Akan Diatonic Sets & Tetrachords

AKAN DIATONIC SETS AND TETRACHORDS

1.1 BACKGROUND

Music theory seems to rest largely on the distinction between pitch and pitch classes. The term Pitch‑Class was introduced by Milton Babbit in the mid. 1950's; in fact, the first genera] exposition elaborating the implications of Pitch and Pitch‑ Class distinction was published by John Rahn in 1980.

RD Morris in his book on Composition with Pitch Classes: Theory of Compositional Design: (1987) develops the basis for Set Theory as proposed by Allan Forte (1973). Clough and Douthett (1993) developed Mathematical tools of Consecutive Integer Sequences and Multiplicity Sequences from Richard Cohns’ Concept of Transpositional Combinations (1982, 1991, 1992, 1995) ‑ A process whereby sets are formed by merging smaller sets related by transposition. It is a procedure that involves the use of Mathematical Applications to realize objectives in Musical Analysis and compliments the understanding of Melodic Procedures of diverse cultures in similar terms.

In their article on "The Scientific image of Music Theory" in The Interpretation of Music (1993:65), Brown and Dempster, maintain that 'If Music Theory is to be taken seriously, it must clarify the nature of music and thereby guide musical activities, whether be they Performance, Composition or Historical Research.

1.1.1 STATEMENT

The controlling assumption is that Akan vocal and instrumental music is constructed on the heptatonic scale as a union of two conjunct tetrachords, organized in two distinct modalities of pitch class sets.

1.1.2 OBJECTIVE ‑

To define and explain necessary mathematical tools of set theory, series and sequences which will form the basis that deals with similarities between a class of Akan heptatonic scales and the usual diatonic set.

1.2 CHROMATIC AND DIATONIC CARDINALITIES

Definitions are taken from Clough and Douthett (1997) with parenthetical numbers appearing after definition headings: where necessary, further explanation/examples of theorems are provided within the framework of a particular Mathematical Application.

1.2.1 Definition ‑ 2.1: Given a chromatic Universe of a class of pitch sets (with c ,as chromatic universe), we say c is the chromatic cardinality. We represent such a universe by the set Uc= {0, 1, 2,…c ‑ 1}, and as usual, we assume the integers are assigned to the notes in ascending order.

1.2.2 Explanation ‑ The word set, is used to represent a group of things having some common property. A set represents a well‑defined collection of objects, things, numbers or symbols. A member of a set is called an element of the set. In the discussion of particular sets, the assumption is always made that the sets under discussion are all subsets of some larger set called the Universal Set, U.

The Chromatic cardinality, Uc (i.e. Universal set Uc), is obtained by translation from chromatic alphabetical pitch order( c ‑ c'), to corresponding numeral pitch equivalents as shown below:

	dB		Eb			GB		AB		Bb
	C#		D#			F#		G#		A#
C		D		E	F		G		A		B	C
0	1	2	3	4	5	6	7	8	9	10	11	0 -

We represent such a universe by the set Uc = {0,1,2,..,c‑1} which is the total collection of all elements under consideration in a given problems.

1.2.3 Definition 2.2 ‑ To indicate a subset of d‑pitch class sets selected from Uc we write:

Dc,d ={Do, D1,D2....D,d -1 }. Here too, we assume the pitch class sets in Dc,d are in ascending order, [i.e. ,D0 < D1,< D2, <…, <DD-1] i.e. D0 is less than D1 etc.,. We say d is the diatonic cardinality of Dc,d

1.2.4 Explanation :

A universal set may have many subsets as shown below:

For C 'mode' of the Akan diatonic set we write D12,7={0,2,4,5,7,9,11} which forms a diatonic set. Another way of describing a set uses the implicit or set‑builder notation. The set of notes in the C major scale may be described by:

{x|x is a note in the C major scale}

Read as "The set of all x ,such that, x is a note in the C major scale". The vertical line is read "such that". The symbol preceding the vertical line, X, designates a typical element of the set which the statement to the right, tells how to find the specific instance of x.

1.2.5 OPERATION WITH SETS ‑ There are three basic operations with sets: Union, Intersection and the Compliment; the operations with sets may be performed in a manner similar to basic operations of addition, subtraction, multiplication and division in ordinary algebra. Union of sets A and B is denoted by the set A U B (read: A union B or A cup B).

The Intersection of sets A and B is denoted by the set A n B

(Read as A intersection B, or A cap B)

A Complement of set A is denoted by the set A (i.e., all elements of U

that do not belong to A.)

Example 32

In the diatonic set D12, 7= {0,2,4,5,7,9,11} two transpositionally equivalent tetrachords divide the major scale, i.e.., H1= {0,2,4,5} and H2 ={7,9,11,0}. The union of these two tetrachords form the entire diatonic set (Hl U H2=D,12, 7) and their intersection is a singleton (Hl n H2 = {O}); i.e. ,0 is an element of both sets.

