Singing involves a specific way of using the voice that creates musical notes and musical intervals. Intervals can be distinguished according to size and whether they are concordant or discordant. Concordant intervals the fourth, fifth, and octave are accepted as axiomatic to melody and thus representative of the basic mater ials from which the various intervals of the Greater Perfect System can be developed.
Aristoxenus, Elementa harmonica, Aristoxenus further distinguishes between composite and incomposite intervals p. The different dieses apply to the three genera mentioned above: the half-tone is used in the diatonic genus; the third-tone is used in the chromatic genus; and the fourth-tone is used in the enhar monic genus. Greek Theory and Cognitive Structure As mentioned, the music theor ies of Pythagoras and Aristoxenus belong to a world remote from our own.
Not only did these theor ists have to grapple with the most basic of principles, but also the music they would descr ibe is a microtonal one that is primarily concer ned with the successive notes of melody rather than the simul- taneous notes of harmony. Despite this or perhaps because of it Pythagorean and Aristoxenian accounts of musical organization give us a glimpse into how the- ories are formed and, more important, the cognitive processes that are basic to these theories.
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In particular, three cognitive processes can be seen at work: categorization, cross-domain mapping, and the use of conceptual models. Were you to lift your eyes from this book and survey your sur roundings, you might well see chairs, lamps, tables, and other books; were you outside, you might see trees, birds, clouds, cars, and bicycles. If you considered the other things that populate your day, you might think of friends and family members, facial expressions and gestures, actions and activities.
Your recognition of these things reflects the categor ies through which we structure our thought: to recognize a book is to identify it as a member of the category book; to recognize a tree is to identify it as a member of the category tree. Categorization occurs in all sensor y modalities and throughout the range of mental activities: we categor ize smells and sounds, thoughts and emotions, skin sensations and physical movement.
Categor ies are not just basic to thought; they also give insight into our thought processes.
At one time it was thought that categor ies reflected the structure of the real world, but recent research has shown that the categor ies humans use are shaped by their interactions with their environments. Our reasons for developing and employing a given category are part and parcel of the category itself: categor ies are not only not given by nature, but also they are subject to change and modification as our thought unfolds.
Two categor ies basic to Pythagorean and Aristoxenian music theory are those for consonant or dissonant intervals. Consonant intervals such as the octave, fifth, and fourth are fundamental to the conceptualization of Greek music: they mark the sta- ble pitches of the Greater Perfect System and are the source of derivation for all fur- ther intervals, both consonant and dissonant. The process of categor ization is also exhaustive: any interval that can be conceived belongs to one of these two cate- gories. This is not to say, however, that consonant and dissonant intervals are given by nature in any simple way, Pythagoras and the blacksmiths notwithstanding.
I should note here that psychoacousticians distinguish between musical consonance, which is a cultural construct framed relative to a particular set of musical practices, and sensory consonance, which is a consequence of how sound waves are processed by the hear ing mechanism which involves the cochlea and the auditory cortex.
Sensory consonance is thus a fairly straightforward product of nature. Although musical consonance has its basis in sensory consonance, there is some freedom in how the sen- sory data are inter preted. Aristoxenians considered the interval a consonance, since it was simply the combination of two smaller consonances.
Pythagoreans, in contrast, classified intervals according to the numer ical ratio for med by their constituent pitches. As explained by the anonymous and thoroughly Pythagorean author of the Sectio canonis fourth century b. Another example of how categor ies shape our understanding of phenomena is provided by Greek theor ists treatment of thirds and sixths. Although thirds and sixths sound fairly consonant, they were nonetheless categor ized as discords. Two factors bear on this classification. First, forming thirds and sixths requires using the movable pitches of the Greater Perfect System at best, a third or a sixth will involve only one of the stable pitches bounding the constituent tetrachords of the system.
Thirds and sixths were intervals that necessar ily varied in size, and so they were placed among the dissonances. Second, in the classification of intervals Greek theory followed a tradition of dichotomous categor ies: there was concord, discord, and nothing else.
Conceptualizing Music: Cognitive Structure, Theory, and Analysis
By contrast, neither of these factors played a part in the music the- ory of early India. Indian music theor ists were consequently free to focus on the qualitative aspect of intervals rather than on their cor respondence with the fixed notes of a tuning system and to construe intervallic relationships as concordant, dis- cordant, or neutral.
