The Pythagorean Comma

The comma has its origin in the Pythagorean Tuning System, which is generated by 12 successive just (perfect) fifth intervals (ratio 2:3).
But the span of 12 just fifths exceeds the span of 7 octaves (each of a ratio 1:2) by a tiny bit - the Pythagorean Comma - leading to the necessity of tempered tuning systems.
As long as we stay in the world of drones and their harmonic overtones everything is relatively simple: From a mathematical perspective it is all a game with the first 20 integers or so and the proportions between them.
But as soon as we also include melody and subsequently harmony, the whole picture becomes more complex, and the most fundamental functions from the harmonic series appears not to meet precisely.
The most fundamental structure in music is the division of the octave (ratio 1:2) in twelve halftones.
In the modern equal temperament all halftone steps are mathematically equal but in this arrangement the just intervals have been sacrificed - specially the thirds and sixths.
The reason for this is that the structure originally derrives from the primary elements of the harmonic series, the octave, 1:2, and the perfect fifth, 2:3.
The span of 7 octaves - 2 to the power of 7 - is almost the same as the span of 12 perfect fifths - 3/2 to the power of 12 - but if you calculate the values you will come to 128 and 129.746 respectively.
So something needs to give way when the 12 tones are arranged within the frames of an octave in a tonal system.
Like the sisters of Cinderella musicians around the globe have been cutting the heels and toes of the musical intervals to make them fit the glass (or gold) slipper - the illusion of a perfect tonal system.

Category:

Music

Tags:

• Pythagorean
• Comma
• Just
• Perfect
• Fifths
• Octaves
• Temperament
• Tuning
• Ratio

[rrr00bb]

A 53ET master scale is a huge improvement (31ET is not as good, but is still a great improvement over 12ET). In 53ET fourths and fifths are essentially perfect, and the major minor thirds are very close as well. The main issue with 53ET is that you basically must abandon anything resembling a piano-like keyboard for dynamic intonation to be feasible; a touch screen that mimics a log-scaled string instrument layout works wonderfully in this context.

[HarmonixHealing]

The use of the term, ‘perfect’, marks out the perfect fifth as belonging to the group of perfect musical intervals which are characterised in Just Intonation by simple, small whole number (integer) pitch ratios and a high degree of natural ‘unity’, or consonance, as distinct from dissonance. Their inversions are also perfect.

[endthedrugwarvision]

@JohnFHendry Hahahah. Funny that CERN is a Saturnalian armageddeonist tool though. You think anyone at CERN gives a fuck about anything decent? The inhumanity of such a project in the midst of pure global dysfunction to the point of mass level extinction is almost a trite aphorism of how Satanism has been perfected. No?

[JohnFHendry]

Excellent demonstration at the end of the comma. You can't stop time and our measuring tape keeps getting bigger. A neutron is no bigger than a proton unless you measure it. If only the researchers at CERN would give up a day to look at the comma and understand the asymmetry of the weak force. The weak force is an oscillator and many laymen get it to understand what time is but physics remains blind and still argues over what the comma proves.

Thanks, JFH^^

[iranief]

I like the video and the notes you wrote.

thanks!