Melodic Similaity - a Conceptual Famewok Ludge Hofmann-Engl The Link +44 (0)20 8771 0639 ludge.hofmann-engl@vigin.net Abstact. Melodic similaity has been at the cente of eseach within the community of Music Infomation Retieval (MIR) in ecent yeas. Many diffeent models have been poposed (such as tansition matices o dynamic pogamming). Howeve, so it seems, all these models exhibit a numbe of shotcomings. The appoach taken in this pape diffes in seveal ways fom pevious attempts. The position is taken that no single model will be satisfactoy in all contexts. Thus, athe than poducing yet anothe (simple) model, the appoach is taken to develop a conceptual famewok wheeupon a numbe of models can be based accoding to specific needs. The paametes which ae to be inputted into a model ae scutinized leading to the definition of atomic beats (smallest time value of a melody), melota (substituting ambiguous definitions of pitch), dynama (patially equivalent to subjective loudness) and chonota (the cognitive coelate to time values). Descibing melodies as chains based upon atomic beats allows fo the mapping of any given melody onto any othe melody via two specific eflections (whee the fist eflection is along the x-axis and the second eflection is mapping this image onto the second melody). The second eflection along a eflection cuve (called similaity vecto) displays two factos of similaity fo each of the paametes (six factos altogethe). The length of the similaity vecto delives infomation about how much two chains diffe in aveage pitch, aveage loudness and aveage time division. The inteval vecto as deived fom the similaity vecto (subtacting the ith component fom the (i+1)th ) poduces infomation of how closely two melodies ae simila in shape. Finally, two specific models will be tested within an expeimental setting.
1 Intoduction In ecent yeas the inteest in melodic similaity has been mushooming. This inteest has been diven by the developments within MIR (music infomation etieval). A typical MIR situation can be descibed as the following: a use wishes to locate a musical piece within a database. The quey can be by vaious means including musical notation, use of metadata (such as atists names) o humming. Typically, a quey by humming (QBH) equies the implementation of similaity algoithms, simply because the input by humming contains geneally eos and excludes the possibility to seach fo pefect matches. A vaiety of similaity algoithm have been put fowad, but they can be divided into fou classes. These classes ae (a) contast models (e.g. Downie, 1999), (b) diffeence models (e.g. Maidin, 1998), (c) dynamic pogamming (e.g. Smith, McNab & Witten, 1998) and (d) Makov chains (e.g. Hoos, Renz & Gög, 2001). Howeve, as pointed out by Hofmann-Engl (2002b), all these appoaches suffe fom a multiplicity of conceptual eos and an inability to undestand melodic similaity as an issue which will equie a sound conceptual famewok, which then can seve as the basis fo similaity models fashioned accoding to the specific needs. This is the main goal of this pape, to establish such a conceptual famewok which cannot only be employed within the setting of MIR but fo the ceative pocess of composing (such as poducing suitable vaiations to a theme) as well as fo musical analysis. Howeve, befoe we ente the discussion about such a conceptual famewok, we will biefly scutinize the paametes implemented in this famewok and intoduce the notation via atomic beats and the tansfomation of melodies. The main section investigating aspects of the famewok will be followed by a shot expeimental evaluation of the appoach taken hee in this pape. 2 The elevant paametes It seems sensible to ask befoe we develop a famewok of melodic similaity, what we intend to input as the elevant paametes. Clealy, the thee aspects pitch, duation and loudness will have to consideed in one way o anothe. Howeve, Hofmann-Engl (1989, 2001, 2002a, 2002b) expessed dissatisfaction with the tems pitch, duation, loudness and melody, as the tems ae highly ambiguous (what one peson consides to be a melody is not egaded as a melody by anothe peson). Thus, we will conside a new teminology consideing the elevant paametes fom a cognitive point of view.
