# Observations about Pythagoras



## millionrainbows

*WIK:
[Pythagoras of Samos* was an Ionian Greek philosopher, mathematician, and has been credited as the founder of the movement called Pythagoreanism. Most of the information about Pythagoras was written down centuries after he lived, so very little reliable information is known about him. He was born on the island of Samos, and traveled, visiting Egypt and Greece, and maybe India, and in 520 BC returned to Samos.[SUP] [/SUP]Around 530 BC, he moved to Croton, in Magna Graecia, and there established some kind of school or guild.
Pythagoras made influential contributions to philosophy and religion in the late 6th century BC. He is often revered as a great mathematician and scientist and is best known for the Pythagorean theorem which bears his name. However, because legend and obfuscation cloud his work even more than that of the other pre-Socratic philosophers, one can give only a tentative account of his teachings, and some have questioned whether he contributed much to mathematics or natural philosophy. Many of the accomplishments credited to Pythagoras may actually have been accomplishments of his colleagues and successors. Some accounts mention that the philosophy associated with Pythagoras was related to mathematics and that numbers were important. It was said that he was the first man to call himself a philosopher, or lover of wisdom, and Pythagorean ideas exercised a marked influence on Aristotle, and Plato, and through him, all of Western philosophy.]

So, what about Pythagoras and his connection to music? Any thoughts, assertions?


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## clara s

Pythagoras was one of the greatest philosophers and mathematicians of ancient Greece.

The real crown of his theory was "tetraktys", the true source of wisdom,
the first 4 natural numbers, connected with various relations.

But pythagorean theorem made him famous for eternity.

Harmony is based on tetraktys, because from these numbers, (1,2,3,4)
the harmonic ratios of intervals of fourth, fifth and eighth can be constructed.

He discovered the relation between the length of the chords
and the tone height they provide.

to find this, he used an instrument he created by himself, called "monochordo".

"World is numbers"

any doubt about this?


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## KenOC

Pythagoras drew a lot of stares
His clothes were old and full of tears
So people sneered
And pulled his beard
But still, he figured out that thing (yes I know this doesn't scan) about the square root of the sum of two squares
(but at least it rhymes)


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## millionrainbows

Clara s said:


> Pythagoras was one of the greatest philosophers and mathematicians of ancient Greece.
> 
> The real crown of his theory was "tetraktys", the true source of wisdom,
> the first 4 natural numbers, connected with various relations.
> 
> But pythagorean theorem made him famous for eternity.
> 
> Harmony is based on tetraktys, because from these numbers, (1,2,3,4)
> the harmonic ratios of intervals of fourth, fifth and eighth can be constructed.
> 
> He discovered the relation between the length of the chords
> and the tone height they provide.
> 
> to find this, he used an instrument he created by himself, called "monochordo".
> 
> "World is numbers"
> 
> any doubt about this?


I have no doubts, but I mentioned Pythagoras elsewhere and there were some who tried to minimize his influence on Western music.

Firstly, Pythagoras lived so long ago that many of the ideas which he might have come up with are "legend", or they have his name attached to them as the 'school' of Pythagoras. Should we let these factors diminish our enthusiasm, or shed doubt on his influence?


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## Guest

Pythagoras is credited with discovering the musical comma. He discovered the musical notes were ratios related to length and frequency. If one took a lyre string tightened at both ends, plucked open we could arbitrarily call it 1. If we halve the length of the string, it will play exactly one octave higher (i.e. it vibrates exactly twice the rate of 1). So the octave can be expressed as a ratio of 2:1 or 1:2. However, if we shorten 1 by a third of its length, it produces a note that plays a fifth higher. The ratio of a fifth is then 2:3 or 3:2. These ratios are called intervals and Pythagoras calculated them thus:

1. C - Fundamental (1) 
2. C# - Minor 2nd (15/16) 
3. D - Major 2nd (8/9) 
4. D# - Minor 3rd (5/6) 
5. E - Major 3rd (4/5) 
6. F - Perfect 4th (3/4) 
7. F# - Tritone (5/7) 
8. G - Perfect 5th (2/3) 
9. G# - Minor 6th (5/8) 
10. A - Major 6th (3/5) 
11. A# - Minor 7th (5/9) 
12. B - Major 7th (8/15) 
13. C' - Octave (1/2)

Just remember that the ratios can be inverted depending on whether you lengthen or shorten the string.

Now, since there are six whole steps in a scale (e.g. in the space of an octave) and a whole step is 9/8, then if we raise that ratio by the power of 6, it should come out to exactly 2. Does it? No. It works out to 2.073 which is only slightly off but the human ear can perceive it and it sounds wrong to us. What is means is that there is a tiny but noticeable drift between enharmonic equivalents such as A# and Bb or E# and F. This is the Pythagorean or ditonic comma.

