Theory d1 Musiic Win Libre

7/18/2019 Theory d1 Musiic Win Libre

http://slidepdf.com/reader/full/theory-d1-musiic-win-libre 1/20

T h e o r y d 1

M u S I I C - W i nM u s i c : I n t e r a c t i v e C o m p u t e r

S y s t e m f o r R e s e a r c h a n dC o m p o s i t i o n 1

I n t r o d u c t i o n

MuSIIC-Win is based on Julio Estrada's Theory d1 on the combinatory potential ofany scale intervals. (ESTRADA 1994) In Theory d1, the notion of scale is alwaysapplicable to pitch and duration. The denomination Theory d1 refers to the continuousnature of all the operations, which are based on transformations at minimal distance,d1. The mathematical reference for those transformations is combinatorics, and itsrepresentation is based on graph theory. In general terms, Theory d1 proposes thefree exploration of scales, not imposing any categories for systems of composition ormusical aesthetics.

Theory d1 is a useful research tool in the following fields:

Musical composition (ESTRADA m1979, 1999) Analysis (ESTRADA 1987) Compositional theory (ESTRADA 1994) Musicology (Ibidem ) Musical pedagogy (LACREMUS 2000) Musical Acoustics (ESTRADA /ADAN 2002)

Theory d1 studies musical scales as a double continuum:

The physical continuum that links rhythm with sound –-as observed by Henry D.Cowell’s research (COWELL 1930, 1969)–-, a principle which allows the integration ofinterval scales of pitch and duration in a unified field.

The mathematical continuum of scales produced by equal divisions of the octave, or

the division in equal parts of a duration unit. The physical continuum of acoustic wave-shape transformations in both frequency

and time domains.

The computer program Theory d1 allows a total of 22 scales ranging from 3 to 24intervals of pitch or duration. The program is expected to develop further andovercome the difficulties of an adequate graphic representation of scales of greater

1 Translation by Alejandra and Víctor Adán.



extension, particularly those approximating 50 intervals. This limit is derived from theidea that, when divisions of a pitch octave tend to surpass 50 intervals –-close to theinterval of the coma ( , ), the ninth of a tone suggested by Gerhardus Mercator(VINTON 1974)—the acoustic sensation of the interval begins to dilute giving way tothe sensation of a continuum, as glissandi . Concerning the practical control ofinformation generated by scales having a great number of intervals, it isunderstandable that the increase in the number of intervals will also increase theircombinatory and processing time, something which will reduce the human capacity ofacoustic discrimination, memorization or practical manipulation.

MuSIIC-Win is the result of the reengineering of MuSIIC for MSDOS (PMMI-IIE-UNAM1990-97). Its appearance has been completely redesigned and made standard to theWindows® environment. It is user friendly, facilitating exploration and interaction. Italso enhances the visualization of the musical materials through graphs and traditionalmusical notation, and allows their audition through a multimedia computer or a MIDIinterface.

T h e o r y d 1

1 B a s i c C o n c e p t s

Interval classes. The theory of the combinatory potential of the scales is based onoperations on interval classes. In the present case, these operations are applicable topitch and duration intervals. Interval classes offer an efficient synthesis for reducingand expanding information to a maximum. They are more efficient than pitch classes,because, among other things, they reduce information of the intervallic groupings to aminimum number of cases. On the other hand, interval classes allow us to considerboth duration and pitch intervals with a single theoretical criterion. Whole numbers areused to represent the dimensions of intervals of pitch or duration, with the smallestinterval being 1. For example, a scale (E) of 5 intervals is expressed as:

E5 (1,1,1,1,1)

Minimal distance transitions: d1. The theory of the combinatory potential of thescales is based on continuous minimal distance transformations: (d1 ). Severaloperations of the theory are based on d1 minimal distance transitions. They are listedhere for reference and will be discussed later:

permutation of two intervals: (1,4) d1 (4,1) substitution of two intervals whose sum is identical: (1,4) d1 (2,5) partition of an interval into two intervals: (5) d1 (1,4) fusion of two intervals in one interval: (1,4) d1 (5) addition of a minimal unit to a group of intervals: (1,4) d1 (2,4) substraction of a minimal unit from a group of intervals: (2,4) d1 (1,4) division of one or various intervals by the same number: (4,4,4) d1

(2,2,2) multiplication of one or various intervals by the same number: (1,2,3)

d1 (2,4,6)

Initial scale. The main structure for all operations with intervals is the initial scale . Thisscale is bounded by the first and last of its adjacent interval. Examples:



E3 (1,1,1)E12 (1,1,1,1,1,1,1,1,1,1,1,1)E24 (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)

An initial scale will contain as many adjacent intervals as terms2.

Scale resolution. The number of adjacent intervals of the initial scale defines theresolution within the continuum of the scales.

Cycle interval of the initial scale. In order to perform theoretical operations, it isnecessary to consider the initial scale as a cyclical structure. The cycle interval is anadjacent interval at d1 minimal distance composed of the first and last terms (n ) of theinitial scale. Numbering its terms from 0 to n , the initial scale can be expressed as:

(0,1, 2, 3,…n )

Using the notation for interval classes, the cycle interval is expressed as follows:

n ) (1,1,…)

Duplication interval. The duplication interval is the sum of all adjacent intervals ofthe initial scale, including the interval of the cycle. Examples:

E5 (5) = (1,1,1,1,1)E11 (11) = (1,1,1,1,1,1,1,1,1,1,1)E19 (19) = (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1)

For pitch intervals, the duplication interval is equivalent to what is traditionally calledthe “octave”. However, we avoid the term “octave” because it does not conform toall possible scale, particularly those where the eighth term is not a duplication ofthe first.

