There is an intimate, if not absolutely essential, relationship between mathematics and music. At the very least, they share a large number of the most fundamental
properties in common, starting with the fact that the equal tempered chromatic scale is a simple logarithmic equation and that the basic intervals are ratios of the smallest integers that allow the brain to keep track of tonics in chord progressions [(68) Theory, Solfege]. Every musician is naturally curious whether mathematics is involved in the creation of music. In composition theory, mathematical symmetry transformations have been a major compositional device since before Bach (Solomon, Larry,). This is not surprising because math applies to practically everything; math is simply a device for describing anything quantitatively. One way to investigate this relationship is to study the works of the greatest composers from a mathematical point of view. Here are a few examples.
Mozart's Formula (Eine Kleine Nachtmusik, Serenade K525)
Professor Robert Levin of Harvard (Levin, Robert,:) lectured on "Mozart's Fingerprints: A Statistical Analysis of his Concertos" concerning a "specific and
sophisticated hierarchy of musical motives that underlies the Mozart concerto form" in December of 1977 at a Bell Laboratories Research Colloquium, at Murray Hill, NJ. I have to thank Brian Kernighan (co-author of "The C Programming Language") for locating the records to this lecture which was still stored in his computer after more than 30 years!
Prof. Levin lectured on a hierarchy of musical motives that were so specific as to be potentially useful for authenticating Mozart's compositions. On the one hand, I was disappointed with the lecture because of my ignorance of music theory; I was expecting an easily understandable musical structure. On the other hand, Prof. Levin awakened my awareness of structure in music, and led me to examine structure in Mozart's music.
If you take just one atom, carbon, you can change the atomic microstructure and get anything from hard, brilliant diamonds to lubricating graphite to light weight golf club shafts, to superconductors, and even buckyballs with amazing properties and uses. It is the differences in the repetitive microstructure of the carbon atoms that gives these materials such different properties, and my expertise was in examining these microstructures.
It was no surprise, therefore, that I immediately recognized the repetitive structure of Mozart's music. For those not accustomed to dealing with structure in music, this repetitive structure is not easily recognizable because it appears to have no obvious relevance to the melodic progression. I have tested this recognition with my musical colleagues and it took most of them a while to recognize this structure as a part of the music. This lack of recognition has historically impeded the pursuit of this microstructure because, for musicians, it seems so trivially simple that it does not deserve attention. One of the best examples of this is the slow movement of Mozart's Piano Concerto No. 21, which is generally considered to be non-repetitive because the incredible emotional content hides the repetitions.
entire piece, so that formal rhythm is 100% repetitive. Mozart's music uses mostly a single repetition (2 units in a row). Bach uses repetitions extensively, but is not mainly confined to a single type like Mozart's. In the Inventions, Bach uses 2 repetitions most frequently (3 units in a row – see Invention #8). Repetitions on larger scales are also important, as Slenczynska, Ruth, (P. 49) wrote: "play all repeats marked by the
composer" - instructions from a seasoned pianist, because the repetitions are there for specific purposes.
These types of repetitive structures are well known among composers, and articles on music analysis and composition are starting to discuss them in greater detail (Brandt). Discussions of pitch sets and symmetry transformations similar to those discussed here have appeared in the literature (Bernard, Solomon).
My structural analysis revealed that Mozart composed practically all of his music, from when he was very young, according to a single formula that expanded his music by over a factor of ten. Whenever he composed a new melody that lasted one minute, he knew that his final composition would be at least ten minutes long. Sometimes, it was a lot longer, because the main part of his formula is a multiplication by a factor of two; so that the multiplication after 10 minutes makes the music 20 minutes long, then 40, etc.!
The first element of his formula was to repeat a "motif". These motifs are very short -- only a few notes, much shorter than you would think of a musical melody — we always think of melodies, not motifs. We see the Taj Mahal, but the individual marble blocks are invisible. These short motifs simply disappear into the melody because they are too short to be recognized; certainly a conscious construct by the composer to hide them.
The motif would then be modified two or three times to produce what the audience perceives as a melody. These modifications consisted of the use of various mathematical and musical symmetries such as inversions, reversals, harmonic changes, clever
positioning of ornaments, etc., as shown below. These repetitions would be assembled to form a section and the whole section would be repeated. The first repetition provides a factor of two, the various modifications provide another factor of two to six (or more), and the final repetition of the entire section provides another factor of two, or 2x2x2 = 8 at a minimum. In this way, he was able to write huge compositions with a minimum of thematic material.
Because of this pre-ordained structure, he was able to write down his compositions from anywhere in the middle, or one voice at a time, since he knew ahead of time where each part belonged. And he did not have to write down the whole thing until the last piece of the puzzle was in place. He could also compose several pieces simultaneously, because they all had the same structure.
This formula made him appear to be more of a genius than he really was, because he could compose so much music, write it down backwards and forwards, compose it
"genius" was simply an illusion of such machinations? This is not to question his genius -- the music takes care of that! However, many of the magical things that these geniuses did were the result of relatively simple devices that we can learn.
