Pythagorean Tuning

 

Origins

 

Pythagorean tuning is a type of just temperament, which is based off a series of tones whose frequencies are related by the ratio of 3:2 and 1:2.  Pythagoras found this ratio of frequencies to be pleasing to and thus used it as a means of tuning.  But the reason why these ratios are pleasing has to do with the fact that 3/2 is one of the solutions to partial differential wave equation. 

 

As an instrument vibrates, it produces various sound waves.  These unique frequencies are established and the set of notes we are familiar with are assigned to these vibrations.  What happens when Pythagorean tuned sounds are being played is that similar overtones, similar sound waves, are being produced as a result of the frequencies being integers multiples (in this case 3/2 & 1/2) of one another.  Simply put, the instrument is set up to have harmonics played on it. * This is why the integer ratio of 3/2 is effective when being used to produce the notes (i.e. this is why Pythagorean found the tones pleasing).

 

As for the 2/1 ratio, this has to do with keeping the tones within the same octave. An octave will have tones which range from 1 to 2 times higher the frequency of the note you start from (the bass note of the octave).  It’s quite clear that by the third tone in the series, the tone whose frequency is 9/4 times above the bass note (3/2*3/2), it is outside the range of the octave (9/4 is greater than 2 or  8/4). So now the tone must be multiplied by 1/2. The new tone will now have a frequency of 9/8 times higher than that of the original tone. Each tone is multiplied as many multiples of ½ as needed to keep it within the octave. The chart below shows original ratios and the adjusted ones.

 

 

Pythagorean Ratios

 

 

A

 

E

 

B

 

F#

 

C#

 

G#

 

D#

 

A#

 

E#

 

B#

 

F##

 

C##

 

B##

 

1

 

3/2

 

9/4

 

27/8

 

81/16

 

243/32

 

729/64

 

2187/128

 

6561/256

 

19683/

512

 

59049/

1024

 

177147/

2048

 

531441/

4096

=3^12/

2^12

 

 

 

 

 

9/8

 

27/

16

 

81/

64

 

243/

128

 

3^6/

2^9

 

3^7/

2^11

 

3^8/

2^12

 

3^9/

2^14

 

3^10/

2^15

 

3^11/

2^17

 

3^12/

2^19

 

 

 

The resulting tones will actually be slightly higher than those, which have been equally tempered (this is the generally accept temperament present in most Western music).  This difference is known as a Pythagorean Comma and is explained below.

 

Pythagorean Comma

 

The following sound files go through the circle of fifths playing the notes of a Pythagorean scale and those of an equally tempered scale simultaneously.  This is of particular interest because of what is known as the Pythagorean comma. This comma is the cents difference that occurs between the final tones in the series:  one is generated using Pythagorean tuning and the other is equally tempered (the difference is approximately 23.46 cents**). All files start with middle C (frequency equal to 261.6255653 Hz.). The more times one goes through the circle of fifth the larger the difference from equally temperament becomes.  The latter two sound files explore this idea further as they go through the circle of fifths two and three times respectively. 

 

 

Sound Files:

 

 

 

 

 

 

The Error Value graph, shown below, shows the cents difference which is heard in  Pythagorean Vs Equal (One Rotation)

 

 

 

If you were to carry on with this process four times, thus getting four commas, the new final tone produced would be approximately one semitone higher than what it should be.  This means that by the fourth go around using Pythagorean tuning results in two tones that should be the same, but are actually almost 100 cents apart.  If you started with a C your four time through the circle of fifths you would end up with a C#.  The following sound file shows this difference.  First the fourth comma is played (middle C, the new tone, and then the two played together).  Next, one semitone is played (middle C, C# or middle C 100 cents sharp, and then the two together).  Finally the two tones are played together.

 

 

Sound File:

 

 

 

 

 

 

 

 

*A more in-depth explanation of the wave equation can be obtained on the Wave Equation page on the home page.

**Note for more information on cents CLICK HERE or click on the Cents Test tab on the homepage.