The notion of the duplication interval in durations is analogous to the traditionalnotion of measure . For example, a scale of dimension E7 is equivalent to a unit ofduration composed of 7 minimal values, or, in the case of the measure, by a groupof the form 7/4, 7/8 or 7/16.

2 S c a l e I d e n t i t y

Identity. All the possible groupings of adjacent intervals can be expressed with apractical convention by keeping a unique order that may allow us to identify them as agroup. Consequently, an identity is a set of intervals ordered in increasing dimensionwithin the duplication interval.

2 All the operations that can be performed with the program are restricted to the realm of an initialscale. In practice, the initial scale can be represented by a total scale , which can be partially ortotally reproduced outside of its scope. The total scale can be that of an instrument or acomposition. When applying Theory d1 it is necessary to reduce the information to the realm ofthe initial scale. (ESTRADA 1994).



Identities arise from the partition of the duplication interval of the scale. This minimaldistance d1 operation is equivalent to the division of a segment in two parts of differentmagnitudes. For example, in the E12 scale, we have the following partitions of theduplication interval:

(12) d1 (1,11) d1 (2,10) d1 (3,9) d1 (4,8) d1 (5,7) d1

(6,6)

The sum of all the intervals contained in an identity will always be equivalent to thedimension of the initial scale. This can be observed in the sum of the intervals shownabove (12). The increasing order in which these pairs of intervals appear allows us toobserve them as an identity.

Any of the pairs of intervals obtained from the identities mentioned above canthemselves be partitioned, generating new identities with three intervals. For example,in the case of (1,11) we will have:

(1,11) d1: (1,1,10), (1,2,9), (1,3,8) , (1,4,7), (1,5,6)

The successive partition of a new interval greater than 1 (which is always the smallestinterval) allows us to obtain the complete set of identities of a scale. The followingexamples illustrate several identities in scales of different dimension:

E5 (2,3)E6 (1,1,1,1,2)E7 (1,1,1,1,1,1,1)E8 (1,1,2,2,2)E9 (2,2,2,3)E12 (3,4,5)E16 (1,1,1,1,1,2,2,2,5)E21 (1,2,3,4,5,6)

The intervals grouped in an identity of pitch or duration can have multiple equivalencesin a musical score:

vertical agglomerate sequence counterpoint free vertical-horizontal accumulation formal structure (scale), etc.

3 C o m b i n a t o r y P o t e n t i a l o f t h e S c a l e s

The total set of identities of a scale constitutes its combinatory potential . In it, identitiesare ordered with respect to their density, i.e. the number of intervals they contain.These density levels are arranged from left to right in increasing order and arebounded at the extremes by the following identities:

L1: initial identity , sum of all the adjacent intervals contained in the duplicationinterval

Ln: Identity of the scale , accumulation of all the adjacent intervals contained in theinitial scale.



The combinatory potential of a scale of dimension E6, for example, will produce a totalof 11 identities, ordered and numbered by the density levels (L), from level L1 to L6.

L1 L2 L3 L4 L5 L6 2 (1,5) 5 (1,1,4) 8 (1,1,1,3)

1 (6) 3 (2,4) 6 (1,2,3) 10 (1,1,1,1,2) 11 (1,1,1,1,1,1)

4 (3,3) 7 (2,2,2) 9 (1,1,2,2)

Connections between identities at minimal distance d1. In the intervallic potentialof a scale, different minimal distance (d1) relationships are represented. The transitionbetween identities of adjacent levels is based on two types of operation:

partition of a density level to an adjacent level of greater density. An interval isdivided in two, for example:

E6 (1,2,3) d1 (1,1,2,2)

fusion of a density level to another adjacent level of lesser density. Twointervals are added into one, for example:

E6 (1,2,3) d1 (3,3)

The operations of partition and fusion allow us to connect identities in a continuousmanner. Continuing with the E6 scale example, one can illustrate a continuoustransition between various identities that cover the whole combinatory potential fromone extreme to the other:

1 (6) d1 2 (1,5) d1 5 (1,1,4) d1 8 (1,1,1,3) d1 10 (1,1,1,1,2) d1 11 (1,1,1,1,1,1)

The following tree-like network is a representation of the set of connections betweenadjacent levels of the potential of scale E6. It illustrates the operations of partition and

fusion. The operations of partition go from left to right while those of fusion go in theopposite direction.



The connections between identities of the same level are obtained through a singletype of operation:

substitution of a pair of intervals by another pair of a different identity. Effectingorder to preserve the level of the identity, the sum of each new pair must beidentical to the sum of the initial pair. For example:

(1,2,3) d1 (1,1,4)

(1,2,3) d1 (2,2,2)

Together, the three connective operations—fusion, partition and substitution—allowthe continuous exploration of the total combinatory potential of the scale’s identities.

