Infinity Research Paper

For centuries, people were confused by the “Sonata No. 25 in G Major, Op. 79: III. Vivace”^[1] of Ludwig van Beethoven and “Chopin: 12 Etudes, Op.25 - No. 9 in G flat, "Butterfly Wings"”^[2] of Fryderyk Franciszek Chopin since they shared extremely similar melodies. This situation is just one example of something that happens a lot in music. This begs the question: how many different compositions can be created? In most people’s eyes, the elements of music such as beats, notes, accidentals, meter signatures, pitch classes are limited. Also, some songs seem similar, which means the harmony or melody is similar. However, that does not necessarily mean musical elements are finite. In fact, there are infinite numbers of possible compositions with infinitely many combinations of musical elements. Also, as time goes on, some of them which are not used often now may be used frequently in the future.

When hearing about rhythm, most people can only use “fast” and “slow” to describe it. Actually, there are infinitely many kinds of rhythmic notes^[3] with fixed values. The most basic unit of rhythm in notation is a whole note. A note half as long as a whole note is called a half note. By the same way, a half-length of a half note is named a quarter note. In order to make the rhythm faster and faster, composers use eighth notes, sixteenth notes, thirty-second notes, sixty-forth notes, and so on. Thus, whenever composers want a faster rhythm, they can divide the existing fastest notes into a half to create a new note. Doted notes^[4] are another mechanism that can be used to manipulate rhythm. Adding a small dot next to a note head can lengthen a note by half its value. For example, since a quarter note is equal to two eighth notes, a dotted quarter note is equal to three eighth notes. Since there are infinitely many note types, there are infinitely many dotted notes. Thus, technically, there can be infinitely many types of notes for composers to use when composing. Apart from notes which have sounds, there are rests which indicate a length of silence. For every note value, there is a corresponding rest. For example, whole notes correspond to whole rests; half notes correspond to half rests; quarter notes correspond to quarter rests, and so on. Since there are infinitely many types of notes, there are infinitely many corresponding rests.

The meter signature, also called the time signature^[5], which appears at the beginning of a composition, provides information about how many beats to count for each value. There are two basic types of meter: simple meter and compound meter. In the case of simple meters, represented by one number over another number, as “A/B,” the upper number of the meter signature specifies the beats per measure, a regular interval separated by bar lines, while the lower number demonstrates which note value gets one beat. For example, a “4/4” signature tells readers that in each measure of the following music, there will be four beats and that the quarter note will count as one beat. “4/4” is only one basic example of simple meters. There are other types of simple meters like “4/8,” which tells people that there are four eighth notes in each measure; “4/16,” which indicates that there are four sixteenth notes in every measure; “8/32,” which demonstrates that there are eight thirty-second notes in one measure, and so on. The upper number can be anything but not a multiple of three. The lower number should be note values discussed earlier in this paper. Since the number of possible upper numbers and lower numbers is infinite, the number of combinations of them is infinite. Thus, there are infinitely many simple meters.

...