Suppose we measure the 5ths in an octave. There's only one 5th in an octave. Two 5ths will pass out of the octave. So, if we start measuring 5ths, we have to find a way to keep the tones within the octave. Once the 5th is out of the octave, its value must be halved to keep it within the octave. Starting at C, for example, the first 5th interval ends at G and we know that the ratio is 3/2 (or 2/3). The next 5th takes us to D and so we would square 3/2 to obtain 9/4 but that passes out of the octave (is greater than 2 and an octave must be exactly 2/1). So we multiply 9/4 by 1/2 to obtain 9/8, which is less than 2 and so is within the octave. Next, we jump up to A which is mathematically obtained by multiplying 9/8 by 3/2 or 27/16 which is within the octave. If we keep going through 5ths until we pass through all 12 semitones (after A, we go to E, B, F#, C#, G#, D#, A#, F and C) we end up with a final value of 262144/531441.

The true octave, however, would yield 262144/524288. Again, the actual length of the string would be somewhat shorter than the true octave string length and so would be sharp. Our differential is the ratio of 531441/524288 which is the ditonic comma (not "diatonic"). That is the Pythagorean comma.

Another comma is called _syntonic_:

If we form a circle of the C major octave where the full octave is 360 degrees exactly, then C=0 and 360, D=320 (360 x 8/9), E=288 (360 x 4/5), F=270 (360 x 3/4), G=240 (360 x 2/3), A=216, B=192, and C'=180. From D to F is a minor 3rd (3 half-steps) with a ratio of 320/270 or 6.4/5.4 even though the true ratio should 6/5 or 324/270. So the actual difference is 324/320 or 81/80. That ratio is called the syntonic comma. From C to G is a perfect 5th of 360/240 or 3/2. However, if we were to measure a perfect 5th in the next octave from D to A or 320/216 ratio, we notice that it is an 80/54 ratio. A true perfect 5th would be 81/54 or 3/2 and so, again, we end up with a discrepancy of 81/80 (oddly, the reciprocal of this number is 0.987654321). The next octave after that would yield the major 6th (F-D) and the perfect 4th (A-D) also off by the syntonic comma. The comma is very noticeable and must be dispersed in some manner.

_Method I _

One way to disperse the comma is through adjusting the major 3rds in the octave. In a 12-tone octave, there are three major 3rds (4 half-steps x 3 = 12 half-steps). A true major 3rd has a 5:4 ratio, that is, if you shorten a string by 4/5 but retain the same tension, the string will play a major 3rd interval higher. If we start at middle C, our three major 3rds would be C-E, E-G#, Ab-C (remember that Ab is the enharmonic equivalent of G#). Since the octave interval must always be an exact 2:1 ratio, the Ab-C major 3rd will be a bit flat. Why?

Since a major 3rd is 4/5, then we would cube that ratio to obtain 64/125 for the full octave. But a full octave is 64/128 (1:2 ratio). Since 64/125 represents a longer string length than 1/2, the major 3rds will be noticeably flatter than in a true octave since a string's length is proportional to the pitch. The discrepancy is the ratio of 125/128 or 0.9765625, which is called a diesis. Each major 3rd interval in the octave must be sharpened slightly by a third of the diesis or about 0.3255208333.

_Method II _

We may also measure the minor 3rds in an octave of which there will be four. Using C major as an example and starting at middle C, our minor 3rds will be C-Eb, Eb-F#, F#-A and A-C'. Since a true minor 3rd would have 5/6 ratio, then the total value of an octave in minor 3rds is 5/6 raised to the power of 4 or 625/1296. The actual value of a full octave is 648/1296 or 1/2. Since the string length of an octave of minor 3rds is somewhat shorter than a true octave resulting in a higher pitch, the minor 3rds will be a bit sharp and must be uniformly flattened. So, the ratio of 648/625 or 1.0368 tells us the total difference in tone and so each minor 3rd interval must be flattened by a quarter of 1.0368 which is 0.2592. While other ratios are called a comma or _diesis_, this 648:625 ratio has never been named for some reason.

_Method III _

In this method, we solve the ditonic comma which also solves the syntonic comma. Pythagoras was said to have solved his comma but we are not told how. We do it today by tempering. The value of the ratio of the ditonic comma 531441/524288 is approximately 1.01364327. So we would flatten our 5ths by 1.01364327/12 or about 0.08447. This is the method for tempering the 12-tone scale.