Knowing Mozart's formula makes it easier to dissect and memorize his
compositions. The first step towards understanding his formula is to be able to identify the motif and analyze his modifications and repetitions. They are not simple repetitions; Mozart used his genius to modify and disguise the repetitions so that they produced music and so that the repetitions will not be recognized.
Another aspect of his compositions is the economy with which he expresses complex ideas; as an example, let's examine the famous melody in the Allegro of his
Eine Kleine Nachtmusik. This is the melody that Salieri played and the pastor
recognized in the beginning of the movie, "Amadeus". That melody is a repetition posed as a question and an answer. The question is a male voice asking, "Hey, are you coming?" And the reply is a female voice, "Yes, I'm coming!" The male statement is made using only two notes, a commanding fourth apart, repeated three times, and the question is created by adding two rising notes at the end (this appears to be universal among most languages -- questions are posed by raising the voice at the end). The response is a female voice because the pitch is higher, and is again two notes, this time a sweeter minor third apart, repeated (you guessed it!) three times. It is an answer because the last three notes wiggle down. The efficiency with which he created this construct is amazing. What is even more incredible is how he disguises the repeated pairs of notes so that when you listen to the whole thing, you would not recognize the repetitions, but hear a single melody.
Let's look at another example, the Sonata #11 in A, K331 (or K300i - the one with the Rondo Alla Turca ending). The basic unit (motif) of the beginning theme is a quarter note followed by an eighth note. The first introduction of this unit in bar 1 is disguised by the addition of the 16th note. This introduction is followed by the basic unit, completing bar 1. Thus in the first bar, the unit is repeated twice. He then translates the whole double unit of the 1st bar down in pitch and creates bar 2. This is the same device used by
Beethoven at the start of his 5th symphony where he gives you the "fate" motif and then
repeats it at a lower pitch. The third bar is the basic unit repeated twice. In the fourth bar, he again disguises the first unit by use of 16th notes. Bars 1 to 4 are then repeated with minor modifications in bars 5-8. From a structural viewpoint, every one of the first eight bars is patterned after the first bar. From a melodic point of view, these eight bars produce two melodies with similar beginnings but different endings. Since the whole eight bars is repeated, he has basically multiplied his initial idea embodied in the first bar by 16! If you think in terms of the basic unit, he has multiplied it by 32. But then he goes on to take this basic unit and creates incredible variations to produce the first part of the sonata, so the final multiplication factor is even larger. He uses repetitions of repetitions. By stringing the repetitions of modified units, he creates music that sounds like a long
melody.
In the second half of this exposition, he introduces new modifications to the basic unit. In bar 10, he first adds an ornament with melodic value to disguise the repetition and then introduces another modification by playing the basic unit as a triplet. Once the triplet is introduced, it is repeated twice in bar 11. Bar 12 is similar to bar 4; it is a repetition of the basic unit, but structured in such a way as to act as a conjunction between the
preceding three related bars and the following three related bars. Thus bars 9 to 16 are similar to bars 1 to 8, but with a different musical idea. The final two bars (17 and 18) provide the ending to the exposition.
With these analyses as examples, you should now be able to dissect the remainder of this sonata. You will find that the same pattern of repetitions is found throughout. As you analyze more of his music you will need to include more complexities; he may repeat three or even four times, and mix in other modifications to hide the repetitions. He is a master of disguise; the repetitions and other structures are not obvious when you listen to the music without analyzing the structure.
Mozart's formula certainly increased his productivity. Yet he may have found certain magical (hypnotic? addictive?) powers to repetitions of repetitions and he probably had his own musical reasons for arranging the moods of his themes in the sequence that he used. That is, if you further classify his melodies according to the moods they evoke, it is found that he always arranged the moods in the same order. The question here is, if we dig deeper and deeper, will we find more of these simple structural/mathematical devices, stacked one on top of each other, or is there more to music? Almost certainly, there must be more, but no one has yet figured it out, not even the great composers themselves -- at least, as far as they have told us. Thus the only thing we mortals can do is to keep
digging.
For further analysis of this Sonata (#11, K331), see Scoggin, Nancy,, P. 224.
Mozart is not the inventor of this formula and similar formulas were used widely by composers of his time. Some of Salieri's compositions follow a very similar formula; perhaps this was an attempt by Salieri so emulate Mozart. In fact a large fraction of all music is based on repetitions. The beginning of Beethoven's 5th symphony discussed
below is a good example and the familiar "chopsticks" tune uses "Mozart's formula" exactly as Mozart used it. Therefore, Mozart simply exploited a fairly universal principle of music composition.
