A Projection-Oriented Mathematical Model for Second-Species Counterpoint
AA PROJECTION-ORIENTED MATHEMATICALMODEL FOR SECOND-SPECIES COUNTERPOINT
OCTAVIO A. AGUST´IN-AQUINO AND GUERINO MAZZOLA
Abstract.
Drawing inspiration from both the classical GuerinoMazzola’s symmetry-based model for first-species counterpoint (onenote against one note) and Johann Joseph Fux’s
Gradus ad Par-nassum , we propose an extension for second-species (two notesagainst one note). Introduction
Guerino Mazzola’s counterpoint model, founded on the concepts of(1) strong dichotomy , which encodes the notion of consonance anddissonance, and(2) counterpoint symmetry , which is the carrier of contrapuntal ten-sion and allows to deduce the rules of counterpoint,has been successful in explaining the necessity of regarding the fourth asa dissonance and obtaining the general prohibition of parallel fifths as atheorem. It also allows to define new understandings of consonance anddissonance, thereby leading to the concept of counterpoint world , i.e.,paradigms for the handling of two-voice compositions represented asdigraphs, whose vertices are consonant intervals and an arrow connectstwo of them whenever we have a valid progression. This, in turn, allowsus to morph one world into another. See the monograph [2] and thetreatise [4, Part VII] for a thorough account.Despite these accomplishments, Mazzola’s model is restricted to thecase of first-species counterpoint, which means that only one note canbe placed against another. Hence, in order to increase the potentialof Mazzola’s model for analysis and composition, it is indispensable toextend it to second-species counterpoint (i.e., two notes against one)and further. Our approach for a first step in this direction is to extendthe notion of counterpoint interval to a 2-interval, i.e., one such that
Date : September 28th, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Second-species, counterpoint.This work was partially supported by a grant from the
Niels Hendrik Abel Board . a r X i v : . [ m a t h . HO ] S e p OCTAVIO A. AGUST´IN-AQUINO AND GUERINO MAZZOLA two intervals are attached to a cantus firmus, the first one coming inthe downbeat and the second one in the upbeat.For our extension, the main idea is that the counterpoint symmetriesin this case do not determine another 2-interval successor, but a first-species interval in the downbeat. The idea behind this is to blend thespecies of counterpoint more easily.2.
General Overview of Mazzola’s Counterpoint Model
Here we quickly survey the key aspects of Mazzola’s counterpointmodel (we refer the reader to [2] and [4, Part VII] for a completeaccount). We consider the action of the group −→ GL ( Z k ) := Z k (cid:111) Z × k (which we call the group of general affine symmetries ) on Z k , whichcan be described in the following manner: T u .v ( x ) = vx + u ;here T u is the transposition by u , and v is the linear part of the trans-formation.We know [1, 2] that, for any k >
4, there is at least one dichotomy∆ = (