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## clara s

Victor Redseal said:


> Pythagoras is credited with discovering the musical comma. He discovered the musical notes were ratios related to length and frequency. If one took a lyre string tightened at both ends, plucked open we could arbitrarily call it 1. If we halve the length of the string, it will play exactly one octave higher (i.e. it vibrates exactly twice the rate of 1). So the octave can be expressed as a ratio of 2:1 or 1:2. However, if we shorten 1 by a third of its length, it produces a note that plays a fifth higher. The ratio of a fifth is then 2:3 or 3:2. These ratios are called intervals and Pythagoras calculated them thus:
> 
> 1. C - Fundamental (1)
> 2. C# - Minor 2nd (15/16)
> 3. D - Major 2nd (8/9)
> 4. D# - Minor 3rd (5/6)
> 5. E - Major 3rd (4/5)
> 6. F - Perfect 4th (3/4)
> 7. F# - Tritone (5/7)
> 8. G - Perfect 5th (2/3)
> 9. G# - Minor 6th (5/8)
> 10. A - Major 6th (3/5)
> 11. A# - Minor 7th (5/9)
> 12. B - Major 7th (8/15)
> 13. C' - Octave (1/2)
> 
> Just remember that the ratios can be inverted depending on whether you lengthen or shorten the string.
> 
> Now, since there are six whole steps in a scale (e.g. in the space of an octave) and a whole step is 9/8, then if we raise that ratio by the power of 6, it should come out to exactly 2. Does it? No. It works out to 2.073 which is only slightly off but the human ear can perceive it and it sounds wrong to us. What is means is that there is a tiny but noticeable drift between enharmonic equivalents such as A# and Bb or E# and F. This is the Pythagorean or ditonic comma.
> 
> Suppose we measure the 5ths in an octave. There's only one 5th in an octave. Two 5ths will pass out of the octave. So, if we start measuring 5ths, we have to find a way to keep the tones within the octave. Once the 5th is out of the octave, its value must be halved to keep it within the octave. Starting at C, for example, the first 5th interval ends at G and we know that the ratio is 3/2 (or 2/3). The next 5th takes us to D and so we would square 3/2 to obtain 9/4 but that passes out of the octave (is greater than 2 and an octave must be exactly 2/1). So we multiply 9/4 by 1/2 to obtain 9/8, which is less than 2 and so is within the octave. Next, we jump up to A which is mathematically obtained by multiplying 9/8 by 3/2 or 27/16 which is within the octave. If we keep going through 5ths until we pass through all 12 semitones (after A, we go to E, B, F#, C#, G#, D#, A#, F and C) we end up with a final value of 262144/531441.
> 
> The true octave, however, would yield 262144/524288. Again, the actual length of the string would be somewhat shorter than the true octave string length and so would be sharp. Our differential is the ratio of 531441/524288 which is the ditonic comma (not "diatonic"). That is the Pythagorean comma.
> 
> Another comma is called _syntonic_:
> 
> If we form a circle of the C major octave where the full octave is 360 degrees exactly, then C=0 and 360, D=320 (360 x 8/9), E=288 (360 x 4/5), F=270 (360 x 3/4), G=240 (360 x 2/3), A=216, B=192, and C'=180. From D to F is a minor 3rd (3 half-steps) with a ratio of 320/270 or 6.4/5.4 even though the true ratio should 6/5 or 324/270. So the actual difference is 324/320 or 81/80. That ratio is called the syntonic comma. From C to G is a perfect 5th of 360/240 or 3/2. However, if we were to measure a perfect 5th in the next octave from D to A or 320/216 ratio, we notice that it is an 80/54 ratio. A true perfect 5th would be 81/54 or 3/2 and so, again, we end up with a discrepancy of 81/80 (oddly, the reciprocal of this number is 0.987654321). The next octave after that would yield the major 6th (F-D) and the perfect 4th (A-D) also off by the syntonic comma. The comma is very noticeable and must be dispersed in some manner.
> 
> _Method I _
> 
> One way to disperse the comma is through adjusting the major 3rds in the octave. In a 12-tone octave, there are three major 3rds (4 half-steps x 3 = 12 half-steps). A true major 3rd has a 5:4 ratio, that is, if you shorten a string by 4/5 but retain the same tension, the string will play a major 3rd interval higher. If we start at middle C, our three major 3rds would be C-E, E-G#, Ab-C (remember that Ab is the enharmonic equivalent of G#). Since the octave interval must always be an exact 2:1 ratio, the Ab-C major 3rd will be a bit flat. Why?
> 
> Since a major 3rd is 4/5, then we would cube that ratio to obtain 64/125 for the full octave. But a full octave is 64/128 (1:2 ratio). Since 64/125 represents a longer string length than 1/2, the major 3rds will be noticeably flatter than in a true octave since a string's length is proportional to the pitch. The discrepancy is the ratio of 125/128 or 0.9765625, which is called a diesis. Each major 3rd interval in the octave must be sharpened slightly by a third of the diesis or about 0.3255208333.
> 
> _Method II _
> 
> We may also measure the minor 3rds in an octave of which there will be four. Using C major as an example and starting at middle C, our minor 3rds will be C-Eb, Eb-F#, F#-A and A-C'. Since a true minor 3rd would have 5/6 ratio, then the total value of an octave in minor 3rds is 5/6 raised to the power of 4 or 625/1296. The actual value of a full octave is 648/1296 or 1/2. Since the string length of an octave of minor 3rds is somewhat shorter than a true octave resulting in a higher pitch, the minor 3rds will be a bit sharp and must be uniformly flattened. So, the ratio of 648/625 or 1.0368 tells us the total difference in tone and so each minor 3rd interval must be flattened by a quarter of 1.0368 which is 0.2592. While other ratios are called a comma or _diesis_, this 648:625 ratio has never been named for some reason.
> 
> _Method III _
> 
> In this method, we solve the ditonic comma which also solves the syntonic comma. Pythagoras was said to have solved his comma but we are not told how. We do it today by tempering. The value of the ratio of the ditonic comma 531441/524288 is approximately 1.01364327. So we would flatten our 5ths by 1.01364327/12 or about 0.08447. This is the method for tempering the 12-tone scale.