This is, instead of using objective o subjective values, we will define intesubjective values. 2.1 Meloton vesus pitch Accoding to models of vitual pitch (e.g. Tehadt, 1979; Hofmann-Engl 1999) a tone does not only poduce one distinct pitch, but a seies of possible candidates which might seve as the pitch of the tone. Musical tones such as a piano a poduce often one stong candidate (e.g. a ) and a seies of vey weak candidates (e.g. d o f ), but othe tones such as tones as poduced by a dum instument do not poduce such a clea distinction between one stong and othe weak candidates. This lead Hofmann- Engl (2001) to define the tem meloton as such: Definition: The meloton is the psychological concept wheeby a listene listens to a sound diecting he/his attention to the sound with the intention to decide whethe the sound is high o low. Tue, this does not delive a quantity we can epesent, and hence we will have to define the value of a meloton somehow without using a physicalistic concept. In this context it seems most appopiate to conside an expeimental setting as intoduced by Schouten (1938). A goup of listenes is asked to tune in a (sinusoidal) compaison tone with vaiable fequency to match a test tone best accoding to each listene s individual judgment. We expect to obtain a distibution of diffeent esponses. In case the majoity of listenes tune into the same fequency unde well defined conditions (compae Hofmann-Engl, 2002b), we will take the logaithm of this fequency as the melotonic value and call the meloton stong. In case thee is no consensus amongst the listenes we will take the mean of the logaithmic fequencies to epesent the melotonic value and call the meloton weak. The concept of meloton is by fa supeio to a concept of pitch, as we find that all tones fetch a melotonic value, and thus dummelodies ae not only conceivable but can also be captued by efeing to the melotonic value. Clealy, we expect that the concept of pitch and meloton will in many cases coincide, but the fundamental diffeence emains that pitch is a cognitive pedicto while the meloton is a cognitive measuement.
2.2 Dynamon vesus loudness In analogy to the tem meloton, we define the tem dynamon: Definition: The dynamon is the psychological concept wheeby a listene listens to a sound diecting he/his attention to the sound with the intention to decide whethe the sound is loud o soft. The expeimental measuement of the dynamic value follows the same idea as did the measuement of the melotonic value. A goup of listenes is asked to tune in a (sinusoidal) compaison tone with vaiable loudness so as to match a test tone best accoding to each listene s individual judgment. We expect to obtain a distibution of diffeent esponses. In case the majoity of listenes tune into the same loudness unde well defined conditions, we will take the logaithm of this loudness as the dynamic value and call the dynamon stong. In case thee is no consensus amongst the listenes we will take the mean of the logaithmic loudness to epesent the dynamic value and call the dynamon weak. Note that elative dynamic values seem to be of geate impotance than absolute values (othewise a piece played back at vaious loudness levels would alte the quality of the piece substantially). We also face the situation that loudness peception depends on the oom chaacteistics and the chaacteistics of the individual listene fa moe than does the peception of pitch. The issue is discussed in detail by Hofmann-Engl (2002b). 2.3 Chonoton vesus duation In ode to ensue an equal teatment of the paametes we expect to be of impotance in the context of melodies, we intoduce the following definition: Definition: The chonoton is the psychological concept wheeby a listene listens to a sound diecting he/his attention to the sound with the intention to decide whethe the sound is shot o long. The measuement of the chonotonic value will have to be conducted this time in a diffeent way. A goup of listenes will be pesented with the test tone fo which we
intend to obtain the chonotonic value. Afte the test tone is head the listene will be asked to adjust a contol tone (by switching it on and off) so as to match the duation of the test tone best accoding to each listene s individual judgment. Howeve, the tem stong and weak chonoton will bea diffeent meaning this time. Geneally, listenes will be asked to segment an audio steam into segments (tones). Segmentation will occu due to sudden dynamic o fequency changes. Should the majoity of listenes segment the audio steam in the same fashion we will accept thei segmentation and talk about stong chonota. Should thee be no consensus, we will incopoate the peak segmentations and talk about weak chonota. We would expect that chonota ae stong in geneal. Howeve, whee tones display time-dependent melota (such as glissandi) o time-dependent dynama (such as cescendi), we might expect highe vaiations of segmentations. 3.4. Melodic chains vesus melodies We ae now in the position to define what we will call melodic chains. Definition: A chain is a sequence of a finite amount of tones. Tue, this definition does not deviate fom the moe conventional concept of what we conside melodies except that the tem melody is highly ambivalent and implies some fom of musical judgment. We denote a chain in the fom: ch = [t 1, t 2,..., t n ] If we ae inteested in the melotonic contents of a chain, we wite: M(ch) = [m 1, m 2,..., m n ] If we ae inteested in the dynamic contents of a chain, we wite: D(ch) = [d 1, d 2,..., d n ] and if we want to depict the chonotonic infomation of a chain, we wite: C(ch) = [c 1, c 2,..., c n ]
