Mozart (Eine Kleine Nachtmusik, Sonata K300)
When I first heard of Mozart's formula, I felt a great excitement, because I thought that it might shed light on music theory and on music itself. You may at first be disappointed, as I was, when you find out that Mozart's formula, as we know it today, appears to be strictly structural. Structural analyses have so far not yet provided much information on how you come up with famous melodies; but then, music theory doesn't either. Today's music theory only helps to compose "correct" music or expand on it once you have come up with a musical idea. Music theory is a classification of families of notes and their arrangements in certain patterns. We can not yet rule out the possibility that music is ultimately based on certain identifiable types of structural patterns. I first learned of Mozart's formula at a lecture given by a music professor. I have since lost the reference -- if anyone reading this book knows of a reference (professor’s name, his institution), please let me know.
It is now known 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. That is, whenever he concocted 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. The first part of his formula was to repeat every theme. These themes were generally very short -- only 4 to 10 notes, much shorter than you would think when you think of a musical theme. These themes, that are much shorter than the over-all melody, simply disappear into the melody because they are too short to be recognized. This is why we do not normally notice them, and is almost certainly a conscious construct by the composer. The theme would then be modified two or three times and repeated again to produce what the audience would perceive as a continuous melody. These modifications consisted of the use of various mathematical and musical symmetries such as inversions, reversals, harmonic changes, clever positioning of ornaments, etc. 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. In addition, his modifications of the original theme followed a particular order so that certain moods or colors of music were arranged in the same order in each composition.
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 look like more of a genius than he really was. This naturally leads us to question: how much of his reputed "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 wonderful things that these geniuses did were the result of relatively simple devices and we can all take advantage of that by finding out the details of these devices. For example, 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 analyze his repetitions. They are not simple repetitions; Mozart used his genius to modify and disguise the repetitions so that they produced music and, more importantly, so that the fact of the repetition will not be recognized.
As an example of repetitions, 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, "OK, OK, I'm coming!" The male statement is made using only two notes, a commanding fourth apart, repeated three times (six notes), and the question is created by adding three rising notes at the end (this appears to be universal among most languages -- questions are posed by raising the voice at the end). Thus the first part consists of 9 notes. The repetition is an answer in 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 (six notes). It is an answer because the last three notes wiggle down. Again, the total is 9 notes. The efficiency with which he created this construct is amazing. What is even more incredible is how he disguises the repetition so that when you listen to the whole thing, you would not think of it as a repetition. Practically all of his music can be analyzed in this way. If you are not yet convinced, take any of his music and analyze it, and you will find this pattern.
Let's look at another example, the Sonata #16 in A, K300 (or KV331, the one with the Alla Turca ending). The basic unit 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. The 3rd bar is just the basic unit repeated twice. In the 4th bar, he again disguises the first unit by use of the 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 8 bars is patterned after the 1st bar. From a melodic point of view, these 8 bars produce two long melodies with similar beginnings but different endings. Since the whole 8 bars is repeated, he has basically multiplied his initial idea embodied in the 1st 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 a final melody that sounds like a long melody until you break it up into its components.
In the 2nd 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 3 related bars and the following 3 related bars. Thus bars 9 to 16 are similar to bars 1 to 8, but with a different musical idea. The final 2 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 piece. You will find that the same pattern of repetitions is found throughout the entire piece. As you analyze more of his music you will need to include more complexities; he may repeat 3 or even 4 times, and mix in other modifications to hide the repetitions. What is clear is that he is a master of disguise so that the repetitions and other structures are not usually obvious when you just listen to the music without any intent to analyze it.
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 just find more of these simple structural/mathematical devices, just stacked one on top of each other, or is there more to music? Almost certainly, there must be more, but no one has yet put a finger on it, not even the great composers themselves -- at least, as far as they have told us. Thus it appears that the only thing we mortals can do is to keep digging.
The music professor mentioned above who lectured on Mozart’s formula also stated that the formula is followed so strictly that it can be used to identify Mozart’s compositions. However, elements of this formula were well known among composers. Thus 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. Thus you will need to know some details of Mozart’s specific formula in order to use it to identify his compositions.
There is little doubt that a strong interplay exists between music and genius. We don't even know if Mozart was a composer because he was a genius or if his extensive exposure to music from birth created the genius. The music doubtless contributed to his brain development. It may very well be that the best example of the "Mozart effect" was Wolfgang Amadeus himself, even though he did not have the benefit of his own masterpieces. Today, we are just beginning to understand some of the secrets of how the brain works. For example, until recently, we had it partly wrong when we thought that certain populations of mentally handicapped people had unusual musical talent. It turns out that music has a powerful effect on the actual functioning of the brain and its motor control. This is one of the reasons why we always use music when dancing or exercising. The best evidence for this comes from Alzheimer's patients who have lost their ability to dress themselves because they cannot recognize each different type of clothing. It was discovered that when this procedure is set to the proper music, these patients can often dress themselves! "Proper music" is usually music that they heard in early youth or their favorite music. Thus mentally handicapped people who are extremely clumsy when performing daily chores can suddenly sit down and play the piano if the music is the right type that stimulates their brain. Therefore, they may not be musically talented; instead, it is the music that is giving them new capabilities. It is not only music that has these magical effects on the brain, as evidenced by savants who can memorize incredible amounts of information or carry out mathematical feats normal folks cannot perform. There is a more basic internal rhythm in the brain that music happens to excite. Therefore, these savants may not be talented but are just using some of the methods of this book, such as mental play. Just as good memorizers have brains that are automatically memorizing everything they encounter, some savants may be repeating music or mathematical thoughts in their heads all the time, which would explain why they cannot perform ordinary chores – because their brains are already preoccupied with something else.
If music can produce such profound effects on the handicapped, imagine what it could do to the brain of a budding genius, especially during the brain's development in early childhood. These effects apply to anyone who plays the piano, not just the handicapped or the genius.