WIDE-SENSE CLAVE: TOWARDS A TRANS-DIASPORIC 
RHYTHMIC GRAMMAR 

 
 
 
 
 
 
 

by 

REILLY GAULT 

 
 
 
 
 
 
 
 
 
 

A THESIS 

 
Presented to the Department of Jazz Studies  

and the Robert D. Clark Honors College  
in partial fulfillment of the requirements for the degree of  

Bachelor of Arts 
 

December 2025 

 



 

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An Abstract of the Thesis of 

Reilly Gault for the degree of Bachelor of Arts 
in the Department of Jazz Studies to be taken December 2025 

 
 

Title:  Wide-Sense Clave: Towards a Trans-Diasporic Rhythmic Grammar 
 
 
 

Approved:  Dr. Juan Eduardo Wolf  
  Primary Thesis Advisor 

 
This thesis begins to develop a systematic cross-cultural framework for wide-sense clave, 

proposing that the rhythmic organizing principles most famously found in Cuban tradition are in 

fact only a single manifestation of a deeper concept shared across Afro-Diasporic musical 

traditions. Building on prior scholarship, the project argues that clave should be understood not 

simply as a regionally bound pattern but as a trans-diasporic logic for evaluating rhythmic 

alignment, orientation, and structural coherence. In part, this thesis is a proposed answer to 

Lehmann’s request for a “hierarchical…contour…for the short-range directionality of the 

clave/timeline period.” (Lehmann, 2002: 71) The first part of the thesis critiques Toussaint’s 

pulse-saliency approach to quantifying meter in African music, resulting in a corrected statistical 

analysis that accounts for clave direction, revealing consistent hierarchical syntactic weights 

across traditional timeline patterns from around the African Diaspora. These weightings form the 

basis of an analytical methodology capable of systematically classifying rhythms, identifying 

clave-neutral or clave-ambiguous rhythms, and potentially extending clave analysis beyond 

typical 16 and 12 pulse rhythmic cycles. This thesis will also propose future avenues of research, 

with particular attention to methodological approaches that may account for clave as a 

harmonically emergent phenomenon. The latter two sections of this thesis translate this 

framework into public-facing and performance-based forms: the beginnings of a comprehensive 

online rhythm repository designed to make clave concepts accessible to musicians and non-

musicians globally, and a senior recital demonstrating wide-sense clave across traditions from 

Cuba, the Dominican Republic, Bolivia, Guinea, Brazil, and multiple regions of Ghana. 

 

  



 

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Acknowledgements 

 

I would like to express my gratitude to Dr. Juan Eduardo Wolf, whose patience was only 

exceeded by his willingness to discuss my theories, entertain my increasingly bold claims about 

the mathematical structure of rhythm, and then play some bomba. I am likewise grateful to my 

CHC Representative and advisor, Carol Stabile, for helping me navigate through the CHC and 

always being so supportive and excited about my progress. I am indebted to the wonderful jazz 

faculty at the University of Oregon, whose flexibility and belief in my niche allowed me to 

pursue one petition-heavy rollercoaster of an undergraduate education. I am likewise indebted to 

the Stamps Scholars foundation, who funded my schooling and my research in Ghana. I am 

deeply grateful to all the musicians, masters, and teachers that I have learned through and from 

over the years, whose playing inspired my obsession with rhythm long before this project had 

even been thought of. To the Good Fight and University Fellowship Church, thank you for 

always pointing me to Christ; the only one who truly lived en clave, died in my place, and was 

resurrected to give me new life. Thank you for always reminding me what and who music, math, 

and clave, ultimately point to. Thanks to all my friends who have endured years of clave 

demonstrations, unsolicited rhythm explanations, showcases, and comparisons, the sounds of 

drumming coming from my room at all times of day, and my countless “wait, play this rhythm 

really quick” moments. To my family, thank you for encouraging me throughout this whole 

process. Thank you for starting me on my unique musical journey. Thank you for always 

cheering me on, bolstering me in my unproductive days, and trusting me when I said that I was 

writing a thesis about “hyperdimensional record players.”  

Reilly Gault 
Eugene, Oregon, Fall 2025 



 

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Table of Contents 

 

Chapter 1: Stumbling Across Clave in Ghana 6 
Chapter 2: What is clave? 11 

Clave Analogies 14 
Chapter 3: The Problem of Counting Without Direction: Revisiting and Correcting Toussaint’s 
Model 19 
Chapter 4: A Methodological Framework for Evaluating Clave Orientation 27 
Chapter 5: Observations for Future Research 33 
Chapter 6: Concluding Remarks 39 
Bibliography 41 

 
  



 

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List of Figures  

Figure 1 Vurkaç’s graphical analogy for clave 16 
Figure 2 Pulses in a 16-pulse clave cycle and their names 18 
Figure 3 Pulses in a 12-pulse clave cycle and their names 18 
Figure 4 Generative Theory of Tonal Music metrical hierarchy 20 
Figure 5 Pater Noster pulse saliency histogram 21 
Figure 6 Common Practice music pulse saliency histogram 21 
Figure 7 German folk song pulse saliency histogram 22 
Figure 8 Afro-Diasporic timeline pulse saliency histogram 22 
Figure 9 Toussaint’s timeline dataset 23 
Figure 10 Toussaint’s timeline dataset with directional labeling 24 
Figure 11 Corrected Afro-Diasporic timeline pulse saliency histogram 25 
Figure 12 Frequency tier values of the 16-pulse clave model 27 
Figure 13 Weight values of the 16-pulse clave model 28 
Figure 14 Pulses of the 16-pulse clave model with primary and secondary pulses labeled 29 
Figure 15 Frequency tier values of the 12-pulse clave model 29 
Figure 16 Weight values of the 12-pulse clave model 30 
Figure 17 Pulses of the 12-pulse clave model with primary and secondary pulses labeled 30 
Figure 18 16-pulse clave model with Cuban son clave in the 3-2 orientation 31 
Figure 19 16-pulse clave model with a rhythm in the 2-3 orientation 32 
Figure 20 12-pulse clave model with a rhythm in the 3-2 orientation 32 
Figure 21 16-pulse clave model with symmetric clave neutral rhythm 33 
Figure 22 16-pulse clave model with asymmetrical clave neutral rhythm 33 
Figure 23 12-pulse clave model with asymmetrical clave neutral rhythm 34 
Figure 24 12-pulse clave model with a second asymmetrical clave neutral rhythm 34 
Figure 25 12-pulse clave model with a third asymmetrical clave neutral rhythm 35 
Figure 26 12-pulse clave model with African ‘standard pattern’ 36 
Figure 27 The seven diatonic modes along with their 12-pulse clave model analysis 38 
 
 
 
 



 

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Chapter 1: Stumbling Across Clave in Ghana 

It was July 20th, 2023. I was in Accra, Ghana, learning Ewe drumming from one of my 

masters, Gasco Atsu Ablordey, whom I had just recently met. We were sitting nearby the 

Performance Studies area of the University of Ghana in Legon, each of us manning an 

atswereshi drum. On this day in particular, Gasco was helping me learn the famed Ewe piece 

known as Gahu. He would play a phrase on an instrument, and I would repeat it until I could 

play it accurately without his help. The phrases started off as simple and became increasingly 

complex as we moved from the supporting instruments to the lead drum. All of this initial 

learning was done in isolation from other instruments; there was no bell or rattle that I could use 

as a time referent. I was stuck with only the phrases that I was given, seemingly in a vacuum.  