4 C o m b i n a t o r y P o t e n t i a l o f t h e I d e n t i t i e s

The point of departure of the combinatorial potential of the identities is the orderedarrangement of its constituents. From this ordered set we obtain all possible

permutations. Some examples of low interval density may illustrate this concept. Theclasses or dimensions of the intervals are represented by the letters a,b,c:

L2: (a,a)L 2: (a,b): (a,b) (b,a)L 3: (a,a,a)L 3: (a,b,b): (a,b,b) (b,a,b), (b,b,a)L 3: (a,b,c): (a,b,c) (b,a,c), (b,c,a), (c,b,a), (c,a,b), (a,c,b)

From a mathematical point of view, the combinatory potential coincides with themultinomial coefficient:

N! / (r1! r2! … r x!)

where r1 is the number of repetitions of each interval of an identity of N intervals(KNUTH 1973, 64).

When exchanging two adjacent intervals, the initial order o f an identity’s constituents ispermuted. In the following example, this exchange can be appreciated as thecontinuous displacement of interval 2:

(1,1,2,3,3,3) d1 (1,1,3,2,3,3) d1 (1,1,3,3,2,3) d1(1,1,3,3,3,2), etc.

Distance measurement among permutations . The minimal distance d1 between

consecutive permutations is used to calculate the overall distance betweenpermutations of one identity. Using the example above, one can measure the distancebetween the identity and the last of the permutations by adding up the number ofminimal distances d1 that separate them while observing how the digit 2 moves threepositions:

(1,1,2,3,3,3) d3 (1,1,3,3,3,2)



5 P e r m u t a h e d r o n s

A new area of combinatorics has given the label combinahedrons to cyclic structuresof combinations. In the case of Theory d1, the term permutahedron is used instead ofcombinahedron since we are dealing only with permutations. A permutahedron is avirtual polygon with a network structure. This structure is generally closed, and allows

transitions from any point, or node, to all the other points connected inside that space.

The permutahedron contains the ordered set of permutations of an identity’s intervals.In the permutahedron, each permutation appears as the result of an operation at aminimal distance d1. For example, the permutahedron of identity 18 (3,4,5) of scaleE12 generates 6 permutations. These are linked by minimal distances d1 and createan hexagonally shaped cycle between the last permutation and the identity:

(3,4,5) d1 (4,3,5) d1 (4,5,3) d1 (5,4,3) d1 (5,3,4) d1 (3,5,4) d1

Converting the above permutations into musical notation expressed with pitches weobtain:

Several observations are useful pertaining to the above procedure:

The operation with interval classes requires the selection of a note as astarting point, from which the intervals will be ordered. By convention, this ismiddle C in MuSIIC-Win .

The conversion of the permutations of the hexagonal cycle of the identity into

musical notation gives chromatic results outside of the diatonic scale. The representation with notes of the scale shows a single note change at each

permutation, equivalent to a permutation of two intervals.

Each of the permutations of the above identity—18 (3,4,5)—is a node linked to twoadjacent permutations. This characteristic is common to all identities composed ofthree different intervals (three interval classes) and is expressed in the followingmanner:

3! (3 factorial) = 1 x 2 x 3 = 6

This combination, of the type (a,b,c) observed from above, is found in 7 of the 12identities of level L3 of the E12 scale:

(1,1,10)(1,2,9) (1,3,8) (1,4,7) (1,5,6) (2,2,8)(2,3,7) (2,4,6)



(2,5,5)(3,3,6)(3,4,5)(4,4,4)

The following graphic representation is applicable to all identities containing three

interval classes. It shows the continuous transition from one operation to another bymeans of changes at a minimal distance d1:

6 P e r m u t a t i o n D i s t r i b u t i o n I n t e r v a l s

The ordered permutation of the intervals conforming each identity denotes thepresence of distribution intervals characteristic of each permutahedron . The exampleof the identity 18 (3,4,5) can easily show the utility of the permutation distributionintervals since the six ordered permutations correspond to the six basic cases ofconsonant harmony. A combinatorial affinity exists between all identities containingthree classes of intervals. Thus, the mathematical foundation of Theory d1 is removedfrom acoustic categories, which, as consonance or dissonance , belong to concretesystems of composition within scale E12.3

7 P e r m u t a h e d r o n ’ s G e o m e t r i c S t r u c t u r e

The ordered distribution of the permutations of an identity generates a geometricstructure: the permutahedron.

4 The majority of permutahedrons generate cycles,which, in some cases, acquire a complex order (the figure bellow depicts apermutahedron of identity 64(1,1,1,2,2,2,3)--scale E12--, with 140 nodes).

For all scales, the cycles generated by the permutahedrons are of three types only:

Link between two permutations (line) Cycle between four permutations (square) Cycle between six permutations (hexagon).

3 Of the seven identities composed of three interval classes (as shown in 2.5), only one would beconsidered consonant.

4 The first case of permutahedrons in music is credited to Nazir Jayrazbhoy in the treatment ofscales from Northern India. (JAYRAZBHOY 1972).



The permutahedron of identity 53 (1,1,1,2,3,4) of scale E12 contains 120 nodes. Thisparticular structure contains two of the three types of cycle, as well as theircombination inside the virtual polygon.

