ISE599: Engineering Approaches to Music Perception and Cognition

Week 10 (26 Mar 2003): Do we see what we hear? Do we hear what we say?



Notes by Yun-Ching Chen, Frances Kao, Andy Lee, Cindy Lee, Arpi Mardirossian, Sean Mo, Xumei Tan, Erdem Unal, and Shivani Yardi on

Music as embodied mathematics: a study of a mutually informing affinity
by Jeanne Bamberger (2003). To appear in
the International Journal of Computers for Mathematical Learning.

Yun-Ching CHEN

The idea of this paper is a little similar to the previous paper, "Turning Music Theory on Its Ear". It also talks about some experiments based on Impromptu. And it gives us more examples on how to use the software.

This paper talks about the relationships in music intuition and mathematical structures. Why does the music sound good? Sounds stable? Or sounds slippery? There are not only reasons in musical theories. Math, in the other way, can also be helpful in explaining this. By using the Impromptu, we'll be able to find the patterns showing the affinity between mathematics and music. To take one step ahead, we'll also be able to analysis the behaviors in music recognition.

Frances (Hui-Yun) KAO

This paper is about the relation of basic concepts of music and mathematics. The authors claim that in the music learning environment, children can easily gain or demostrate the knowledge of math fundamentals, such as fractions, ratio, and relation. The experiment environment is again Impromptu, in this way the paper is more like an extension of the previous reading. The target group of the experiment is 6th grade students, and the results suggest that there are really some concepts of mathematics embedded in music, among which the most general aspects that those two disciplines are related are perception and patterns. Although some of the concepts are not easy for children to learn (or conciously learn), in the music environment they can easily apply those knowledge. It also shows that Impromptu makes an effective computer environment.

It's interesting to see those experiments with children, and to find that how different representation ways of Impromptu more efficiently show the different relation of notes/beats/rhythm than conventional notation. But I am kind of curious to know that why authors came up the idea of using Impromptu to help mathematics learning.

Some terms:
-- Transpose: changing a pitch collection, but keeping the internal relations the same. It preserves scale degree and intervals.
-- Fragmentation: a kind of transformation technique to retain cohesiveness while make variation.
-- Coda: extended tail.

Andy (Ming-Chang) LEE

In this paper, Jeanne Bamberger tried to link the human perception to mathematics learning. In the beginning, Jeanne quoted from Leibnitz' words, "Music is the arithmetic of the soul, which counts without being aware of it." I like this quote very much. Recently, I happened to read two books: one is about human perception, and the other one is about Prime number. In Oliver Sacks early work, The man who mistook his wife for a hat, he mentioned some examples that many *simple-minded* people have special abilities in painting, calculation, acting, music perception etc. These patients may have some problem to do basic division, like 4 divided by 2, however, some of them can communicate with very large prime number, more than 12 digits, in a short time or they can memories every page of Grooves dictionary of Music and Musicians. In the past, these abilities are mysterious because any normal person can't do that. The author mentioned about the "two worlds" in human brain, abstract and concrete world. These brain-impaired patients may have lost all their ability in abstract function, but their concrete function is gained more. For example, one of the patients who can memories any 300 digits number told Oliver that he can't explain why but he can *see* the number. For the prime number communication case, twin can play a game by speaking to each other a very large prime number, which is still hard to be done in any nowadays computer. However, as I mentioned in the beginning, I read the other book about prime number, "In code, a mathematical journey" by Sarah Flannery. In this book, the author tried to introduce her idea about encryption. She mentioned about some prime number related history, from Fermat's Little Theorem, Euler's equation, Carmichael's number, to Leonard Adleman's RSA coding. The interesting thing is, to determine if a number is a prime number is easier than before, but to generate a large prime number is still difficult. However, Olivers patient who can't do basic division can tell if a number is prime or even generate a prime. Another patient who can memorize a whole music dictionary can't even read any score, but he can understand the complex skill of Bach's music pieces. This is very amazing.

Back to Jeanne Bambergers paper, the reason I mentioned about the concrete concept is: its still mysterious how human perceive music. However, as we saw in the prime number case, there should be some deep theory applied in humans mind but still haven't been discovered. Maybe music itself is one way or a clue to find out the answer of the mystery. Music is not only the foundation of human life as math does, but it should relate to the evolutionary secret that how nature solves difficult problems as millions of years pass.