X/Y ) of Z k such that there is a unique p ∈ −→ GL ( Z k ) and p ( X ) = Y and p ◦ p = id Z k , which is called the polarity of the dichotomy. The dichotomies withthis property are called strong , and represent the division of intervalsinto generalized consonances X and dissonances Y .Next we consider the dual numbers Z k [ (cid:15) ] = Z k [ X ] (cid:104)X (cid:105) = { x + (cid:15).y : x, y ∈ Z k , (cid:15) = 0 } in order to attach to each cantus firmus x the interval y that separatesit from its discantus . Thus for a strong dichotomy ∆ = ( X/Y ) wehave the consonant intervals X [ (cid:15) ] := { c + (cid:15).x : c ∈ Z k , x ∈ X } and the dissonant intervals Y [ (cid:15) ] = Z k \ X [ (cid:15) ]. Considering the group −→ GL ( Z k [ (cid:15) ]) := { T a + (cid:15).b . ( v + (cid:15).w ) : a, b, w ∈ Z k , v ∈ Z × k } , The discantus can be understood in the sweeping ( x + y ) or the hanging ( x − y )orientations, but we will only use the sweeping orientation from this point on. ECOND-SPECIES COUNTERPOINT 3 there is a canonical autocomplementary symmetry p c ∆ ∈ −→ GL ( Z k ) suchthat p c ∆ ( X [ (cid:15) ]) = Y [ (cid:15) ] andand leaves the tangent space c + (cid:15). Z k invariant.With this preamble it is possible to state a classical paradox for first-species counterpoint theory: all the intervals c + (cid:15).k used in a first-species counterpoint composition or improvisation are consonances.Hence, how can any tension between the voices arise, if at all? Maz-zola’s solution is inspired in the fact [6, p. 33-35] that it is not that thepoint c which is to be confronted against c + k , but it is the consonantpoint ξ = c + (cid:15).k who will face a successor η = c + (cid:15).k . The ideais to deform the dichotomy ( X [ (cid:15) ] /Y [ (cid:15) ]) into ( gX [ (cid:15) ] , gY [ (cid:15) ]) through asymmetry g ∈ −→ GL ( Z k [ (cid:15) ]), such that(1) the interval ξ becomes a deformed dissonance, i.e., ξ ∈ gY [ (cid:15) ],(2) the symmetry g is an autocomplementary function of( gX [ (cid:15) ] , gY [ (cid:15) ])which means that p ( gX [ (cid:15) ]) = gY [ (cid:15) ],and thus we can transit from ξ to a consonance η which is also adeformed consonance, i.e., η ∈ gX [ (cid:15) ] ∩ X [ (cid:15) ]. Since we wish to have themaximum amount of choices, we request also that(3) the set gX [ (cid:15) ] ∩ X [ (cid:15) ] is of maximum cardinality among the sym-metries that satisfy conditions 1 and 2.The elements of this latter set are the admitted successors .3. Dichotomies of -intervals For the purposes of the second-species counterpoint, we need now analgebraic structure such that two intervals can be attached to a basetone. In the spirit of the model presented in the previous section, wetake all the polynomials of the form c + (cid:15) .x + (cid:15) .y ∈ Z k [ X , Y ] (cid:104)X , Y , X Y(cid:105) = Z k [ (cid:15) , (cid:15) ]where (cid:15) ≡ X mod (cid:104)X , Y , X Y(cid:105) , (cid:15) ≡ Y mod (cid:104)X , Y , X Y(cid:105) , c is thecantus firmus and x, y are the intervals ( x is for the downbeat and y is for the upbeat). An element ξ ∈ Z k [ (cid:15) , (cid:15) ] is called a 2 -interval . If∆ = ( X/Y ) is a strong dichotomy with polarity p = T u ◦ v , then X [ (cid:15) , (cid:15) ] := Z k + (cid:15) .X + (cid:15) . Z k OCTAVIO A. AGUST´IN-AQUINO AND GUERINO MAZZOLA is an dichotomy in Z k [ (cid:15) , (cid:15) ]. We choose this dichotomy because therules of counterpoint demand that the interval that comes on the down-beat to be a consonance. A