Mathematics of African Dance Rhythms

>> From The Library of
Congress in Washington, D.C. >> David Plylar: Good
evening, everybody. Thank you so much for being
here at the next event in our Video Discotheque Series,
which has already been a nice — we've had a nice run of
events this week and actually since April 12th was
when we started. I just wanted to let you know
about a couple of upcoming events and then just a short
introduction about tonight's talk. We invite you to come
tomorrow at 12 p.m., we have a Veterans History
Project talk at noon in the Whittall Pavilion, that'll
be kind of a panel discussion. In the evening in this
same room is a screening of House Party at seven o'clock. And then on Saturday we have a
Symposium that starts at one, and then we have the Gloria Gaynor
events that start in the evening. Now you may or may not have
tickets to those events, they sold out very quickly.

If you don't have a ticket we do
have the possibility of getting in, there will be a space-available line
that will start forming for each of those events if you'd
like to still attend, and we'll try to get
everybody in that we can. Tonight I'm really pleased to
introduce Martin Scherzinger, he's a Professor of Media Culture and Communications at
New York University. He's going to speak to us about
various issues of mathematics, of African dance rhythms. And one thing I wanted to
mention is what does this have to do with Disco? One thing that we wanted to do
with some of these extra talks is to expand the context of what we're
talking about with different types of dance musics and how they are
still relevant contemporaneously and to Disco and to other cultures.

So I'm really pleased
to introduce Martin. He's a fascinating person,
also a composer and performer, so it's great to welcome you. Thanks, Martin. [ Applause ] [ Music ] >> Martin Scherzinger:
Okay, hi, everybody. I'm going to let this roll visually
while I introduce the subject. What you have in front
of you is a performance, a reenactment from a scholar who is
doing some research in the region from the upper Volta, and the dance that you see is an ugbeckor
[Assumed Spelling] dance, which is traditionally
a war dance in Ghana. And I just want to talk a little
bit, just to introduce this, about how the rhythms that you hear
in the background are constructed. So we have a number of
different kinds of drums. We have a Kagan drum,
we have a Kidi drum, we have an atsemiru
[Assumed Spelling] drum, and as well as a gankogui,
which is the bell pattern that you hear prominently. Just very briefly and
schematically all of these rhythms are
a little more complex than the way I'm going
to introduce them. They tend to be organized in what Western people would
understand is different metrical frameworks, okay? That means that the Kagan drum
would do something like this, and I'm just going to
ask you to take a look up at the board if you can.

You can think of this as a
short, long, short, long, short, long, short, long. So Kagan, Kagan, Kagan, Kagan. Sort of in sense organizing things
in threes, one, two, three, one, two three, one, two,
three, one, two, three. At least from the perspective
of a Western ear. A drum like Kidi would be doing
something like, dit, do, dit, dit, do, dit, do, which is
organized a bit more like, okay? So something like that
before it repeats, right, which is in the West says something
like that would be one, two, three, one, two, three, as opposed to one,
two, three, two, two, three, okay? So we layer these two different
drums more or less differently. Now again both patterns are
more complex in practice, but this is their signature sort
of placement in which they work. The bell pattern that you hear
is a very common bell pattern, a 12-8 bell pattern, which I'm just
going to play now in slow motion so that you can hear that part of
the ensemble for special attention.

[bells] It's a seven-stroke
bell pattern, one, two, three, four, five, six, seven. It happens in a field or in what we
might call proct [Assumed Spelling] time in a field of 12. Subdivided as two plus two plus one,
two plus two plus two, plus one. Okay, so let me just
illustrate that this way, okay? Um, bum, bum, ba, bum, bum, one,
two, one, two, one, one, two, one, two, one, two, one, one, two, one,
two, one, one, two, one, two, one, two, one, two, one, one,
that kind of thing, okay? I want to just draw that one up
in notation to give you an example of a striking, what we
might call a correlation between an African practice and a
Western technical instrument, okay? We've got one, two, one, two,
one, one, two, one, two, one, two, one, two, one, okay? Everybody sort of get
the notation, right? For those of you in the back
it's not crucial and try to work with the ear, but visually
I put it this way because what's striking
about it is two things.

The first is that there's
two longs, a short, then three longs and a short, right? So there's something a little
asymmetric about this, right? In other words, when
you have a field of 12 and we have 12 pulses here. Why? Because we have that and then
there's a space, space, space. If I count up the dots
I get 12 micro pulses against which seven are struck. So we have seven elements in a
field of 12, that's clock time for which seven stand out
for special attention, okay? The relationship of seven and
12, think of days of the week, think of the annual way
in which we organize time. This is something that is
contemplated a lot globally, okay? But the thing that I want
to draw your attention to is by being a little bit
asymmetrical we can sort of split the pattern
into two halves.

The first half, which is long,
long, short, and the second half, which has the three
longs and a short. In other words, the
boundaries happen in a slightly asymmetrical
way, okay? So with 12 you can multiply and
subdivide and so on into three by four, four by three, six by two,
two by six, or which is important because it can do all those
things, but it can do another thing, which is common in African music
which is to say you take your 12, you take your symmetry and then
you off center the symmetry by one. So you might call it N
minus one, N plus one. So if we've got 12, six plus
six, now we have six minus six, six plus six, five
plus seven, right? And if we count up the dots,
one, two, three, four, five, and then one, two, three,
four, five, six, seven — I should have put another dot
in here, ran out of paper — we have exactly that structure.

Now just something to
bear in mind here is that this is not just a rhythmic
structure, but if you were to put this into a pitch field,
in other words, A, B, C, D, E, F, like on a piano, we get
exactly the structure of the modern piano, right? Instead of long we have a
tone, so it's long, long, short, long, long, long, short. We have a tone, tone,
half tone, half tone, tone, tone, tone, half tone. It's the white notes
on the piano, okay? If I were to fill in the green
notes, we'll confuse colors tonight, that is the black notes
on the piano, okay? Something to think about why? Because the piano is designed in
such a way, in the 18th Century when they sort of came to standardize the modern industrial
piano it's designed in such a way that it's capable of
very easily modulating from one pitch area
to another, okay? So you can move from C major to,
say, G major with very little — with tweaking the system
just a little bit.

In time you can also
modulate from one place to another place fairly easily, and
I'll explain that in more detail as we get into this
lecture a little later. So that is an important sort of
feature and common characteristic that it has as a mathematical
structure that incarnates in African rhythmic practice
as a timeline pattern and in Western practice as a
technical device for music making, which has been, you know,
had a great monopoly of thinking and music in the West. Not an insignificant
instrument, the piano, especially if we consider
its afterlife in MIDI [Assumed Spelling],
which is a software that basically incarnates the
assumptions of the piano into code. Okay, so these are just some of the
features that I want you to remember as we move on through this lecture. That we have here a sense
of staggered downbeats, different rhythmic cycles that we can say have a
different kind of downbeat. We have the principle of N minus
one or N plus one, asymmetry, right? And we have another
principle that I just want to quickly draw your attention to,
which is that the rattle pattern, the rattle player is doing something that supplements the
gankogui in the following way.

If your gankogui is doing this —
going to keep these pens nearby, I'll need them — if your
gankogui is doing this, dum, bum [beats] your upshashir
[Assumed Spelling] — just using a [inaudible]
bottle here — your upshashir player is
going to more or less for most of the pattern inhabit the spaces of what the gankogui
player has occupied. So if that is a space after the
beat, okay, so [beats] [rattle]. These are the black
notes of the piano, okay? One, two, one, two, three, one, two,
one, two, one, two, three, okay? One, two, one, two, three,
one, two, one, two, one, two, one, two, three, okay? I may have rushed a little
bit quickly through things and I'll recap them again as we
move further into this lecture.

But for now I want to jump to — that was the northwestern
part of Africa — now I want to jump to the southeast. Take a look at this dance,
called Nyungwe, Mozambique. [ Music ] Okay, very interesting piece
of music making, right? In South Africa before the days of
liberation, in the age of Apartheid, we used to say this was music
called for one man, one note. You had a set of pipes, maybe three
or four, with a different name and you would play
one note at a time in which other people would
play different notes, okay? That particular piece is called
Uzungwa Gona [Assumed Spelling], which means something
like White Man Sleeps, and there's been an interesting
set of compositions based on it. This is an old transcription
from when I was still there, so I apologize for
that, but what I want to quickly do is show you how
this music is put together, okay? Again, we're in Mozambique.

The first part is called
korirambu [Assumed Spelling], the second pakirakabambo
[Assumed Spelling], and then enbitae [Assumed
Spelling], okay? And the way this is put together is that the first part
will sing something, blow something, and then breathe. So it's sing, blow, breathe, sing,
blow, breathe, sing, yahoo, yahoo, yahoo, yahoo, yahoo, yahoo, yahoo. Okay, that's the first line,
yahoo, yahoo, yahoo, yahoo, yahoo, yahoo, yahoo, yahoo. The second part, pakira [Assumed
Spelling], yahoo, yahoo, yahoo, yahoo, yahoo, yahoo,
yahoo, yahoo, yahoo. And then kabambo [Assumed Spelling],
yahoo, yahoo, yahoo, yahoo, yahoo, yahoo, yahoo, yahoo, yahoo. And enbitae, yahoo, yahoo, yahoo,
yahoo, yahoo, yahoo, yahoo, yahoo. Okay, for give my terrible
singing, but you get the point. So what happens here is that
what's interesting about this is if we design this in such a way that
we just call it that they sing, ya, and then this bland note fu, right? And the second, the third
time point is a breath, okay? And we sort of line up our singing
and our blowing in that kind of way.

