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A Brief History Of The Pythagorean Comma
and how it pertains to Hammond organs!


In discussing the theory of operation of a Hammond organ, the terms "harmonics" and "intervals" come up very often - more so than in any other instrument - this is because you can't control the volumes of the harmonics produced by any other instrument. However I often hear people confusing these two terms as is easily done - and the explanation of them is not only a fascinating topic but is also well outside the scope of my Hammond organ page. So...

Musicians generally understand intervals better than harmonics - very simply an interval is just the distance between two notes - but the history of Western music is so complex and has carried so much baggage from it's earlier methods that even intervals are hard to describe.

The best way to describe intervals was invented by Guido of Arezza in 1020 and is known as the Sol-Fa system or more recognizably "Doh-Re-Mi" - you can describe the intervals of a major scale by the notes "Doh, Re, Mi, Fa, So, La, Ti, Doh" or in more practical terms these days as the numbers one to eight (this is how an "octave" get's it's name - musicians will of course realize that there are now twelve notes in an octave although a major scale still uses only eight of them).

This eight note "scale" is based on the laws of physics - by dividing a string or length of pipe or other sound making device into different lengths you can get the different notes. The exact same physical laws apply in electronics - however, if you try to play any music that's more complicated than a nursery rhyme or "Happy Birthday" the physics begin to let us down.

Pythagoras' Comma is a strange quirk of nature that wreaked havoc with musicians, composers and especially instrument makers for hundreds of years - it's always existed but it only became a big problem about 500 years ago when people wanted to write more complex music. But it was discovered much longer ago...

Pythagoras was the first person to devote a great deal of time and effort to work out a "theory of music" and much of what we use now is based on his research. He noticed the mathematical relationship between notes and worked out ways of applying them to musical instruments. This is where the harmonics come in.

Pythagoras worked out that if you take a string - like a guitar string for example - and exactly halve it, the half-string will produce a note that is an exact replica of the original note, but higher up - an octave in fact. This is called the "first harmonic".

If you halve that string, you will get a harmonic that is called the "dominant", which is a fifth interval away from that original string. Guitarists will note that the first harmonic is marked on the guitar with two dots (the twelfth fret) and the second harmonic (the dominant) is the fifth fret.

As well as continually halving this string the dominant can be produced by dividing the original string by two thirds - you'll see the fifth fret is two thirds of the way from the bridge to the nut of the guitar. If you keep dividing the string by 2/3 you will get (eventually) all the notes of the scale - and many more - in theory this sequence goes on infinitely which is what causes problems musically.

After you have cleverly produced the twelve tones that make up all the notes we use in music (the black and white notes) we run into a very strange predicament that virtually no modern musicians are even aware of but which caused hundreds of years of strife for our predecessors - this dilemma is called "Pythagoras' comma".

If our original string was a "C", the harmonic sequence produced by dividing it up will be C,G,D,A,E,B,F#,C#,G#,D#,A# right? It gets more dissonant (in relation to the starting note) as it goes but they're all still real notes at least. The next note in that sequence however is E#. Now whereas the A# we mentioned is the same note as Bb - the E# we need to continue our sequence mathematically or physically is not the same as F - it is a note in between E and F which doesn't exist on a piano, guitar, saxophone, flute or any other Western musical instrument - except fretless ones technically.

This thirteenth note is known as the "Pythagorean comma" - and of course if you keep going you'll get B#, F##, C## etc. and all other kinds of notes that don't exist - thank goodness because they're completely incompatible with the twelve that we do have!

How this theory applies in practice is that, whereas C is the fifth interval of the key of F, its also the third interval of the key of G#, the second interval of the key of Bb, the fourth interval of the key of G etc. However technically, theoretically, mathematically, physically - in fact in every way other than "Western musically" - all these C's are actually slightly different.

Particularly the third and sixth intervals of any scale are quite dodgy and were frowned upon and even made illegal at different periods of history! So it's only really possible to tune an instrument properly (according to the laws of physics outlined by Pythagoras) in one key - and this is what was done for many centuries.

Incidentally this is why I have so much trouble tuning the B string of a guitar in standard tuning - I always tune it flat because it sounds better (because you've just tuned the G string and B is the third interval of G - one of the dodgy intervals) And while that works fine for playing in G - it causes havoc when using that B as anything other than the third of G. This is also partly why I like open tunings so much on the guitar - you can tune the thirds and fifths a little bit flat - as long as you're playing in that one key (or using a capo) you're actually more "in tune" than a standard tuned guitar.

When Western music wanted to use more notes at once and use different keys and chords in the same piece of music that the laws of physics and nature let us down. The obvious and simple solution is to shuffle the notes around a bit so that the E# which is very, very nearly an F actually is an F - then the actually infinite progression of fifths would be limited to twelve different notes and would join up and become the "circle of fifths" that we're familiar with.

This was very much easier said than done though in the days before electronic tuners, oscilloscopes or even high precision hand tools. Dutch mathematician Simon Stevin first worked out that, in order to make the cycle of fifths resolve itself each note had to be 1.059463094 times the frequency of it's neighbour before 1600 but it took until 1722 before the theory was able to be put into practice.

In this year Johann Sebastian Bach published his "Well-Tempered Clavier" (tempering was the name given to the process of shuffling the notes around until you could play in any key - and a Clavier was an early name for any type of keyboard instrument) with pieces for all twelve major keys and all twelve minor keys. We don't know if he tempered his own Clavier (or how) or if he had it done but as they say, the rest was history.

All the other instruments were modified so that they could be tuned to a keyboard instrument and this tuning became known as Equal Temperament and most musicians now don't realise that it was ever any different.

So the point of this story was that harmonics are the natural physical effects of dividing a string (or a sound wave frequency) by different amounts - whereas intervals are man made adjustments of the laws of physics that were invented to allow the playing of music in more than one key.

Equal Temperament didn't solve everything - the harmonics themselves still obey the laws of physics as described by Pythagoras - we've just moved the intervals around - so you can still hear dissonances and discrepancies between the artificial intervals and the natural harmonics - but, other than going back to playing in only one key there's nothing we can do about that...

Where this applies to Hammond organs is that, while the various tones produced by the different tonewheels are described as "harmonics" and function like real, physical harmonics, they're really intervals that are adjusted to work with any note in any key - this is essential on a Hammond because there is no other instrument in which the "harmonics" are so pronounced and audible - or controllable.

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