Cindy (Chia-Ying) LEE

This paper shows how music can help children learning mathematics. By using Impromptu, children can see the rhythm-the onset and duration of beats, and beats in different hierarchy. From playing and setting different set of drum beats, children can learn ratio, reciprocal relation and lowest common multiple. The ratios are formed between slower and faster beats. For a certain amount of time, the duration and frequency of beats are reciprocal. If we have two different beats proceeding, they will meet on their lowest common multiple. The mismatch of beats also generates a tension and it is resolved when the two beats meet. From music examples, children can extend to general mathematics concepts.

Besides the learning from rhythm, melodies also help children to understand patterns. In mathematics, sometimes we have to deal with abstract things which contain pieces and the relations among them. Here children have melodies as examples. By making fragmentation or transposition of a melody, children learn the meaning of patterns and transformation. In the experiment of constructing melodies, the author also observed that children are able to hear the stability of pitches and make use of different structural relations (such as transposition and fragmentation) to complete a tune.

Arpi MARDIROSSIAN

This reading discusses how music and the software Impromptu can be used to illustrate to students the importance and a possible use for basic mathematical ideas such as ratio, proportion, and common multiples.

This reading first introduces the software Impromptu and shows how there is both graphical and numerical representation.

This reading then discusses a few examples of children working with Impromptu to compose music. With each example, it is brought to our attention as to how a student uses this music forum to either understand or use some mathematical concept. The main concept that the children dealt with were when dealing with beats was least common multiples.

Zhenyao (Sean) MO

This is a very interesting paper, or I could say this is the paper I like most among all the papers I read in this course. It talks about the relationship of music and mathematics. Some experiments were designed and discussed in this paper, all about the affinities between music and math, people's perception of both, and people's ability of express.

I think this is a very important research direction. Usually people found it very difficult to explain why a masterpiece sounds so good, how to produce such excellent works, etc. Of course there are some rules, but even you follow all these rules, the result still lack something: the "spirit".

But if you can perceive music as something mathematical, then maybe, the inexplicable could be explained in a strict and formal math language. Then will comes the day of computer composer: a smart and well-trained computer could produce music as good as Beethoven, Bach ...

Xumei TAN

This paper is related to the previous paper. In working with impromptu, there are two basic aspects that initially encourage students to make practical use of structures shared by music and math: structures of music and the way those structures are represented in impromptu. For the first aspect, the paper talks about beats and temporal relations, which seems quite problematic. Several graphs show how beats are represented in impromptu and how the temporal relations and proportions are represented. Impromptu also has numeric representation, using number to show the proportion of durations. The paper compares these representations with conventional notation and finds that each captures some features while ignoring or minimizing others. The idea of multiple representations is very important in music education.

Then this paper shows some examples about how to work with children. There are several experiments done with children. For example, they ask the children to use impromptu to represent the temporal relations and find that children are exploring and describing relations among particular kinds of phenomena in a specific situation. The kids have seen the relation before and they will also learn it in school math, but the researchers are quite sure that students cannot generalize these relations in the way expressed in the formal mathematics. The multiple representations can actually help students understand the basic relations involved. There are more experiments about beats and more interesting things are the experiments about composing. The kids can find the pattern and reorder the blocks. When observing how they 'compose', the researcher find that they are trying to find the ending and the tonal center. Next, the topic moves on to the mathematical structure in transposition and invariants and the structure of melody. Finally, this paper conclude that the result from limited number of experiments suggest some significant points in music theory and music education.

Erdem UNAL

This paper is a collection of studies over students that try to show that music can be a learning context in which basic mathematical ideas can be elicited and percieved as relevant and important. Perceptions of music provides a path into mathematical relations such as fractions and proportions and basically this paper is trying to focus on this idea.

Using the software impromptu the author made some experiments about rhythmic structures of music samples over some students. Students were able to find out some fractions of beat easily when explained what is to be done. Some experiments were about 4-8-12-16 beats and their mathematical relations between each other. Some ideas implemented about least common multiplier that shows where two different beats meet in the same music.

Some students ask some interesting questions like, even the beats of two melodies are same how do they sound so different. So the authors explained the pitch contour, that one was different than the other one.

One another question come from another student, even most of the notes of a two stream are same except for the last two notes, how do these two melodies sound so different, arising another question, what makes something sound ended, anyhow?

And the final experiment was to make children segment a melody piece into blocks. Interesting answers occured all covering the discussed materials from previous experiments. Author focused on one students answer.

Phillip WU

MAIN POINTS: The author noticed that mathematical relationships play a role in the musically novice students' perception and composition of musical coherence. Software named Impromptu was used to show how mathematic works in music. The most general aspect of the affinity between mathematics and music might be the perception and articulate study of patterns.

Shivani YARDI


Posted 12 Mar 2003
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