We can say that korirambu
does this kind of thing, yafu, yafu, yafu, yafu, okay? Pakira, you will notice,
starts with a breath, okay? Actually, you know what,
I'll keep it all in red. Starts with a breath and
then begins with a ya, okay? So breath, ya, and so on. Okay, enbitae, down at the
bottom, begins with the fu. Et cetera, okay? So again we have this notion
that each of them seems to start at a different part of the
beat, that's on the first, that would be on the second and
I put it in inverted commas, and this would be in some sense
on the third, two beats missing, and then on the third where
you start with your ya. Okay, what's fascinating
about that is that we've got that same multi-metric situation. In other words, the down beat
for each performer would happen at a different place, okay? What's interesting musically is that what you hear is not exactly
what's going on meteorically because what comes to the
ear is a different thing, it's a kind of phantom
baton that is designed out of all the ya's being
put together, right? So the ya gets thrown around
between different performers.

So you're kind of going
ya, ha, ya, he, ha, ha, being thrown around two different
performers, and your flute part, your pipes are doing the same thing
but within the spaces of that. Okay, so that's the first thing. So you're generating melodies out of
this is the formic principle, right, this is the zoetrope, where you spin
a photographic image fast enough and the eye cannot not
see movement, right? This is music of full
stops that becomes lines, but nobody is fully creating those
lines, they get passed around. This technique was used by various
composers in the '60s and '70s when ethnographic recordings made
their way to the United States and to Europe and very often
to great avant-garde effect.

Somebody like Luciano Bario [Assumed
Spelling] takes a very similar style from horn music that was recorded
by Sinclair Arum [Assumed Spelling] and rocks music in a heterophonic
style, which became very, very interested — people in
the West became very interested in this and, in fact, philosophers
like Dulars [Assumed Spelling] and Guatari [Assumed Spelling], kind of left wing philosophers
thought this was a way of organizing political life and
then that became a kind of metaphor for organizing for political
life in many philosophies and political philosophies to
follow, which is interesting, all the way up until today. It's kind of hidden genealogy
for this kind of thinking. How do you hold this
multi-metric situation together? If I ask just this room to
go yafu, and we all start at a different time we are really
not going to do a great job and I can anticipate that. And the way you do it is to
organize your dance step in a way that everybody is equally in
and not in on the beat, okay? So if it's going yafu, yafu, okay,
so one, two, three, one, two, three, one, two, three, one, two, three,
one, two, three, one, two, three, one, two, three, right, there's
three elements, yafu and breath.

Your dance happens in four's, so
it's yafu, yafu, yafu, yafu, yafu, yafu, yafu, yafu, yafu,
yafu, yafu, yafu, okay? You're on the beat and off the
beat at different times, right? And because of that the beat floats
amongst the different performers and is able to bring them all
to the same kind of place. Now as you saw from the actual
dance it's a lot more complicated than that, but when you're
teaching a four year old to get it right you make sure
that you're dancing in four's, when the music is structured
in staggered three's. So everybody with me? Okay, so that's more or less
another lesson we take away because though the music sounds
very different it has some things that are similar. What are the similar things? Multiple entries, right? Different metric downbeats,
if you like. Interlocking, which means be
in the spaces of each other to create melodies
that float amongst you. In the case, remember the gankogui
upshashir in between the gankogui? And the third principle, which
one might just call polyrhythm because in a sense my feet are
moving in four's, one, two, three, four, but this is moving in
three's even though the three's are staggered, okay? So those are the three sort of
broadly speaking principles that I'd like to introduce to you
to now before we change to a completely different
time and place.

[ Music ] Okay, probably one of the most
famous samples in the history of music or history of
popular music in America, okay? The Amen Break, which was a
B-side tract called Amen Brother by the Winston's. This is now 1969, this
was a short fragment of what was called The Break, which
is when the music starts playing and for moments the rhythm
is exposed and in the words of Steinski the music takes
its coats off and you get down and get funky and so
on and so forth, right? That, in addition to James
Brown's Funky Drummer, was probably the most frequently
sampled music in the 1970s. As I say, this particular moment
I'm going to try to tie back to some of these African principles, and
then I'm going to try to make a more or less speculative argument
about how these things got here and what happened along the way, how
did we translate some of the values from one place into
another place, right? And that's not always a happy story,
there's some, how shall we say, tragedy involved in this moment of great transition
across the Atlantic. But let's just get straight into it and ask some questions
of funk music, right? What are some of the
techniques of funk? And I want to introduce you
just to two very simple ones.

The first one is sometimes called
shifting or pushing something on to the metric offbeat, right? Which is just a question of if my
beat is here I emphasize the space between [beats] okay? The second is shaping,
sometimes called shaping the beat or extending it in time,
but extending it somewhat. So if the beat is here
[beats] I now go, okay? We're going to get to how
that is actually sublimated into that particular rhythm that
you heard, but for now I just want to play some very obvious examples
of both so we can get this slowly.

Here is Rufus Chaka Khan, 1974, Tell
Me Something Good, shifting the beat to the metric offbeat, okay? [ Music ] Okay, everybody get the point? What makes this feel the way it
does is precisely this shifting the on and the offbeat. You could actually step on both and it would give you
decent movement, right? It doesn't script you or conscript
you into one way of dancing, you can be on or off the beat. Same year, 1974, here's
a way of shaping time.

Actually, let me just
remind you what — you know, what, we'll do another
shifting time, but to remind you that this isn't just a question
of like pulses interlocking. You can have entire rhythmic
configurations kind of being pushed over by half a beat, that sort of
classic 0.5 or something, right? So the classic funk kind of
drumming often does this. So if you listen to the following
passage, this is Tower of Power, Scrub Case 1974, you will hear a
short rhythmic motive that sort of go dum, da, da, da, da,
dup, da, da, dup, da, da, da, dum, ta, dum, dum, dem, dem. And then it will delay by 0.5
that same dup, da, da, da.

Okay, so if you can hear that, four measures of funk
drumming, Tower of Power, 1974. [ Music ] Everybody get the point? [ Music ] Okay, so the entire sort
of, as it were, the passage, the rhythmic configuration,
dup, da, da, da, dum, ta, dum, that gets pushed over by 0.5. Did anybody hear that? Okay, let's do it one more time. [ Music ] Okay, so I came down. [ Music ] Where that little rhythmic
fragment started again, but the drummer waited just half
a beat and that would lead us into the rhythm of the one.

Okay, it's as if it
came a little early, came a half beat early, right? And that is very sort of
more sophisticated way of shifting the beat, okay? In Disco music, of course,
this shifting became a kind of dominant feature, it almost
became a characteristic sound which was associated very
often with the high hat, so think Donna Summer
and so on and so forth. The prominent high hat that
happened within the spaces. This is just a digital
audio workstation sort of showing how you place the
high hat within the context of an ongoing beat using
Disco as you're trying to invoke Disco and stylistic. [ Music ] Now we have it here. [ Music ] Okay, now we've entered into
the reign of Donna Summer. [ Music ] Okay, so we've sort of got the idea
of interlocking or shifting, right, being in the space
of something else, the shift of 0.15, 0.5 in a beat.

What I want to now turn
to is the shaping sound, which is to say instead of having
just the beat go you have another beat going, but that's
like stretched. [beats] Okay, and this made its way
also both through funk into Disco and then ultimately into the
mainstream sort of dance musics of the late '80s and into the '90s. I want to give you one
example, this is from Tim Moss, so we're now already in the
'90s of techno music just to again get the conceptual
point across.

It starts out with a symph that is very prominently
giving you a slow beat against which a quicker
one or the primary set of beats is going to join. [ Music ] What, we're musically doing this
I'd say one, two, three, one, two, three, one, two, three, one,
two, three, one, two, three. Because the real beat is heavy. [ Music ] Then the DJ will bring
in the underlying beat. [ Music ] Okay, in the music world
people sometimes call this grouping dissonance. As far as I'm concerned, we could
call it a metric dissonance. It's like two elements or two
parts of the layering are aspiring to different ways of counting time,
right, but you do it simultaneously and they go in and out
of phase with each other.

Think about building a brick
house with different size bricks or something, clearly they're going
to go out of phase and come back into phase with one another. By the time we get to the late
'90s this becomes quite a refined technique for the needle
droppers and the refined DJs. This is some IDM music, which began or so-called intelligent dance
music coming out of Detroit, but making its way through
to the UK and so on. Mark Butler has written a
remarkable book on this. I'd like to show you that what
we have here is both a shaping and a shifting of the beat, okay? So take a listen to
this passage of music. [ Music ] Okay, now listen to the way he
drops the needle into this texture.