excellent Victor

you are named a Prominent Pythagorean
in the steps of Philolaus and Eurytus

ps you've got definitely the two of the three requirements in your life,
your music and your philosophy
I do not know about the hot cup of tea


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## clara s

millionrainbows said:


> I have no doubts, but I mentioned Pythagoras elsewhere and there were some who tried to minimize his influence on Western music.
> 
> Firstly, Pythagoras lived so long ago that many of the ideas which he might have come up with are "legend", or they have his name attached to them as the 'school' of Pythagoras. *Should we let these factors diminish our enthusiasm, or shed doubt on his influence?*


definitely not

his whole philosophy had strong characteristics of mysticism
and this brought fame very rapidly.

Also this brought a strong public reaction, 
which ended to the destruction of his school

very complicated life and philosophy


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## clara s

KenOC said:


> Pythagoras drew a lot of stares
> His clothes were old and full of tears
> So people sneered
> And pulled his beard
> But still, he figured out that thing (yes I know this doesn't scan) about the square root of the sum of two squares
> (but at least it rhymes)


one small objection

his clothes were clean and white, as was the uniform of his school,
to match the purity of his philosophy


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## millionrainbows

But when I tried to associate Pythagoras with the 12-note division of the octave we currently use (although ours is equally tempered), and, as mentioned above, the stacking of twelve fifths over seven octaves gets us almost back to our starting note (the remainder being called the ditonic or Pythagoran comma), I was met with a withering firestorm of protest. 

It was pointed out to me that Pythagoras only used a seven note scale, and had no need for twelve. Why, then, is this remainder of the circle of fifths called the "Pythagoran comma" if he did not divide the octave into 12 parts?


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## Guest

I'm not so sure the "scholars" know what they are talking about. How can we be sure the Greeks didn't have 12 TET? Here is something that has always bugged me:

The Greeks had a little thing called isopsephia which is when each letter in the alphabet is given a numerical value. Those words that have the same numerical value are considered identical concepts--be that as it may. Now, we know today that the value of a half-step or semitone is calculated to the 12 root of 2 or 1.0595 (approximately), that is, if you raise 1.0595 to the 12th power, it will equal 2 which is the perfect octave. Strangely, the Greek god Apollo--who is the god of music--is actually called Apollon in Greek and using isopsephia, his name has the value of 1061. One of Apollo's titles is Pythias (from which Pythagoras is derived--"Pythias speaking") which adds up to 1059. That's a little close for a coincidence.

Hermes was the brother of Apollo and made a lyre of a tortoise shell and gave it to him indicating a musical relationship between the brothers. In isopsephia, Hermes is 353. The author of the _Corpus Hermeticum_ was said to be Hermes Trismegistus or "thrice-great Hermes." What does that mean? Well, if we multiply 353 by 3, we get 1059. If we shorten the string by a 3rd, we get 706/1059 or 2/3, the ratio of the perfect 5th. Were the Greeks encoding the secret of 12 TET?


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## Che2007

Victor Redseal said:


> I'm not so sure the "scholars" know what they are talking about. How can we be sure the Greeks didn't have 12 TET? Here is something that has always bugged me:
> 
> The Greeks had a little thing called isopsephia which is when each letter in the alphabet is given a numerical value. Those words that have the same numerical value are considered identical concepts--be that as it may. Now, we know today that the value of a half-step or semitone is calculated to the 12 root of 2 or 1.0595 (approximately), that is, if you raise 1.0595 to the 12th power, it will equal 2 which is the perfect octave. Strangely, the Greek god Apollo--who is the god of music--is actually called Apollon in Greek and using isopsephia, his name has the value of 1061. One of Apollo's titles is Pythias (from which Pythagoras is derived--"Pythias speaking") which adds up to 1059. That's a little close for a coincidence.
> 
> Hermes was the brother of Apollo and made a lyre of a tortoise shell and gave it to him indicating a musical relationship between the brothers. In isopsephia, Hermes is 353. The author of the _Corpus Hermeticum_ was said to be Hermes Trismegistus or "thrice-great Hermes." What does that mean? Well, if we multiply 353 by 3, we get 1059. If we shorten the string by a 3rd, we get 706/1059 or 2/3, the ratio of the perfect 5th. Were the Greeks encoding the secret of 12 TET?