polarity for this dichotomy, which is anal-ogous to the one for the first-species case, is p c = T c (1 − v )+ (cid:15) .u + (cid:15) .u ◦ v because p c X [ (cid:15) , (cid:15) ] = T c (1 − v ) ◦ v. Z k + (cid:15) .pX + (cid:15) .p Z k = Z k + (cid:15) .Y + (cid:15) . Z k = Y [ (cid:15) , (cid:15) ]and it is such that p c ( c + (cid:15) . Z k + (cid:15) . Z k ) = c + (cid:15) . Z k + (cid:15) . Z k , which means p c fixes the tangent space to cantus firmus c as well.We also check the following formula for future use: p c + c = T ( c + c )(1 − v )+ (cid:15) .u + (cid:15) .u ◦ v = T c (1 − v )+ c (1 − v )+ (cid:15) .u + (cid:15) .u ◦ v = T c ◦ T − vc ◦ T c (1 − v )+ (cid:15) .u + (cid:15) .u ◦ v = T c ◦ T c (1 − v )+ (cid:15) .u + (cid:15) .u ◦ v ◦ T − c = T c ◦ p c ◦ T − c . Species Projections
If we represent the polynomial c + (cid:15) .x + (cid:15) .y as a column vector, thethe candidates to (non-invertible) species projections are g : Z k [ (cid:15) , (cid:15) ] → Z k [ (cid:15) ] cxy (cid:55)→ (cid:18) s sw s sw (cid:19) cxy + (cid:18) t t (cid:19) = [ sc + t ] + (cid:15) . [ s ( w c + x + w y ) + t ]for we want to keep it as simple as possible and that the second partof the interval to influence the first part of the successor, but not thesecond one. We do not require the transformation to be bijective forwe want it to be able to swap from second-species to first-species ifnecessary . For the converse swap the standard rules of counterpoint suffice: we can ar-bitrarily define the third component of the 2-interval. This is coherent with thelocal application of counterpoint rules in Fux’s theory, and also with the particular
ECOND-SPECIES COUNTERPOINT 5
Let X [ (cid:15) , (cid:15) .y ] := Z + (cid:15) .X + (cid:15) .y . We might define a speciesprojection of a 2-interval ξ = c + (cid:15) .x + (cid:15) .y as one such that(1) the condition c + (cid:15) .x / ∈ gX [ (cid:15) , (cid:15) .y ] holds,(2) the square(1) Z k [ (cid:15) , (cid:15) ] g −−−→ Z k [ (cid:15) ] p c (cid:121) (cid:121) p c ∆ Z k [ (cid:15) , (cid:15) ] −−−→ g Z k [ (cid:15) ]commutes, where p c ∆ := T c (1 − v )+ (cid:15) .u ◦ v is the canonical polarity of ( X [ (cid:15) ] /Y [ (cid:15) ]), and(3) the cardinality of gX [ (cid:15) , (cid:15) .y ] ∩ X [ (cid:15) ] is maximal among theprojections with the previous properties.The reason for the second requirement is that when it is fulfilled then p c ∆ ( gX [ (cid:15) , (cid:15) ]) = g ( p c X [ (cid:15) , (cid:15) ]) = gY [ (cid:15) , (cid:15) ] , thus p c ∆ is an autocomplementary function of gX [ (cid:15) , (cid:15) ].5. Algorithm for the Calculation of Projections
As with the first-species case, if for a projection of the form g = T (cid:15) .t ◦ (cid:18) s sw s sw (cid:19) we define g ( t ) = g ◦ T (cid:15) .s − w t + (cid:15) .t then the relation T t ◦ g = g ( − t ) ◦ T s − t + (cid:15) .t , holds, and hence contrapuntal projections can be calculated with can-tus firmus 0 and successors can be suitably adjusted [2, Theorem 2.2].Therefore, we can set t = 0 and work with intervals of the form ξ = (cid:15) .y + (cid:15) .z . For (1) to commute, it is necessary and sufficientthat(2) t + su (1 + w ) = u + vt . For (cid:15) .y / ∈ gX [ (cid:15) , (cid:15) .z ] we need y = sp ( (cid:96) ) + t + sw z idea of projection that stems from the fact that, in order to analyze a fragment, we“disregard” notes on the upbeat [3, pp. 41-43]. OCTAVIO A. AGUST´IN-AQUINO AND GUERINO MAZZOLA for some (cid:96) ∈ X . Hence, for some (cid:96) ∈ X we have(3) t = y − s ( p ( (cid:96) ) + w z ) . Remark . Letting w = 0 in (2) and (3), they reduce to the first-species case. Thus, taking s = v and (cid:96) = y both are satisfied and hencewe conclude that there exists at least one second-species counterpointprojection.We only need to work with the following set gX [ (cid:15) , (cid:15) .z ] = (cid:91) x ∈ Z k g ( x + (cid:15) .X + (cid:15) .z )= (cid:91) x ∈ Z k ( sx + (cid:15) . ( sw x + sw z + t + sX ))= (cid:91) r ∈ Z k ( r + (cid:15) . ( w r + sX + w sz + t ))= (cid:91) r ∈ Z k ( r + (cid:15) .T w r + w sz + t ◦ sX )to calculate the following cardinality | gX [ (cid:15) , (cid:15) .z ] ∩ X [ (cid:15) , (cid:15) .z ] | = (cid:88) r ∈ Z k | T w r + w sz + t ◦ sX ∩ X | . When (3) holds, this reduces to(4) | gX [ (cid:15) , (cid:15) .z ] ∩ X [ (cid:15) , (cid:15) .z ] | = (cid:88) r ∈ Z k | T w r + y − sp ( (cid:96) ) ◦ sX ∩ X | . From now on we only need to adapt mutatis mutandis
Hichert’salgorithm [2, Algorithm 2.1] to search projections that maximize theintersection.We must remark that (2) and (3) are perturbations of the conditionsto find the counterpoint symmetries for the first-species case. These,together with (4), show that the conditions for deducing a counterpointtheorem [2, Theorem 2.3] hold again, which yields the following result.
Theorem 5.2.
Given a marked strong dichotomy ( X/Y ) in Z k , the -interval ξ ∈ X [ (cid:15) , (cid:15) ] has at least k and at most k − k admittedsuccessors. Algorithm 5.3.
Here χ ( x, y ) is the function that returns the cardi-nality T x .yX ∩ X . Input:
A strong dichotomy ∆ = (
X/Y ) and its polarity T u .v . Output:
The set of counterpoint projections Σ y,z ⊆ H for each (cid:15).y + (cid:15).z ∈ X [ (cid:15) , (cid:15) ]. ECOND-SPECIES COUNTERPOINT 7 for all y, z ∈ X do M ← , Σ y,z ← ∅ . for all s ∈ GL ( Z k ) do for all (cid:96) ∈ X do for all w , w ∈ Z k do t ← y − s (( v(cid:96) + u ) + w z ). if t + su (1 + w ) = u + vt then if w = 0 then S ← kχ ( t , s ). else if w ∈ GL ( Z k ) then S ← k else ρ ← gcd( w , k ) S ← ρ (cid:80) kρ − j =0 χ ( jρ + t + w z, s ). if S > M then Σ y,z ← (cid:26) T (cid:15) .t ◦ (cid:18) s sw s sw (cid:19)(cid:27) . S ← M . else if S = M then Σ y,z ← Σ y,z ∪ (cid:26) T (cid:15).t ◦ (cid:18) s sw s sw (cid:19)(cid:27) . return Σ y,z . Example . The first (valid ) example of second-species counterpointin the Gradus ad Parnassum [3, p. 45] is ξ = 2 + (cid:15) . (cid:15) . , ξ = 5 + (cid:15) . (cid:15) . , ξ = 4 + (cid:15) . (cid:15) . ,ξ = 2 + (cid:15) . (cid:15) . , ξ = 7 + (cid:15) . (cid:15) . , ξ = 5 + (cid:15) . (cid:15) . ,ξ = 9 + (cid:15) . (cid:15) . , ξ = 7 + (cid:15) . (cid:15) . , ξ = 5 + (cid:15) . (cid:15) . ,ξ = 4 + (cid:15) . (cid:15) . , ξ = 2 + (cid:15) . The first example is the student’s attempt to write a second-species discantusby himself, but he makes two mistakes near the end of the exercise, namely thesteps from the sequence 7 + (cid:15) . (cid:15) .
11, 5 + (cid:15) . (cid:15) .
9, 4 + (cid:15) .
11 + (cid:15) .