Note, that the intoduction of chains, melota, dynama and chonota does not mean a adical change o beak fom taditional appoaches. Howeve, it has been a shift towads cognitive measuement and an equal teatment of the paametes, which we consideed to be elevant. 3 Atomic beats and atomic notation It is inteesting to obseve that conventional musical epesentation does little justice to an adequate epesentation of time. This is, the x-axis of a typical scoe does not coelate to the actual time flow. The autho agues that some of the cognitive misconceptions about melodic similaity ae due to this fom of misepesentation of time. Hopefully, this will become moe obvious as the text develops. We will now intoduce the concept of atomic notation by means of an example. Let us conside the following c-chain: In a fist instance we can wite: C(ch) = {1/4, 1/8, 1/16, 1/16, 1/4} Howeve, if we conside that the 1/16 th notes ae the smallest chonota (=duations) and that all othe chonota ae multiples of the 1/16 th value, we can denote the same c- chain in the following manne: C(ch) = [4, 4, 4, 4; 2, 2; 1; 1; 4, 4, 4, 4](1/16) This is, the c-chain consists of 12 atomic beats taking 1/16 th to be the atomic beat. The fist value (a quate note) stetches ove fou atomic beats, the 1/8 th ove two
atomic beats and so on. The quate note is fou times longe than the atomic beat and hence the c-chain fetches fo the fist fou atomic beats the value 4, then the value 2 and so on. As we will see, this fom of epesentation will enable us not only to define melodic tansfomations but also to establish a conceptual famewok of melodic similaity. Howeve, befoe we will do so, we pesent the geneal fom of a c-chain, given as: C( ch) [ c, c,..., c ; c, c,..., c ;...; c c,..., c ]( 1 / a) = 11 12 1m 1 21 22 2m2 n1, n2 nm n whee C(ch) is a chonotonic chain in atomic notation, c i1, c i2,... c im i the ith chonoton in atomic notation, m i the numbe of atomic beats coveed by the ith chonoton, n the length of the the c-chain and 1/a the atomic beat. Dealing with melodic chains in geneal, we obtain: ch = [ t, t,..., t ; t, t,..., t ;...; t, t,... t ]( 1 / a ) 11 12 1m 1 21 22 2m2 n1 n2 nm n whee ch is a melodic chain in atomic notation, t i1, t i2,..., t im i atomic notation, m i the numbe of atomic beats coveed by the ith tone, n the length of the chain and 1/a the atomic beat. Each tone consists of a meloton, dynamon and chonoton. We will give an example. The opening of the theme fom Mozat s a-majo sonata (Köchel N. 331) is: The chain ch is given as: ch = [(c#, 3/16, p), (d, 1/16, p), (c#, 1/8, p), (e, 1/4, p), (e, 1/8, p), (b, 3/16, p), (c#, 1/16, p), (b, 1/8, p), (d, 1/4, p)] Note, we ae dealing hee with a scoe athe than a tanscipt of a sound souce. Hence, we ae dealing with pedicted values athe than measued values. Still, fo the pupose of illustation, we will assume that the scoe does epesent appoximated measued values. Howeve, this is not tue fo the dynama. Clealy, Mozat did not intend the piece to be pefomed without any dynamic vaiation, but following 18 th centuy notational pactice, Mozat did not feel the need to be moe specific about the dynamics of the piece. Rating dynama on a scale fom 1 to 9 (1 = soft, 9 = loud), we
might assume that the following chain might be an appopiate intepetation of the scoe: ch = [(c#, 3/16, 4), (d, 1/16, 1), (c#, 1/8, 2), (e, 1/4, 3), (e, 1/8, 2), (b, 3/16, 4), (c#, 1/16, 1), (b, 1/8, 2), (d, 1/4, 3)] The atomic beat of the except is 1/16. Thus the except falls into 22 atomic beats. We obtain the thee chains: M(ch) = [c#, c#, c#; d; c#, c#; e, e, e, e; e, e; b, b, b; c#; b, b; d, d, d, d](1/16) C(ch) = [3, 3, 3; 1; 2, 2; 4, 4, 4, 4; 2, 2; 3, 3, 3; 1; 2, 2; 4, 4, 4, 4](1/16) D(ch) =[4, 4, 4; 1; 2, 2; 3, 3, 3, 3; 2, 2; 4, 4, 4; 1; 2, 2; 3, 3, 3, 3](1/16) This fom of notation peseves the infomation of all thee paametes equally well elating them to quantasized time events. The main advantage of this method might only become fully appaent in the context of melodic similaity, howeve the tansfomation theoy as pesented next will enable us to see some useful aspects of this notation. 4 MELODIC TRANSFORMATIONS Thee ae seveal ways of intoducing melodic tansfomations. Howeve, the autho decided it would be most appopiate to efe to an example, and then to explain the undelying concept in moe detail.