We expect the beat. [ Music ] That's where we expect the beat. [ Music ] So it's both shifted the beat
or turned the beat around, as the saying goes, and it's shaping
the beat by de, dum, de, dum, de, dum, which is the slower
beat within a beat. It's like having two cycles of a different size rotating
simultaneously, you know, a 24-hour day and a 28-hour day
somehow, simultaneously unfolding. Okay, and by 1999 — I'm going
to give one more example, this had gone thoroughly mainstream,
so when people were dancing raves and so on and so forth we often
had these kinds of techniques.

So I'm actually going to try to draw
this up on the board because it's — try listening to the different
layers because it's shaping and shifting on numerous levels. I'll draw it up and speak
across the actual track. So here we go, Almac Twins, Almac
Head [Assumed Spelling], sorry. [ Music ] You've got the beat. [ Music ] Okay, but the symph is prominently
off that beat in this phase. [ Music ] Okay, I'm going to go back,
this is what happens next, so that we call shifting
of the effect. What was once just a gentle high
hat is living up to the beat and Disco has now become
a prominent [inaudible]. [ Music ] Listen to what happens now. [ Music ] It's becoming more prominent. [ Music ] Sort of doing its own thing, and there's another layer
which is doing this. [ Music ] Okay, in fact, the drum,
the main sort of beat that hold it together they remove
so that you can hear all this. This is like that break that
you used to have in funk music, now you remove the basic element and
just allow all these rhythmic sort of ambiguities to come
to the forefront and then dancers often speak of
flowing and trancing, whatever.

You know, the language there is
interesting how to describe this. But what I want to show
you just very quickly is that the first thing
is that prominent symph in the spaces happened in
the space of the beat, right? It was somehow punching in between. The second layer went across that in
three's, so this was all in two's, and then this went
into three's, okay? But this one went chun,
ja, chun, ja, chun, ja — this one actually went into the
space of the space over here, okay? So you have all these various
elements that are suddenly starting to sound a little bit
like an African ensemble. Very little harmonic change and
all sorts of rhythmic complexity that the DJs are bringing and
filtering and figured rounding in ways that are attentive
to the audience's reactions. So how did we get there? Well, let me start very
quickly by going back to Africa. This time just South Africa. And again it's hard to measure
exactly how these things transferred from one place to another, but
what I do want to say is — and it's hard to map because
the place, South Africa, for example I'm South
African and so on, obviously a hybridized
South African, hybridized African, if you like.

And so, too, is the
music deeply hybridized. So it's quite difficult to
look at contemporary music and find just this
irreducible stamp of Africanus because it's become part of
the record industry, the radio, emergence of radio, the
globalization of culture, very quickly hybridized all of this. So what I'm trying to get at
is some of the pre-hybrid forms of music making which were
equally complex and non-static, but there is a sense in
which they functioned across a different type of modality.

But even in the hybrid
music of South Africa, this is back to when I
grew up, turning the beat around was a common thing. It is almost like even in
popular music it was modeled on Western music you
hear this sort of ease with which the beast
gets turned around. Here is Inzie Zamoya [Assumed
Spelling] playing a Marabi music, in fact, this piece of music is
called Marabio [Assumed Spelling]. Sorry, attempt to hit that button. [ Music ] That's where you can move. [ Music ] Moving into all sorts
of dance things that are like MiteMite [Assumed Spelling],
which is associated with [inaudible] and I'll talk about that when we
talk about [inaudible] next time, but that's like to walk like
a duck and so on and so forth. So this is a sort of association
that goes along with this, but it's clear that turning the
beat around is not a big deal in the South African popular music
even of the modern hybridized sort.

Okay, now a speculation,
how did this get here? And I want to just take a sort of
detour through certain soundscapes, such as the sites such as the Bronx where arguably we can find the
origins of Hip-Hop and argue that the actual sounds that were out
there in the streets, in the parks, in the dancehalls, on stairways
and so on and so forth, actually carried an imprint of this
music by way of the Africarribean. Okay, this is actually an argument
that's very well made by a colleague of mine called Mark
Katz [Assumed Spelling]. If anybody wants to read up
about that exact connection. I think it's a compelling argument. I lived in New York for a
long time now and when I lived up in Harlem it was clear that if
you left your apartment on a Sunday or Saturday or a Friday,
maybe even a Thursday in summer these soundscapes
are very prominent, right? There's different competing groups that still have traces
of all of this.

And it seems to me that that sonic
geography has something to do with what people felt was a
certain appropriate way of beating. So here's an example from a
novel, written by Allen Jones, this is in the 1950s, the book
is called The Rat That Got Away, and here he's describing
the soundscape of a certain sort of
area in the projects. The Patterson Houses at night
were alive with activity, he says, and alive with sound. Music was everywhere, coming
off people's apartments and on project benches. One side of the street he
would have people who brought out portable turntables
with two big speakers. That's already in the 1950s. Cool hoke [Assumed
Spelling] only starts, you know, a good 10 years later. And on the other side you could
hear some brothers singing a Frankie Lymon song.

But the one constant every
night without fail was the sound of Puerto Ricans playing
their bongos in local parks and playgrounds. The steady beat of those
drums was background music to my living reality. Renee Scroggins, 1970s, describes
the Bronx in very similar terms. Every summer in St. Mary's Park
you would have some Latin gentleman in the park with some Coke bottles,
a cowbell and a set of congos and it was our summer sound.

Okay, so that's by way of
speculation about one place where something moved from the
African by way of the Africarribean into the soundscapes of the United
States and then into popular music of various sorts, from funk to
Hip-Hop through Disco and so on. That break beat was the most
sampled or played with on vinyl on a certain kind of turntables. Of all sounds it's
probably the most popular. And the argument is that when
you make a recording this is the phonograph effect, when
you make a recording of something it's different
to being live.

If you're live you can adjust to
your audience, you can look at them, you can slow down, speed
up, you can extend the break if they're getting down and so on. How do you do that if
you've got it on vinyl because on vinyl you always
keep things shorter, right, even if you're like recording Chopin
or something like that, you know, live I can do that, I can
really hold that silence, right, and people will wait. A phonograph always
compresses it, you can't — you don't see that silence
so, therefore, you shorten it. And so the break was
relatively short. Well, how do you extend that
if people are getting down? And you're just basically
creating a party. Well, one thing you can do is have
two turntables and needle drop or pull it back, which is their way, that particular technique
emerged from.

So it was a kind of production,
creation, and then became a kind of music stylistic of its own. In fact, Colverk [Assumed
Spelling] used to toast over this, which is a Jamaican thing
to do, perhaps to cover up some of the little mistakes. So it's an interesting story
about how this all comes into the soundscapes
of our parks and so on. So I want to get a
little bit into this, into the claves [Assumed Spelling]. Demonstrate, which is the
Africarribean that I'm referring to here, and demonstrate
how these take up residency, these patents take up residency
in various contemporary styles.

And you should be able
to recognize them, I'm trying to in some sense
give some irrefutable sort of styles here. But here is the sound
of the 3-2 sound claves. And when you think of the claves
think of that gankogui [inaudible] that we did a little bit of work
on, that looked like the piano, that's the one I'd like you
to think of when you think of the sound claves
and then we will talk about the relationship it
takes to the African original. [ Music ] Okay, it's three on one side
and two on the other, right? So one, two, three, four [beats]
that kind of thing, right? Remember there's black
notes on the piano, right? Five things happening in a field
of 12 or five things happening in four big beats, five
beats in four, okay, trying to make five things
happen in that space. This one happens in this particular
way, so it's one, two, three, one, two, one, two, three,
one, two, okay? If I speed that up just a little
bit, so dun, dun, dun, dun, dun, dun, dun, dun, dun
— I find it here.

[ Music ] This is the block party call to … [ Music ] Okay, I hope we all know that
particular piece of music, okay? There were many variants of this
particular pattern and I'm not going to have the time to
do it all, right? But we have in a sense three on
one side and then two on the other. The beat falls sort of in
between all these things, right? And then sometimes on these things.

So there's the beat, dah, dah, dah,
dah, dun, dun, bum, bum, bum, okay? That's kind of how
this pattern works. If I were to just to delay this one
just a little bit I get the rumba, okay? If I delay this one just a
little bit I get the bossa nova. I can also delay both a little bit and I get also a very
interesting pattern. So the original song
pattern, something like this. Okay, here's the beat. [beats] Rumba, I'm going to
delay the third of those. [beats] Everybody hear
that subtle difference? Okay, so instead of I'll play
them one after each other. [beats] Okay, bossa nova, now I'm
going to delay the other side of it.

[beats] Again, I'll play now
the sun [Assumed Spelling] and then the bossa nova. [beats] Sorry, that was
the wrong way around. [beats] I could do both. [beats] Now I'm almost playing
exactly five equally spaced things in the field of four, but we
are locked into four-four time, which is undermining
that a little bit. If you slow down, for example,
the rumba pattern from [beats] but slow it down [beats]
you get something like this. [ Music ] So the bass drum. [ Music ] Okay, I raise it because when
we look at Hip-Hop, you know, often in the popular
media or at universities. You know, I teach at a university. We often look at the voice, the
importance of voice, politics, culture, storytelling,
imagery, et cetera. What we don't often do is just
listen to the pure grammar of how are these beats put together
and where do they come from, right? This is clearly a slowed
down kind of rumba, right? And the girding through the
bass drum, the feel of the song.