A few days later, the second time I met with Gasco, he had brought with him another 

student around my age, from the University of Ghana, who had agreed (willingly or unwillingly) 

to play a bell pattern for the duration of my lesson. To my joy, and Gasco’s surprise, I managed 

to play all the lead drum phrases and variations in the correct alignment with the bell phrase, on 

the first try. This is something that is notoriously difficult for foreigners to achieve, and Gasco’s 

reaction reflected that humorously. After that moment, he began to affectionately refer to me by 

the nickname “International Drummer!” I was initially shocked by his response, it didn’t seem 

like much of a difficulty to me. The relationship between the interlocking lead drum and bell 

phrases felt intuitive and natural to me; in fact, I didn’t give it much thought at the time, my 

hands just began to move when I felt they needed to. 

After working on Gahu, we moved to other pieces like Atsiagbekor and Borborbor, 

which I picked up just as easily, receiving more praise and excitement from my master each 

time. One day Gasco brought five others with him (as witnesses, I later found out), sat me down, 



 

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and told me that he wanted to teach me something very difficult. “Agbadza,” he explained, “is a 

rhythm that is very difficult, even for natives to learn.” At the University of Ghana, they teach a 

‘watered-down’ version of Agbadza to the students over a period of three months. Then, looking 

me directly in the eyes, he said, “I want to teach you the real Agbadza, the indigenous Agbadza.” 

I shuddered with excitement and anticipation at this honor and challenge, it seemed I even had an 

audience! 

This time, we started with the bell pattern already playing, and Gasco played a lead drum 

phrase on the sogo that I tried my best to imitate. At first, it caught me off guard, it was oriented 

in a way that I didn’t expect, but once I got used to it, it aligned perfectly with my already 

present intuitions. Gasco beamed with glee and those watching cheered with surprise and 

excitement and began dancing Agbadza. After a few more lessons specifically on Agbadza, 

Gasco honored me further by inviting me to perform Agbadza with his cultural troupe at two 

Ewe funerals, with me playing sogo, the lead drum. 

The reason I share this is not to show my own musical prowess, but to highlight how 

something deeply ingrained in my musical intuition enabled me to pick up these difficult 

rhythms and phrase relationships with ease. This instinct, as I later found out, came from my 

enculturation in clave-based music. 

At the time, I happened to be reading a paper by Mehmet Verkaç titled A Cross-Cultural 

Grammar for Temporal Harmony in Afro-Latin Musics: Clave, Partido-Alto and Other 

Timelines. I came across this quote: 

Identifying clave direction is a challenging task for anyone who has not been 
raised with clave–based music, and sometimes even for those who have. 
Moreover, those who have been “raised with clave” often find it difficult to 
explain why clave works the way it does. It is not uncommon to hear that “it’s in 
the blood.” However, given the current scientific understanding of the interaction 



 

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of nature and nurture, it is far more likely that clave is a rational principle that can 
be described in a quantitative manner. (Vurkaç 2013: 49) 

I grew up listening to a wide variety of Caribbean music, and I started seriously learning to play 

it late in High School. Unbeknownst to me, this early engagement had already started structuring 

my musical intuition around clave-based music. I never would have imagined that all the years of 

listening to music, learning different rhythms, clapping along to salsa, and slowing down 

YouTube videos would start to embed me in a sense of temporal grammar that I would candidly 

encounter in Africa.  

When I read Vurkaç’s quote, the phrase “it’s in the blood” resonated immediately. 

Musicians in both Latin and West African traditions commonly explain their musical fluency as 

something innate or ineffable. Yet my own experience contradicts that narrative. I had not been 

raised within an Ewe community; I had no cultural lineage tying me directly to Agbadza; and 

yet, the logic of the interlocking patterns felt immediately understandable to me. 

I also brought with me on my trip Godfried Toussaint’s book The Geometry of Musical 

Rhythm: What Makes a "Good" Rhythm Good? In this book, which will be discussed in further 

detail later, Toussaint employs numerous mathematical methodologies to attempt quantifying 

various musical phenomena pertaining to rhythm. At some point after reading Vurkaç’s paper, 

and continuing with my musical endeavors in Accra, I read chapter 18 of Toussaint’s book, and I 

found something that shocked me. Toussaint attempted to make a pulse-saliency analysis of 

select Afro Diasporic timelines: a histogram intended to show which rhythmic positions are the 

most salient among timeline patterns, or which pulses function as strong anchors. But when I 

worked through his data set, I found a significant oversight with major implications. About half 

of the timelines that he included in his comparative set were oriented backwards relative to the 

others. This effectively flipped their apparent saliency and produced a misleading cross-pattern 



 

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comparison. In other words, his histogram was comparing some patterns that were head-first, and 

others that were tail-first. The results mixed two incompatible directions and produced false 

symmetries. That mistake mattered. Toussaint was trying to understand the intuitions of 

musicians in a quantitative way, but his starting point in that analysis failed to even incorporate 

the intuitions he was striving to quantify!  

After my initial shock, this mistake became an inspiration and compass for my 

subsequent work. It suggested a very concrete starting place: fix the orientation problem, expand 

the database, formalize a consistent way to measure pulse saliency while recognizing timeline 

directionality, and then test whether this model aligned with performers intuitions, including my 

own. 

By August 29th, 2023, those corrections and additional ideas had coalesced into a 

working model. I sketched my framework in a notebook and quickly validated it against all the 

Ewe pieces I had learned with Gasco, and all the Ga and Akan pieces I had learned with one of 

my other masters, Daniel Brown. My model matched my own, and my masters, musical 

intuitions perfectly! A couple weeks later, when I moved north to Tamale to study Dagbani 

drumming, I tested the model again, in a completely new musical environment. Once more the 

model perfectly matched my own intuitions, as well as the intuitions of my masters. That 

moment was when I realized that what I had stumbled upon was not just a possible explanation 

for my isolated experiences in Accra, but the beginnings of a possible broader framework, a way 

to formalize something that musicians have been describing as “natural,” “intuitive,” or “in the 

blood” for generations. 



 

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The rest of this thesis emerges from that realization. Before presenting the model itself, it 

is necessary to clarify relevant foundational concepts. The following section introduces the 

definitions, terminology, and theoretical assumptions that comprise the proposition itself. 



 

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Chapter 2: What is clave? 

According to Kevin Moore, clave has a reputation of being one of the “most infamously 

confusing topics in all of music.” (Moore, 2011: 12) In his dedicated book Understanding Clave 

and Clave Changes, Moore seeks to demystify the concept of clave as it pertains specifically to 

Cuban tradition. David Peñalosa, in his books The Clave Matrix and Rumba Quinto, likewise 

makes a significant contribution to understanding Cuban clave in popular and traditional 

contexts. Because the relevant popular discourse has been largely publicized strictly in the Cuban 

context, I have elected to adopt much of the terminology that Moore and Peñalosa use, for the 

sake of clarity and consistency.  