Let us conside the beginning of the 1 st vaiation of the Mozat piece we wee talking about. This vaiation opens as: As befoe, we have no sufficient dynamic infomation. We assume that the following d-chain epesents an acceptable intepetation of the piece: D(ch v ) = [4; 3; 0; 2; 4; 2; 3; 2; 0; 2; 3; 1; 4; 3; 0; 2; 4; 2; 3; 2; 0; 2]. It is inteesting to note that in this case standad and atomic notation poduce the same chains. This is, because now all chonota ae 1/16 notes. We futhe obtain the m-chain M(ch v ) = [b#; c#; -; c#; b#; c#; d#; e; -; e; f#; e; e; b; -; b; a#; b; c#; d; -; d] and the c-chain C(ch v ) = [1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1]. Now tansfoming ch into ch v, will equie some fom of tansfomation pocess. We follow the mechanism as poposed by Hofmann-Engl (2001, 2002b) wheeby ch will be mapped onto ch v via a chain of eflection points. In ode to illustate this, we will conside the m-chain of the theme compaed to the m-chain of the fist vaiation. We find: M(ch) = [c#, c#, c#; d; c#, c#; e, e, e, e; e, e; b, b, b; c#; b, b; d, d, d, d](1/16) andm(ch v ) = [b#; c#; -; c#; b#; c#; d#; e; -; e; f#; e; e; b; -; b; a#; b; c#; d; -; d](1/16) We will depict the chains in log fequencies (whee a est will be mapped onto a 0 value) Thus we obtain: M(ch) = [2.75, 2.75, 2.75; 2.77; 2.75; 2.75; 2.81, 2.81, 2.81, 2.81; 2.81, 2.81; 2.71, 2.71, 2.71; 2.75; 2.71, 2.71; 2.77, 2.77, 2.77, 2.77] and M(ch v ) = [2.73; 2.75; 0 ; 2.75; 2.73; 2.75; 2.79; 2.81; 0 ; 2.81; 2.85; 2.81; 2.81; 2.71; 0 ; 2.71; 2.69; 2.71; 2.75; 2.77; 0 ; 2.77] Mapping the values of each atomic beat of ch onto C(ch v ), we obtain: R m (ch) = [2.74, 2.75, 1.38, 2.76, 2.74, 2.75, 2.80, 2.81, 1.41, 2.81, 2.83, 2.81, 2.76, 2.71, 1.34, 2.73, 2.70, 2.71, 2.76, 2.77, 1.39, 2.77] This eflection chain as such makes it possible to map two m-chains onto each othe. Moeove it delives some infomation on how closely the two m-chains in question ae invesions to each othe (the moe staight the eflection line the close
the tansfomation to an invesion). Clealy, in ou case we can see that Mozat did not have the concept of an invesion in mind when he devised the fist vaiation. In fact we will late see, that his intention must have been vey diffeent. We now conside the c-chains. Rewiting them in log 2 notation, we get: C(ch) = [1.6, 1.6, 1.6; 0; 1, 1; 2, 2, 2, 2; 1, 1; 1.6, 1.6, 1.6; 0; 1, 1; 2, 2, 2, 2] C(ch v ) = [0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0; 0] and Mapping both c-chains onto each othe will equie the eflection chain: R c (ch) = [0.8, 0.8. 0.8, 0, 0.5, 0.5, 1,1,1, 1, 0.5, 0.5, 0.8, 0.8, 0.8, 0, 0.5, 0.5, 1, 1, 1, 1] We saw that the moe staight the eflection chain in the context of m-chains, the close is the tansfomation elated to an invesion. The same is tue in the context of c-chains, but what exactly is an invesion of a c-chain? We conside the c-chain C(ch 1 ) = [4, 4, 4, 4, 2, 2, 4, 4, 4, 4](1/16) and the c-chain C(ch 2 ) = [1, 1, 1, 1, 2, 2, 1, 1, 1, 1](1/16). We get in log 2 : C(ch 1 ) = [2, 2, 2, 2, 1, 1, 2, 2, 2, 2] and C(ch 2 ) = [0, 0, 0, 0, 1, 1, 0, 0, 0, 0]. We obtain the eflection chain: R c (ch) = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]. This is a staight line, and hence C(ch 2 ) is the invesion to C(ch 1 ). This is, chonota which ae longe than the values of the eflection line, will be split into smalle chonota, and chonota which ae shote than the values of the eflection line, will be fused into longe chonota. As fa as the autho is awae, inveting c-chains is not a commonly known o used tansfomation. As we can obseve, Mozat s vaiation and theme ae not close chonotonic invesions. We finally map the dynamic chains onto each othe. We found ealie the possible intepetations fo the theme and the fist vaiation to be: D(ch) = [4, 4, 4; 1; 2, 2; 3, 3, 3, 3; 2, 2; 4, 4, 4; 1; 2, 2; 3, 3, 3, 3] and D(ch v ) = [4; 3; 0; 2; 4; 2; 3; 2; 0; 2; 3; 1; 4; 3; 0; 2; 4; 2; 3; 2; 0; 2] Assuming that ou dynamic values ae based on a logaithmic scale, we obtain the eflection chain: R d (ch) = [4, 3.5, 2, 1.5, 3, 2, 3, 2.5, 1.5, 2.5, 2.5, 1.5, 4, 3.5, 2, 1.5, 3, 2, 3, 2.5, 1.5, 2.5] As we can see, the d-chains ae also not close invesions to each othe eithe. Howeve, geneally we calculate the eflection points as:
