The Pythagorean Tetractys, or how multiplying by 2 & 3 gets you all the notes you need, with stops at the Pythagorean Comma and Equal Temperament along the way.
thank you! this also gives some hints about how multiples of 2 and 3 somehow defines most rythmic configurations, even hidden in the milisecond inconsistencies of seemingly even spaced grooves
I knew about the Pythagorean comma from theory books but never grasped how it was derived. Until now. This unpacks it incredibly clearly and does indeed uncover the ultimate simplicity underlying our tonal system. Thanks, Dan!
2. Now, what has the West done with the Pythagorean comma? They spread it evenly... but the Comma is a Microtone. A whole Tone is made of Nine commas in Byzantine music, in Ottoman music and even in Arabic music. In Ottoman music, between C and D there are 9 commas and at 1 comma above C there is a small sharp. At 4 commas above C is the large sharp. Symmetrically, at 1 comma below D there is a little flat and at 4 commas below D is the large flat. So there are 4 accidentals beteen C and D. Hence there are no chords played because of extra beats of frequency but there is an enormous amount of tetrachords and pentachords possible that gave rise to 400plus modes or Makams at the apex of development of Ottoman music still studied, performed and used for composition.
@@DanTepferMusic I went to Istanbul a few years ago and I saw how the vocalists for Classical Ottoman music train...the trainer plays the tanbur [34 frets/ first octave] or the kanun. The vocalist must reproduce each musical phrase with laser mark pitch accuracy. This gives them such precision that the irregularities of the speaking timbre are no longer there. When they pronounce the syllables in pitch one thinks it s an instrument that pronounces words and not a human trying to imitate instrument pitch. Portamento is not used.
Yay! This make total sense. Thank you for sharing your findings on the Pythagorean Tetractys! I'm going to share this with my piano technician. Have you already done thirds?
Thank you Dan. I came across your work from your blog post converting pulse into pitch. I have been looking at this stuff for a while and your computer programming of these harmonic experiments and sharing them are wonderful and greatly appreciated. I'm looking that the harmonic system with to relation to pulse, rhythm and movement. Polyrhythmic expression. Your work helps me out greatly. Well done!
When most of teachers brushed Pythagoras off as mostly just having to do with music, my brain discounted it. It is the music of the universe. Those doubling patterns are powers of 2.
3.The accidentals of Ottoman music do not match at all any of the Western accidentals. A kanun[ lap zither like]has a system of levers under each course of 3 strings tuned in unison for each pitch. There are 9 or 10 levers raising or lowering the tension by a minute amount related to the comma. The octave division can be seen on an Ottoman tanbur. The higher octave is shorter on the neck of the instrument so not all divisiins of the first Octave are reproduced.
This is awesome from so many angles! Amazing video Dan. Out of curiosity, what programming language are you building this app with? To me it looks like OpenFrameworks or Processing if I had to guess!
Excellent! But it seems you are not aware that this order of numbers is also included in the Pythagorean Tetractys. Wikipedia didn't mention this, but you can see how Plato used it in Timaeus (36a) - it was certainly known to the Pythagoreans.
@@DanTepferMusic It's called "Plato's Lambda" and I'm familiar with it mostly through books... what I see online useful for a quick review? One page is under the title "More about the Tetractys" and another one "Plato's Lambda - historical background". I hope it helps.
No, it doesn’t. This is entirely about fractional relationships between frequencies, not about absolute numbers. If our measurement unit for frequency were different, it would simply shift the names of the notes; it wouldn’t change the underlying relationships at all.
@@DanTepferMusic I meant the values of the notes being a certain hz. The same sound would be different hz if people chose a different duration for a second. So A does not need to always be 432hz as a value. It is in my view more appropriate to start with a tone you think should be A because it sounds a certain way and divide backwards.
Well yes, of course. But that’s not really relevant. If whales heard music as we do, but with their range of hearing, all the above would just be shifted downwards by a number of octaves. These ideas even apply to the much much slower orbital ratios of planets in planetary systems. It’s not about the absolute numbers, it’s about their relationship to each other.
@@MatthewSuffidy I think I finally understand your point: you are saying that 432hz is an arbitrary value. I happen to agree with you, but my video here has really nothing to do with the 432Hz topic. It has to do with how to build a scale of frequencies from whatever your starting frequency happens to be. Could be 440Hz, could be 432, could be 0.5Hz - it just doesn't matter.
@@DanTepferMusic Well it also means that a note you call for example 'A' is a certain pitch due to what someone called a second in duration. But yes I think it mostly does not matter.