Given the diversity of meanings attributed to the word clave, it is important to define 

precisely how I will use the term in this thesis. The following definitions from Vurkaç help begin 

to lay this out: 

The word “clave” has several meanings, so it is important to specify which I am 
addressing here. Three of these meanings are well–known: (1) claves, the 
instrument; (2) clave in the harmonic sense, as in “the key of C major” (in 
Spanish); and (3) a small family of patterns, which is what musicians typically 
have in mind for clave. (Vurkaç 2013: 38) 

A fourth category that I would like to define is what I will refer to in this thesis as Cuban clave. 

Lehmann offers a helpful definition: “[Cuban] clave operates as a sort of “deep structure” of 

melo-rhythmic organization within the psychological, cognitive and aesthetic reality of Cuban 

music.” (Lehmann 2002: 11) This specific definition and understanding of clave is what authors 

like Peñalosa and Moore have primarily focused on. 

The fifth and final category that I propose is what Vurkaç refers to as “clave direction” 

and “wide-sense clave,” Lehmann refers to as “a certain structural rhythmic universality or 



 

12 
 

essence,” and Fiol refers to as “the hidden rhythm.” Vurkaç offers a very clear explanation of 

this broader concept: 

Clave direction is a concept and a rhythmic–regulative principle. It is analogous 
to the key of a piece of music, but instead of governing tonality, it governs fine–
scale local timing… Clave direction gives a composer or improviser a set of 
preferred timing options and the appropriate places within each phrase to place 
them…along with acceptable ways to break these rules… Just as the key 
determines the tonal center of the piece, the clave direction is the overarching 
determinant of the timing preferences for the piece, but like its tonal counterpart, 
it allows for variety in musical expression, and even tricks, puzzles and multi–
layered playfulness in its execution. In order to make the most artistic and 
culturally sensitive use of this principle, I offer an approach to understanding 
clave direction that I call wide–sense clave. In this wider sense, clave–the–
concept denotes not just the existence of specific patterns, but the relationships 
that any pattern may have with a family of associated patterns. [emphasis mine] 
(Vurkaç 2013: 38) 

Fiol reflects upon a profound realization of this reality, similar to my own in his anecdote: 

…my Afro-Cuban background allowed me to converse with them [master 
drummers] in a musical tongue far more ancient and profound than any bits of 
speech we could have exchanged. That language, the hidden rhythm uniting West 
Africa's musics with those fashioned in its Diaspora, is called clave in Cuba. My 
already developed knowledge of the clave's underlying structure made it possible 
to intuit the best way to riff atop Olatunji's masterful rhythmic bed. In that 
moment of near familial recognition, when I felt us all in some kind of zone, I 
began to understand how intimately Afro-Diasporic music has always remained 
connected to the "Motherland." [emphasis mine] (Fiol 2007: 1) 

Lehmann articulates his own understanding of this phenomenon: 

…a deep-structural grammar of ‘clave’ has perhaps more cognitive salience, and 
thus analytic-theoretical relevance, than usually recognized. I propose that such a 
grammar can apply to musical situations beyond the particular context of Cuban 
musical expression, and to idioms without an externalized timeline… ‘clave’ – 
rather than being just one equal component in a polyphonic musical texture – is 
experienced as a fundamental, internal structural dynamism, which for the 
enculturated listener and performer imbues the music with a particular kinetic 
energy and contrapuntal organization, extending into both temporal and tonal 
dimensions…the notion of a ‘clave’ as an African cognitive construct or “belief” 
points beyond its particular conceptualization in Cuban culture. ‘Clave’, I assert, 
has important theoretical implications to the body of African musical practices in 
general… [emphasis mine] (Lehmann 2002: 12) 



 

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Unless specified otherwise, this broad structural meaning is how I use the term clave in this 

thesis. The main idea is that there is a shared “hidden rhythm,” a “cognitive construct,” a 

“contrapuntal organization” that exists in a wider sphere of Afro-Diasporic musical tradition, 

with Cuba representing only one instantiation of this broader phenomenon. Clave isn’t a merely 

surface level abstraction or outside concept being imposed onto African music. It has to do with 

the very DNA of how Afro-Diasporic music is organized and generated. Furthermore, as 

Lehmann notes, “if the ubiquity of this structural phenomenon as one of the most enduring 

Africanisms in the music of many traditions in the diaspora may be taken as an indicator, we 

may be justified attributing more than superficial significance to its presence.” (Lehmann 2002: 

22) 

For this reason, it is especially important to make certain that the distinction between 

Cuban clave and wide-sense-clave is understood by the reader. The claim here is not that a 

“Cuban-centric” manifestation of clave and clave concepts should be imposed on the rest of the 

diaspora. Indeed, no one cultural manifestation should be seen as prescriptive of the whole. 

Rather, the claim is that there exists a shared structural principle that manifests uniquely in 

different cultures but remains ubiquitous at the level of temporal organization. The use of Cuban 

terminology to describe this trans-national phenomenon should not be confused with a claim of 

“Cuban-centrality.” (Lehmann 2002: 143) 

With the conceptual ground laid, we can now turn to an analogy that lays out clave in a 

more intuitive sense and touches on the concept of clave direction. I refer to this illustration as 

my “hyper-dimensional record player.” 



 

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Clave Analogies 

Imagine that you have a record player that allows you to vertically stack vinyl records, 

such that each record is spinning at the same rate, and the hyperdimensional stylus is making 

contact with each record in the same location simultaneously. Now imagine that each vinyl 

contains a recording of a single percussion instrument, playing an unchanging rhythmic pattern 

that repeats with each revolution of the record. If there are five records being played at once, and 

they all contain differing instruments and rhythms, the sounding result should be some sort of 

composite rhythm sounded by a seeming ensemble of five different musical voices. Now, to alter 

the sound of this ensemble, you are allowed to remove one of the records of your choice, while 

the remaining four continue to spin and play. This is somewhat easily done, it doesn’t matter too 

greatly which instrument you choose to remove, or when you decide to remove it. Say that 

you’ve decided to remove the top disc, and after doing so you can hear the subtle difference in 

the sound of the ensemble. Now, say you decide that you would like to place your vinyl back on 

the stack of spinning records. How can you possibly situate the record in your hands, so that, 

when you decide to place it back on the stack, it is aligned properly with the other records as they 

spin? 

Upon further examination, you notice that the face of the record that you’re holding is 

divided into two halves, one half with a red hue and the other with a blue hue. With a quick 

glance at the stack of still-spinning records, you now notice that, even though all the records are 

playing recordings of different instruments and rhythms, each has a blue tinted half and a red 

tinted half. You quickly realize that all the blue tinted halves are vertically aligned on the stack, 

as are the red tinted sides. Intuitively, you assume that if you place your record on the stack in 

such a way that its red and blue sides vertically align with all others, then the resulting rhythm 



 

15 
 

would be aligned. With this in mind, you place your record on the stack in the proposed 

orientation, and it begins spinning again, with its rhythm perfectly aligned. 

