientation where the minimal meters
can be identified in a straight line, mapping the pulses duple meter to the left and triple meter to theright pathways provides an additional benefit of a representation which differentiates the meters
visually as distinct duple and triple meters and pathways. Representation of a deep meter on a ski-hillgraph such as <232> means the relationships between the pulses forming duple and triple meters canbe observed more closely and studied as isomorphic representations of the embodied experience ofmathematics. The integers in Figure 1c) <232> represent the depth of the meter in the metric hierarchy
experienced from listening to this piece. In the deep meter <232> the span pulse is worth two of its
next fastest (adjacent) pulse in the metric hierarchy, six of the next fastest pulse, and twelve unit or the5548

fastest pulse. Thus, the ski-hill graph enables the listener to articulate the embodied mathematicalexperience of musical meter without having to default to traditional Western notational understandingsof meter (10).
A summary of the temporal evolution of the 12 s Ikoro drum recording in shown in the timeline in
Figure 4. The data documented (bottom to top) in Figure 4 represents observations of the following
elements: the rhythm in traditional notation; integers for C3 (beat-class theory); a timeline; the onsets
as ten sets of two pulses as X’s; the listener’s subjective accents; an indication of the two pitches as L
(Low) and H (High); the numbers for counting meter are based on the number of pulses heard for each
timepoint (as indicated by the pulses stacked above in the metric hierarchy section); the top four lines
are the pulses a listener can observe from the fastest (bottom) to slowest (top) in the metric hierarchy.
Above the lines is an indication of the metric displacement the listener hears and the minimal meters
are listed to the left as pairs of pulses in a vertical stack. The metric hierarchy is represented through an
indication of the minimal meters through a pulse stack with 2 representing duple meter and 3
representing triple meter and numbers for counting meter. An approximation of the pitch is illustrated(L is approximately C3 and H is approximately E3). The notionally isochronous time point sets
(pulses) in the metric hierarchy section of Figure 4 are those which can be observed from hearing therepetition of the rhythm of the Ikoro drumming. Through projection and entrainment the parallelism(MPR 1) experienced through listening to the rhythms, initiates inclusionally-related sets of pulses inratios of 2:1 or 3:1, to form minimal and deep meters.

Figure 7 Timeline and initial analysis (‘note-taking’) for clues to analyse the rhythm, meter and pitch.
From listening closely to the Ikoro drumming solo and quantifying the rhythms, meter and pitch
through visualizations and sonifications of Cohn’s beat-class theory, many important details ofpotential cultural significance were discovered that would normally go unnoticed. For instance,
through ski-hill graphs it was possible for a listener to map all of the pulses that formed part of their
psychoacoustic experience of meter and mathematics. From quantifying the pulses through pairing in
2:1 or 3:1 ratios, minimal meters could be mapped to a metric hierarchy in distinct duple and triple
metric pathways to form deep meter. The meter, rhythms and pitch were then mapped to cyclic graphs
so as to further quantify the Ikoro drumming solo through examining the cyclical hierarchy, periodicity,
sets, ratios, and divisions through polygons to reveal an otherwise hidden metric and tonal structure.
The value and potential of Cohn’s mathematical music theory to quantify details about Igbo culture,including music and everyday sounds, is immense. In this way Igbo cultural practices andcontributions, which are at risk of disappearing through a rapidly changing world, can be preserved for
future generations. Recordings, spectral analysis, written accounts and photographs are all important
for the preservation of a culture, however, the importance of the role of the listener in preserving theaural tradition through quantification of music and sound for future generations, should no longer beoverlooked.

4. Conclusion
The Ikoro drum plays a central role in Igbo culture. Thus, an important part of preserving their5549

musical heritage is to document the musical and acoustical characteristics of the Ikoro drum. These
characteristics have been studied using acoustical signal analysis and Cohn’s beat-class theory.
The musical characteristics of the Ikoro drum include the meter, rhythm, and pitch. If this rhythm
were to be analysed and notated solely using traditional staff notation, or, as a linear arrangement, a
number of important structural elements would go unnoticed. Cohn’s beat-class theory, therefore, canprovide more accurate accounts of music such as this study of the music of the Igbo’s of Nigeria.
Thus, although the listener quantifying the Ikoro pitch couldn’t hear A3 a machine detected it and thesix-note scale was important to study. In a mathematical environment it was possible to study bothmeter and rhythm with pitch where before none of the music was literally notated only ‘felt’ or
projected and entrained to. Most of the pulses were experienced in the imagination unseen and in asense unheard but ‘felt’.
In approaching musical analysis through visualisation and sonifications of Cohn’s beat-class theory
the listener articulates their own quantifiable experience of music through the scientific examinationof music, such as in our study of the music of the Igbo’s Ikoro music of Nigeria. The capacity for the
ski-hill graph, to delineate between duple and triple meters provides a powerful reason to include
ski-hill graphs in the analysis of any music wherever duple and triple meter are evidenced in thepsychoacoustical experience of the listener. We would argue that Cohn’s mathematical music theory
should be included in classroom music textbooks and online resources so that the cultural significanceof music such as the Igbo’s can be appreciated and preserved for future generations.

References

1

Chijoke Onuora (2016) Ikoro drums among the Igbo: Iconology and design symbols.
https://www.atdin.org.ng/wp-content/uploads/2018/04/UNN-FAJH49.pdf Viewed 24 May 2019.
2
Nwezi, M. (1999) Strategies for music education in Africa: Towards a meaningful progression fromtradition to modern. International Journal of Music, os-33 (1), 72-87
3Cohn, R. (1998). Music Theory’s New Pedagogability. Music Theory Online, 4 (2).
4Cohn, R. (2018a: In press). Meter. Rehding, A. & Rings, S. (eds.) The Oxford handbook of criticalconcepts in music theory. Oxford: Oxford University Press.
5
Cohn, R. (2016d). Teaching atonal and beat-class theory, modulo small. Brazilian Journal of Music andMathematics, 1 (1), 15–24.
6Cohn, R. (2018b). Scaling up to Atonality: The Pedagogy of Small Cyclic Universes. Montiel, M. &Gómez, F. (eds). Visualizing and sonifying mathematical music theory with software applications:
Implications of computer-based models for practice and education. In Theoretical and Practical Pedagogyof Mathematical Music Theory. Music for Mathematics and Mathematics for Musicians, From School to
Postgraduate Levels. World Scientific Press.
7
Cohn, R. (2001). Complex Hemiolas, Ski-Hill Graphs and Metric Spaces. Music Analysis, 20 (3),295-326.
8
Milne, A. J. (2018). Linking sonic aesthetics with mathematical theories. The Oxford Handbook of
Algorithmic Music, 155-180.
9
Milne, A. J. (2019a). XronoMorph: Investigating paths through rhythmic space. In New Directions inMusic and Human-Computer Interaction (pp. 95-113). Springer, Cham.
10
Calilhanna, A. M. (2018). Teaching Musical Meter to School-Age Students Through The Ski-Hill
Graph (Master's thesis, University of Sydney).

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