Cycle of four permutations: 1 d1 2 d1 7 d1 4 Cycle of six permutations: 1 d1 3 d1 9 d1 19 d1 10 d1 4

Note: as an example of a simple link between permutations, see MuSIIC-Win , identity75 (1,1,1,1,1,1,1,1,2,2) of scale E12.

The geometric structure of a permutahedron can reappear identical in other identitiesof the same scale or even in identities of scales of different dimensions. 5

8 C o n t i n u u m o f S c a l e s

The theory of the combinatory potential of intervals observes the set of all scales as avast continuum which can be traversed (from one scale to the next) at transitions ofminimum distance d1.

The operations required to go from one scale to its two adjacent scales are based onthe substraction or the addition of one interval. Several identities of the scale E12illustrate the idea of transition to both adjacent scales, E11 and E13.

5 The combinatory structure of identity 53 is unique in the scale E12, but it is identical, forexample, to identity 63 (1,1,1,2,3,5) of scale E13, as well as to all other identities of scales ofbigger dimensions containing 6 intervals, four of them being equal. This case can generally beexpressed as (a,a,a,b,c,d).



Identities of the same level L of density:

addition: L1: E12 (12) d1 E13 (13)substraction: E12 (12) d1 E11 (11)

addition: L4: E12 (3,3,3,3) d1 E13 (3,3,3,4)

substraction: E12 (3,3,3,3) d1 E11 (2,3,3,3)

addition: L6: E12 (1,1,2,2,2,4) d1 E13 (1,1,2,2,3,4)substraction: E12 (1,1,2,2,2,4) d1 E11 (1,1,1,2,2,4)

Identities of adjacent density levels:

addition: L1: E12 (1, 11) d1 N2: E13 (1,1,11)substraction: E12 (1,11) d1 E11 (11)

addition: L4: E12 (3,3,3,3) d1 N5: E13 (1,3,3,3,3)substraction: E12 (3,3,3,3) d1 E11 (3,3,5)

addition: L6: E12 (1,1,2,2,2,4) d1 N7: E13 (1,1,1,2,2,2,4)substraction: E12 (1,1,2,2,2,4) d1 E11 (1,1,1,2,6)

The combinatorial affinity between identities of scales with different dimensions offersa second perspective to the operations of addition and substraction.

Having as example identities containing three different intervals we can illustrate thegradual transit through minimal distances d1 to various other identities of scales whichshare the same combinatory potential since they are composed of three classes ofinterval (3!). The following cases show a continual variation of the intervals containedby various successive identities:

E6 (1,2,3) d1 E7 (1,2,4) d1 E8 (1,3,4) d1 E9 (2,3,4) d1 E10 (2,3,5) d1 E11 (2,4,5) d1 E12 (3,4,5) d1 E13 (3,4,6) d1 E14 (3,5,6) d1 E15 (3,5,7) d1 E16 (3,5,8) d1 E17 (3,6,8) d1 E18 (3,6,9) d1

E19 (4,6,9) d1 E20 (4,7,9) d1 E21 (4,7,10) d1 E22 (4,7,11) d1 E23 (4,8,11) d1 E24 (4,9,11)



9 M u l t i p l i c a t i o n a n d D i v i s i o n O p e r a t i o n s

The connections at minimal distance d1 between identities containing intervals withequal proportions offer an additional perspective to the continuum of the scales. Inthese cases, the transition at minimal distance d1 requires one to maintain an identicalproportion between the intervals of the first and the second identities while changing

from one scale to another. The relationship between identities proportionately equalrequires that both coincide in at least one of the prime numbers making up the numericbase of both scales.

For example, consider the factors of E12, E6 and E24. E12 is the product of 2x2x3, E6is the product of 2x3 and E24 is the product of 2x2x2x3. Since they have 2 as acommon factor, several identities of these scales can be related through theoperations of division or multiplication:

division of the intervals of identity 15 (2,4,6) of scale E12 by two, we obtain theequivalent identity: 6 (1,2,3) of scale E6 at distance d1.

multiplication of intervals of identity 15 (2,4,6) by two, we obtain the equivalentidentity: 47 (4,8,12) of scale E24 at distance d1.

In all these cases the intervals of pitch and duration are identically notated.

These connective relations allow us to understand how scales of a greater dimensioncontain several similar identities found in other scales of smaller dimensions. Forexample, the identities of the combinatory potential of scale E24 that are divisible by 2and 3 can be found in 14 of the 22 scales of smaller dimension. While these identitieshave different digits, their proportions are identical:

3: E32x2: E42x3: E6

2x2x2: E83x3: E92x5: E102x2x3: E122x7: E143x5: E152x2x2x2: E162x3x3: E182x2x5: E203x7: E212x11: E22

1 0 C o m b i n a t o r y E x p l o r a t i o n

Theory d1 proposes a new way of organizing intervals through the reduction of allscales to a combinatory potential of identities. The continuous exploration of thediscontinuous space of scales is achieved through transformations at minimal distanced1. These offer a subtle form of transit within a vast universe that can be seen from themicro-view of the permutahedron or from the macro-view of the continuum of scales.Both suggest one to consider a scale as a variegated set of permutahedrons interconnected through forms that bring about flexibility and continuity to discontinuousmaterials.