This is analogous to how clave-based music functions. Every asymmetrical rhythm (any 

rhythm that doesn’t contain two equal halves) has a half, to maintain the analogy, that is colored 

relatively red, and a side that is colored relatively blue. In order to properly orient any such 

rhythms being played simultaneously, their ‘red’ and ‘blue’ sides should vertically line up with 

each other. In Cuban terminology, this alignment is called played en clave. If any rhythm is 

misaligned such that its ‘blue’ side is aligned with all the other ‘red’ sides, its alignment is said 

to be crossed, or cruzado. This is akin to playing out of tune, rhythmically speaking. This 

concept of rhythmic polarity and bisectional alignment is what’s known as “clave direction” and 

understanding it intuitively is vital for the authentic performance of such traditions. (Vurkaç 

2012: 1, Lehmann 2022: 24) 

Vurkaç offers another analogy to help demystify this concept from a different angle: 

…a graphical analogy is provided below to present the idea in an alternative 
manner. Imagine a fictional culture in which visual decorations, no matter how 
elaborate, are generally arranged in a pattern of alternating round and pointy 
designs. Assume for the sake of argument that stars and circles have traditionally 
been used to provide the alternating pattern, and that the arrangement in Figure 
[1], line one, is called the “forward” pattern. A pattern with circles first and then 
stars would be “reverse.” Now imagine that an artisan wishes to expand his or her 
options beyond stars and circles, and introduces a rhombus. However, in order to 
remain true to tradition, he or she must consider whether a rhombus is more or 
less pointy than a star or a circle. The artisan concludes that a rhombus is less 
pointy than a star (Figure [1], line two), but more pointy than a circle (Figure [1], 
line three). The artisan has managed to stay within the tradition while introducing 
new elements. If other shapes are introduced, the keepers of the tradition who 
have internalized the rules of the design idiom would intuitively recognize the 
direction of a new example such as those in Figure [1], lines four and five. 



 

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Figure 1 Vurkaç’s graphical analogy for clave 

For clarification, the first three lines of Figure 1 are in the “forward” direction, and lines four and 

five are both in the “reverse” direction. The artisan introducing new shapes recognized the 

triangle in line 4 as more pointy than the pentagon and recognized the ellipse in line 5 as more 

pointy than the circle. 

Bringing this back to clave: the alternating “round” and “pointy” shapes in the analogy 

correspond to the red and blue halves of a vinyl disc spinning on the hyperdimensional record 

player. What this second analogy emphasizes is that the degree of “redness” or “blueness” is 

determined only in relation to the opposite half. One side isn’t absolutely red; it is simply more 

red than the other. Likewise, the other half isn’t inherently blue; it is just more blue than its 

counterpart. 

In other words, clave direction, like directionality in the shape analogy, emerges from 

relative contrast, not from a fixed property of each half. In Cuban terminology, instead of these 

halves being referred to as red and blue, or round and pointy, they are called the “three side” and 

“two side.” (Lehmann 2002: 24) 

A clave cycle can be understood, in general terms, as a recurring rhythmic span whose 

two halves are asymmetrical. In the hyperdimensional record-player analogy, each spinning disc 



 

17 
 

represents one full clave cycle. In Vurkaç’s shape analogy, a clave cycle consists of one pointy 

shape and one round shape. The requirement of asymmetry is essential: without a meaningful 

contrast between the two halves, no direction can be inferred. A record that is entirely blue, or a 

pair of identical circles, offers no basis for comparison. This kind of perfect symmetry eliminates 

the possibility of directionality altogether. For the purposes of this thesis, I will only be focusing 

on clave cycles divided into either twelve or sixteen relatively equal subdivisions, which, 

borrowing from Toussaint’s terminology, I will refer to as pulses. (Toussaint 2020: 104) Each of 

these pulses occupies a distinct position in the cycle and therefore merits a label. For the sake of 

clarity, I will lay out here a (largely Cuban) naming scheme for each of them that I have adopted 

from Moore and Peñalosa, with my own modifications and additions. (Moore 2011: 73, 2012: 

77-85)  

Let us begin with a 16-pulse clave cycle. Pulses are referred to in symmetric pairs; each 

pulse is grouped with its corresponding pulse in the opposite half of the cycle. Because our cycle 

is 16 pulses long, each half contains 8 paired positions. To remain consistent with Toussaint’s 

methodology, I will number these pulses 0 to 15, with those 0 to 7 being in the first half, and 

those 8 to 15 being in the second half. Thus, pulse 1 (𝑝𝑝1) corresponds to 𝑝𝑝9, 𝑝𝑝2 corresponds to 

𝑝𝑝10 and so forth. Within each pair, one pulse will be designated primary, and the other is 

secondary. This distinction will become much more relevant, and will be defined, when my 

model is presented. 

Pulses p0 and p8 are called the downbeats or frontbeats. I will refer to these with the letter 

‘F.’ The next pulses, p1 and p9 are called the quinto pulses, named after this pulse’s prominent 

featuring in quinto phrasing in Cuban rumba. These will be abbreviated as ‘Q.’ Pulses p2 and p10 

are called quemado, seco, or slap, named after the position of the slap stroke in the common 



 

18 
 

conga marcha pattern; I will denote them as ‘S.’ The pulses p3 and p11 are very important. They 

are called bombo, after the bass drum on a drum kit. This pulse is critical to many Latin music 

genres; I will refer to them with the letter ‘B.’ Pulses p4 and p12 are common in American music, 

usually known as the backbeat, or “two and four.” These will be marked with ‘K.’ The following 

pulses p5 and p13 are usually called either bota or cáscara and will be abbreviated ‘C.’ Pulses p6 

and p14 are also of vital importance; many Cuban musicians pair them with the bombo pulses. 

These pulses are called ponche and I will refer to them with the letter ‘P.’ Finally, p7 and p15 are 

often called platillo, named after a common rhythmic figure in Charanga Habanera’s music. I 

will denote these final pulses as ‘O.’ This is visible in Figure 2 below: 

 
Figure 2 Pulses in a 16-pulse clave cycle and their names 

Much of this terminology is applicable to a 12-pulse cycle as well, except that there are only six 

pulse pairs rather than eight. Pulses p0 and p6 are the frontbeats (F), pulses p1 and p7 are the 

quinto pulses (Q), pulses p2 and p8 are the bombo pulses (B), pulses p3 and p9 are the backbeats 

(K), pulses p4 and p10 are the ponches (P), and pulses p5 and p11 are the platillos (O). This 12-

pulse model is shown in Figure 3 below: 

 
Figure 3 Pulses in a 12-pulse clave cycle and their names 

Having now established a conceptual framework and clarified terminology, we can turn to 

Toussaint’s crucial work, and the key error that led to the development of my own model. 



 

19 
 

Chapter 3: The Problem of Counting Without Direction: Revisiting and 

Correcting Toussaint’s Model 

In Chapter 18 of The Geometry of Musical Rhythm, Meter and Metric Complexity, 

Toussaint begins by laying out four antiquated definitions of meter listed by Richard Cohn, and a 

fifth from G. Cooper and L. B. Meyer. (Toussaint 2020: 103) They are as follows: 

1- “Beats are … grouped into a regular repeating pattern of strong and weak. 
This is the meter.” 

