Where Math Meets Music

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Where Math meets Music

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Ever wonder why some note combinations sound pleasing to our ears, while others make us cringe? To understand the answer to this question, youll first need tounderstand the wave patterns created by a musical instrument. When you pluck a string on a guitar, it vibrates back and forth. This causes mechanical energy to tra

through the air, in waves. The number of times per second these waves hit our ear is called the frequency. This is measured in Hertz (abbreviated Hz). The more w

per second the higher the pitch. For instance, the A note below middle C is at 220 Hz. Middle C is at about 262 Hz.


Now, to understand why some note combinations sound better, lets first look at the wave patterns of 2 notes that sound good together. Lets use middle C and the

above it as an example:

Now lets look at two notes that sound terrible together, C and F#:

Do you notice the difference between these two? Why is the first consonant and the second dissonant? Notice how in the first graphic there is a repeating pattern

3rdwave of the G matches up with every 2ndwave of the C (and in the second graphic how there is no pattern). This is the secretfor creating pleasing sounding

combinations: Frequencies that match up at regular intervals (* - Please see footnote about complications to this rule).

Now lets look at a chord, to find out why its notes sound good together. Here are the frequencies of the notes in the C Major chord (starting at middle C):

C 261.6 Hz

E 329.6 Hz

G 392.0 Hz

The ratio of E to C is about 5/4ths. This means that every 5thwave of the E matches up with every 4thwave of the C. The ratio of G to E is about 5/4ths as well. Th

of G to C is about 3/2. Since every notes frequency matches up well with every other notes frequencies (at regular intervals) they all sound good together!

Now lets look at the ratios of the notes in the C Major key in relation to C:

C 1

D 9/8

E 5/4

F 4/3

G 3/2

A 5/3


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The jump between C and C# is 15.56 Hertz, the jump between C# and D is 16.48 Hertz. Although the Hertz jump is not equal between the notes, it is an equal jumpexponent number and it sounds like an equal jump to our ears going up the scale. This gives a nice smooth transition going up the scale.

To tell you the truth, these are approximate ratios. Remember when I said the ratio of E to C isabout 5/4ths? The actual ratio is not 1.25 (5/4ths) but 1.2599.

isnt this ratio perfect? Thats a good question. When the 12-note western-style scale was created, they wanted not only the ratios to be in tune, but they also wante

notes to go up in equal sized jumps. Since they couldnt have both at the same time, they settled on a compromise. Here are the actual frequencies for the notes in

Major Key:

You can see that the ratios are not perfect, but pretty close. The biggest difference is in the C to A ratio. If the ratio was perfect, the frequency of the A above middle

would be 436.04 Hz, which is off from'equal temperament' by about 3.96 Hz.

The previous list shows only the 7 notes in the C Major key, not all 12 notes in the octave. Each note in the 12 note scale goes up an equal amount, that is, an equa

amountexponentiallyspeaking.

Here is the equation to figure out the Hz of a note:

Hertz (number of vibrations a second) = 6.875 x 2 ( ( 3 + MIDI_Pitch ) / 12 )

The symbol means to the power of. The MIDI_Pitch value is according to the MIDI standard, where middle C equals 60, and the C an octave below it equals 48. A

example, lets figure the hertz for middle C:

Hertz = 6.875 x 2 ( ( 3 + 60 ) / 12 ) = 6.875 x 2 5.25 = 261.6255

The next note up, C#, is:

Hertz = 6.875 x 2 ( ( 3 + 61 ) / 12 ) = 277.1826

And the next note, D, is:

Hertz = 6.875 x 2 ( ( 3 + 62 ) / 12 ) = 293.6648

Another important feature of the scale is that it jumps by 2 times each octave. The A below middle C is at 220 Hertz, the A above middle C is at 440 Hertz, and the A

that is at 880 Hertz. This means that you can move notes into different octaves and still have themsound consonant. For instance, lets take the case of middle C and

again, except move G into the next octave. We still have middle C at 261.6Hz, but G is now at 784 Hz. That gives a ratio fromG to C of about 3/1 (twice the original r

3/2). The waves still meet up at regular intervals and they still sound consonant! Another nice feature of having an equal exponential jump is that you can start a scale

note you wish, including the black keys. For instance, instead of C,D,E,F,G,A,B, you can start on, say, D# and have D#,F,G,G#,A#,C,D as your scale with the same g

sounding combinations of frequencies.

At a certain point frequency ratios are too great to sound consonant. It takes too many waves for themto match up, and our ears just cant seemto find a regular pa

what point is this? The simple answer is when the ratios numerator or denominator gets to about 13. For instance, C# has a frequency ratio to C of about 18/17ths. T

just too many waves before they meet up, and you can tell that immediately when you play themtogether.

So now youre thinking that we have a scale that goes up in even steps and has reasonably accurate ratios, were all set, right? Actually, there are a lot of disse

opinions on the subject. Remember those not-quite-accurate ratios? One reason for this was for instruments to be able to be tuned once, and sound reasonably

all keys. Some of the grumpier musicians still complain, though, saying that equal temperament makes all keys sound equally bad. If you tune to just one particu

you can get those ratios perfect (since the human ear can detect a difference of 1Hz, being off by several Hz can be a problem!).

Maybe more importantly, though, is that there are a lot of undiscovered frequency combinations that cant be played in the confining 12-note system. Many alter

scales used in India have up to 22 notes per octave. If youre not satisfied with the standard western scale, there are lot of alternative tuning methods available,

'Just Intonation' and 'Lucy Tuning'. With modern digital equipment, these alternate tunings have become much easier to implement. We should hear some new a

incredibly interesting music come out of these tuning methods as they are gradually accepted into the mainstream.

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Note Perfect Ratio to C Actual Ratio to C Ratio off by Frequency in Hz

Middle C 261.6

D 9/8 or 1.125 1.1224 0.0026 293.7

E 5/4 or 1.25 1.2599 0.0099 329.6

F 4/3 or 1.333 1.3348 0.0015 349.2

G 3/2 or 1.5 1.4983 0.0017 392.0

A 5/3 or 1.666 1.6818 0.0152 440.0

B 17/9 or 1.888 1.8877 0.0003 493.9


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Where Math Meets Music