No they were not.


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## Che2007

Victor Redseal said:


> Pythagoras is credited with discovering the musical comma.


By whom?



> These ratios are called intervals and Pythagoras calculated them thus:
> 
> 1. C - Fundamental (1)
> 2. C# - Minor 2nd (15/16)
> 3. D - Major 2nd (8/9)
> 4. D# - Minor 3rd (5/6)
> 5. E - Major 3rd (4/5)
> 6. F - Perfect 4th (3/4)
> 7. F# - Tritone (5/7)
> 8. G - Perfect 5th (2/3)
> 9. G# - Minor 6th (5/8)
> 10. A - Major 6th (3/5)
> 11. A# - Minor 7th (5/9)
> 12. B - Major 7th (8/15)
> 13. C' - Octave (1/2)


Except that all of that would have been totally alien in the time period you are talking about.


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## Che2007

We have no evidence of any scale at all to be attached to Pythagoras, or any other musical procedure for that matter. Also the idea of stacking 3:2s would have been a complete anachronism to the ancient greeks.

Ancient Greek music theory is very interesting however, particularly in the differences between Aristoxenus and Ptolemy. If you are interested in being able to hear the scales that they suggest, I heartily recommend the program ZynAddSubFX which is an easy to use and accurate midi program that can reproduce exact intervals.


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## Che2007

millionrainbows said:


> Why, then, is this remainder of the circle of fifths called the "Pythagoran comma" if he did not divide the octave into 12 parts?


It is called the pythagorean comma because it relates to the pythagorean school not because it was a discovery of Pythagoras'. It also is to do with medieval music theory and canonics. I did list you all those books/articles on the subject.


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## Che2007

If anyone wants to do some reading on this subject I suggest Andrew Barker (The Science of Harmonics in Classical Greece), Thomas Mathiesen (Apollo's Lyre), Richard Crocker (Pythagorean Mathematics and Music) or one of the many entries about early music theory in Cambridge History of Western Music Theory.

Richard Crocker (Pythagorean Mathematics and Music) relates particularly strongly to this discussion.


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## Guest

Che2007

Methinks you be takin this thread a bit too seriously. I can't speak for others here but I personally don't believe there ever was a Pythagoras (or a Jesus or a Buddha or a Krishna or a Moses or a Zoroaster or a Mohammad). I mean, 500 BCE? Get real.


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## Che2007

Victor Redseal said:


> Che2007
> 
> Methinks you be takin this thread a bit too seriously. I can't speak for others here but I personally don't believe there ever was a Pythagoras (or a Jesus or a Buddha or a Krishna or a Moses or a Zoroaster or a Mohammad). I mean, 500 BCE? Get real.


Ok, I see where you are coming from  I read an interesting article claiming that the figure of Pythagoras might be an amalgam of lots of different figures. I thought that was an interesting take on it.


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## millionrainbows

millionrainbows said:


> Why, then, is this remainder of the circle of fifths called the "Pythagoran comma" if he did not divide the octave into 12 parts?


Nobody has fully answered this question; the attribution was explained, but not the musical part of it. Why would anyone be stacking fifths to 12? 
(This is a discussion question, not a reading assignment)


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## millionrainbows

There is an entry in the Harvard Dictionary of Music called "The Pythagoran scale."

No one here has attributed the stacking of 3:2s to Pythagoras; certainly not I.

The Harvard Dictionary of Music mentions Pythagoras under the entry of 'circle of fifths'

Apparently, The Harvard Dictionary of Music is making some assumptions in the interest of clarity. On the other hand, there is no agreement in this thread about even the *existence* of Pythagoras!

Talk about a derailed, bogged-down, useless discussion!

This thread is not about clarifying or presenting information in a discussion format (go read a book), but has been appropriated for the purpose of negation. It's useless to anyone, now.


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## Guest

Che2007 said:


> Ok, I see where you are coming from  I read an interesting article claiming that the figure of Pythagoras might be an amalgam of lots of different figures. I thought that was an interesting take on it.


I've always believed that to be the case--there were probably a number of people who got rolled into one and collectively named Pythagoras. In that sense, there is no difference between attributing something to Pythagoras or to a Pythagorean school. Either way, we're referring to a pool of thought that could have come from any number of persons or group of people.


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## millionrainbows

Victor Redseal said:


> I've always believed that to be the case--there were probably a number of people who got rolled into one and collectively named Pythagoras. In that sense, there is no difference between attributing something to Pythagoras or to a Pythagorean school. Either way, we're referring to a pool of thought that could have come from any number of persons or group of people.


So, it would be nifty if we could discuss any ideas attributed to Pythagoras or "Pythagoreanism" without getting bogged down with these essentially unproductive and irrelevant questions as to whether there was an actual figure named Pythagoras.