The scales and identities are considered from their combinatory potential, where thenumber of permutations or connections represents the mathematical richness of thealternatives. This richness pertains only to structural criteria, which are important inevaluating the results obtained. In principle, a scale or an identity with multiple linkswill be an important point of reference to relate with other points in space. This ideacan be observed in the coincidence between the connective richness of thecombinatory of an identity, with other identities of its own level and of adjacent level,and its connective richness at the interior of the permutahedron .

The synthesis of rhythm or pitch intervals offered by the notion of identity is a usefulresource for processes of acoustic discrimination, since these processes can be aidedby the information obtained by the combinatory potential of scales and thepermutahedrons . The structural affinity between interval structures and the operationsof transition at a minimal distance d1 reinforce the perception of relations betweenidentities and the transformations they undergo inside the permutahedron . It isinteresting to consider how the permutations of an identity’s intervals in thepermutahedron seem to produce a same sound or a same kind of rhythm while havinga different order.

1 1 S p a t i a l a n d T e m p o r a l P o t e n t i a l o fS e q u e n c e s

The practical application of the combinatory potential of scale intervals does withouttraditional conceptions of harmony and counterpoint, as well as synchronicorganizations or rhythm. This is an eminently neutral conception inviting one to freelyexplore the combinatory potential of intervals. The vertical projection of sequentiallygenerated intervals is here understood as a resource that requires to be developedindependently. There are two basic forms of elaboration of vertical structures that arecharacteristic of music based on scales:

Sound simultaneities, which are equivalent to vertical groupings. Rhythmic simultaneities, which are equivalent to the superposition of pulses of

one or more measures.

Sequential orderings are the basis for the elaboration of spatiotemporal relations inmusic. Operations applied to sequences are among the first forms used to organizemusical materials and, in contrast to vertical orderings, are present in all cultures. Theinvention of sequences is the most elemental form of evolution in music. In thisprocess, the operations are performed by the successive selection of distinct elementsthat correspond to the terms of a given scale, even when the scale is not usedconsciously (as it may happen in different manifestations of traditional music).

The adjacency between sequentially selected terms is the first piece of informationthat allows one to obtain data to be projected in the vertical dimension 6, since thevertical projections per se cannot be without the temporal dimension. This leads one to

6. The same idea can be found in the bases of the tonal harmonic system, which tries to reconcilethe harmonic information and the combinatory between pitches of the diatonic scale even whenthese lead to a significant reduction of the possibilities and to an organization based on thirdswhich is eminently discontinuous



think that the elaboration of the vertical dimension stems from more abstractoperations, as well as from processes of memorization of the temporal evolution.

1 1 . 1 P r o j e c t i o n o f t h e H o r i z o n t a l - V e r t i c a l

The relations between the horizontal and vertical structures can be performed

consecutively from spontaneous calculations, but not simultaneously. The simultaneityrequires a previous analysis that leads to an understanding of the interaction that thespatiotemporal evolutions can maintain. We know that the success of the traditionalsystems resided in their demand for memorizing the spatiotemporal relations under theform of schemes of sequential links of vertical groupings. By contrast, the richness ofthe combinatory potential of the scales does not give way to memorization due to itsvastness, nor would it be understandable within a single system. Thus, this workproposes considering three levels for approaching the horizontal-vertical relations:

Theory: neutral base of the combinatory potential of the scales. System: set of selective decisions created individually. Style: characteristic mechanisms used by a particular individual, such as

perception, imagination, memory and others.

Thus, the theory serves as a fundamental reference, allowing the mental processes ofspontaneous construction to be the generative base of a personal system. Finally, astyle will be associated with this system as a result of the preservation of the liberty ofthe creative process.

In a general way, the theoretical foundation of this work aspires for the exploration ofthe combinatory potential of the scales to be the result of a personal search, free ofprecepts. The method proposed starts by analyzing the results obtained by eachindividual’s free creative processes. From this analysis one can derive the data thatallows one to observe the autonomous tendencies of the individual style.

1 1 . 2 S p a t i o t e m p o r a l R e l a t i o n s o f A d j a c e n c ya n d C o n t i n u i t y

The process of creating a sequence has implications at both the spatial and temporallevels. The relationships between space and time tend to become complex as chainsof simple sequential operations unfold over time. Various factors intervene in definingthese relationships: perception, knowledge, calculation, prediction, memory,imagination or affect, among others. Together, these factors appear to influence, in animperceptible and decisive manner, the accumulative tendency that the sequenceacquires at the psychic level.

When one creates, listens or imagines a sequence, the mind registers the priority ofthe continuous relationships through the order of the pulses or pitches. Besides the

relationships of adjacency, other relationships are inevitably created due to theprogressive accumulation of the sequence. This accumulation is not necessarilyrelated to the composer’s proposed musical structure, but the psyche’s tendency torecognize sequences point by point tends to create a mnemonic imprint of therhythmic and/or sonic spectrum.

The creation of a personal system requires one to consider the perception of time as apriority, as well as the relationships of adjacency in the sequences created. Theperception of these minimum distance relationships provides the most important data



pertaining to sequential invention because it allows the subsequent reproduction of theinformation, partially or totally.

1 1 . 3 S p a t i o t e m p o r a l P o t e n t i a l o f S e q u e n c e s

A simple sequence of pitches or pulses generated within a given scale can be studied

through different sweeping methods. These are a group of original analyticalapproximation techniques used to study the combinatory potentials of intervalsequences. The information obtained from the sweeping of a sequence is useful inmusical creation under two complementary perspectives: that of sequential analysis,and that of its repercussion on the possibilities of vertical projections.