2- “This pattern of stressed and unstressed beats results in a sense of metrical 
grouping or meter.” 

3- “Meter provides the framework that organizes groups of beats and rhythms 
into larger patterns of accented and unaccented beats.” 

4- “Meter is the arrangement of rhythms into a pattern of strong and weak beats.” 
5- “Meter is the measurement of the number of pulses between more or less 

regularly recurring accents. Therefore, in order for meter to exist, some of the 
pulses in a series must be accented—marked for consciousness—relative to 
others.” 

Toussaint surmises “… the notion of regularity of beats appears to be one of the salient features 

in most definitions of meter.” Given this understanding, Toussaint seeks to apply these concepts 

of meter to several questions, one of which is extremely relevant to this discussion: “the question 

of whether meter exists in African rhythm.” (Toussaint 2020: 103) Here, Toussaint only focuses 

on rhythms that fit inside a 16-pulse cycle. 

Toussaint uses what he terms a pulse saliency histogram to quantify meter. The method 

involves stacking a sample set of rhythms of equal cycle length such that their pulse positions 

align vertically, then counting the number of onsets, pulses that are “played” rather than rested, 

at each pulse position. The resulting frequencies indicate how often each pulse is activated across 

all included rhythms. By examining these values, Toussaint identifies which pulses occur most 

consistently and organizes them into a hierarchy of relative importance, a framework he calls 



 

20 
 

hierarchical meter (Toussaint 2020:104). Using this approach, he presents pulse saliency 

histograms for several Western musical repertoires and then compares them to a histogram 

generated from a selection of African “asymmetric timelines” in order to explore the question of 

meter in African music. 

As a comparative constant Toussaint introduces the theoretical General Theory of Tonal 

Music (GTTM) hierarchy which is as follows:  

Instead of measuring frequency on the y-axis like Toussaint’s other histograms do, this 

graph measures “metrical weight.” In essence, this graph is a theoretical framework for 

hierarchical meter, and the y-values are simply the priorities that are associated with each pulse, 

and not yet the “empirically observed frequencies of occurrences.” (Toussaint 2020: 107) 

  
Figure 4 Generative Theory of Tonal Music metrical hierarchy 

The first sample that Toussaint draws upon for a pulse saliency histogram is Palestrina’s 

Sixteenth-Century motet, Pater Noster as compiled by Joshua Veltman. The resulting graph is 

shown below in Figure 5:  



 

21 
 

 
Figure 5 Pater Noster pulse saliency histogram 

As Toussaint says, “The correspondence between this histogram and the GTTM hierarchy in 

terms of the ranks of saliencies of its pulses is visually striking.” He continues by showing a 

histogram compiled by Palmer and Krumhansl of a large sample of Common Practice music 

from composers J. S. Bach, Mozart, Brahms, and Shostakovich. This is shown below in Figure 6: 

 
Figure 6 Common Practice music pulse saliency histogram 

Once again, this histogram bears a striking resemblance to the GTTM hierarchy, save the 

exception that p12 is more frequent than p8. The final histogram of Western music that Toussaint 

offers is one compiled by Huron and Ommen of a select number of German folk songs. The 

graph is shown below in Figure 7: 



 

22 
 

 
Figure 7 German folk song pulse saliency histogram 

Once again, this graph very closely matches the priority hierarchy presented by the GTTM 

model. When Toussaint moves to his pulse saliency histogram for his selected African timelines, 

the results are quite different. The result is shown in Figure 8 below: 

 
Figure 8 Afro-Diasporic timeline pulse saliency histogram 

This graph doesn’t look like the other three histograms, or the GTTM hierarchy for that matter, 

at all. In fact, it seems that there is no apparent or stark pattern shown by these 34 timelines. 

However, Toussaint concludes that it still resembles a similar pattern to the GTTM hierarchy and 

Western music samples because it retains the “same perfectly alternating contour.” (Toussaint 

2020: 109) In other words, all histograms so far contain the same oscillation in frequency values 

or priorities. If we use the symbols “+” for an increase in value, “-“ for a decrease in value, and 



 

23 
 

“0” for a maintenance of value, then, starting with p0 the contour for Figures 4-8 would all be [-

+-+-+-+-+-+-+-+]. 

 
Figure 9 Toussaint’s timeline dataset 

A critical problem emerges, however, when we examine Toussaint’s dataset of 34 African 

timelines, shown in Figure 9. Although the collection initially appears coherent, a closer analysis 

with a clave-sensitive framework reveals a substantial flaw: nearly half of the timelines are 

oriented incorrectly. In other words, many of the patterns in Toussaint’s sample are effectively 

presented “backwards,” with their structural direction reversed relative to how they function in 

their originating traditions. This misorientation is not a minor or cosmetic error; it fundamentally 

alters the internal relationships between pulses and disrupts the very patterns whose metric 

properties Toussaint seeks to quantify. Because pulse saliency depends on aggregating vertically 



 

24 
 

aligned onsets across multiple rhythms, any widespread reversal of direction produces 

misleading frequency distributions and, by extension, an inaccurate hierarchy of metric weight. 

Thus, before one can meaningfully interpret the resulting histogram, or draw conclusions about 

the nature of meter in African music, the underlying data must be correctly oriented. For visual 

aid, Figure 10 applies the hyperdimensional record-player analogy to Toussaint’s dataset, 

marking the “red” and “blue” sides of each timeline to indicate their respective clave 

orientations.  

 
Figure 10 Toussaint’s timeline dataset with directional labeling 

This visualization makes clear not only which rhythms are reversed, but also how their 

misalignment disrupts the symmetry that should be preserved across the sample. By clearly 



 

25 
 

indicating each timeline’s directional halves in this way, the figure highlights the structural 

inconsistencies introduced by Toussaint’s orientation choices and allows the reader to see 

immediately how the dataset fragments into two incompatible groups which cannot be 

meaningfully aggregated because their directional frameworks are not aligned. In some cases, the 

same rhythm is included multiple times, but in both orientations. Rhythms 17 and 18 are an 

example of this, as are rhythms 23 and 25. 

The basic solution is quite simple: align the rhythms properly and create a new 

histogram. The following figure is the resulting histogram that I calculated once I fixed the 

directionality issue and expanded the data set to include more well-established Afro-Diasporic 

timelines, without duplicates.  

 
Figure 11 Corrected Afro-Diasporic timeline pulse saliency histogram 

As shown in Figure 11, the corrected graph differs dramatically from the one presented in Figure 

8. Whereas Toussaint’s original histogram produced a largely ambiguous and internally 

inconsistent distribution, the corrected version reveals a pattern that is substantially more 

coherent and articulate. Once the rhythms are properly oriented, the aggregated onset frequencies 

no longer cancel or obscure one another; instead, a clear hierarchical contour emerges, with 



 

26 
 

distinct peaks and troughs that reflect the structural priorities found across the traditions 

represented in the dataset. In other words, the correction transforms what previously appeared to 

be an inconclusive pattern into one that displays a meaningful and interpretable organization. As 

a side-observation, this histogram newly yields the contour [--+-++-+-+-+-+-+]. 