Ideas such as: 
The Pythagoran scale
The circle of fifths
The Pythagoran comma

Does anybody have a problem with that?

Moderator input and guidance is welcomed!


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## Guest

Well, I did start a thread on the circle of fifths some time ago but nobody appeared to be interested.

http://www.talkclassical.com/37807-drawing-circle-5ths-without.html


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## Che2007

millionrainbows said:


> Nobody has fully answered this question; the attribution was explained, but not the musical part of it. Why would anyone be stacking fifths to 12?
> (This is a discussion question, not a reading assignment)


As it says in the Crocker article I mentioned:
"When Aristoxenus indicated that the octave could be divided into six equal whole-tones, he was taken to task by an anonymous writer who demonstrated what every pythagorean knew, that the sum of six tones each 8:9 missed being an octave by the rather small quantity 524288:531441... The demonstration was originally carried out, as we just saw, to show that six whole-tones did not exactly divide an octave. The pythagoreans drew from this the conclusion that the octave would have to be divided in some other way."

He goes on to explain:
"The early pythagoreans projected this tone (9/8) inside the fourth; it went twice and a little over, this latter quantity being called the "limma" or remainder. Its actual size was determined in the usual way, by constructing a series of two tones and comparing the result with a fourth."

He clarifies the procedure thusly:
"This division of the octave, obtained not by a "cycle of fifths" but by projecting the tone 8:9 inside the fourth, is perhaps the oldest one used by the pythagoreans and in some ways the most characteristic"

He gives some historical context:
"This division of the octave went out of style only in the Renaissance; indeed, this "pythagorean" division seemed more popular with the Franks than with the Greeks themselves, for already by 400 B.C. other divisions had been proposed, and soon there was a host of rival scales, each offering its own musical or arithmetic advantages. In dividing up the octave, the pythagoreans encountered problems different
from those we have already seen; at the same time they found opportunities to use other operations which-even though perhaps older-were becoming popular around 400 B.C. These operations involved placing a third term-a "mean"-between the two terms of a given ratio."

My opinion is that the pythagorean scale (1:1, 256:243, 32:27, 4:3, 3:2, 128:81, 16:9, 2:1) is probably one of many scales from its period but we don't have any description of others until later. I would think it is likely that chromatic and enharmonic genera were around just as early as the pythagorean scale.

When people like Carl Huffman call the scale a stack of 5ths we should remember that he isn't a musicologist by trade and is in error. It is derived from genera, which is a division of the 4th.

A key thing to remember with the above scale is how it was conceived. It is the central octave in the greater perfect system, which also includes a further tetrachord above (hyperbolian) and a further tetrachord below (hypaton) as well as the proslambanomenos a further tone below that. These pitches were being conceived of very differently from our own. Also, note that the procedure is about division of the octave not the stacking of intervals as pitch classes: it is unclear to me whether Ancient Greek theorists understood octave equivalence as the equivalence between two pitch classes. I would be interested to find out.


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## Che2007

millionrainbows said:


> No one here has attributed the stacking of 3:2s to Pythagoras; certainly not I.


You did in the Harmony vs Counterpoint thread. Unfortunately it got redacted.


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## millionrainbows

Victor Redseal said:


> Well, I did start a thread on the circle of fifths some time ago but nobody appeared to be interested.
> 
> http://www.talkclassical.com/37807-drawing-circle-5ths-without.html


Maybe this was before the theory sub-forum? This would be a good place for it.


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## millionrainbows

WIK: The Pythagorean comma can be also defined as the difference between a Pythagorean apotome and a Pythagorean limma (i.e., between a chromatic and a diatonic semitone, as determined in Pythagorean tuning),* or the difference between twelve just perfect fifths and seven octaves,* or the difference between three Pythagorean ditones and one octave (this is the reason why the Pythagorean comma is also called a _ditonic comma_).

Somehow this concept of stacking fifths has wormed its way into the common lexicon. I didn't do it.


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## Che2007

millionrainbows said:


> WIK: The Pythagorean comma can be also defined as the difference between a Pythagorean apotome and a Pythagorean limma (i.e., between a chromatic and a diatonic semitone, as determined in Pythagorean tuning),* or the difference between twelve just perfect fifths and seven octaves,* or the difference between three Pythagorean ditones and one octave (this is the reason why the Pythagorean comma is also called a _ditonic comma_).
> 
> Somehow this concept of stacking fifths has wormed its way into the common lexicon. I didn't do it.


There is nothing wrong with stacking 5ths and conceiving of notes as pitch classes. It just wasn't the procedure of ancient greeks pythagorean or not.


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## millionrainbows

The equal-tempered scale we use today, with fifths being off by only two cents, seems more related to the Pythagoran scale than any of those later attempts to get good major thirds. This makes Pythagoran ideas seem more relevant to us today, not less. Emphasizing the similarities is much more productive than emphasizing the differences.