Example 33:

The Akan heptatonic scale may also form a union of three conjunct tetrachords as shown below:

In observing the diagram below we find:

Example 35:

Let Uc = {0,1,2,3,4,5,6,7,8,9,10,11}

H1 = {0,2,4,5}

H2 = {7,9,11,0} and H3 ={4,5,7,9}

From the above,

(a) H1 U H2 U H3 ={0,2,4,5,9,11}

(b) H1 n H2 n H3 ={0}

(d) A’ n B’ n C ={0}

Where H1 = {1,3,6,7,8,10,11},

and H2 = {1,2,3,4,5,6,8,10}

The symbol {0} denotes a null set, which indicates the set, is as empty as possible.

Corollary

Operations by Postulates: Many of the postulates for sets are the same as found in ordinary algebra:

1. Commutative laws: A U B = B U A

A n B = B n A .

2 Associative law: (A U B) U C = A U (B U C) (A n B) n C = A n (B n C) (Harsharbeger and Reynolds: 1985:27)

1.3 HYPERTETRACHORDAL STRUCTURES

1.3.1 Definition 2.6 ‑ A pitch class set (pcset) Dc, d has a hypertetrachordal structure if there exists two transpositionally equivalent subsets H1 and H2 both step related to Dc,d whose union is the entire parent set and also whose intersection is as empty as possible, as earlier established.

Hence we write Hl U H2 = Dc,d and Hl n H2 contains at most one element.

1.3.2 ‑ EXPLANATION

Pitch Class Sets in consecutive integers (normal order) provide a basis for the development of certain fundamental procedures from which analytical observations about structure can be deduced.

Accordingly, two pitch class sets will be said to be equivalent, if and only if, they are reducible to the same prime form, by transposition or by inversion followed by transposition, therefore, if they are not reducible to the same prime form they are not equivalent.

The operation of transposition is of fundamental importance to tonal music, therefore configurations which may appear dissimilar in many respects, can be equivalent at a more basic level of structure. (Forte 1973:37). The concept of Transpositional Combination is therefore an indispensable tool in set theory and serves as a powerful means of connection to Mathematical Applications.

In the usual diatonic set D12, 7 ={0,2,4,5,7,9,11}; for the two transpositionally equivalent tetrachords (TE) we write ‑ Hl = (0,2,4,5) {0,2,4,5} and H2 = {0,7,9,11} or {7,9,11,0}. This therefore divides the C 'mode' into two parts. The union of these two tetrachords is the entire diatonic set H1 U H2 = D12,7 - their intersection is a singleton written as H1 n H2 = {0}.

Assuming that the tetrachords must divide the diatonic set into two transpositionally equivalent parts whose union is an entire pitch class set, then intersection must be non‑empty (since d is odd); therefore the intersection is either a singleton or empty depending on whether the cardinality of the parent set is odd or even.

1.3.3 Definition 2.7: Assuming Dc.d (i.e.., D12,7) has a hypertetrachordal structure. The clen(s)- (chromatic length) between the hypertetrachords will be called gap(s): that is, a gap is a non‑zero clen from the "end" of one hypertetrachord to the “beginning” of another. This Theorem involves step‑related hypertetrachords and will prove useful in the process of developing hyperscale axioms. Example: For hypertetrachord H1= {0,2,4,5,} and H2 = {0,7,9,11} ={7,9,11,0} in the C major scale or 'mode' the gap is 2; i.e.., the clen (chromatic length), from F to G, or 5 to 7 is 2. (Clough & Douthett 1997:70).

1.3.4 CASE 1 [COROLLARY]

Assume the gap is c/d. Since the gap is even and the spans of the hypertetrachords are equal, the span of each hypertetrachord is (c/d )/2. Therefore, the hypertetrachords can be written as

H1 = {D0,D1…,1/2(c-c/d)

and H2 = {D1…,cc/d }

1.3.5 CASE 2: Assume the gap is c/d+ 1. Since the gap is even, the span of each hypertetrachord must be (C - c/d ‑ 1)/2; therefore the hypertetrachord can be written as:

H1 = {0, Dl,…, l/2(cc/d-1 )} and

H2 = { I /2(Cc/d‑1), 1/2(cc/d-1) +D1,…cc/d-1}

For any x as an element of Uc |x| denotes the ceiling of x and |x| the floor of x as the greatest integer in x hence |c/d|. This section can be skipped with no loss to continuity.