With this corrected histogram in place, each onset now has a clearly defined statistical 

prominence, reflecting how frequently it appears across correctly oriented timelines from the 

African Diaspora. These frequency derived priorities form the basis for assigning each pulse 

position a numerical weight, a value that captures its relative structural importance within the 

cycle. Once these weights are established, they can be utilized to accurately determine the 

directionality of any individual rhythm. In essence, this histogram provides the beginning of a 

stable statistical grammar, and the model articulated in the following section uses this grammar 

to determine whether a given rhythm aligns more correctly with the “3-2” direction or “2-3” 

direction, or if it’s ambiguous or clave-neutral. This is the “vertical tension” contour that 

Lehmann is referring to, the hierarchical prominence values, the weights “created for the short-

range directionality of the clave/timeline period.” (Lehmann 2002: 71) What follows is an 

explanation of how these weights are calculated and employed to evaluate clave direction in a 

precise and reproducible way.  



 

27 
 

Chapter 4: A Methodological Framework for Evaluating Clave Orientation 

In my corrected histogram, which I will refer to as the 16-pulse clave model, each pulse 

position is assigned a unique frequency value based on its percentage in the top row of Figure 11. 

These percentages indicate how often each onset position occurs across the entire sample of 

correctly oriented timelines: for example, p₄ appears in 25% of the sampled rhythms, whereas p₉ 

appears in only 4%. To convert these raw frequencies into analytically useful categories, the 

percentage values are grouped into seven priority tiers (1-8). The boundaries of these tiers are 

determined by the standard deviation of the frequency distribution, allowing pulses with similar 

statistical prominence to be grouped while still preserving meaningful distinctions between them. 

These bounds are further refined to ensure that no pair of corresponding pulses receives the same 

priority value, a necessary adjustment in cases such as the cáscara pulses p₅ and p₁₃, whose raw 

frequencies initially grouped them in the same tier. This procedure yields a set of uniquely 

weighted pulse positions suitable for evaluating the directional structure of any given rhythm. 

The given frequency tier value of any pulse will be referred to as 𝑓𝑓(𝑝𝑝𝑘𝑘), where k is the 

pulse position. For example, 𝑓𝑓(𝑝𝑝0) = 6 and 𝑓𝑓(𝑝𝑝11) = 1. Each pulse within the 16-pulse cycle 

has an individual frequency tier value. The 16-pulse clave model can be shown as an array of 

these values, as shown below: 

 
Figure 12 Frequency tier values of the 16-pulse clave model 

The weight of any pulse position 𝑝𝑝𝑘𝑘, denoted 𝑤𝑤(𝑝𝑝𝑘𝑘), is defined as the absolute difference 

between the frequency tier of that pulse and the frequency tier of its corresponding pulse in the 

opposite half of the cycle. Using the terminology established earlier, if we want the weight of the 



 

28 
 

quinto (Q) pulse (either 𝑝𝑝1or 𝑝𝑝9), we compare the frequency tiers assigned to both positions; the 

resulting weight is simply the magnitude of the difference between them. In equation format: 

𝑤𝑤(𝑝𝑝𝑘𝑘) = | 𝑓𝑓(𝑝𝑝𝑘𝑘)  −   𝑓𝑓(𝑝𝑝𝑘𝑘+8) | , 0 ≤ 𝑘𝑘 ≤ 7 

where: 

𝑓𝑓(𝑝𝑝𝑘𝑘) is the frequency tier (priority value) of pulse 𝑝𝑝𝑘𝑘, 

𝑝𝑝𝑘𝑘+8 is its corresponding pulse in the opposite half of the sixteen-pulse cycle. 

This yields the table of weight values shown below: 

 
Figure 13 Weight values of the 16-pulse clave model 

Because weight values are shared by corresponding pulses, the array of weights forms a mirrored 

distribution. In terms of determining clave direction, a weight value represents the degree to 

which the presence of that pulse influences the directional analysis of any given 16-pulse 

rhythm. According to this model, the pulses exerting the strongest influence are S and B, each 

with a weight of 6. They are followed by K with a weight of 5 and Q with a weight of 4. P 

contributes only a weight of 2, while F, C, and O exert the least directional influence, each with a 

weight of 1. 

I previously noted that within any corresponding pulse pair, one pulse is designated 

primary and the other secondary, depending on which is more “clave-aligned” in its traditional 

context (Moore 2012: 81). In my model, this distinction is determined by comparing the 

frequency tier values assigned to each pulse: the pulse with the higher tier value is classified as 

primary. For example, the first-half quinto at 𝑝𝑝1, denoted 𝑄𝑄1, has a higher frequency tier than its 

corresponding second-half quinto at 𝑝𝑝9, denoted 𝑄𝑄2. Accordingly, 𝑄𝑄1 functions as the primary 



 

29 
 

quinto, and 𝑄𝑄2 as the secondary quinto. Primary and secondary pulses will be marked using the 

following notation: 𝑝𝑝𝑝𝑝𝑝𝑝and 𝑝𝑝𝑠𝑠, where 𝑤𝑤 is 𝑤𝑤(𝑝𝑝𝑘𝑘). The resulting array of labeled pulses is shown 

below: 

 
Figure 14 Pulses of the 16-pulse clave model with primary and secondary pulses labeled 

This framework aligns perfectly with the categorizations proposed by both Moore and Peñalosa 

for the quinto, bombo, ponche, and backbeat pulses. At the same time, it extends their work by 

incorporating all remaining pulse positions and, through the use of weight values, organizing 

them into a comprehensive hierarchy of relative importance. It’s important to note that both 

Moore and Peñalosa focus solely on the Cuban context, whereas the rhythms sampled for this 

model span the entire African diaspora. The resulting coherence is striking, indeed suggesting 

the presence of a deeper, unifying temporal grammar.  

The very same process can be done with 12-pulse cycles, with some minor adjustments to 

the equations to account for the difference in cycle length. The results of my calculations with 

12-pulse cycles are shown below. First are the frequency values: 

  
Figure 15 Frequency tier values of the 12-pulse clave model 

Following are the weight values: 



 

30 
 

 
Figure 16 Weight values of the 12-pulse clave model 

Lastly are the labeled pulses: 

 
Figure 17 Pulses of the 12-pulse clave model with primary and secondary pulses labeled 

Here, I will outline the process for using this data and model to calculate the clave direction of 

any rhythm. For clarity, I will use the 16-pulse model, although the procedure is identical for the 

12-pulse model. The first step is to define the rhythm in terms of its onsets, the pulses that are 

‘played’ rather than rested. Each onset is then assigned its corresponding frequency tier value, 

and these values are summed to produce the first total, which I refer to as the ‘3–2 Value,’ 

following Cuban directional terminology.  

Next, a second sum is calculated, simulating the rhythm as if it were oriented in the 

opposite direction. In this calculation, each pulse contributes the frequency value of its 

corresponding pulse on the opposite half of the cycle, for example, p₁ adds the value of p₉, and 

p₁₂ adds the value of p₄. This total is referred to as the ‘2–3 Value.’ Comparing the two sums 

determines the rhythm’s directional orientation: the higher sum indicates the clave direction of 

the rhythm. 