p i = t + t 2 1i 2i whee p i is the eflection point at the place i, t 1i the value of the fist chain at the place i and t 2i the value of the second chain at the place i. A eflection chain will have the fom: R ( ch ) = [ p, p,..., p ] 1 2 n We will now conside a vecto notation of the eflection chains. 5 Melodic vectos and eflection matices Instead of depicting melodic chains as a sequence of tones in atomic notation, thee lies a geat advantage in epesenting them in fom of (n+1)-dimensional vectos, simply because eflections via eflection matices is mathematically well defined. Thus, a melodic chain consisting of n atomic beats, will be epesented in the fom: M whee tone, t n the nth tone and 1/a the atomic beat. t1 t2. = 1/ a. t n 1 v M is the melodic vecto, t1 the fist tone in atomic notation, t 2 the second Reflecting the melodic vecto M of the dimension n+1 onto the melodic vecto M of the dimension n+1 will equie the following eflection matix:
1 0.. 0 2 p1 0 1 2 p2... R =... 0 0 1 2 p n 0 0 0 1 whee R is the eflection matix and p 1, p 2.. p n the eflection points of the eflection chain as defined peviously. This is, we find: 2 p1 t1 2 p2 t2. R M = = M. 2 pn t n 1 whee t 1, t 2.. t n ae the tone components of the melodic vecto Fo a moe detailed desciption of the algeba undelying these eflection matices compae Hofmann-Engl (2002b). M. 6 A concept of melodic similaity Just as we intoduced melodic eflections above, so will we now intoduce a concept of melodic similaity by efeing to the Mozat example. The m-chains of the theme and the vaiations wee: M(ch) = [2.75, 2.75, 2.75; 2.77; 2.75; 2.75; 2.81, 2.81, 2.81, 2.81; 2.81, 2.81; 2.71, 2.71, 2.71; 2.75; 2.71, 2.71; 2.77, 2.77, 2.77, 2.77] and M(ch v ) = [2.73; 2.75; 0 ; 2.75; 2.73; 2.75; 2.79; 2.81; 0 ; 2.81; 2.85; 2.81; 2.81; 2.71; 0 ; 2.71; 2.69; 2.71; 2.75; 2.77; 0 ; 2.77]
Reflecting (=inveting) the theme along the x-axis, we obtain: -M(ch) = [-2.75, -2.75,- 2.75;- 2.77;- 2.75;- 2.75;- 2.81,- 2.81,- 2.81,- 2.81;- 2.81,- 2.81;- 2.71,- 2.71,- 2.71; -2.75;- 2.71,- 2.71;- 2.77,- 2.77,- 2.77,- 2.77] Reflecting -M(ch) onto M(ch v ), we obtain the similaity chain S m (ch, ch v ): S m (ch, ch v ) = [0.02, 0, 2.75, 0.02, 0.02, 0, 0.02, 0, 2.81, 0, 0.04, 0, 0.1, 0, 2.71, 0.04, 0.02, 0, 0.02, 0, 2.77, 0] We illustate this in figue 1. 3 2.5 2 1.5 1 0.5 0 Fig. 1. The x-axis epesent the atomic beat and the y-axis the coesponding value of the melotonic similaity chain, whee Mozat s theme and vaiation ae compaed As we can see, the lines each a peak each time the fist vaiation of the Mozat theme is on a est. All othe times, we find small deviations between the theme and the vaiation. Vaiation and theme even coincide in 9 atomic beats. Clealy, theme and vaiation fetch high melotonic similaity. If we fom the similaity chain compaing the c-chains, we obtain: S c (ch) = [1.6, 1.6, 1.6; 0; 1, 1; 2, 2, 2, 2; 1, 1; 1.6, 1.6, 1.6; 0; 1, 1; 2, 2, 2, 2] As we can see S c (ch, ch v ) and C(ch v ) ae identical. This is, because all chonotonic values of the vaiation fetch the value 0. As we can see in the figue below, we obtain a patten, wheeby the chonotonic similaity is highest on the 4 th atomic beat and lowest on the 7 th to 10 th atomic beat. Clealy, the theme and the vaiation seem to be little simila in tems of chonotonic similaity.
