Sedgwick
© Copyright 2003-2017 Stephen Sedgwick
Equal Temperament came about with the lute. The frets were calculated by the minor semitone 18:17. The lutenist, composer, theorist, singer, teacher and  mathematical genius Vincenzo Galilei (ca1525-1591), derived the Rule of Eighteen. Dividing the string length by 18 would give you the first fret position. Then  dividing again the string length this up to the first fret would give you the second fret, so on and so fourth. Today we use a more precise calculation of 17.817  although still refered to as The Rule Of Eighteen. This meant that the lute could not play with the clavichords and organs of the time since they were not in  equal temperament. With Equal Temperament if you go around in 5ths or 4ths you will end up back where you started in the scale perfectly.  The Pythagorean scale was built on pure 5ths (3/2) or called a 3:2 ratio. Unfortunately with the Pythagorean scale if you go round the circle of 5ths you will  end up back where you began, but this time the note is not quite the same, it is sharper. To compensate this problem people would go round in 5ths and then  stop at a convenient 5th like Eb for example and then start working backwards from C in 4ths (4:3). When you reach Eb going backwards in 4ths the two Eb  ratios are different. The difference between these notes is called the Pythagorean Comma. From here people explored different tunings by flattening the 5th.  By using the perfect 5th, 3/2 ratio you can work out a C Major scale from the C up, G, D, A, E, B, F#, C#, G#. Using the perfect 4th, 4/3 to work out the last  few notes starting back at the root C... F, Bb, Eb. When the ratios are worked out, using the magic formula you can then convert the ratios into cents value. Cents = 3986*logRatio. It is like converting imperial  to metric. With the cents value it is clear to see how the Pythagorean scale compares to the E qual Temperament scale.
As a project for this subject I wanted to make a guitar with a Pythagurean tuning in C Major? As I didn't have time to make the whole instrument from scratch,  I managed to get my hands on a cheap damaged Romanian classical guitar. Nothing was wrong with the construction but the polish was damaged and  chipped. I removed the finish from the front and put a couple coats of an oil finish to improve the sound from the thick yellow lacquer that was on before. Then  I planed off the fingerboard to replace it with the Pythagoras fingerboard. But before cutting the slots I had to work out the maths behind it. 
Pythagoras Guitar
Looking at the interval between the Pythagorean scale you can see the two distinctive semitones that make up the scale. The Chromatic Semitone 114cents The Diatonic Semitone 90cents Equal Temperament Semitone 100cents
Now putting this onto a guitar fingerboard seems difficult at first, and you might expect the frets to be all over the fingerboard, but it worked better than  expected. Since there are two semitones, there are two fret slots, one for the chromatic semitone and one for the diatonic semitone. After mapping out the  frets on paper first, it worked out that I did not have to cut so many slots. 
I was very pleased with the out come of the Pythagorean guitar. The fingering was just like a normal guitar so you didn't have to think about moving your  fingers further back or forward when fretting. At first you don't notice much difference in the sound/pitch. But then your ears get use to it and it is quite  amazing to hear the perfect 5ths and then realise how out of tune the equal temperament 5ths are. Unfortunately big chords are out of the question and  strumming widely will only cause dissonance. The guitar should still be at London Metropolitan University as a teaching aid. but I fear that it has probably  gone for a walk.  One idea is to have a guitar with changeable fingerboards for the different tunings and keys. The problem with this is that the fingerboard is not 100% fixed to  the guitar as if it were glued. This means there would be quite a loss in the quality of the tone and the sustain. Another method would be to use some midi  system where you could program different tunings and not worry about bizarre fretting. There is probably something like that out there already. The ideal  thing would be to make a set of Pythagorean guitars, one for each key, but who would pay for such a thing? 
Note  C C#  D D#  E F F# G G# A A# B C
Position  Root  Minor 2nd  Major 2nd  Minor 3rd  Major 3rd  Perfect 4th  Augmented 4th  Perfect 5th  Minor 6th  Major 6th  Minor 7th  Major 7th  Octave 
Ratio  1/1 2187/2048  9/8 32/27  81/64  4/3 729/512  3/2 6561/4096  27/16  16/9  243/128  2/1
Py 0 114 204  294  408  498  612  702  816  906  996  1110 1200 
E T  0 100  200  300  400  500  600  700  800  900  1000  1100 1200 
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While at university I studied 'Scale, Tunings and Temperaments'. This is a hard subject to grasp first time but when you understand it, you can start thinking  on a new level. There are many books and a number of web sites that explain the different temperaments and tunings in much greater depth. Here I have  gone over the elements of the Pythagorean scale and how it is built. I have not gone into the full depth to keep you awake. 
PY ET  0 0 114 100  90  100  90  100  114 100  90  100  114 100  90  100  114 100  90  100  90  100  114 100  90  100 
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Sedgwick
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© Copyright 2003-2017 Stephen Sedgwick
The Pythagorean scale was built on pure 5ths (3/2) or called a 3:2 ratio.  Unfortunately with the Pythagorean scale if you go round the circle of 5ths  you will end up back where you began, but this time the note is not quite the  same, it is sharper. To compensate this problem people would go round in  5ths and then stop at a convenient 5th like Eb for example and then start  working backwards from C in 4ths (4:3). When you reach Eb going backwards  in 4ths the two Eb ratios are different. The difference between these notes is  called the Pythagorean Comma. From here people explored different tunings  by flattening the 5th. 
Pythagoras Guitar
I was very pleased with the out come of the Pythagorean guitar. The  fingering was just like a normal guitar so you didn't have to think about  moving your fingers further back or forward when fretting. At first you  don't notice much difference in the sound/pitch. But then your ears get  use to it and it is quite amazing to hear the perfect 5ths and then realise  how out of tune the equal temperament 5ths are. Unfortunately big  chords are out of the question and strumming widely will only cause  dissonance. The guitar should still be at London Metropolitan University   as a teaching aid.
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While at university I studied 'Scale, Tunings and Temperaments'. As a  project for this subject I wanted to make a guitar with a Pythagurean  tuning in C Major? As I didn't have time to make the whole instrument  from scratch, I managed to get my hands on a cheap damaged  Romanian classical guitar. Nothing was wrong with the construction but  the polish was damaged and chipped. I removed the finish from the front  and put a couple coats of an oil finish to improve the sound from the thick  yellow lacquer that was on before. Then I planed off the fingerboard to  replace it with the Pythagoras fingerboard. But before cutting the slots I  had to work out the maths behind it.