The magnitude of the difference between the sums quantifies the strength of this 

directional implication, with smaller differences signaling greater clave ambiguity, and larger 

differences indicating a clearer, more pronounced orientation. This method allows for a 

systematic and replicable way to assess rhythmic directionality, providing both a qualitative and 



 

31 
 

quantitative measure of clave orientation and perhaps the unspoken intuitions that form it. 

Represented mathematically, the process looks like this: 

𝑜𝑜𝑘𝑘 = �
1    , 𝑖𝑖𝑖𝑖 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑘𝑘 𝑖𝑖𝑖𝑖 𝑎𝑎𝑎𝑎 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 (𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝)
0            , 𝑖𝑖𝑖𝑖 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑘𝑘 𝑖𝑖𝑖𝑖 𝑎𝑎 𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 (𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠) 

𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷 =  

⎩
⎪
⎪
⎪
⎨

⎪
⎪
⎪
⎧ 3/2               ,�𝑜𝑜𝑘𝑘

15

𝑘𝑘=0

× 𝑓𝑓(𝑝𝑝𝑘𝑘) > �𝑜𝑜𝑘𝑘  ×  𝑓𝑓(𝑝𝑝(𝑘𝑘+8) 𝑚𝑚𝑚𝑚𝑚𝑚 16 )
15

𝑘𝑘=0

2/3               ,�𝑜𝑜𝑘𝑘  ×  𝑓𝑓�𝑝𝑝(𝑘𝑘+8) 𝑚𝑚𝑚𝑚𝑚𝑚 16 �
15

𝑘𝑘=0

> �𝑜𝑜𝑘𝑘

15

𝑘𝑘=0

× 𝑓𝑓(𝑝𝑝𝑘𝑘)

𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁𝑁        ,�𝑜𝑜𝑘𝑘

15

𝑘𝑘=0

× 𝑓𝑓(𝑝𝑝𝑘𝑘) =  �𝑜𝑜𝑘𝑘  ×  𝑓𝑓(𝑝𝑝(𝑘𝑘+8) 𝑚𝑚𝑚𝑚𝑚𝑚 16 )
15

𝑘𝑘=0

 

𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆ℎ = |�𝑜𝑜𝑘𝑘

15

𝑘𝑘=0

× 𝑓𝑓(𝑝𝑝𝑘𝑘) −  �𝑜𝑜𝑘𝑘  ×  𝑓𝑓�𝑝𝑝(𝑘𝑘+8) 𝑚𝑚𝑚𝑚𝑚𝑚 16 �
15

𝑘𝑘=0

|   

We can also see the process visually by looking at my 16-pulse clave model: 

 
Figure 18 16-pulse clave model with Cuban son clave in the 3-2 orientation 

The analyzed rhythm is typed into the INPUT row, with an x marking an onset. The frequency 

values for each onset are shown in the “3-2 Value” row, and the opposite values for each onset 

are shown in the “2-3 Value” row, with each rows sum shown in the final column. The output for 

this particular rhythm, known as son clave in Cuba, is 3-2, meaning that it is currently aligned in 



 

32 
 

the 3-2 orientation with respect to the model. Below is the input and output for a rhythm 

classified in the 2-3 direction: 

 
Figure 19 16-pulse clave model with a rhythm in the 2-3 orientation 

Below is my 12-pulse clave model, calculated in the same exact way as my 16-pulse clave 

model: 

 
Figure 20 12-pulse clave model with a rhythm in the 3-2 orientation 

This model for analyzing clave direction provides a robust and consistent framework for 

understanding rhythmic orientation across the African diaspora, reconciling empirical 

observation with encultured musical intuition. In the following section, I highlight two 

observations that suggest avenues for further research, approaches that could deepen this 

framework by examining not only statistical patterns, but also the emergent properties of the 

clave cycle itself. 



 

33 
 

Chapter 5: Observations for Future Research 

In Vurkaç’s conclusion, he mentions the existence of clave ambiguous patterns as an area 

for future research. (Vurkaç 2012: 60) Of course, this seems obvious at first. In fact, any cyclical 

rhythm with symmetric halves yields a neutral clave direction when run through my model. This 

is because the 2-3 and 3-2 summations end up being the same, because only corresponding 

pulses are chosen as onsets. An example is shown below, in a 16-pulse cycle. 

 
Figure 21 16-pulse clave model with symmetric clave neutral rhythm 

What, then, about asymmetrical rhythms? This is the issue Vurkaç appears to gesture toward in 

his remarks. In practice, asymmetrical patterns tend to favor one clave orientation over the other, 

except in cases where the asymmetry falls exclusively on pulse positions with the lowest weight 

values. An example of this, in the 16-pulse clave model, is provided below: 

 
Figure 22 16-pulse clave model with asymmetrical clave neutral rhythm 

Here, the asymmetry lies in 𝑝𝑝7 and 𝑝𝑝13. Pulses O and C, as we know from earlier, each have a 

weight value of 1, meaning that they have a relatively weak effect on the direction of the rhythm. 

It makes sense that the resulting analysis would be so close to 0, so close to neutral.  



 

34 
 

However, in a 12-pulse clave model, I managed to find a class of asymmetric rhythms 

who achieve a neutral or near neutral analysis without relying on low weight values. The 

following figure highlights one prime example: 

 
Figure 23 12-pulse clave model with asymmetrical clave neutral rhythm 

Twelve is not only divisible by two, but also by three. This rhythm, while not showing symmetry 

with regard to equal halves, shows symmetry with regard to equal thirds. Perhaps it is for this 

reason that it gives an output of neutral? This possibility seems far-fetched considering the fact 

that the model is calculating clave direction based on bisectional symmetry and not ‘tri-sectional’ 

symmetry, but it is highly unlikely that this is mere coincidence. Indeed, all other “tri-

sectionally” symmetric rhythms yield either neutral or near neutral analyses. Below are two other 

examples: 

 
Figure 24 12-pulse clave model with a second asymmetrical clave neutral rhythm 

 



 

35 
 

 
Figure 25 12-pulse clave model with a third asymmetrical clave neutral rhythm 

This observation not only hints at a deeper structure merely implied by the statistical analysis, 

but it also holds up to how these rhythms are employed in traditional contexts, both as neutral 

timelines or neutral solo phrases, commonly played on either side of the clave. It may well be 

possible to generate various frequency arrays that consistently satisfy this property, but pursuing 

that task lies beyond the scope of this study and remains open for future research. 

A second observation that I have encountered concerns the striking structural parallels 

between European scalar systems and Afro-Diasporic rhythmic practice. These two domains, 

often treated as conceptually separate, share underlying principles of asymmetry, hierarchy, and 

patterned tension and release, suggesting that they may be governed by similar generative logics. 

