Monday, December 21, 2009

Intervals and Ratios


    I've always been fascinated by the crossover of music and math.  In this article, I will attempt to explain how intervals can be expressed as ratios. This can also be looked at as an introduction to understanding the mathematical nature of music.  The ratios found in this article are based on the very same ratios that Pythagoras discovered by dividing a vibrating string, so that is where I will begin.
     First, I should clarify that what I mean by an interval is simply 2 pitches being sounded simultaneously or in sequence.  Second, it should be known that expressing intervals as ratios is much easier than it sounds.  It is just another way to express something musical. In my opinion, the more ways you have of looking at something, the greater the possibilities of expressing it exist in your imagination (we can be more creative about something if we can look at it in a number of different ways, as opposed to just one way).  Third, the term "vibrating string" refers to a string whose purpose is to sound a note, i.e. a piano string, or cello string, or guitar string, etc.

    Pythagoras discovered the phenomenon of interval ratios by hanging different sized weights, proportionate to each other, to the ends of strings of equal make and length.  Later, he developed what was called a "monochord," which consisted of a string anchored at both ends on top of a box with a sound-hole with a movable bridge between the anchors which essentially divided the string into two smaller strings.

    Pythagoras used this instrument to experiment with different ratios of the string.  He would position the bridge (the thing in the middle) at different places and pluck the string on both sides, which would sound an interval.  In experimenting this way, Pythagoras made quite an intriguing discovery.  He found out that the most pleasing intervals to the ear were the result of even divisions of the string, that is whole number ratios (3:4 is a whole number ratio as opposed to 3.76:4.5).

    If a vibrating string is divided in half, that is the bridge on the monochord is positioned exactly halfway between the two anchor points, the notes on either side will be the same.  This is called a unison and has a ratio of 1:1 because each string is the same length.

    If one side is doubled in length so that it is twice as long as the other, you also get the same note, though the shorter side sounds one octave higher.  All simple intervals fall between these two extremes.

    1:1, or the unison, is considered the most consonant interval, 2:1 the next most consonant, 2:3 the next most consonant after that.  What you will find is that as the numbers go higher, so does the level of dissonance.  There are a number of theories as to why this is so.  Personally, I believe it to be because of the mind's preference for rational numbers, but it could be any number of reasons.

    As was just stated, after the octave (2:1), the next most consonant interval is 2:3, or the fifth.  If the bridge is positioned so that it is two thirds of the way across the string, the interval produced by sounding the string on both sides of the bridge is a fifth.

    I'm sure you get the basic idea by now.  However, the ratio doesn't just represent the proportionate lengths of strings, but also the ratio of vibrations between the two pitches.  For example, 2:3 means that for every time one of the pitches that make up a fifth goes through 2 cycles of its vibration, the other goes through 3.  Pretty cool, huh.

    The other interval ratios are as follows:

3:4 - 4th
4:5 - Major 3rd
5:6 - Minor 3rd

    By raising the lower tone in an interval up an octave, the interval is said to be inverted.  If you remember from a previous example, doubling a number in the ratio raises it an octave.  A 4th is an inverted 5th, a minor 6th is an inverted major 3rd, a major 6th is an inverted minor 3rd, etc.  The inversions are as follows:

If you invert 2:3 (a 5th), you get 3:4 (a 4th)
If you invert 4:5 (a Major 3rd), you get 5:8 (a Minor 6th)
If you invert 5:6 (a Minor 3rd), you get 6:10, which reduces to 3:5 (a Major 6th)

    Since intervals can be expressed using whole number ratios, then chords, which consist of three or more pitches sounded simultaneously, should be able to be expressed in the same manner.  Let's look at the Major triad, for example.  In the key of C, it consists of the notes C, E, and G.

C : E : G

    C to E is a major 3rd (4:5)

C : E
4 : 5

    C to G is a 5th (2:3)

C : G
2 : 3

    Based off of this information, we can see that the major triad can be expressed with the following ratio:

C:  E  :  G
1 : 5/4 : 3/2

    This ratio contains fractions and so it doesn't sit well within our framework of whole number ratios, so let's clean it up a bit.  Let's start back at the major 3rd interval of C to E, which is 4:5.  That is a whole number ratio so it is fine and dandy.  Now, instead of using the interval of a 5th from C to G, let's take the minor third from E to G, which is 5:6.  When we combine these two ratios, as you can see below, we get a much cleaner, better understood ratio that fits within our framework of whole number ratios.

C : E : G
4 : 5 : 6

    I hope you found this article interesting, and if not, maybe I'll catch your attention next time.  Maybe you don't like to look at music from such a highly mathematical perspective, and that is fine.  Music is what you make it, though it can't be denied that music is very mathematical in nature.  That is just how it is, but it doesn't mean you have to look at it that way.  My intent was not to say that you should think about music in mathematical terms or not, but rather to introduce to you the mathematical nature of the art itself; to offer another perspective from which to draw inspiration.





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