1.3.6 - EXPLANATION-One of the most important and widely applicable types of efficient algorithm is based on the divide-and-conquer approach. The strategy in general, is to approach a problem of the size Uc={0.1.2.3…,11} where d7 is an element of Uc, by directly dealing with a small value of d, (i.e. the tetrachord segment), this provides the initial condition for the resulting recurrence relation. Breaking the general problem of the size of d,(diatonic set), into smaller problems of the same types and approximately the same size as |c/d|, where d7 is an element of c12 (b). As in the case of the fore-mentioned hypertetrachords derived from the diatonic set, since they are equal each hypertetrachord is (c-|c/d|)/2 ; hence they are written as expressed above.

c = d + (d ‑ 2)

c = 2(d‑1) also , 2(d‑1)‑c = 0

This implies that there are two step ‑ intervals with clen (Chromatic length) one, and 2 intervals with chromatic length 2. Since there are precisely two step‑intervals with clen 1, each hypertetrachord must have one, and they must be between corresponding pitch set classes in the hypertetrachords, therefore, D12,7 has precisely one ambiguity, i.e., the naked tritone or augmented 4th

But now Dc,d or D12,7 has hypertetrachordal structure comprising H1 and H2 respectively as a pair of hypertetrachords of a pcset ‑Span (H,) = Span, (H'), that is both spans are equal. The basic assumption is that Dc,d = {Do, D1…, DD-1} is a set of distinct elements out of Uc = {o,1…, c‑1} where d< c and generally Dc,d Satisfies Do < D1 <...<D1, < …,<DD-1(Clough, Douthett 1997:74).

Therefore, it follows that d = c /2+ 1

and c = 2d‑ 2

i.e. c = 2(d‑1)

1.4 AKAN DIATONIC SET AND THEIR AXIOMS

To begin with, we re‑state the controlling assumption the Akan vocal music is constructed on the diatonic set as a union of two conjunct tetrachords organized in two distinct modalities of pitch class sets. This has been the guiding generalization throughout this chapter the object of which has been to define and explain Mathematical tools of set theory, sequences and series that deal with similarities between a class of Akan diatonic sets and the usual diatonic set.

From the evidence of extensive recordings of Ghanaian music, it appears the scale systems in use in Ghana are diatonic in concept and that microtones are not integral elements of melodic structure (Nketia 1962:34). Ethnic groups, which use the pentatonic scale, include the Adangme, Dagbani and Frafra, while those who use variety of the Heptatonic scale include the Builsa, Konkomba and the Akan.

As J.H. Nketia (1963:32) and B.A. Aning (1969:98) have confirmed, all Akan songs including Nnwonkoro songs and Akan Guitar Band songs are based on the diatonic set. There is further evidence to support the contention that the scale is organized on the basis o tetrachordal segments within which the melodic phrases are built, these have already been defined and explained in terms of set theory as a union of two hypertetrachords which form the entire diatonic set (H1 U H2 = D12,7 ).

The basic assumption was that, Dc,d = {D0, D1…, DD-1} is a set of distinct elements out of the universal set Uc= {0,l,..c-1 }, where d<c and generally Dc,d satisfies D0 < Dl…,<DD-1 (Clough, Douthett 1993:). The result is that while Dc,d contains no contradiction in terms of equivalence, Dc,d has precisely one ambiguity which is the naked tritone of the augmented 4th, F to B. Based on this assumption it also follows that d = c/2 + 1 and c = 2d – 2; it can also be expressed as c = 2(d ‑ 1).

Clough and Douthett (1997) discuss a variety of inherent properties of the usual diatonic set that when connected with those of the Akan diatonic set has parallels. They define diatonic tone system as a Maximally Even scale system that has precisely one ambiguity, meaning that, a set of diatonic intervals (hypertetrachords) contain exactly one enharmonically equivalent pair. The seven‑tone heptatonic scale Dl2,7 is of particular importance because it shares affinities with the Akan heptatonic seven‑tone set earlier described.

In summing up on the basis of earlier definitions and classifications, the usual diatonic set can be connected to the Akan heptatonic set by means of the following axioms (Douthett and Clough: 1997) with which it shares similarities.

(a) Dc,d has precisely one ambiguous tritone (An Augmented fourth ‑F to B)

(b) C = 2(d ‑ 1)

(d) Dc has precisely two intervals of clen (Chromatic length) of one.

We say Dc,d is a diatonic set, if it satisfies any one of the stated conditions in the Theorem. (Clough, Douthett 1997:88); indeed, the axioms further strengthen the definition of the diatonic set. It is my hope that this initially established link will serve as a premise to stimulate further inquiring into its Theoretical implications and possible applications.

Every piece of Mathematics is about the consequence of a particular set of rules or axioms. As with the basic laws of a country or the rules of a game, some set of axioms prove to be more useful or more interesting than others, but the only essential quality is to be clear and consistent without contradiction (Turner 1976:5). Both children and adults gain satisfaction from solving problems, perceiving connections, understanding structures and learning new techniques. Mathematics and music have been presented as similar because both are concerned with linking together abstractions with making patterns of ideas (Storr 1992:180).