Of course, there are numerous other categor ies important for Pythagorean and Aristoxenian music theor y, including those for pitches, intervals, and numer ical ratios. These categor ies and others are basic to the sort of systematic account of musical phenomena provided by these theor ies indeed, it is simply not possible to have a theory of music, or of anything else, without first having categor ies.
For example, one way to think about the elusive concepts of electr ical conductance is in terms of a hydraulic model: flipping the light switch tur ns on the juice, and elec-. Sectio canonis, in Harmonic and Acoustic Theory vol. By this means we take what we know about a fairly concrete and familiar source domain the flow of water and other liquids and map it onto a rather abstract and unfamiliar target domain: that of electr icity. As a wealth of research on analogy and metaphor has shown, the process of mapping structure from one domain to another is basic to human under- standing.
One place cross-domain mapping is evident is in the Pythagorean and Aristox- enian construal of interval.
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Because musical pitches are ephemeral and virtually intangible, relationships between pitches musical intervals represent something of a challenge to understanding. One way to meet this challenge is to map structure from the physical world onto music, a process evident in Nicomachuss story of Pythagoras and the blacksmiths. Pythagoras hears harmonious sounds, traces their origins to the blacksmiths hammers, and then proceeds to conduct various exper- iments using weights equivalent to those of the hammers. These exper iments lead, among other things, to a highly pragmatic objectification of musical pitch, as pitches are translated into physical objects that can be weighed, studied, and preserved.
By performing a mapping from the concrete physical domain proper to the black- smiths hammers onto the domain of musical sound, Nicomachuss story allows us to structure the latter domain in ter ms of the for mer. Of course, musical notes are not physical objects that can be weighed, studied, and preserved they remain ephemeral and virtually intangible. Nonetheless, we are so accustomed to the map- ping between concrete physical objects and musical sound that we sometimes have to be reminded that notes are not endur ing physical objects.
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Aristoxenuss construal of musical interval involves a slightly different mapping. As we have seen, according to Aristoxenus, when the voice moves intervallically, it appears to stand still at a given place a musical pitch and then pass over an inter- val of space a musical interval before coming to rest at another place another musical pitch. Underlying this account is a mapping from the familiar domain of two-dimensional space onto that of music.
This mapping allows us to apply the methodology of measur ing space to music. The difference between two linear mea- sures yields a third measure; similarly, the difference between the intervals of a fifth and a fourth yields the interval of a tone. Since linear measures can be easily divided into equal halves or thirds or fourths, the musical tone can be similarly divided, something impossible from the Pythagorean perspective.
On closer inspection, the Pythagorean and Aristoxenian construals of interval are indeed incommensurate. From the Pythagorean perspective, pitches are physical objects, and an interval descr ibes the relationship between these objects. From the Aristoxenian perspective, pitches are breadthless points that simply mark out an expanse of two-dimensional space, and an interval is the expanse itself.
Each map- ping gives an account of interval, but each leads to a different conceptualization of musical structure. This point can be generalized for music theory as a whole: map- ping structure from a nonmusical domain onto music is a way of creating musical structure, and different mappings will lead to different accounts of musical structure. They can also lead to conditional statements: if the interval is an octave, then it is a consonance; if a pitch is an object, then its properties are measurable. Propositions like this are basic to conceptual models, which act as guides to reasoning and inference.
In their sim- plest for m, conceptual models consist of concepts in specified relationships, which pertain to a specific domain of knowledge. For an example of a conceptual model, let us return to the classification of con- sonant and dissonant intervals presented in the Sectio canonis, according to which all consonant intervals have either multiple or epimor ic ratios, and all dissonant inter- vals have epimer ic ratios.
anerriabour.gq This classificator y system relies on a conceptual model that organizes concepts related to interval, concord, discord, and the three classes of ratios. The simple patter n of inference that follows from this model is that if an interval has a multiple or an epimor ic ratio, it is a concord; if it has an epimer ic ratio, it is a discord.
Integral to this model are the products of categor ization and cross-domain map- ping. Two types of categories are involved in the model: those pertaining to music the categor ies of concord and discord and those pertaining to number the mul- tiple, epimoric, and epimer ic ratios. Cross-domain mapping cor relates the two types of categor ies by construing musical interval as a relationship between two objects namely, musical pitches to which magnitudes in the for m of numbers can be assigned. Specific classes of ratios can then be used to distinguish between the musi- cal categor ies.
The robustness of this particular conceptual model is reflected in the debate over the status of the octave plus a fourth that continued into the Middle Ages.