The simplest form of Mozart's formula appears in the famous "Twinkle, Twinkle, Little Star" song where the motif is a single note which is repeated. This tiny melody embodies most of the basic rules of composition, and was composed before Mozart was born. Since Mozart undoubtedly heard it as a child, it is possible that he started
composing by adopting it as a model, and eventually used it for almost all of his
compositions. This hypothesis explains why Mozart used this formula from his earliest composing days — it is the first melody most youngsters hear at an early age. Mozart
may have initially based his compositions on this formula and, as he developed it, discovered that he didn't need anything else, especially because it enabled him to compose everything in his head without having to write anything down.
Beethoven & Group Theory (5th Symphony, Appassionata, Waldstein) The use of mathematical devices is deeply embedded in Beethoven's music.
Therefore, Beethoven is the best place to dig for information on the relationship between mathematics and music. I'm not saying that other composers did not use mathematical devices. Practically every musical composition has mathematical underpinnings and every famous composer has used incredible mathematical devices to compose. However, Beethoven stretched everything to extremes and such extremes are most useful because the underlying principles can be identified with certainty.
We all know that Beethoven never studied advanced mathematics, yet he used
group theory type concepts to compose this famous symphony (Bernard, Jonathan W.,, and search "group theory" or "symmetry in music" on the internet). In fact, he used what crystallographers call the Space Group of symmetry transformations! Group Theory governs many advanced technologies, such as quantum mechanics and nuclear physics that are the foundations of today's technological revolution. At this level of abstraction, a crystal of diamond and Beethoven's 5th symphony are one and the same! I will now explain this.
The Space Group that Beethoven used has been applied to characterize crystals, such as silicon and diamond, and is the basis for analyzing useful properties of crystals. It's like the physicists needed to drive from New York to San Francisco and the
mathematicians handed them a map! That is how we perfected the silicon transistor, which led to integrated circuits, the computer, and the internet. So, what is the Space Group? And why was this Group so useful for composing this symphony?
Mathematicians found that groups consist of Members and Operations, such that if you perform an operation on a member, you get another member of the same group. A familiar group is the group of integers: -1, 0, 1, 2, 3, etc. One operation for this group is addition: 2 + 3 = 5. Note that the application of the operation + to members 2 and 3 yields another member of the group, 5. Since operations transform one member into another, they are also called Transformations. A member of the Space Group can be anything in any space: an atom, a frog, or a music note. The atom and frog reside in our 4-
dimensional space-time. The music note operates in any musical dimension such as pitch, speed, or loudness. The Operations of the Space Group relevant to crystallography are (in order of increasing complexity) Translation, Rotation, Mirror, Inversion, and the Unitary Operation. These are almost self explanatory (translation means you move the member some distance in that space) except for the Unitary Operation which basically leaves the member unchanged. However, it is subtle because it is not the same as the equality transformation, and is therefore always listed last in textbooks. Unitary Operations are
generally associated with the most special member of the group, which we might call the Unitary Member. In the integer group noted above, this member would be 0 for addition and 1 for multiplication (5+0 = 5x1 = 5); this demonstrates that figuring out the unitary operator is not simple.
Let me demonstrate how you might use this Space Group, in ordinary everyday life. Can you explain why, when you look into a mirror, the left hand becomes a right hand (and vice versa), but your head doesn't rotate down to your feet? The Space Group tells us that you can't rotate the right hand and get a left hand because left-right is a mirror operation, not a rotation. Note that this is a strange transformation: your right hand becomes your left hand in the mirror; therefore, the wart on your right hand will be on your left hand image in the mirror. This can become confusing for a symmetric object such as a face because a wart on one side of the face will look strangely out of place in a photograph, compared to your familiar image in a mirror. Although the right hand
becomes a left hand, a mirror cannot perform a rotation, so your head stays up and the feet stay down. Curved mirrors that play optical tricks (such as reversing the positions of the head and feet) are more complex mirrors that can perform additional Space Group operations, and group theory will be just as helpful in analyzing images in a curved mirror.
The solution to the flat mirror image problem appeared to be easy because we had a mirror to help us, and we are so familiar with mirrors. The same problem can be restated in a different way, and it immediately becomes much more difficult, so that the need for group theory to help solve problems becomes more obvious. If you turned a right hand glove inside out, will it stay right hand or will it become a left hand glove? I will leave it to you to figure that one out (hint: use a mirror).
Let's see how Beethoven used his intuitive understanding of symmetry
transformations to compose his 5th Symphony. That first movement is constructed using a
short fate motif consisting of four notes. The first three are repetitions of the same note. Since the fourth note is different, it is called the surprise note and Beethoven's genius was to assign the beat to this note. This motif can be represented by the sequence 5553, where
3 is the surprise note and the bold indicates the accent. This is a pitch based space group;
Beethoven used (and was aware of) a space with at least three dimensions: pitch, time, and volume. I will consider only the pitch and time dimensions in the following
discussions.
Beethoven starts his 5th Symphony by first introducing a member of his group: 5553. After a momentary pause to give us time to recognize his member, he performs a translation operation: 4442. Every note is translated down. The result is another member of the same group. After another pause so that we can recognize his translation operator,