Peñalosa notes: 

It is worth noting that the African standard pattern and the European major scale 
both exhibit the same structural properties. If one pulse is considered the 
equivalent of a half-step and two pulses a whole step, the correspondence is 
evident… The correspondence of these two patterns is not mere coincidence. Both 
the rhythmic properties of Africa and the harmonic principle of Europe are based 
on simple whole number ratios whose workings were articulated by the Greek 
philosopher and mathematician Pythagoras. The two most important rhythmic 
ratios in African-based rhythm are 3:2 and 3:4. In European-based music the 
perfect 5th is generated by the harmonic ratio of 3:2 and the perfect 4th by 3:4. 
A.M. Jones’s seemingly simplistic statement that “rhythm is to the African what 
harmony is to the European” takes on deeper meaning when one considers how 
both rhythm and harmony are generated by the phenomenon of proportional 
interactions. (Peñalosa 2012: 220) 



 

36 
 

The rhythm that Peñalosa refers to as the “African standard pattern” is shown below in my 12-

pulse clave model: Notice how closely this rhythm matches the pattern of primary pulses in 

Figure 17. In fact, the only difference is the presence of 𝑝𝑝5, which only has a weight value of 1. 

 

Figure 26 12-pulse clave model with African ‘standard pattern’ 

In general, structural symmetry between European harmony and African rhythm is not 

something that has gone unnoticed. In fact, Toussaint dedicates a whole chapter of his book to 

the phenomenon. (Toussaint 2020: 45) The observation that I offer here emerges from this 

conceptual premise and integrates it with a principle of harmonic theory encountered during my 

studies as a jazz major at the University of Oregon. 

As a jazz major, I was trained to distinguish among the various modes of a diatonic scale 

and to understand how each is generated, both theoretically and in practice. In this context, a 

mode refers to a rotation of a scale: a reorientation in which a different pitch is treated as the root 

while the underlying sequence of whole steps and half steps remains unchanged. For example, 

the C-major (Ionian) scale is spelled C–D–E–F–G–A–B–C, and its intervallic structure can be 

represented numerically, using 1 to denote a half step and 2 to denote a whole step, as [2-2-1-2-

2-2-1]. If we rotate this pattern so that the scale begins on D while preserving the same interval 

sequence, we obtain D–E–F–G–A–B–C–D, which corresponds to the Dorian mode and is 

expressed numerically as [2-1-2-2-2-1-2].  



 

37 
 

I was also trained to conceptualize the seven modes as forming a gradient ranging from 

“bright” to “dark” qualities. Although these descriptive terms are inherently somewhat 

subjective, they at least provide a useful framework for identifying the two poles of this modal 

spectrum. The “brightest” mode, Lydian, can be generated by stacking six perfect fifths above a 

chosen root pitch until seven distinct pitch classes are obtained, then compressing those pitches 

into a single octave. Conversely, the “darkest” mode, Locrian, is produced by stacking six 

perfect fourths above the root. Because a perfect fourth is the inversion of a perfect fifth, this 

procedure can equivalently be understood as stacking six perfect fifths below the root, in contrast 

to the upward stacking used to derive Lydian.  

To obtain the next “brightest” mode beneath Lydian, one stacks five perfect fifths above 

the chosen root and one perfect fifth below it. This combination yields the Ionian mode, 

popularly known as the major scale. Progressing along the gradient toward increasingly “dark” 

modes simply involves successively relocating additional perfect fifths from above the root to 

positions below it, a process that continues until reaching Locrian, in which all six perfect fifths 

lie below the root.  

Given the established relationship between the African Standard Pattern and the Ionian 

mode, I wondered whether a comparable pattern might emerge between this “brightness” 

gradient and the relative clave direction strengths of the corresponding rhythms. Even at first 

glance, the rhythms implied by the Lydian and Locrian modes I recognized as being 

diametrically opposed when it came to clave direction. This raised an intriguing question: might 

the other modes exhibit a similarly ordered relationship? If such a correspondence existed, it 

might suggest a deeper structural principle linking harmonic organization and rhythmic 

orientation. Below is a table that lays the modes out in order from “brightest” to “darkest,” along 



 

38 
 

with their numerical expression and clave alignment output as given by my 12-pulse clave 

model: 

 
Figure 27 The seven diatonic modes along with their 12-pulse clave model analysis 

Astoundingly, there does seem to be some sort of correlation between this ordering of modes, 

and the clave direction strength of each mode’s rhythmic counterpart. The “brightest” modes 

have the highest strength in the 3-2 clave direction, and the “darkest” modes have the highest 

strength in the 2-3 clave direction. The modes between them appear to have clave direction 

strength values that consistently reflect their relative position in the “brightness” gradient. 

This observation shows that there may indeed be a deeper connection between harmony 

and rhythm than previously thought, especially as it relates to the quantification of clave 

direction. Similar to my previous observation, it may very well be possible to reverse engineer 

this correlation to generate arrays of frequency values that satisfy the specified conditions, but 

likewise that task lies beyond the scope of this study and remains open for future research. 

 



 

39 
 

Chapter 6: Concluding Remarks 

In this thesis I sought to develop a systematic cross-cultural framework for wide-sense 

clave, proposing that the rhythmic organizing principles most famously found in Cuban tradition 

are in fact only a single manifestation of a deeper concept shared across Afro-Diasporic musical 

traditions. In doing so, I addressed a gap in previous scholarship, correcting the errors in 

Toussaint’s pulse saliency approach, expanding Peñalosa’s and Moore’s frameworks beyond 

Cuba, and offering a model whose consistency has been tested against an enormous amount of 

rhythmic material.  

The resulting 12-pulse and 16-pulse clave models that I have offered provide a robust 

method for identifying primary and secondary pulses, determining rhythmic orientation, and 

understanding how rhythmic direction emerges from asymmetric contrast within a fixed cycle. 

These models have been validated against my experiences in learning Ewe, Ga, Akan, and 

Dagbani drumming, demonstrating that wide-sense clave is not limited to Cuban music, but 

manifests in shared structural patterns across the African Diaspora. Unlike Vurkaç’s model, 

which treats all pulses as equally important, leading to inconsistencies and rhythmic 

disagreements with established tradition, my model recognizes an inherent multi-level priority 

hierarchy, leading to a consistent quantification of clave direction that also considers encultured 

musical intuition. Unlike Lehmann’s model, which centers on only one selected rhythmic 

pattern, my model recognizes clave not as a single rhythm, but as a nuanced framework that can 

manifest itself strongly in many rhythmic forms.  

Limitations do remain. The database of rhythms that I analyzed as a foundation for my 

model, while diverse and encompassing, is not exhaustive. The current model’s statistical 

grounding is only a first step towards reverse engineering emergent musical properties. Future 



 

40 
 

research might expand to developing models for other cycle lengths, such as 6, 8, 9, 18, or 24, 

using the same philosophy that frames my 12-pulse and 16-pulse models. Lehmann and 

Toussaint have done some good work on 24-pulse cycles that I want to dig into more in future 

research. Developing my publicly available online repository of rhythms and their clave 

mappings will make these concepts more accessible for a wider audience, which I anticipate will 

further discussion and research opportunities. 

Ultimately, this research points to a unifying temporal grammar that underlies Afro-

Diasporic music, connecting Cuba to Ghana and beyond. By revealing the structural logic 

embedded in these musical traditions, this thesis contributes to a deeper understanding of how 

clave operates across cultures, demonstrating that the intuition musicians often describe as “in 

the blood” can, in fact, be systematically described, analyzed, and shared. Wide sense clave is 

not only a conceptual tool, but also the key to understanding the rhythmic heartbeat that unites 

the African Diaspora. 

 

 



 

41 
 

Bibliography 

 

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