Music and Math: Rhythmic Relationships, Meter, and Fractions

Happy Pi Day! My normal blogging schedule has me posting on Sundays, but in honor of Pi Day, and the importance of math in music, I am posting on a Tuesday.

One of the interesting things I discovered recently about Pi is that, just like music, every culture has figured out Pi. It's expressed by the 14th Greek letter Pi, Chinese mathematicians calculated it to the 7th digit using geometrical measurements, Indian mathemeticians calculated it to the 5th digit by around 5 A.D. The earliest approximations were discovered in Babylon (1900-1600 BC- 25/8 = 3.125) and Egypt (1650 BC- (16/9)2 = 3.1605). I just found that to be extremely interesting as I see music as being a universal constant as well- all cultures throughout history have had some form of music.

Even though I have to admit to not being the best at math, it is hard to deny that math is part of every part of music- from the different lengths of sound waves which are adjusted to create the "correct" sound when tuning, to the speed of the beat that we tap our feet to (calculated in beats per minute). For this primer level post, we're looking at rhythm and fractions.

Fractions

First off... quick definition time for fractions. The numerator is the top number, the denominator is the bottom number.

Time Signature and Meter

At the beginning of most music, there are two numbers stacked on top of either other- looking very much like a fraction. This is your time signature. I have a tendency to use the terms "time signature" and "meter" interchangeably, but they are separate terms. The time signature is the term for both numbers at the beginning:

Meter refers to the rhythmic, recurring pulse. In simple terms, the meter is usually the top number in your time signature, however there is such a thing as "compound" meter- for example, feeling 6/8 time in a large two: 1,2,3, 4,5,6, stressing beats 1 and 4. Interestingly enough, you can also have compound fractions, but that is not for me to discuss. The concept of compound meter goes beyond primer level... I'm just throwing interesting tidbits out there. Anyway, meter is how the music FEELS, and the time signature helps dictate that. Back to time signature... the top number tells you how many beats are in a measure, and the bottom number tells you which note gets one beat.

You can have however many beats you want in a measure (or a bar). Typically, the higher the number, the more you will want to subdivide and feel the larger beat. This gives the musician the ability to create movement within the beat without changing tempo. The bottom number tells you which note gets one beat. The typical denominators are 4 (quarter note), 2 (half note), or 8 (eighth note). Again, the note names are reflective of the denominator- a quarter is one fourth, represented by a 4, a half is one second, represented by a 2, and an eighth is one eighth, represented by an 8.

One of the myths we tell our students is that the quarter note gets one beat. ONLY IF THE DENOMINATOR IS 4! The note that gets one beat changes depending on the time signature! What doesn't change is the relationship of the notes to each other.

Rhythmic Relationships

Rhythm is also based in fraction relationships. The whole note takes up a whole measure in 4/4 time- also called common time. So, it starts with 1 note for an entire measure. The half note gets half of that value, so again in 4/4 time, it would get 2 beats (2 is half of 4). One quarter of 4 is 1, and the quarter note receives 1 beat in 4/4 time. One eighth of 4 is 1/2 of 1, so eighth notes receive 1/2 of one beat, etc. The relationship never changes, even though the beat value might.

6/8 Time

If you're in 6/8 time, the eighth note receives 1 beat. One quarter is twice as much as one eighth, so in 6/8 time, the quarter note receives two beats. 1/2 is twice as big as 1/4, so the half note gets twice as many beats as the quarter note, so in 6/8 time, it gets 4 beats. The whole note is where we began, and it gets twice as many as 1/2, which is too large for 6/8 time- it gets 8 beats, and 6/8 only allows 6 beats per measure. The note that fills an entire measure in 6/8 is the dotted half note. Dotted rhythms don't fit into our relationship tree particularly well, but it is still built on a fraction- the dot adds half the beat value back to the note. So... in 6/8 time, if the half note receives 4 beats, the dotted half note receives 6 beats (4+2 (1/2 of 4)= 6).

Cut Time

Also called 2/2 time, cut time divides everything in half. The half note gets 1 beat. 1/4 is half of 1/2, so the quarter note receives a half beat. 1/2 is half of the whole, so the whole note receives 2 beats in cut time.

Thank you so much for reading! Now go practice!