2 1.5 1 0.5 0 Fig. 2. Hee, the x-axis epesents the atomic beats and the y-axis the coesponding values of the chonotonic similaity chain, whee Mozat s theme and vaiation ae compaed. Befoe we will discuss moe geneal featues of the similaity chains, we will also poduce the dynamic similaity chain. Howeve, it must be stessed that this is somewhat poblematic as the dynamic chains ae the esult of intepetation by the autho athe than based on Mozat s scoe. Even moe, going by Mozat s instuctions, both the theme and the vaiation ae to be pefomed piano. Accodingly, all values of the similaity chain would fetch the value 0. Howeve, if we consideed ou ealie intepetation as acceptable, we obtain the following similaity chain: S d (ch) = [0, 1, 4, 1, 2, 0, 0, 1, 3, 1, 1, 1, 0, 1, 4, 1, 2, 0, 0, 1, 3, 1] As shown in figue 3, the similaity between theme and vaiation is smallest on the atomic beats 3 and 15. This is a est in the vaiation (dynamon value 0) is compaed to a tone on an down beat. Howeve, we find, unlike in the context of chonotonic similaity that the dynamic similaity chain still fetches a 0 value on 6 atomic beats. Hence, we assume that the dynamic similaity is highe than the chonotonic similaity. This does not supise: Mozat clealy oveides the oiginal hythm in his fist vaiation by implementing a monotonous c-chain, while the dynamics would emain somewhee simila, because the m-chains ae simila and because the accents will be somewhat the same as both pieces ae witten in a 6/8 time. 4 3 2 1 0 Fig. 3. Again, the x-axis epesents the atomic beats while the y-axis epesents the coesponding values of the dynamic similaity chain, whee Mozat s theme and vaiation ae compaed
We finally point out two specific featues of similaity chains: (A) A staight line such as S(ch) = [1, 1,..., 1] indicates in the context of melotonic similaity a tansposition, in the context of dynamic similaity a volume change and in the context of chonotonic similaity a split o fusion of chonota. (B) The cuvie the line (e.g. S(ch) = [1, 4, 2, 9...], the moe the two compaed chains diffe in shape. As shown by Hofmann-Engl & Pancutt (1998), melotonic similaity can be pedicated by efeing to the tansposition inteval and the inteval diffeence (shape). Futhe (Hofmann-Engl 2002a) poduced data which indicate that the distance of the chontonic similaity chain is the sole pedicto fo chonotonic similaity. The points of a similaity chain ae given as: p = p p s i 1 i 2 i whee p si is the ith point of the similaity chain, p 1i is the ith point of the fist chain and p 2i is the ith point of the second chain 7 Similaity and inteval vecto Mathematically, the similaity chain is the esults fom the composition of two eflections. Given the two m-vectos following eflections: M 1 and M 2. We map M 1 onto M 2 via the M 2 1 0.. 0 t + t 0 1 0 t + t... =... 0 0 1 t + t 0 0.. 0 1 11 21 12 22 1 0.. 0 0 t11 0 1 0 0 t12.... =.... 0 0 1 0 t 1 0 0.. 0 1 1 1n 2n n M 2 Isolating the last column of the fist matix in the subspace n, we obtain the similaity vecto S :
S = t + t t + t.. t + t 11 21 12 22 1n 2n The length of the similaity vecto seves as a pedicto fo the following similaity featues: Melotonic (pitch) similaity: The longe the similaity vecto, the moe the two chains diffe in aveage pitch. In case all components of the similaity vecto fetch the same value, we ae dealing with a tansposition. Dynamic (loudness) similaity: The longe the similaity vecto, the moe the two chains diffe in aveage loudness. In case all components of the similaity vecto fetch the same value, we ae dealing with a volume change. Chontonic (hythmic) similaity: The longe the similaity vecto, the moe the two chains diffe in density (while one chain may consist of long duations, the othe chain may consist of shot duations). In case all components of the similaity vecto fetch the same value, all chonota ae split into chonota of the same atios (e.g. quate into two eighths, an eighth into two sixteenths etc.). In ou example above we did not only conside the oveall distance of the similaity chains but also the cuviness of the chains. At fist, we might think that this cuviness could be measued by the angle between the diagonal and the similaity vecto. Howeve, this is not the case, as the following example will illustate. Given the two m-vectos: m 1 1 = m = 1 and x 1 2 whee x is a vaiable, we find that the similaity vecto fetches the value: S = 0 x 1 Now, we find that fo x > 1, that the angle between the similaity vecto and the diagonal D = (1, 1) is:
angle( S, D) = cos ( 1 0) + ( x 1) 1 + 1 ( x 1) = cos x 1 1 = cos = 45 2 ( x 1) 2 1 1 1 2 This is tue fo any value of x>1. Howeve, the cuviness of the similaity vecto will be affected by the value of x and hence, the angel is not a suitable measuement. Instead of the angle, we will intoduce the inteval vecto in the subspace n-1 as: I = s s s n s s 2 1 3 2 s n 1 whee s 1, s 2,..., s n ae the components of the similaity vecto. The length of the inteval vecto seves as a pedicto fo the following similaity featues: Melotonic (pitch) similaity: The longe the inteval vecto, the moe the two m-chains diffe in thei inteval sequences. Note, inteval diffeence captues by how many cents two intevals diffe, while contou captues diections only. As we will see, contou is not a similaity pedicto, while inteval diffeence is. Dynamic (loudness) similaity: The longe the inteval vecto, the moe two d- chains diffe in thei dynamic inteval sequences. Chontonic (hythmical) similaity. As we will see in the next section, expeimental evidence indicates that the chontonic inteval vecto is not a similaity pedicto. Note, chontonic and hythmical similaity ae not identical. This is, hythmical similaity incopoates both duations and accents. Chontonic similaity incopoates duations only. Accents ae to be seen a dynamic aspect and hence they ae coveed by dynamic similaity. Finally, we have not consideed chains of diffeent length. Hofmann-Engl (2002) poposed the following appoach. Let L be the length of the chain ch and L the length of the chain ch, we can wite:.. L = a L Basing a similaity pedicto upon the facto a, we can coelate similaity in the following fashion:
S ln 2 a Note, the lengths of the similaity and inteval vectos ae coelated to similaity. Howeve, similaity models might be based upon the lengths but will not necessaily be identical with the lengths. We will now conside some expeimental findings. 7 Expeimental findings Hofmann-Engl & Pancutt (1998) conducted two expeiments testing melotonic similaity and Hofmann-Engl (2002) undetook one expeiment testing chontonic similaity. Dynamic similaity emains untested. The esults will be biefly summaised in the following subsections. 7.1 Two expeiments on melotonic similaity Both expeiments wee conducted ove a mixed sample of 17 and 20 people. The stimuli wee melodic fagments consisting of 1 to 5 tones of equal length (isochonous). A fagment a was played followed by a fagment b and the paticipants wee asked to ate the similaity on a scale of 1 to 9. The stimuli included (a) tanspositions, (b) contou changes, (c) inteval changes, (d) tempo changes and (e) invesions. The following esults emeged: No ode effect was obseved (significance level 95%). This is, it did not matte whethe the fagments wee played in ode a - b o in ode b - a. Tempo change was ignoed by the paticipants (even when the tempo was changed by a facto 6). Clealy, paticipants ecognized tempo changes, but decided to ignoe them. In the fist expeiment a coelation between tansposition and similaity of 2 = 0.72 (p < 0.005) was established. In the second expeiment length of fagments and tansposition inteval wee manipulated simultaneously. Multiple egession evealed a coelation of 2 = 0.79 (with p(inteval) < 0.001 and p(length) < 0.01). Inputting the data into a model based upon the similaity vecto poduced a coelation of 2 = 0.92 (p < 0.003). Inveting fagments did not poduce significantly diffeent similaity atings (significance level 95%). Inputting inteval diffeence and contou diffeence evealed that contou diffeence is not a significant pedicto (p > 0.2).