It is essentially important
and throughout the first decade of the 21st Century I
challenge anyone in the room to find me seriously sort of popular
Hip-Hop that didn't have a mark of some of this kind of claves
pattern built into the underground of that particular sound. So how would you render this
pattern in an African context? What is different about these to the
ones that we started with, right? And I want to get into
this a little bit slowly, but let's just first have
a demonstration of it.

This is Mario Puchae
[Assumed Spelling] again. This is the African
Icefield [Assumed Spelling]. [ Music ] Okay, it's interesting because
the beat should really be here. [ Music ] Okay, it's a slightly
different feel, right? So consider the one goes like
this [beats] that's our song, and this one goes [beats]. Okay, a little bit
different to [beats] subtle, but it's a very different feel. And what makes them different
primarily is that the one is what in the West is called four-four
time or binary time and the other is in a kind of ternary time.

It's going one, two, three, two,
two, three, three, two, three, three, two, three, four, two, three. And the other is going one, two,
two, two, three, three, four, two. And you're trying to fit the beats
on and off the beat in the same kind of way, but in a different
metric framework. Is everybody with me? And notice that this three-two
claves is exactly what we had when we knocked off right
in the beginning the spaces between the gankogui pattern. Here was that first
pattern you heard. This is now the bell
pattern, gankogui bell. [beats] [rattle] Africanized. [beats] [rattle] So I'm not
going to even try to make that final translation,
but you get the point. It's entrained differently, the
way the placement of the beat is in a slightly different spot
within the temporal spans that are provided by the music.

Okay, this manner of rendering that
three-two claves, quote-unquote, and it's completely back formation
now because we're now going back to the earliest music that we
have traces of, is part and parcel of something that's been sort of at
work in African, how shall we say, temporalities for a
very, very long time. I'm going to take you back to a Bucker [Assumed Spelling]
ensemble, one might call it. Here's a string instrument called
the venaluma [Assumed Spelling]. This is now Central Africa,
and I want you to hear first that particular three-two
and it's going to come in the form of a rattle, okay? And the string instrument
is going to double it, so the string instrument will
feel that pattern in the same way. Listen carefully. [ Music ] You might be sort of [inaudible] by the stick pattern
that's in the foreground. [ Music ] One, two, three, four, five, six. One, two, three, four, five.

One, two, three, four, five, six. [ Music ] Can everybody hear that? Okay. [ Music ] I think for the purposes of the
argument tonight I just want to make two points about that. There's a third layer that
I wanted to introduce us to, but for now I'm going
to suspend that. What we have is the simultaneous
unfolding of two patterns, both of which are structured through this N minus one,
N plus one technique. [sounds] Right, that's the
venaluma and the shaker. One, two, one, two, three, one,
two, one, two, one, two, three. So it's five plus seven. One, two, one, two, three, one,
two, one, two, one, two, three. [sounds] That's how you're
dividing up a field of 12. But you also have the Dketo [Assumed
Spelling], which is the sticks that you heard, not the clapping,
the sticks which went like this. They went down below,
short, long, long. [beats] Okay, we need to be asking
a lot of questions right now because this is now going
one, two, two, two, two, two.

One, two, two, two, two, two, two. Six times. But what it's doing is
as it sets up one beat for the first half it reverses
that for the second half, right? And it does so in a way that
recapitulates the N minus one, N plus one logic but on
a much longer time scale. In this case 24, okay, if
you add up all the three's and two's here we get 24. So you have two cycles, both
organized through N minus 1, N plus one, running
simultaneously and going on a phase with each other. Okay, so you need to keep your
wits about you playing this music. There is no overriding beat
or if there is one it is music to find the beat by or even
music to rotate the beat by, which is something that
we will get to hopefully at the end of this lecture.

How do Westerners hear this music? Here's a fragment from Herbie
Hancock's album, Headhunters, in which his drummer, Bill
Summers, takes a short fragment of African music and begins a song
called Watermelon Man that way. [ Music ] Okay, where did he find this? He found this on a UNESCO
collection album that was recorded by Simpha Har Aram [Assumed
Spelling], whom I mentioned at the beginning who is an
Israeli ethnographer who went, commissioned by the French
Government to go teach people in Central Africa to play the horn. And he discovered, of course,
that people knew how to play horns in some ways much better than he
did and he kind of found Jesus and stayed there for a
number of years and made some of these fascinating recordings,
which we now have circulated, right? This particular piece,
there's a very famous article by Steven Feld [Assumed Spelling]
written about how it circulates in a global frame, and
we can talk about that as a slightly separate issue
about when things become monetized and when they stay in a gift
economy but we'll leave that aside for the purposes of tonight.

What I want to draw attention
to is the original recording which was listened to by
Bill Summers, here it is. [ Music ] Okay, once again the one
note, one flute, et cetera. These are by Bengali [Assumed
Spelling] singers or players. In this area the rain forest is
so thick that people often have to find their way by
ear rather than by eye, so it's a very interesting
soundscape and we have a lot of sort of intermediate linguistic coding
through sound that might be music, as language might just be
spatial, location and so on. So definitions of music
are challenged by this, but nevertheless that's the sample. If we went deep into it you'd hear
all the interlocking parts come in and so on. But what I want you to
do is take a listen again to what Bill Summers hears in this
music and how he feels the beat and then compare it, okay? And then I want to make a
point about that difference, even with the same music.

[ Music ] Where is the beat? [ Music ] Where is he feeling the beat? [ Music ] Anybody want to give it to me? I'm giving it to you. [ Music ] That's the beat, right? [ Music ] Everything is conspiring
to that, right? Okay, now again I don't sing
very well, but the way he more or less hears it in that band is
bah, bah, bah, beepa, bah, bah, bah, bah, bah, beepa, dah, dah, dah,
dah, beepa, dah, dah, dah, okay? The way it is entrained in the original context would be
more something like bah, bum, ba, beepa, bum, bum, ba,
ba, beepa, okay? Very different way in which it is
heard as a metric phenomenon, right? And I think that difference
has something to do precisely with this N minus one, N plus one
thing I've been talking about, which is to say if you look
carefully how this melody is structured, dah, de, dah, de,
dum, bum, bum, ba, beep, bum.

Okay, something of
the ba, ba, bum, bum. [beats] Something of that claves
Africanized pattern is embedded in the way the melody is structured, which makes it symmetrically
malleable, okay? That is something now that I
want to turn to in more detail and now this sort of
math game is going to start a little bit
more intensely, so far it's just been
like introduction. But let me draw your
attention to this, right? Here's our claves pattern, here's our gankogui
pattern, excuse me, right? And these are the time
points, zero, one, two. In set theory in math you start,
you count from one to 12 by starting with zero, so forgive me, that's
a convention, but it's zero, one, two because nothing
has happened yet, like time 0.0, right,
we start at zero. So zero, one, two, three,
four, five, six, seven, eight, nine, 10, 11, right? That is 12 time points,
this is clock time, it's as if it's going one
to 12, but zero is 12, okay? so zero, one, two,
three, four, five — well, where do the beats
fall on this, right? I remind you now, dum, dah,
dah, dah, dah, dah, dum, right? That is the pattern.

The beat can fit anywhere
within that pattern. Why is that? Let me play the pattern
and let me play you some of the possible beats, okay? [beats] Easy enough. Why? I align the first
two notes with the beat and then I keep the beat going and
this thing knocks on and then off, just like we heard in that
music of Central Africa, right? Or I can start by placing the beat in between those notes,
the first two, right? Now instead of starting
together I go [beats]. It's as easy to do. As many things are off the beat
and on the beat within the cycle. I could also extend the beat,
quote-unquote, shape it, have a longer beat instead
of having dong, dong, dong, have it dum, dum, dum, okay? [beats] I could take that
place and shift it over one and have the longer beat
but one pulse later.

[beats] Basically, I'm giving you a
phenomenology of an algorithm here. No matter where I put the beat and
no matter how short or long I make that beat the pattern has
the capacity to entrain that beat with equal validity. Now that might not have quite of
made sense and here's the sort of formulation of it, but what
happens when you entrain — these are the binary times, on, off,
on, off, on, off or off, on, off, on, off, on, on, off, off,
on, off, off, on, off, off, or shifted one on,
off, off, et cetera, and this is the third permutation.

Each of those coincides and uncoincides the
same number of times. So to design a pattern
like that takes some doing. There are one or two within a span
of 12 where you have seven pulses, where you can design a pattern
that entrains with equal validity for every possible
time signature, okay? There aren't that many. It's actually kind
of a math problem, which is why I'm absolutely
convinced this isn't just music of the body, but music of the mind. That's probably the biggest
lesson to take away from today. I won't explain this level except
very briefly for those of you who are tuned into
this kind of talk. The zero, three, six, nine is moving
across the time steadily from zero, to three, to six, to nine
in ternary time, okay? That's what that means. The bold on zero and nine is
when the beats are on and off.