Sequential analysis: detailed analysis of the sequential intervals createdspontaneously. This generative and selective process can provide interestinginformation about the subjective tendencies of the sequence.

Analysis of accumulation of intervals: This is important when considering theinherent possibilities of projecting the sequential structures vertically.

The sweeping method consists of registering the gradual advance of each and every

one of the adjacent intervals in a sequence. This process is done by looking at oneinterval at a time for the entire sequence, then two intervals at a time, then three andso on until we reach the length of the sequence. This sweeping method allows one toanalyze each sequence from three different levels: initial, linear and cyclical .

1 1 . 4 I n i t i a l , L i n e a r a n d C y c l i c a l S w e e p i n g

The three kinds of sweeping are exemplified with one sequence belonging to scaleE12. The sequence is expressed in four different forms:

Numeric sequence of the terms employed:0 3 5 1 2 4 3

Series of adjacent intervals with their respective directions (ascending or descending):+3 +2 -4 +1 +2 -1

Pitch sequence:

C C# F C# D E D#

Sequence of durations:

1 1 . 4 . 1 I n i t i a l S w e e p i n g

The sequence of terms can be converted into a small group of identities as a result ofan ordered reading of the sequence in multiple steps. The careful description of theinitial sweeping will allow us to easily understand the other two forms of sweeping .



First we record the accumulation of the terms of the scale generated by the sequence:

00 30 3 50 3 5 1

0 3 5 1 20 3 5 1 2 40 3 5 1 2 4 37

We then accumulate the terms of the sequence in the scale. This accumulation allowsus to order the terms of the sequence within a scale, although in terms of intervals, thisrequires one to consider their direction. For example, expressing the previoussequence in terms of intervals we have:

0 3 5 1 2: (3 2 -4 1)

Once the terms have been accumulated within a scale they offer a new order:

0 1 2 3 5

Therefore, the order of the descending interval (-4) is eliminated, resulting in thefollowing succession of terms and their intervallic relationships:

0 1 2 3 5: (1 1 1 2)

This operation converts the directional intervals of the series (ascending anddescending) into non-directional expressing the distance between adjacent terms ofthe scale. In other words, we have moved from a temporal to a spatial order. Here wesee the relationship between the time-ordered interval sequence and the spatialordering of the intervals that result from the accumulation of the terms of the

sequence:

0: 0 (0)0 3: 0 3 (3)0 3 5 (3 2): 0 3 5 (2 3)0 3 5 1 (3 2 -4): 0 1 3 5 (1 2 2)0 3 5 1 2 (3 2 -4 1): 0 1 2 3 5 (1 1 1 2)0 3 5 1 2 4 (3 2 -4 1 2): 0 1 2 3 4 5 (1 1 1 1 1)0 3 5 1 2 4 3 (3 2 -4 1 2 -1): 0 1 2 3 4 5 (1 1 1 1 1 )8

Each identity requires its intervals to add up to the dimension of the scale, so it isnecessary to add an additional interval (the cycle interval ) to the above intervalcollections. This procedure converts one term in the sequence into an interval, one

7 Similar processes have been developed in Memorias for a keyboard (Estrada 1971), applicableto melodic or percussive instruments.

8 In this case the last result does not produce a new adjacent interval since the sequence repeatsthe term 3. In addition, the sequence is contained between terms 0 and 5 from the third point onin the sequence, so the rest of the results will be contained within the partition of the interval (5) intwo or more segments: (3 2), (1 2 2), etc.

http://c/cygwin/home/Alejandra/projects/Musiic/musiic_3.2e/manual/2.xTeoria%20d1/2.11Potencial%20espacio%20temporal.htm%23_ftn5






interval into two intervals, three intervals into four, and so on. This will give us theinitial sweeping of the sequence:

from 0 to 1 interval: (0): (0+12) = (12)from 1 to 2 intervals (3): (3+9) = (3 9)from 2 to 3 intervals (3 2): (2+3+7) = (2 3 7)

from 3 to 4 intervals (1 2 2): (1+2+2+7) = (1 2 2 7)from 4 to 5 intervals (1 1 1 2): (1+1+1+2+7) = (1 1 1 2 7)from 5 to 6 intervals (1 1 1 1 1): (1+1+1+1+1+7) = (1 1 1 1 1 7)from 6 to 6 intervals (1 1 1 1 1): (1+1+1+1+1+7) = (1 1 1 1 1 7)

The initial sweeping of this sequence results in a total of 6 interval identities which arefound between levels L1 and L6.9

1 1 . 4 . 2 L i n e a r S w e e p i n g

The linear sweeping extends the initial sweeping previously discussed and allows oneto obtain a wide range of identities by considering all the adjacent intervals containedin the sequence. Since we have already explained some of the steps of the sweeping,

we here explain only the most relevant:

We cover the entire sequence by selecting a window that groups adjacent terms. Thewindow starts at density 1 and covers the entire sequence moving one term at a time.Then a window of density 2 is used, jumping one term at a time so that the windowsoverlap, and so on until the density of the entire sequence is reached.

