By now you should be familiar with the concept that musical notes are complex tones consisting of not a single frequency but rather a harmonically-related sequence of frequencies. We associate the pitch of a note with the fundamental frequency of this harmonic sequence. Surprisingly, as we heard in the demo in class, this fundamental component of the harmonic sequence might be completely absent from the note; yet, we still ascribe the pitch to that frequency because the remaining harmonics point to that one fundamental frequency.

In this page I want to explore, from a quantitative viewpoint, what is meant by the mucical concept of an interval. Starting from that concept we examine some of the strategies that have been used to build musical scales eventually outlining our current practice of using the so-called equal tempered scale. I am not delving into the musical niceties of this process rather I am concerned with showing that the musical terms have quantitaive meanings. I am not going to go through all of the messy fractional math that I did in class. If you are interested in pursuing this process the outline that I give here should be sufficient to get you started.

One point that I should make before starting is that there are certain musical intervals that are fairly universally considered consonant by very different cultures. What this implies is that there must be a fundamental reason why these tones sound good together to people from different eras and different parts of the world. You will appreciate in the section on The Scientific Theory of Consonance why certain tone pairs separated by intervals such as a fifth or a fourth sound pleasing for physical reasons not just because they have been conditioned culturally.

Pythagoras

The Greek mathematician Pythagoras (yes, Mr. Right-Angle Triangle) was the first to observe (or at least observeandwrite down) the connection between mathematics and pleasant musical intervals. He performed the following experiment (or at least something like this experiment). He noticed that when you pluck a string and then divide it (fret it) in half and pluck it again you get notes that sound good. He even realized at some level that these were the same notes (they are just an octave apart). Being very into mathematics, he tried dividing the string at the one-third point and plucking the longer portion. Again he found that the note of the full length string and the 2/3 length string sounded good together. Because you understand the nature of the standing waves on a string you can do some calculations that Pythagoras was not able to perform. Lets look at the frequency difference between the fundamenatal of the full length string (lets say it has a length L) and the string fretted 2/3 of the way along (of length 2L/3). Our formula for the standing waves is given by f_n= nv/2L where v is the speed of waves on the string. The fundamental is given by n = 1 (remember n = 1,2,3...) thus the pitch associated with the full length string is f_L= v/2L. For the 2/3 length string the only change is that where we had L in the formula we replace it with 2L/3! Thus the frequency of the fundamental of the 2/3 length string is 3v/4L = 3f_L/2 = 1.5xf_L. In other words the frequency of the shorter string is one and a half times higher than the full length string. If you played these two notes (on a guitar you can approximate this by playing the open note and then by fretting at the 7th fret) you might (if you were musical) recognize this interval as a fifth. Thus, what I have done is translate the musical notion of an interval into a numerical frequency relationship between the fundamental of the complex tones.

You could try the next step that Pythagoras embarked on which was to divide the string length into 4 and pluck the longer portion. This exercise corresponds to an interval of a fourth between the full length string note and the 3/4 length string note (on a guitar you can approximate this by playing the open note and then by fretting at the 5th fret). If you follow the same math path that I did in the presvious example you should be able to find that a perfect fourth corresponds to multiplying the lower frequency by 4/3=1.3333.

Again you can go further and find the interval corresponding to dividing the string by 5 or by 6 always plucking the longer portion. The notes eventually get less consonant for high ratios as you would expect. Pythagoras and the Greeks found the relationship between music, which was aesthetic and mystical, and nice neat whole number mathematics to be amazing.

Scales Based on Perfect Intervals

One idea for creating a musical scale was to base all the notes on the nice neat interval of a fifth. The basic scheme involved starting with the low note of a scale and then ascending in fifths to generate the frequencies of all the other notes. Whenever going up a fifth took a note out of the octave range one simply divided by 2 to get the note back into the octave range. To relate the notes to the notes of our musical scale use the "circle of fifths" to figure out which note is being generated at each step. I went through the exercise in class--it's a bit messy but essentially straightforward in principle. This method sounded like a good means of creating a scale; however, the mathematics is not perfect and a couple of problems crop up. First, in the full 12 tone scale 2 different fractions are created for the interval corresponding to a semitone. Second, if one goes completely around the cicrle of fifths back to the original note (many octaves up) and then translate down the correct number of octaves you find that the frequency of the original and final notes do not agree. They are off by over 1% a noticeable difference.

A one paragraph description does not really do justice to this complicated subject. However, you should have some insight into the problems that crop up when trying to base a scale on perfect intervals. The way that modern music has skirted these problems is to adopt the equal tempered scale.