Two of them are on, two are off. If I shifted and rotate it to
four, seven, 10, one, two are on, two are off, a different two. And the same thing, there's three
on and one off in the lowest layer. If I go look down the columns I also
get it maximally distributed, okay? So the on and offishness with
seven pulses within a field of 12 is maximally
metrically ambiguous, okay? Remember that this is also
the structure of the piano, which has the capacity to
modulate from one place to another and for you to be able to position,
find yourself in the music, you know where you are in the music,
musically speaking, not the words.

Because of certain properties
that it has mathematically. The colophony of the diatonic
system, which has to do with every interval appearing
a different number of times. That's probably a little jargonish, but for similar reasons this
particular mode of entrainment or of patterning permits any
of many types of entrainment or maximal metric ambiguity. Okay, I'm going to go into a
slightly more speculative mode. I'm hoping that I haven't
lost folks just yet. What you see on the left is
Pingala's Metaprastar [Assumed Spelling], which is the Mountain
of Jewels which he thought up many thousands years ago which
is also known as Pascal's Triangle, but it also emerged in the context
of a certain pneumonic device for memorizing Sanskrit passages
that were opaque and so on and so forth and you
didn't know the meaning but you could memorize
through rhythm.

And he designed this Metaprastar, which gives you the
binomial coefficients. It also gives you the fubinache
[Assumed Spelling] series if you slice through it in different
ways and so on and so forth. So that's India, you
know, what is it? Is it 6,000 years ago, I can't
quite remember the date right now for that. It's on my notes, but
I don't want to look. Pascal, we don't know if
it's just a translation or if he actually reinvented
it, could have reinvented it. And what I want to draw your
attention to is this layer over here, one, three, three,
one, just for the time being.

There's a Sanskrit sort of
recitation [foreign language] which was supposed to be a
way of memorizing that layer of Pingala's Metaprastar. I want to compare it to the African
rhythmic patterns that you have on the righthand side, but let
me go there in more detail. One of the clapping
patterns that's associated with enbitozamazima [Assumed
Spelling] music is very similar to the gankogui pattern
that we listened to. The gankogui remember had
seven in the field of 12 and they looked a little
bit like this. Okay, the gankogui and I've rotated
it a bit, okay, started over here. Dum, bom, ba, bom, bom,
bom, bom, ba, bom, okay? One of the clapping patterns
that holds an ensemble in Zimbabwe together, which we will
get to, has this kind of character. It fills in exactly
that, but it fills in one of the beats over there, okay? So the pattern that you clap
while this ensemble is going on sounds something like this. [claps] Okay, why is
this pattern interesting? The first thing is
we have in the mbira, world of mbira an equal problematic
kind of way in which people are all on potentially on different
downbeats and they're interlocking like this.

So when you're playing
the other person is quiet. It's a very difficult
scene to hold together. So having a pattern that is maximally metrically
ambiguous helps. But it's more, what if you
get lost in this ensemble? What if for a minute
you're just one beat off? Where is that beat when all
the sonic input that's coming from the other members of the ensemble is pointing
you in the wrong direction? Well, here's one technique
for doing that and, again, there are two possible
rotations here in math. This one is the one
that's used in Zimbabwe. If I ask the question,
which Pascal kind of asked, about how many ways are there
of ordering say two elements in a field of say three spaces? So binary, think zeroes and one's
or think dits and dahs, Morris code, or think longs and
shorts, African patterns.

How many ways are there of ordering
say two, zero and one elements in a field of three spaces? Okay, so what I mean is we've
got two elements, long and short. We've got three spaces to put them
in, and there's a number of ways, a fixed number of ways that
those can be ordered, right? We would call this two to the
N, right, two things binary to the power of something,
in this case three, okay, two to the power of three.

Like this is what we would do. How can you create a sonic
pattern that imprints that mathematical value, okay? So that if you get lost
in the ensemble you know in an instant where you are. You don't want to wait,
where is the beginning? This whole beginning thing was
already problematized by the music, itself, so how do you
position and find yourself? Well, consider that the two ways, the ways in which you can
order two elements in a field of three would include, let's
say longs and shorts, okay? We could have three longs,
we could have three shorts.

Everybody with me? You could have two longs and a
short and then we have to recognize that there are three
ways of ordering that. Long, short, long and
short, long, long, right? So we have three longs over here. Here we have two longs and a
short, so we go long, long, short. Long, short, long. Short, long, long. Just spelling it out. The other third permutation would
be having two shorts and a long.

Let's just do the same thing. Short, short, long. And now we'll rotate
that in its threefold way and we get long, short, short. So short, short, long. Short, long, short. Long, short, short, okay? Notice that this is one, plus
three, plus three, plus one. It's two to the power of three,
which equals one, two, three, four, five, six, seven, eight, okay? That's what's going on and these
are known as binomial coefficients. How would you encode that in sound? Well, let's take a
look at this pattern, let's just call it
off or three elements. The first three we can
call them long, long, long. The second three, long, long, short. The next one, next three
is long, short, long. Let's go up here, the next
one is short, long, short. Then we have long, short, short. We have short, short, short. And then we have short, short,
and we've got to spin it around this side, long, and
finally short, long, long.

One, two, three, four, five,
six, seven, eight groupings. And what are they? Long, long, long is over here. Short, short, short is over here. Long, long, short. You can figure it out, but
they're distributed right across that pattern. This means that as soon as you hear
any three pulses you are exactly precisely located within
that pattern. So your clappers are not just
sort of having fun and, you know, they're actually guiding
in a sort of metronomic way where you fit in to the pattern. Given that a big conductor pulse
would just be absolutely disruptive, it would simply pick out one part of the performance that's
actually multi-metric. Let me skip this, but the composer,
Steve Rush [Assumed Spelling], actually took one of these similar
related pattern and turned it into minimalist — it was one of the founding moments
of minimalist music.

I'd be happy to entertain that
in question time, but let's — it circulates at this point
as an ethnographic text. He read a lot of transcriptions
through A.M. Jones, the Reverend Jones who had done a
lot of work in Africa and so on. There's a long story to be
told here about how things get. It's not just the Africarribean,
there's different stories, and I haven't done
that justice but I want to get you attuned to
some of the issues. In fact, he wrote that piece and
there's a sketch that he wrote and he considered it
to be too African, he wanted it to sound more
original, but he nevertheless stayed with the African pattern. Why? Because in a permutation
setting you can't mess with these change of digit
and everything falls away and the interest is
considerably less.

So briefly I want to just talk about
African mathematics in general. I'm not an expert in
the entire field, but I do want to draw your attention
to it, just a couple of features of current research in this. Such as that into the
archaeological sort of remnants of mathematical practice. In other words, when did
we as a species begin to think mathematically, like when
is the first baboon fibula not used as a tool but used
as a counting device? And the oldest ones emerge as
expected, in my view, from Africa. This eshongo [Assumed Spelling] bone that you see here is
8,000 years old. There's an argument about whether
it's 9,000 or 7,000, you know, it's not my debate, but
it's a long time ago and it predates a lot
of numeric thinking. There's also the Lebombo [Assumed
Spelling] bone in South Africa. What I just want to draw
your attention to is not only like if you look at the
markings that are on here, it's not just that we have
a prime quadrant on the left which equals 60, we have certain
kinds of ratios in the middle, and you can figure those out.

But this one interests me
because it's 11, 21, 19 and nine. This looks like decimalism in the
form of N minus one, N plus one, to me, but I can just speculate
at this point about how that kind of thinking gets to be
amongst us as a species. Okay, so I want to end with two
more genres of music to really get down more seriously into the way in which African patterns
articulate very interesting mathematical worlds. Remember I talked about shifting
and shaping or being off the beat or turning the beat around, and
the other one extending the beat. In Africa there are
considerable traditions that do both of these things. I want to turn you now to
a kind of a region called, today called Uganda, that in
some sense is a tragic place because when Motona Bota
[Assumed Spelling] came into power in 1961 I think it is they
soon after, he had the cabaco and the luberi [Assumed
Spelling] court sort of destroyed and terminated, right? The cabaco is the former
chief, as it were, it's a terrible word,
but cabaco, okay? And in this court there was
considerable music played, 24-7.

It was a semi-urbanized area,
said to be the source of the Nile, there are competing
sources of the Nile, but it had a certain
kind of stature. It was in pre-Colonial times. However, because of the way Colonial
rule works very often what you do is politics, we're in
the politics capitol, what you do is you amplify
some part of your enemy, right, you separate and divide and rule.

In direct rule you give your
chieftans a little more power than they might have had and then
they become corrupted in some sense or are seen later by the resistance
fighters as complicit, right? We have this debate
constantly at play. Unfortunately, this meant that
the complicit court was then ended and all the instruments were
burned and so on and so forth. So it's a fairly tragic environment. But, fortunately, there's some small
positive affects of Colonialism, if one can say it that way. And Hugh Tracy [Assumed
Spelling], who was commissioned to actually make recordings for the
Rhodesian Government at the time, kind of reeked out, a little bit like [inaudible] and
really got into it.