1 term: 0, 3, 5, 1, 2, 4, 32 terms: 0 3, 3 5, 5 1, 1 2, 4 33 terms: 0 3 5, 3 5 1, 5 1 2, 1 2 4, 2 4 34 terms: 0 3 5 1, 3 5 1 2, 5 1 2 4, 1 2 4 35 terms: 0 3 5 1 2, 3 5 1 2 4, 5 1 2 4 36 terms: 0 3 5 1 2 4, 3 5 1 2 4 3

7 terms: 0 3 5 1 2 4 3

We then order the terms of the sequence within the scale to measure the adjacentintervals generated by each of the interval groups previously derived, and eliminatethe direction of the descending interval:

0 interval0 3 5 1 2 4 3 = 1 interval (0+12)

1 to 2 intervals0 3 (+3)3 5 (+2)

5 1 (-4): 1 5 (+4)1 2 (+1)2 4 (+2)4 3 (-1): 3 4 (+1)

9 Observe how the same identity is repeated at the end, since no new term of the scale or newinterval is introduced.



2 to 3 intervals0 3 5 (+3+2)3 5 1 (+2-4): 1 3 5 (+2+2)5 1 2: (-4+1): 1 2 5 (+1+3)1 2 4 (+1+2)2 4 3 (+2-1): 2 3 4 (+1+1)

3 to 4 intervals:0 3 5 1 (+3+2-4): 0 1 3 5 (+1+2+2)3 5 1 2 (+2-4+1): 1 2 3 5 (+1+1+2)5 1 2 4 (-4+1+2): 1 2 4 5 (+1+2+1)1 2 4 3 (+1+2-1): 1 2 3 4 (+1+1+1)

4 to 5 intervals:0 3 5 1 2 (+3+2-4+1): 0 1 2 3 5 (+1+1+1+2)3 5 1 2 4 (+2-4+1+2): 1 2 3 4 5 (1+1+1+1)5 1 2 4 3 (-4+1+2-1): 1 2 3 4 5 (1+1+1+1)

5 to 6 intervals:

0 3 5 1 2 4 (+3+2-4+1+2): 0 1 2 3 4 5 (+1+1+1+1+1)3 5 1 2 4 3 (+2-4+1+2-1): 1 2 3 3 4 5 (+1+1+0+1+1)

6 to 6 intervals:0 3 5 1 2 4 3 (+3+2-4+1+2-1): 0 1 2 3 3 4 5 (+1+1+1+0+1+1)

We order the interval sets obtained and add the complementary interval to obtain eachof the identities contained in the sequence:

0 interval: (0+12) = (12)

1 to 2 intervals(+3) = (3 9)(+2) = (2 10)(+4) = (4 8)(+1) = (1 11)(+2) = (2 10)(+1) = (1 11)

2 to 3 intervals(+2+3) = (2 3 7)(+2+2)= (2 2 8)(+1+3) = (1 3 8)(+1+2) = (1 2 9)(+1+1) = (1 1 10)

3 to 4 intervals:(+1+2+2) = (1 2 2 7)(+1+1+2) = (1 1 2 8)(+1+1+2) = (1 1 2 8)(+1+1+1) = (1 1 1 9)

4 to 5 intervals:(+1+1+1+2) = (1 1 1 2 7)



(1+1+1+1) = (1 1 1 1 8)(1+1+1+1) = (1 1 1 1 8)

5 to 6 intervals:(+1+1+1+1+1) = (1 1 1 1 1 7)(+0+1+1+1+1) = (1 1 1 1 8)

6 to 6 intervals:(+0+1+1+1+1+1) = (1 1 1 1 1 7)

The linear sweeping leads us to obtain a total of 16 interval identities, which includethe total set of 6 identities resulting from the initial sweeping. This new result presentsa spectrum that encompasses a good part of levels L1 through L6 of the potential ofscale E12:

N1: (12)N2: (1 11) (2 10) (3 9) (4 8)N3: (1 1 10) (1 2 9) (1 3 8) (2 2 8) (2 3 7)N4: (1 1 1 9) (1 1 2 8) (1 2 2 7)

N5: (1 1 1 1 8) (1 1 1 2 7)N6: (1 1 1 1 1 7)

1 1 . 4 . 3 C y c l i c a l S w e e p i n g

This kind of reading views the sequence as a cycle, connecting the last term with thefirst. This approach makes the theoretical treatment of the sequence similar to a scale.The cyclic sweeping extends the linear sweeping with results obtained by the newconnection between the end and the beginning of the sequence. The whole process issummarized in the following steps:

We obtain each of the groups of terms of density two, three,…, until all the intervalsfound in the cyclical sequence are included. Observe how the terms at the beginningof the sequence reappear at the end due to the cyclical connection:

0 3 5 1 2 4 3:3 5 1 2 4 3 0

5 1 2 4 3 0 31 2 4 3 0 3 5

2 4 3 0 3 5 14 3 0 3 5 1 2

3 0 3 5 1 2 4

We group the terms of the sequence making up the cycle to order them according totheir adjacency within the scale.

We measure and order the adjacent intervals formed.

We add the cycle interval to construct each identity.

The results obtained are similar to those of the linear sweeping.

