『Pythagorean System 』
Invented by Pythagoras,582-496 B.C.,Superman of ancient Greek,known of Pytagorean theory.
Maybe for him,the studies of mathematics,astronomy,philosophy and music are all same categories.
Clarity of pure 5th is truly a comfortable sound.
Pythagorean scale derives from the succession of this pure 5th.
（F←C→G→D→A→E→B ※←lower pure 5th, →upper pure 5th）
This is the same character with the tuning system of violin,also of Japanese sho(笙）、koto（筝）.
Gregorian chant was sung in this scale: the pitch of the intervals out stands the beauty of the melody.
When this succession of pure 5th[702 cents] is continued 12 times,it will go over 24 cents in one cycle.
If this system is set up to 12 notes of keyboard instrument,the last 5th will be 678 cents,
obviously narrow and out of tune.
Also 4 major 3rd that is slightly narrow than pure 3rd is effects from this narrow 5th.
This means,that it is impossible to set up Pythagorean scale in every keys.
Melody sounds very comfortable with major 3rd of Pythagorean scale.
Still,major 3rd with lots of howling is a characteristic of this system.
This makes a slightly high in harmony,feel uncomfortable,feel tension or sometimes broad sound.
Many tuning systems are invented in time of Bach. Characteristics of Pythagorean scale comes out often in the keys at the place of #&b.
※It is very interesting,when this system is turned 180°,it looks similar to the graph of Kirnberger Ⅰ.
In this time and still after,many composers must have used this character in their music.
Chisako Murakami translated it into English from pianotuning.jp
Special thanks to Ms. Midori Kita.