He wasn't just doing
it as a job anymore, and discovered this great
universe of cultural making and made some recordings before
the luberi court was ended. And so we have a few
fragments of Hugh Tracy asking to demonstrate how this music works. Now this is interlocking music, but
of a highly virtuosic character. What happens in this music is
that across a single xylophone, known as the amadinda
[Assumed Spelling], a performer will play
a certan pattern. Bong, bong, boo, be, bom, bom,
bom, bom, bom, bom, bom, bom. Very simple, but with octaves,
not what you might expect. And the reason is because
on the other side of the instrument another
performer is playing in the spaces of that first performer, a
pattern of a different length. So you're now nesting into
one pattern of another, right? And this happens at very high speed,
sometimes 600 pulses per minute. This is very seriously high speed. And what happens as a
result is that because of this interaction
notice this is actually from Kubeka's [Assumed Spelling]
work, he made a depiction of this.

Kubek is another interesting
Viennese scholar who went and moved to Africa and so on, settled there. He made this depiction
back I think it's in 1970s around his research,
goes back to the '50s. A very interesting guy. Here is umwawuzi [Assumed Spelling], here is umwanazi [Assumed
Spelling] playing on opposite ends of this xylophone called the
amadinda [Assumed Spelling], right? And what happens, of course, if
you play piano, think of two people on the opposite ends of a piano
playing within the same register and never touching
each other, right? You can't touch anyone because
technically you're going to dampen the wood, right? So you're always going to be off
when the other player is down, but you're occupying the same what
in music is called the register or the same area of
your keyboard, right? You're both coming
down in that same area.

It's not like base and treble,
you're actually both in the middle. The music that comes to the
ear is radically different to the music that's
actually being played. The motor image of the performer
doing this is very different from the audial image, which
is construing this again using that phonic zelatrope
[Assumed Spelling] technique by putting patterns together
that are like phantom patterns. And these are taken out
by a third performer, umaqanazi [Assumed Spelling], and the part is called the
ookonira chunak [Assumed Spelling], and pulled out for
special attention as a kind of resultant or inherent pattern.

So the concept of inherent
pattern or resultant pattern, which emerges in a lot
of Western composition, again of the avant-garde sort. I will not bore you
with those details, but that is actually an
African concept of ookonira, to bring something out that
nobody is playing, right? Take a listen to a short fragment, as I say these are compromised
recordings, very short, didactic almost, at a long time ago. [ Music ] Your ear cannot follow that line,
it can only hear the results. [ Music ] And that's the knocking
brought out, the ookonira, where he's doubling
the resulting pattern. [ Music ] Okay, we'll stop that one there.

Very briefly, the music gets even
more virtuosic with the akadinda, which is to say now you can
also sometimes place two beats within the space of one and so long as you never touch
the other performer. Take a listen to how this sounds
in its more virtuosic form. [ Music ] Second part will come in. So far so good. Now the spaces of the
spaces will be filled. [ Music ] Sometimes two beats, sometimes one. [ Music ] Okay, one wants to continue, okay? The very last music that
I want to introduce you to is that of the [inaudible]. I've spent considerable
time with this music. I learned how to play this before
I learned how to play the piano, which is why I find the piano
so fascinating as you can tell. But here it is and this is the last
of the points I'm going to make, because I want to get
into what's actually going on in those inherent patterns? Right, where is the beat? It's very, very difficult for a
Westerner to place a beat there because in some ways the question
concerning the beat is itself put to question, right? And that's perhaps what's
at stake in this music, if you just hear water
flowing or something like that you're not doing enough
with your ears to open them to this field of possibility.

Okay, here are what one might call
three keyboard interface designs, right? How is the technology designed? And the reason I take this as a
very interesting starting point for thinking through history
and thinking through math is because in some ways once the
interface design gets settled into a fixed structure it
carries a kind of history and it doesn't get
changed as quickly. When things become standardized
it tells you something about that culture and
how it standardizes, how it construes the body,
right, as an interacting medium. So machine learning, all of
this, how is the body construed in relation to the machines
with which it interacts. We will always [inaudible]. The point is not that there's
natural people over here and technological — no, we
are always technological. We automate fire, we're
technological, we sit back, okay? We can watch things happen for us. So this is not the point. The point is how do
different machines, how do different algorithms
comport the body in its behavior? So take, for example, Qwerty, the
typewriter, right, Q-w-e-r-t-y, where did that come from? Why Qwerty? This not how the piano is designed,
which is also a typewriter, it's a typewriter for sound, it's a
music inscription device that works like a phone, too, lots of
slices that make melodies.

Why are typewriters
designed that way? They seem both efficient
and inefficient, right? They're inefficient
because you're kind of going back and forward, right? Why don't we just stroke through,
like t-h-e should be, you know, it shouldn't be like ta, ta, ta. Why is it back and forth? Why does that interlock? Because the hammers would
come flying into the same sort of end point, same focal point,
and you didn't want them jamming. So you don't want people playing
like a piano because the piano is in parallel hammers,
this is in hammers that all have the same destination. You didn't want that clogging up. So you build in an inefficiency and make sure you interlock
it a lot, right? So we get Qwerty. And then what happens is the theory
of ergonomics or whatever it was that the engineers were
working with have to take in account the 10 fingerishness
and then they have commitments about what these fingers can do. And the commitments are
usually these four in the West, these four are really good,
they're facile, and this one is kind of bulky and stupid, maybe
it just has two, you know? Whatever the reasons are, ergonomics
and so on, you construe these as being more facile or quicker.

And so early pianism, the
piano, actually didn't even — eliminated the form, right,
you were supposed to do this. Plus it emphasized righthandishness. So we have in the piano high
notes on the righthand side, low notes on the left-hand side. The very sort of, how should we
say, scalar way of organizing pitch as well as time, but that's
a different issue, okay? So instead of the circular one or symmetrical one this
is a scalar one, right? So from left to right, it's
a kind of darkness over here.

And why did you put this slow stuff, why did you put the base
notes in the left hand? Because the argument was that
the left hands are slower because we're all righthanded. Well, I mean that's the
sort of bias of technology. One needs to look at the values
built into technological designs because if you happen to be
lefthanded it's tough luck, okay, because there ain't that
much music that really — that's not entirely true,
but generally slow movement in the left hand and
then faster up here. And plus on typewriters it was
like the space bar, which was huge and bulky, and you really eliminated
one, you didn't even need it, but there seemed to be so transient,
you know, you can just hit that space bar easily
with the thumb, right? Same thing with pianism, it
comports the body in a certain way and then we respond
and we get really good at typing and playing piano.

And so the model is, the engineers who are designing these systems
are flattered and they think, yes, we must have been right
about the body. So we are the order of complete for the incomplete
algorithm in some funny way. But what I want to draw
your attention to is that that's not the only way of
designing music inscription devices. So if you just take look at the
metepa [Assumed Spelling] over here or the [inaudible] over
here, the difference is that your low notes are more or
less in the middle of the instrument and the high notes fan
off to the side, right? So it's kind of reflective
of the symmetry or semi-symmetry of
the body, itself. I'm not arguing this is better,
it's just a different way of organizing your digitorum, right? And what's interesting is,
I was going to bring it but it's too complicated to
sort of play in the context of this environment, but if you
were to play like a scale on this, because there's seven notes
before the octave recurs in this instrument.

You'd do something very strange,
you'd have to start here, jump here, jump all the way across there,
back there, back there, back there, back there, back there, back there
to play just da, da, da, da, da, da. You'd be going — it's
very cumbersome to play. However, jumping by a fifteenth or by octave is very
easy to play, okay? So it's construing the
sounding forms that come out of this interaction with
the body are very different as a result of that. No less facile, no less
eloquent and so on and so forth. The other important thing
about the mbira [inaudible] is that there's a division of labor
between two performers, right? So where one performer is playing
on the beat, the other player plays in the spaces of that beat, right? So this is not — it's
like the amadinda, you don't play simultaneously,
you play and nest your performance
in this kind of way.

So where you're on, the
other performer is off, okay? Let's listen to actually
a solo performance by Gwanzuto Gwenzie [Assumed
Spelling] just opening to hear some of the rhythmic complexity that
emerges within this environment. [ Music ] Okay, again, pretty
interesting patterns emerging out of this instrument. They say the mbira players that
there's more that comes back from the instrument than what they
put in, okay, that something starts to happen that's not under
their direct supervision and it's music associated
with the ancestral spirits and [inaudible] music of
the ancestral spirits.

I want you just to focus in —
this is going to be the last point for the night and I do appreciate
you staying the course — notice that the left-hand thumb
is the one that is charged with two manuals, while the right
hand has only got one manual. This is a reversal of the
psychology of asymmetry in the West, that we're all righthanded and our technology is designed
around righthandedness. This is in some sense
constraining the left hand as being the dominant hand, okay? That's the first thing to take into
account, there are different ways of understanding the human body
and we can become just as virtuosic in different arrangements. But the second point is the final
point, is I want you to take note of just what could possibly
happen when you're only working with the left hand going up and down
between these two manuals, okay? And remember that there's
a second player in the spaces of the first player. And this is where I'm going
to ask you to take a look at one more diagram because it's
perhaps the most important point of the night, okay? Which is if the first player, the arrows up indicate what
one player is doing, okay, so the arrows up are when one player
is going ding, dong, ding, dong, ding, dong, ding, dong,
ding, dong, ding, dong, okay? So let's say that that is a blue
pattern over here, going up, down, up, down, up, et cetera, okay? The other player — so that's
arrows up, so I'll just make sure that that's the same as what
you see on the PowerPoint.