A model (compae Hofmann-Engl, 2001) based upon the similaity and inteval vecto poduced a coelation of 2 = 0.74 within the fist expeiment The second expeiment was designed to test specific cases (such as multiple coelation between tansposition, length and similaity, hence the data wee not inputted into the model. 7.2 Expeiment on Chontonic Similaity This expeiment was conducted ove a mixed sample of 18 people. The stimuli wee hythmical fagments consisting of 1 to 9 duations (e.g. 6 eighths notes compaed to 3 quate notes). All tones had the same loudness and fequency. A fagment a was played followed by a fagment b and the paticipants wee asked to ate the similaity on a scale of 1 to 9. The stimuli included (a) split atio (e.g. splitting a quate into two equal pats (eighth notes) and duing anothe tial splitting a quate into a atio 7:3 which is appoximately a dotted eighth and a sixteenth note), (b) evesal (i.e. playing a hythmical patten backwads), (c) complexity (i.e. compaing a simple hythmical sequence with a complex sequence), and (d) tempo changes. The following esults wee obtained. One of the tials poduced an ode effect (significance level 95%). Howeve, the ode effect disappeas when setting the significance level at 96%. Split atio: It was found that the moe equal the split atio of a duation is the smalle ae the similaity atings. This is, consideing the split atio 1:x, we obtain minimal similaity fo x=1 and inceasing similaity fo inceasing values fo x. It also was found that the length of the fagments is a second pedicto (the moe duations ae compaed the highe the similaity). Multiple coelations poduced 2 = 0.77 with p(split atio) < 0.001 and p(length) < 0.02. Revesing hythmical sequences did not poduce any effect (t-test, p < 0.05). Complexity: It was shown that tials compaing simple pattens with complex pattens poduced a significantly lowe similaity ating (t-test, p < 0.004). Tempo change: It was found that changing tempo in the context of chontonic (hythmical) similaity affects the similaity atings. This is, the lage tempo change the smalle the similaity ( 2 = 0.77, p < 0.001). Implementing the data into a model (compae Hofmann-Engl, 2002a), poduces a coelation of 2 = 0.79, p < 0.001. Howeve, the inteval vecto appeaed to have negative influence and hence it was omitted.
8 Conclusion In this pape we agued that existing appoaches to the phenomenon of melodic similaity ae insufficient. Instead of eplacing these models by yet anothe model, we pesented a novel musical epesentation in fom of atomic chains. These chains enabled us to tansfom any given chain into any othe chain. We then intoduced the concept of the similaity and inteval vecto which can be consideed as the theoetical famewok fo melodic similaity. Finally, we pesented some expeimental data which ae in suppot of this appoach. In tems of ceativity we ague that no specific model will be needed as long as the compose is awae of the factos which detemine melodic similaity. Howeve, in ode to poduce a moe compehensive knowledge of melodic similaity, much exp eimentation (such as testing dynamic similaity) will be needed. Refeences 1. Downie, J.S. (1999). Evaluating a simple appoach to music infomation etieval: Conceiving melodic N-gams as text. Gaduate Pogamme in Libay and Infomation science. London, Ontaio, Univesity of Westen Otaio: 179. 2. Hofmann-Engl, L. (1989). Beitäge zu theoetischen Musikwissenschaft. M 65+1, TU Belin 3. Hofmann-Engl, L. & Pancutt, R. (1998). Computational modelling of melodic similaity judgments - two expeiments on isochonous melodic fagments. Online: http://www.chameleongoup.og.uk/eseach/sim.html 4. Hofmann-Engl, L. (1999). Vitual pitch and pitch salience in contempoay composing. In Poceedings of the VI Bazilian Symposium on Compute Music, Rio de Janeio 5. Hofmann-Engl, L. (2001). Towads a cognitive model of melodic similaity. In: Poceedings of the 2 nd annual ISMIR, Bloomington 6. Hofmann-Engl, L. (2002a). Rhythmic Similaity: A theoetical and empiical appoach. In Poceedings of ICMPC 7, Sydney 7. Hofmann-Engl, L. (2002b). Melodic tansfomations and similaity: a theoetical and empiical appoach. PhD thesis, Keele Univeisity 8. Hoos, H., Renz K., Gög, M.(2001). Guido/Mi - an expeimental Musical Infomation Retieval System based on GUIDO Music Notation. In Poceedings of ISMIR 2001, Bloomington, Indiana. 9. Maidin O, D. (1999). A Geometical Algoithm fo Melodic Diffeences. In: Melodic Similaity - Concepts, Pocedues, and Applications (ed. Hewlett & Selfidge-Field). Computing in Musicology, vol 11.
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