1. They arealso forbidden steps in the projection model!
OCTAVIO A. AGUST´IN-AQUINO AND GUERINO MAZZOLA
Some counterpoint projections for the successors are g = (cid:18) (cid:19) , g = T (cid:15) . ◦ (cid:18) (cid:19) , g = T (cid:15) . ◦ (cid:18) (cid:19) g = g , g = g , g = T (cid:15) . ◦ (cid:18) (cid:19) ,g = (cid:18)
11 0 00 11 8 (cid:19) , g = g , g = g , g = g . Let us examine in little bit more of detail the first transition. Notethat η = 11 + (cid:15) . (cid:15) .
11 is a consonance, and that g ( η ) = (cid:18) (cid:19) = (cid:18) (cid:19) , which justifies the fact that the 2-interval 5 + (cid:15) . (cid:15) . Comparison with Fux’s Approach
Fux states the following in relation to second-species counterpoint(emphasis is our own) [3, p. 41]:The second species results when two half notes are setagainst a whole note. The first of them comes on thedownbeat and must always be consonant; the secondcomes on the upbeat and it may be dissonant if it movesfrom the preceding note and to the following note step-wise . However, if it moves by a skip, it must be conso-nant .We made a program that compares the performance of a first-speciesmodel that takes into account Fux’s restrictions against the projectionmodel. More explicitly, taking a second-species step(0 + (cid:15) .k + (cid:15) .t , c + (cid:15) .k )such that we can proceed (in first-species) from 0 + (cid:15) .k to c + (cid:15) .k ,we verify the following cases:(1) the upbeat interval t of the first 2-interval is allowed to bedissonant only when it connects a valid progression of conso-nances stepwise, i.e., 0 + t is between 0 + k and c + k and itis separated at most 2 semitones from them and(2) if t is consonant, we duplicate the cantus firmus and check if(0 + (cid:15).k , (cid:15).t ) and (0 + (cid:15).t , c + (cid:15).k ) are valid first-speciessteps. ECOND-SPECIES COUNTERPOINT 9
The results appear in Table 1 for cases 1 and 2. We must stressthat the projection model was not restricted in case 1 to stepwise dis-sonances but it allowed any dissonance in the upbeat.Number of steps Case 1 Case 2Total 1994 2592Valid only for Fux model 9 178Valid only for the projection model 1447 860Valid in both models 301 1464
Table 1.
Data for comparison of Fux’s model with re-strictions for second species against the projection model.We note that the number of cases the projection model cannot ex-plain and only Fux can is relatively small: they amount to 2 .
9% and17 .
1% for cases 1 and 2, respectively. Thus we can conclude that thevast majority of what is forbidden in the projection model is also for-bidden in Fux’s model, or that we have successfully extended Fux’shandling of dissonance and consonance for second species. Even if thiscould be ascribed to the fact that the projection model admits 87 . . . References
1. Octavio A. Agust´ın-Aquino,
Counterpoint in k -tone equal temperament , Jour-nal of Mathematics and Music (2009), no. 3, 153–164.2. Octavio A. Agust´ın-Aquino, Julien Junod, and Guerino Mazzola, Computationalcounterpoint worlds , Springer, Heidelberg, 2015.3. Alfred Mann,
The study of counterpoint , W. W. Norton & Company, 1965,Translation of fragments of
Gradus ad Parnassum by J.J. Fux.4. Guerino Mazzola,
The Topos of Music , 2nd ed., vol. I: Theory, Springer, Heidel-berg, 2017.5. Alejandro Nieto,
Una aplicaci´on del teorema de contrapunto , B. Sc. thesis,ITAM, 2010.6. Klaus-J¨urgen Sachs,
Der Contrapunctus im 14. und 15. Jahrhundert , Beiheftezum Archiv f¨ur Musikwissenschaft, vol. 13, Franz Steiner Verlag, 1974.
Instituto de F´ısica y Matem´aticas, Universidad Tecnol´ogica de laMixteca, Huajuapan de Le´on, Oaxaca, M´exico
E-mail address : [email protected] School of Music, University of Minnesota, MN, USA
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