The second player is playing in
the spaces of the first player, but let's say the second player is
doing something also very simple but a little different, okay? Instead of going ding, dong, ding,
dong, ding, dong, ding, dong, ding, dong, he's going dong, ding, dong,
dong, ding, dong, dong, dong, dong. We would call that two-four time,
dong, ding, dong, ding, dong, one, two, one, two, one, two. And then this one, so one, two,
three, three-four time, okay? So I'm going to do like a
three-four time in between and this one is going
to be in black, happening in the spaces
so we'll start here.

Okay, et cetera, okay? So what happens is this player is
going down, up, down, down, down, up, down, down, up — and this
one up, down, up, down, down. But the ear is picking up that
layer, so you might be playing like this but the ear is only
picking out the bass line, okay? Because the ear, this is the
film, this is the movie, right? Where these are the slices of
sound that are coming to the ear. And so what happens is the pattern
— I'm going to get a red pen, actually, I'm going to
have to redraw this diagram because I didn't give
myself enough space. But what I do want to do is show you that this is what the
ear is listening to. First, that particular
configuration, and in the top it's listening
to the other notes, okay? So I'm going to do this
just one more time, okay? So we're going up, down,
up, down, up, down. And the other one is going down,
up, down, down, up, down, down. Correct? Okay, everybody got that? This I claim is what we hear, right? And this pattern could be written as
two beats absent, one beat absent, two beats absent, none, okay? So what I hear is something like
dum, da, dum, da, dum, da, ta, dum, ta, dum, ta, ta, dum, ta, dum.

So instead of hearing bing, bong,
bing, bong, bing, bong, bing, bong, which is the first player or hearing
dum, da, dum, da, dum, da, dum, dum, dum, what I hear is — put these
down, too many pens — [beats]. Even though each player is doing
something much simpler than that. From the perspective
of Western theories of meter what was binary time
before, dum, dum, dum, two-four or one dum, dum, dum, dum,
three-four, becomes ternary, becomes dum, da, dum, da, dum,
da, ta, dum, ta, ta, dum, ta, dum. Okay, that's the first thing, so
there's a transformation of time that happens that neither
player is in, okay? And that an emergent pattern
that appears out of this.

Why is this fascinating? Because if you change
the smallest detail in your movement totally new
patterns emerge that are not under your direct supervision. So if you're doing up,
down, up, down, up, down and now suddenly you do what
you teach people from the 101 of improvisation on this instrument,
not up, down, up, down, up, down, but down, up, down, up, down, up — it transforms the pattern
completely.

Why is that? Take a look at the second square. The first arrow down is doing
exactly like the diagram up here, it's down, up, down,
down, up, down, down. Again, down, up, down,
down, up, down, down. That player has not changed at all. The second player instead of
going up, down, up, down, up, down is going down,
up, down, up, down, up. The emergent pattern over
here, which that's the form, is very different to that form. And all I've done is
change my sticking. So from a vertical
motion I've produced — this is like typing into a system
and the code gets scrambled, right? You're not typing what
you think you're typing. This is ventriloquized
music, has something to do with ancestors appearing, okay? This is not how you
would design a system for ancestors to appear, okay? And the final point is the
very last point of the night, is that the curious thing about
these patterns is that if you look at what emerges from here,
dum, ta, dum, ta, ta, dum, and you were to look
at this pattern does that appear anywhere
in this pattern? The answer is yes, but
six time points further, where you see the arrow here, dum,
ta, dum, ta, dum, ta, ta, dum.

So a pattern that is fixed has
managed to dislocate itself or be set adrift from the
meter and has shifted over or metrically modulated to a place
elsewhere in that pattern, right? But not in a way that
you might expect. This isn't I'm going to start
a canon and then you're going to follow me like six beats
later with the same music. But rather I'm going to simply
change the smallest detail in the way I'm moving my form. And suddenly it throws this new
voice into alignment, misalignment. In other words, it's not just
the transformation of pattern that happens, but it's
transformation that recoups similar to it, right? A similar pattern reappears
as a phantom pattern, but dislocated from
its reigning beat.

Small wonder that the
mbira players are going to say they hear something more
than what they are putting in, okay? And what they hear,
what these patterns are, just like in the ocarina [Assumed
Spelling] is when they start to sound like a flute which has
to do something with the harmonics and something to do with
these inherent patterns, the ancestors come. Something is speaking
because now you're typing into a typewriter making complete
sense, it scrambles the code into a higher order coherence.

It's talking in sentences
but they're not yours. And I'll end there. Thank you. [ Applause ] It's eight-thirty so
technically we're at the end, but I'm happy to entertain questions
even if they don't make their way into whatever the format
this gets delivered in. I think the first question
was there, so I'm just going to do this, okay? I've seen five questions, I'm
going to go one, two, three, four, five, is that okay? All right, thanks. >> I think you skipped over, I
mean you skipped over a few slides, but there was one that
had like [inaudible]. >> Martin Scherzinger: Yes.

>> So I was wondering? >> Martin Scherzinger: Okay,
those are symmetrical patterns that are drawn in sand in Angola, a tradition that barely exists
anymore, to sonar patterns where you have to make your way
through a series or a gridded kind of set of matrix and then find
your way and create patterns of imagination and
so on and so forth, but they have to be
symmetrical in these ways. So it's a trick, it's a
trick, it's a math trick. This was actually a secret kind
of group, quite patriarchal, and so this is one of the tragedies
though of African mathematics because the way in which you got
important information was not necessarily through this
principle of copyright, which is how one does it in the
West, but by abolishing all evidence so that you can retain authority within smaller kind
of secret societies.

So it's one, it's an area
of great interest and also in some sense tragic because how
do we reconstruct these great mathematical traditions
when part of the project was to eliminate the archive? And in the West we can talk
about copyright as a way of giving authority to
certain voices and that, too, is a system that has great
affordance but also limits, there are serious limits,
especially if we look at how some of the biographies of
these sounds work, right? We have this music from
Africa that is authored there, but then when it's taken up by Western musicians suddenly
it becomes copyright worthy, right? That's another very important
question, but I'll leave it at that.

That's where that's from, Angola. Gerhart Kubek [Assumed Spelling]
writes about it beautifully. Gergli Legetti [Assumed Spelling],
who is a Western composer, also part of the avant-garde, he
was like a highly funded composer of Europe, drew great inspiration
from these diagrams, as well, when he Africanized
his piano techniques and that's a very interesting new
kind of piano sound that comes out of his because
of this encounter. Hi. >> Can you talk a little bit about South African inspiration
[inaudible] where did ancestors come up with these rhythms [inaudible]
and everything around them, somewhere out in the bush,
an orchestra in itself. I'm just curious if there's things,
mathematical connection with things that are in nature that sequence in
with these rhythms that are used? >> Martin Scherzinger:
That's such a nice question, but it really is too
big for me to engage. And I have to say that
my focus is quite narrow.

It's deliberately narrow
because of my own biography and my own biographical sort
of relationship to the land, which means I grew up in
Apartheid and one of the dangers of organizing people according
to cultural identities was that it very quickly
becomes hierarchized. So I felt very suspicious
about that from the beginning. I could see in some sense as a child
unfolding before me great injustice simply because of the way people
were being understood to be. And so for me understanding Africans
as having very connected to dance and embodied and so on was a way
of disavowing the intellectual work that was built into this. So when the question of
environment comes up I'm a little — I back off a little bit only because
of that, because it's the idea that people are more
natural or something.

However, there's very
interesting work being done that. I mean Steven Feld [Assumed
Spelling] has done some very interesting work, looking at how soundscapes impact
how people organize time. And you know anybody
who has been to Africa, I mean it's an incredible
soundscape, and just the simple rhythms of
open space are something to behold. So it wouldn't surprise me if
there's some kind of connection. The archaeology and all of those
kinds of questions are extensive, you know, how tuning
systems emerged. Most of my work that I've done
in this region has nothing to do with rhythm, partly because
I shy away from the idea that Africans have a special
endowment for rhythm or something, and I use the pun intendedly because there's this
cliche that comes with it. In fact, [inaudible] music
is more noteworthy for me because of its profound harmonies
and the way harmony works through space, which is
a kind of fractal logic, which would have just been
a little too difficult for an open forum like this. It's a more strictly
mathematical set theoretical world and I would have had to start there
and I would have maybe got a point or two across in that time.

So I didn't do it, but imagine
there's no rhythm in African music, imagine there's no dance,
and now you're just left with certain tuning practices
and harmonies and you get results that are even more
mathematically compelling. I'd be happy again to talk right after if you're interested
in that part. So I was always drawn to how this
music is thought because the people, you know, we were raised in
interesting ways in South Africa. I was raised by partly by my
older sister and a maid, right, so I didn't ever understand this
as being a kind of distinction, that I was encouraged to in the
classroom, in our education system. Of course, I benefitted
from white education, but at the same time my cultural
wires got a little crossed. I forget who, was it you, sorry? >> The last recording
that you played you had on the screen a picture of I
guess a finger piano, a kalimba, and you called it a solo,
was that a single person? >> Martin Scherzinger: Yes.