0 3, 3 5, 5 1, 1 2, 4 3,3 0: 0 3 = (3 9)



0 3 5, 3 5 1, 5 1 2, 1 2 4, 2 4 3,4 3 0: 0 3 4 = (1 3 8)3 0 3: 0 3 3 = (3 9)0 3 5 1, 3 5 1 2, 5 1 2 4, 1 2 4 3,2 4 3 0: 0 2 3 4 = (1 1 2 8)4 3 0 3: 0 3 3 4 = (1 3 8)3 0 3 5: 0 3 3 5 = (2 3 7)0 3 5 1 2, 3 5 1 2 4, 5 1 2 4 3,1 2 4 3 0: 0 1 2 3 4 = (1 1 1 1 8)2 4 3 0 3: 0 2 3 3 4 = (1 1 2 8)4 3 0 3 5: 0 3 3 4 5 = (1 1 3 7) ** 3 0 3 5 1: 0 1 3 3 5 = (1 2 2 7)0 3 5 1 2 4, 3 5 1 2 4 3,5 1 2 4 3 0: 0 1 2 3 4 5 = (1 1 1 1 1 7)1 2 4 3 0 3: 0 1 2 3 3 4 = (1 1 1 1 8)2 4 3 0 3 5: 0 2 3 3 4 5 = (1 1 1 2 7)4 3 0 3 5 1: 0 1 3 3 4 5 = (1 1 1 2 7)3 0 3 5 1 2: 0 1 2 3 3 5 = (1 1 1 2 7)0 3 5 1 2 4 3,3 5 1 2 4 3 0: 0 1 2 3 3 4 5 = (1 1 1 1 1 7)

5 1 2 4 3 0 3: 0 1 2 3 3 4 5 = (1 1 1 1 1 7)1 2 4 3 0 3 5: 0 1 2 3 3 4 5 = (1 1 1 1 1 7)2 4 3 0 3 5 1: 0 1 2 3 3 4 5 = (1 1 1 1 1 7)4 3 0 3 5 1 2: 0 1 2 3 3 4 5 = (1 1 1 1 1 7)3 0 3 5 1 2 4: 0 1 2 3 3 4 5 = (1 1 1 1 1 7)

The cyclical sweeping generates 17 identities, adding only one more (marked with adouble asterisk) to the results obtained with the linear sweeping.

Even though this third form of reading extends the spectrum of the combinatorypotential, the example presented also shows that the results are not too different fromthe linear sweeping.

1 1 . 5 T i m e a n d M e m o r y

In summary, the sweeping analysis provides data that maintains a strong congruencewith the information generated freely and which the constructive creativity can employthrough speculation on the interplay between the time of the sequence and the verticalstructures.

1 2 M u S I I C - W i n 3 . 2

MuSIIC- Win 3.2 is the newest version of Theory d1 and it contains an additionalapplication on the field of acoustics through the continuous transformation of waveshapes (ESTRADA/ADÁN 2002, MúSIIC-Win 3.2).10

MuSIIC-Win is a product of PAPIME-UNAM's project for Educational Innovation andImprovement of the University of Mexico, UNAM, 1997-2000, 2003-2006 assigned tothe Escuela Nacional de Música (ENM). It is a project of LACREMUS (MusicalCreation Laboratory). Julio Estrada served as project director.

10 Theory d1 has been the reference point for a research on the spatial distribution of musicsources (SOLÍS 2000).



MuSIIC-Win ’s design, software engineering and system programming underWindows® was done by mathematician Max Diaz, with the assistance of ErikSchwarz, mathematician, and Víctor Adán, composer and computer programmer,with a National Research System (SNI) fellowship under Estrada’s research project.

B i b l i o g r a p h y

COWELL, Henry D., New Musical Resources , New York, 1930, new edition 1969.

ESTRADA, Julio. Théorie de la composition: discontinuum-continuum . Université deStrasbourg, France, 1994, 932 pages. Ph. D thesis.

ESTRADA, Julio / ADÁN, Víctor, International Society of Musical Acoustics (ISMA)Mexico City, 2002

JAYRAZBHOY, The Rags of North Indian Music, Their Structure and Evolution , Faber& Faber, London, 1971, 222 pages.

JOHNSTON, Ben, in Dictionary of Contemporary Music , John Vinton, Editor, NewYork, 1974, “Microtones”, pp. 483-484.

KNUTH, Donald, Fundamental Algorithms , Addison Wesley, 1973.

LACREMUS, Laboratorio de Creación Musical, J. Estrada, director, PAPIME, 1997-2000, Escuela Nacional de Música, UNAM, Mexico City 2000, 24 pages.

MUSIIC-WIN 3.2, Theory d1, DVD, trilingual versions menú and users manual(Español, French, English), Escuela Nacional de Música, UNAM, Proyecto PAPIME,

México 2006.

PMMI, Proyecto Música, Matemáticas, Informática, Instituto de InvestigacionesEstéticas, director J. Estrada, Instituto de Investigaciones en Matemáticas Aplicadas yen Sistemas, co-director M. Peña, PAPIIT-DGAPA, UNAM, México, 1990-1997.

SOLÍS, Hugo, “El espacio físico como variable estructural en música”, LACREMUS2000, ENM, UNAM, México (see on www.julioestrada.net)

http://www.julioestrada.net/




Documents

Theory d1 Musiic Win Libre