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## Che2007

millionrainbows said:


> The equal-tempered scale we use today, with fifths being off by only two cents, seems more related to the Pythagoran scale than any of those later attempts to get good major thirds. This makes Pythagoran ideas seem more relevant to us today, not less. Emphasizing the similarities is much more productive than emphasizing the differences.


hmmm... I think if you did want to make some theoretical connection with the ancient Greeks you would be better talking about Aristoxenus' geometric divisions of the tetrachord.

Just-tuned 5ths in the diatonic scale provide something of a perfect storm of modulatory limitation which seems pretty at odds with modern ideas of tonality...


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## millionrainbows

WIK: The Pythagorean comma can be also defined as the difference between a Pythagorean apotome and a Pythagorean limma (i.e., between a chromatic and a diatonic semitone, as determined in Pythagorean tuning),* or the difference between twelve just perfect fifths and seven octaves,* or the difference between three Pythagorean ditones and one octave (this is the reason why the Pythagorean comma is also called a _ditonic comma_).



> Just-tuned 5ths in the diatonic scale provide something of a perfect storm of modulatory limitation which seems pretty at odds with modern ideas of tonality...


The equal tempered scale has fifths which are only 2 cents sharp from being just. Modern tonality welcomed this with open arms. (the stacking of) These Pythagoran fifths lend themselves perfectly to ET, being only 2 cents off.
Where the discrepancies really arise is with older ideas of tonality, a period where better major thirds were being sought.
I question your use of the term "modern ideas of tonality." Modulation occurs mainly by movements of fifths. The stacking of fifths really has more to do with the preservation of the fifth within a span of octaves, not an attempt to achieve "just" intonation. 
Let's talk about stacking fifths rather than "just" intonation.


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## Che2007

millionrainbows said:


> The equal tempered scale has fifths which are only 2 cents sharp from being just. Modern tonality welcomed this with open arms. (the stacking of) These Pythagoran fifths lend themselves perfectly to ET, being only 2 cents off.
> Where the discrepancies really arise is with older ideas of tonality, a period where better major thirds were being sought.
> I question your use of the term "modern ideas of tonality." Modulation occurs mainly by movements of fifths. The stacking of fifths really has more to do with the preservation of the fifth within a span of octaves, not an attempt to achieve "just" intonation.
> Let's talk about stacking fifths rather than "just" intonation.


You misunderstand me. What I mean is that in a diatonic scale built of a chain of perfect 5ths doesn't display modulatory invariance which is the most important aspect of equal temperament. This is because when you modulate in the pythagorean diatonic the wolf 5th moves in relation to the tonic. As such, each key sounds distinct and some keys are totally unusable.

The fact that just 5ths are only around 2 cents different from an ET 5th doesn't mean that pythagorean tuning and equal temperament are analogous. They sound different and have very different attributes.

Also, modulation happens by all sorts of intervals. For example, I was analyzing Glazunov 5 mvt 1 and the modulations are mainly by chromatic 3rds.


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## millionrainbows

Che2007 said:


> You misunderstand me. What I mean is that in a diatonic scale built of a chain of perfect 5ths doesn't display modulatory invariance which is the most important aspect of equal temperament. This is because when you modulate in the pythagorean diatonic the wolf 5th moves in relation to the tonic. As such, each key sounds distinct and some keys are totally unusable.


I'm not interested in Pythagoran scales, or perfect fifths. I'm interested in stacking fifths because it produces 12 notes, and I trace this 12 division back to Pythagoran ideas and procedures.



Che2007 said:


> The fact that just 5ths are only around 2 cents different from an ET 5th doesn't mean that pythagorean tuning and equal temperament are analogous. They sound different and have very different attributes.


The Pythagoran scale and its fifth are more closely related to ET than these tempered tunings seeking better thirds, because the stacking of fifths produces notes that are only 2 cents off. That's all I'm interested in, seeing that connection. Other writers have mentioned this as well.



Che2007 said:


> Also, modulation happens by all sorts of intervals. For example, I was analyzing Glazunov 5 mvt 1 and the modulations are mainly by chromatic 3rds.


So?


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## Mahlerian

millionrainbows said:


> I'm not interested in Pythagoran scales, or perfect fifths. I'm interested in stacking fifths because it produces 12 notes, and I trace this 12 division back to Pythagoran ideas and procedures.


This is manifestly untrue. Only stacking equal-tempered fifths produces 12 notes. Non-tempered fifths will produce notes for C-flat and E-sharp that are separate from B natural and F natural.


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## millionrainbows

millionrainbows said:


> I'm not interested in Pythagoran scales, or perfect fifths. I'm interested in stacking fifths because it produces 12 notes, and I trace this 12 division back to Pythagoran ideas and procedures.


Note that I didn't say "stacking 'just' fifths."



Mahlerrian said:


> This is *manifestly untrue. (Wow!-Ed.)* Only stacking equal-tempered fifths produces 12 notes. Non-tempered fifths will produce notes for C-flat and E-sharp that are separate from B natural and F natural.