>> Playing a single instrument? >> Martin Scherzinger: Yes, so that's Gwanzuto Gwenzie [Assumed
Spelling], a remarkable player, absolutely remarkable player, who is
basically imitating how it would be to play with two players. So I mean he's just a virtuoso, but
if you put him together with someone like Ford Kwinda [Assumed
Spelling] I mean things are really just unbelievable. You have to tune into it for a
long time before you really get at what's going on there. These are all night spirit
possession ceremonies and so on. After awhile it does feel like everything is just
sort of automated, right? It's other worldly in that sense,
but Gwanzuto is just a brilliant, brilliant, brilliant player and he
made this recording because he came and took a visit to
the United States.

And you know that's what you do,
you have your CD in your pocket and then you do your
concert and so on. But traditionally you
never play with one player, I mean in the context of mbira or an
all-night ceremony you always play with a second player
who is locking in, so that the whole thing
becomes much more mysterious and phantom patterns can emerge. But, yes, a great player. I think, yes, sorry? >> How are we spelling, if I heard
you correctly, the Angolan tusona? >> Martin Scherzinger:
Yes, t-u-s-o-n-a. So a short question. >> And also the mbira in that other
slide was I think on the right, I wonder where the
other one [inaudible]? >> Martin Scherzinger: Yes, okay, that is called matepe,
m-a-t-e-p-e, matepe. Matepe is, in fact, from the
perspective of fractal logic that I've been working with, an even more interesting
instrument than the [inaudible]. It is less played than
the [inaudible] because it's basically what was
once known as the Korekrre people, which is a dialect of
Shauna [Assumed Spelling]. It's actually an independent
language, but history has it that …

>> Korikori? >> Martin Scherzinger:
K-o-r-e-k-r-r-e. Very few matepe playing
families left in the world, but that is I think the Korekrre
region is up in northeast of Zimbabwe or modern day Zimbabwe, an area that once controlled
the salt pan, which for a landlocked land
is gold, I mean you need salt. And it was once in my view a
very dominant area culturally and then became in some sense
impoverished through the forces of history, let's just call it that. And as a result of that the
instrument is marginal today even in Zimbabwe, which is really
interesting and kind of problematic because it is I think the source
of much of the fractal reasoning in harmonies, but that was
a different conversation but it's a fascinating instrument.

Okay, one more question? >> How different is the
matepe to the mbira? >> Martin Scherzinger:
The mbira [inaudible]? >> Yes, because you said in the
center you have your low notes and the high notes on the side? >> Martin Scherzinger: Yes, so
that basic principle is the same, but you'll see all of the
notes, the note configuration — there are many more notes here,
first of all, in most of them and then they're organized
very differently. [ Inaudible ] >> Martin Scherzinger: If
you can play the matepe that would be the place to go
because the mbira [inaudible], here's the tragedy, remember
I mentioned radio, too, where they were recording made
for Africans to keep Africans in separate developed space,
you know, so radio, too? Well, the mbira [inaudible]
the one on the right happens to be the instrument
that was the most played in the Colonial center then called
Salzbury and the engineers didn't go that far, and so there's a great
archive of mbira [inaudible] and the matepe sort of
gets written out of history because it's a little
too far away to travel so there's fewer historical
recordings.

[ Inaudible ] >> Martin Scherzinger: Yes, Stella
Chuwachi [Assumed Spelling]. >> Is she still playing? Because I saw her here,
there was some program, and I can't remember the
name and it was all women. She and [inaudible] and others
and it was really extraordinary. And they say she had to fight
to be able to play the mbira because it's traditionally
played by a man.

>> Martin Scherzinger: Yes, it's
a sort of patriarchal thing, but there have been some
great woman performers. I mean Beulah Dyoko, who died
recently, but she traveled with some really great
players in the '90s. Beulah D-y-o-k-o, she died
recently, but a magnificent player. I think perhaps better than Stella. Stella is kind of a
little more hybridized. But, yes, it's a constantly
evolving tradition. But the one thing that's interesting
is that, as I said earlier, as a technology it didn't
change much, like the piano, it sort of settled down towards
the end of the 18th Century and then it reached
its talus, right? I mean it tweaked a little bit,
but the basic technical art lay, the interface design remained the
same and it's that that becomes such an instructive
microscope into history. >> You talked about how
the music was created. My question is what did
the dancers do with this? Can you look at one
of these diagrams and know what the dancers
would be doing? >> Martin Scherzinger: Okay, so
the dancers do different things in different places, obviously,
but take for example — I'll just have to be
brief so I'm just going to give you an interesting
sort of schematic.

Remember we started with the
gankogui pipes from Mozambique, which was the second thing we did,
but the dancers there remember move in a way that the instrumentalists
are not playing, right? So they're going one,
two, three, one, two, three, yafu, yafu, yafu, yafu. And then once that is locked in,
once that felt beat is locked in, then there are all sorts
of dancers with steps, such as 50 Boro [Assumed
Spelling], which is a football. But you're feeling it in
four, so you're not doing — it's a bit boring, but you
are feeling it in four. In the context of the mbira,
which is sort of on, off, on, off, on, off, binary, right? Daka, daka, daka, daka,
daka, daka — and that's kind of basic principle.

I mean here's a diagram
that makes that apparent. This is if everybody is
playing exactly the same thing. So ding, dong, ding,
dong, ding, dong. And the other one ding,
dong, and so but what comes to the ear is da, ta,
da, ta, da, ta. Interestingly, what
the [inaudible] does, which guides the dance step is not
going da, ta, da, ta, da, ta — now the music is in four, one,
two, three, four, one, two, three, four — it's binary, that's why,
but the [inaudible] goes choo, choo, choo, choo, choo, choo, choo,
choo, it's in ternary, right? And the dancers are going
to move in ternary time. So it's kind of cutting across
that again and, of course, the dancers become complex but
the basic felt pulse is at, how should we say, at
a kind of cross-current with the dominant musical
input, okay? Again, I don't want to
generalize this Continent. There are certain features,
you know, it's interesting because even just the Nyungwe,
which is close to the [inaudible].

I mean we've just got a
simple Colonial border, almost a straight line
separating them, so they're related but they've inverted
that relationship of in the one we've got four against
three, in the other we've got three against four, broadly speaking
when it comes to the dance. But the question of dance
is much bigger than this and the choreographic supplement
is an important component of felt time, right? So I mentioned it sometimes and
not always just for time reasons, but it's a great question. >> Is there anything I can
look at or read that talks more about the dancing, any research? >> Martin Scherzinger:
Yes, so my colleague, Kafi Agawu [Assumed Spelling],
he's a [inaudible] musicologist, he write about Mozart and he writes
about Africa, which is the way to do it because then you
understand more the affinities and not just all the differences. Kafi Agawu, A-g-a-w-u,
Princeton University, has written a book called
Representing African Music, where he actually raises the
argument that we need to be looking at the dance steps if we want to really understand how
the felt metrics work.

So, yes, what I was trying to
suggest is that there's a number of possibilities and layers that vie
for syntax formation and, of course, what the body is doing is
one layer within that sphere. Yes? >> Can you go back to
the slide [inaudible]? >> Martin Scherzinger: Okay, this
one is interesting because … >> Not that one. >> Martin Scherzinger:
Oh, not that one. Oh, this? >> No, no, the secret, you talked about a secret communication
pattern? What are those? I must have missed that sentence because when I look
at that I see animals. >> Martin Scherzinger: Yes,
yes, yes, that's interesting. So but what they are is symmetrical,
they're sort of geometric shapes that are uni-linear, you cannot lift
your stick or your pen if you like, as you create a symmetrical pattern through a series of
gridded dots, right? And you then emerge with a pattern
and, yes, of course, that's a lizard over there and so on
and so forth, yes. >> So the dots are rocks on
the ground or there's …

>> Martin Scherzinger: Holes
in the sand, holes in the sand. >> Okay. >> Martin Scherzinger: And then
you've got to step your way through it without lifting your
stick and come up with a design. If you think about this
mathematically there's some tricks that you have to know. >> Okay, I understand that now. I wasn't sure if these were
dance steps or notes or what? >> Martin Scherzinger:
No, this I didn't talk about because it's visual culture,
all this is is an introduction to look, here's that bone, the
ushango [Assumed Spelling] bone, and there's evidence that there's
a sort of numerocic thinking in the region that predates
most other regions of the world, at least in terms of
archaeological evidence.

I have a softer theory
of this and that is that given certain problem
sets people in different parts of the world might come
up with the same answer. So I'm not saying that
the piano, which came long after the gankogui pattern,
I'm not saying that it derived from necessarily, I'm
saying thinking in a certain set theoretic way with
certain problems and aspirations in mind might lead you
to the same conclusions, but I'm certainly not saying that
mathematics was invented in Europe. >> Now that I know what this is
I'm thinking [inaudible] part of the world. >> Martin Scherzinger: I think that's probably
the next talk that's coming up with The Library of
Congress, but, yes, so fantastic. I think we should probably
close things up. Thank you for the great questions
and thank you for your time. [ Applause ] >> This has been a presentation
of The Library of Congress. Visit us at loc.gov..

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