The Equal Tempered Scale

In the equal tempered scale the "perfect" intervals such as a fifth or a fourth are sacrificed slightly to make a mathematically consistent system of notes. The basic premise is that the interval of a semitone must be of only one value. In our musical system of 12 semitones in an octave this means that the following condition must be true:

a¹²x f = 2 x f

or, eliminating f

a¹²= 2

 

The multiplicative factorarepresents the step in frequency that corresponds to a semitone. Because there are 12 semitones in an octaveato the 12th power equals 2 orais the 12th root of 2. If you can wrestle the answer from your calculator the value ofaturns out to be

a= 1.059463.

You can see that this choice does not lead to perfect intervals by examining the case of a fifth. A fifth is 7 semitone steps up in the scale; thus, to figure out the equal tempered interval corresponding to a fifth we need to calculatea⁷.

a⁷= 1.059463⁷= 1.4983

which does not exactly equal 1.5, the interval for a perfect fifth. The exercises below make sure you know how to do the appropriate math associated with this multiplicative factor. Again make sure you know how to solve these problems with the calculator you will be using in the quizzes or on the final. Note also that the scale is intrinsically logarithmic; each semitone step up is multiplicative not additive.

Cents

There is one further wrinkle that often crops up in practice, particularly with modern electronically controlled devices in which the frequencies can be very accurately controlled. That wrinkle is the concept of cents. The cent is just a further division of the semitone into 100 parts (cents). Again the steps are multiplicative not additive which makes the mathematics a bit trickier. Lets call the symbol for a centc. If there are 100 cents in a semitone and there are 12 semitones in an octave then it is clear that

c¹²⁰⁰= 2

Again some calculator wrestling lead to the fact that a cent is given numerically by

c= 1.00057779

To test this quickly calculate c100.

c¹⁰⁰= 1.00057779¹⁰⁰= 1.059463

which is the value of the semitone step as expected. Again try the problems to make sure you understand how to work with these quantities.

Questions

1. Make sure that you can use your calculator to find the 12th root of 2 to get the value of the equal tempered multiplier. Also make sure you can get the 1200th root of 2 to get the multiplier of the cent.

2. For the equal tempered scale the semitone interval is one fixed frequency ratio equal to 1.059463. This number is the 12th root of 2 which means that if I multiply this number by itself 12 times I get 2 (use your calculator to try this!). To see how the equal tempered scale is constructed write out the twelve notes in the chromatic scale C, C#, D, D#,… Under each note write the frequency ratio of that note to the frequency of the tonic note C. For example, to get C# from C I would multiply the frequency of C by 1.059463. To get D from C# I would multiply the frequency of C# by 1.059463. This second step is the same as multiplying the frequency of C by (1.059463)2. Keep on going in this fashion from C to (shining?) C. Just to get you started the first terms should look like this

   C C# D ….

   1 1.059563 1.12267 …

   Now if the frequency of the first note C was 100 Hz then the frequency of C# would be 1.059563x100=105.96Hz etc. Fill these numbers in for all of the chromatic notes. Note that at the low frequency end the difference between frequencies is small whereas at the high end the difference is much larger. This is because the equal tempered scale is a logarithmic rather than a linear scale. [Note the frequency of C is not really 100Hz I just chose that frequency to make some of the numbers easier].

3. Your group—Ron Resonance and The Basilar Membranes—after years of successfully switching to the latest trends in the music industry is trying to decide the next wave in musical taste. Based on a combination of bad management, too much coffee, and a horribly flawed focus group study, it is decided that the next hot area will be a musical form based on dividing the octave into a 18 note scale. As the only group member to survive with your numeracy intact after the group’s experience with Gothic Speed Metal you are given the task of calculating the frequency intervals between notes on this new scale. If all 18 notes are equal intervals apart what is the appropriate multiplicative factor to move from one note to the next (i.e. a number like the 1.0596 for the 12 tone chromatic scale). Write out the frequency ratios of the ascending notes. Are any of them close to the intervals that are consonant such as the fifth and fourth?

4. What is the frequency of the note that is 30 cents higher (sharper) than a frequency of 262 Hz? What is the frequency of a note 30 cents lower (flatter) than 262 Hz?

5. What is the difference in cents between two notes with fundamental frequencies of 350 Hz and 354 Hz? Hint: Mathematically you are trying to solvecⁿ350 = 354 to find the value of n.\