A Fifth Of Circles 
      
   
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 What was Music like in the time of Pythagoras?: 
    The Greek philosophers held music in highest regard.   Its study was deemed essential to the development of a well-rounded individual.   In the Ancient Greek tongue, word sounds had different meanings at different pitches, and so melodies were mostly a derivative of the sound patterns of words and song.   Accompaniment to the songs and oratory of Ancient Greece probably amounted to notes or pairs of notes that corresponded to the note patterns in the spoken or sung part.   Harmony developed later.  The primary instruments of Ancient Greece were the aulos, a double-reeded pipe, and the kithara:  the lyre that we all associate with Greek poetry and song.   
 
What does Pythagoras have to do with Music?: 
    Pythagoras is reported to have begun thinking about pitch when he heard the sound of iron being worked on anvils.   The different tones being produced caused Pythagoras to wonder why the tones were different.  So Pythagoras began trying to produce different tones and his experimentation lead him to discover the effect of 'tautness' and 'length' on the notes produced by a vibrating string.  Pythagoras discovered that different lengths of wire at the same tautness produced different notes, and that the same is true of strings of the same length but different tautnesses.    Pythagoras believed that the universe was definable in terms of the simple ratios of the numbers,  such as 1:2, 1:3, 2:3, 1:4, and so on, and so he naturally tried to apply this to his study of pitch.   
 
What did he find out?: 
   He discovered that different lengths of string at about the same tautness, whose lengths are in simple ratios to one another sound pleasing together.  Take an aribitrary string  (In our case we will choose one tuned to 'C' at 261 Cycles per Second) and pluck it,  Then take a similar string half as long, making a simple ratio of 1:2.  This shorter string will vibrate at about 523 Cycles per Second, (twice as often) generating 'C' an octave above.   When plucked together, the two notes produce a pleasing sound: .   This is the sound of the 'octave'.  The vibration patterns of the two notes correspond twice every single wavelength of the longer string.  This correspondence makes the note pleasing.  The most pleasing of harmonies are those formed by differences of whole numbers.  As shown, a pair of notes an octave apart requres two strings whose ratios are 1:2.  The note 'F' below a particular 'C' requires half again as much string (a ratio of 3/2), and the 'G', a third more (a ratio of 4/3).  Naturally, the notes we find pleasing are the ones that constructively reinforce each other, and those are the ones that are related by whole number ratios.   
 
To Summarize: 
    Pythagoras discovered that lengths of string at simple ratios of length to one another produced sounds that were pleasing when played together.  It is simple to see that notes that are simple ratios to one another in frequency or wavelength will come into correspondence at regular intervals, producing these pleasing sounds.  This became the basis for our subsequent construction of musical scales.   
 
So what? 
    Well, the simple ratio between the note 'C' and the note 'F' is 3/2 wavelength.   This was one of the original note relationships investigated by Pythagoras.  This interval is called in music a 'Fifth' because counting the starting and ending note there are Five notes (not counting sharps and flats) between the two notes.  (Just for general useless information that helps to make everything make sense, I will tell you that the same note played with itself is called a unison, a note played with the note 8 notes above, when you count itself, and the end note, is called an octave, one less than that, a seventh, and so on.) 
    If you take a note, and find it's fifth (a simple ratio of 3/2 it's wavelength) and then find the fifth of that note, and the fifth of that note, and so on you will, with very little error, generate all 12 notes of the modern equal tempered scale, including sharps and flats.  This is called 'The Circle of Fifths" and gives rise to the name of this page, and is why I argue that Pythagoras is the Father of Modern Tuning.  
 
From here whence?: 
    Not all simple ratios are represented by our current musical scale.  The current common (equal tempered) musical scale emphasizes the ratios 4/3 and 3/2 at the expense of other ratios, which are only approximated.   The 'Just' scale lacks a robust system of sharps and flats, but incorporates many more simple ratios between notes, while maintaining the basic scale structure.   Many other scales are possible. 
   Music and sound has a profound effect upon our mystical, emotional and social selves.   It is shown that sounds that are tuned to, or harmonious with the rhythms of our brainwaves can have profound clarifying and relaxing effects.  In the near future I will add some pages to assist with understanding of tonality and alternate tone systems, and their effect upon us.