What is "manifestly untrue?" Stacking fifths? This is commonly known as the circle of fifths. The circle of fifths is the result of "stacking," or projecting, the interval of a fifth. This eventually divides the octave into 12 notes. I'm not interested in 'just' fifths, or 'just' intonation.

If I trace it back to Pythagoras, that's my business, but other writers have seen this connection, as well. A fifth is a fifth. I'm not interested in these discrepancies resulting from the inability to close the octave with 'just' 2:3 fifths. Everybody knows that is mathematically impossible.

Besides, you're not interested in making this connection, or you would have offered an alternative. You're stuck in negation, not the affirmation of ideas. I 'm interested in good, healthy discussion, not toxic interchanges.


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## Che2007

millionrainbows said:


> I'm not interested in Pythagoran scales, or perfect fifths. I'm interested in stacking fifths because it produces 12 notes, and I trace this 12 division back to Pythagoran ideas and procedures.


*I don't know how many times I need to repeat this before you get it*: division of the octave by 12 is not a pythagorean idea or procedure. The chromatic gamut wasn't fulfilled until the 17th century and really as a settled backdrop until the 18th century. Saying that is rooted in earlier history is trivial, nonspecific and, in the terms you are using, misleading.


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## Che2007

millionrainbows said:


> The Pythagoran scale and its fifth are more closely related to ET than these tempered tunings seeking better thirds, because the stacking of fifths produces notes that are only 2 cents off.


No, many intervals are quite discrepant. For example, an Equal Temperament semitone is 100c and the pythagorean has two semitones: C-Db=113.7c or E-F=90.22c. That is quite a difference! How about the interval G#-Eb (the wolf 5th in pythagorean tuning)? That would be 678.5c in pythagorean compared to the 700c of ET.

You should know, ET is actually a meantone tuning. Meantone tunings spread the diatonic comma out across the scale in different ways, ET does it in a completely flat profile. To my mind, that is pretty different from pythagorean tuning.


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## Che2007

millionrainbows said:


> I'm not interested in 'just' fifths, or 'just' intonation... Besides, you're not interested in making this connection, or you would have offered an alternative. You're stuck in negation, not the affirmation of ideas. I 'm interested in good, healthy discussion, not toxic interchanges.


Since this thread is ostensibly about Pythagoras (although it is clearly a not so guarded attack on yours truly) why have you veered off into talking about ET anyway? Are you not really interested in the Pythagorean tradition?

Back to topic: what do you guys think about Ptolemy's debunking of Aristoxenus' mathematical irregularities? He shows that Aristoxenus' proofing method of his postulate that a 4th amounts to 2 and a half tones is incorrect mathematically. This draws out a pretty stark distinction between the two theorists: On the one hand, Aristoxenus is concerned with concordance as a rough estimate of an interval, his method seeming to require a level of compromise to succeed. On the other hand, Ptolemy is concerned that the mathematics to underpin such a claim be accurate.

To me, both theorists have something to add. Aristoxenus' intuitive modelling of relationships has a very 'real-world' feel to it, while Ptolemy has a particular clarity and elegance. I think this debate is much the same as those still alive in theory today, between pragmatism and formalism. Thoughts?


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## millionrainbows

"The Pythagoran sysyem is based upon the octave and the fifth, the first two intervals of the harmonic series. Using the ratios of 2:1 for the octave and 3:2 for the fifth, it is possible to tune all the notes of the diatonic scale in a succession of fifths and octaves, or, for that matter, all the notes of the chromatic scale." --_J. Murray Barbour: Tuning and Temperasment: A Historical Survey_

"In a sense, the rise of equal temperament can be seen as a partial resurgence of the old Pythagoran doctrine, since the Pythagoran tuning also produced good fifths (and fourths), wide major thirds, and narrow minor thirds."--_David B. Doty: The Just Intonation Primer_


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## millionrainbows

Che2007 said:


> *I don't know how many times I need to repeat this before you get it*: division of the octave by 12 is not a pythagorean idea or procedure. The chromatic gamut wasn't fulfilled until the 17th century and really as a settled backdrop until the 18th century.


The concept is traceable, and I'm not alone in seeing this connection. Again, I am not interested in minute discrepancies or overly-literal historical quibbles.

~"The Pythagoran sysyem is based upon the octave and the fifth, the first two intervals of the harmonic series. Using the ratios of 2:1 for the octave and 3:2 for the fifth, it is possible to tune all the notes of the diatonic scale in a succession of fifths and octaves, or, for that matter, all the notes of the chromatic scale." --_J. Murray Barbour: Tuning and Temperasment: A Historical Survey_

"In a sense, the rise of equal temperament can be seen as a partial resurgence of the old Pythagoran doctrine, since the Pythagoran tuning also produced good fifths (and fourths), wide major thirds, and narrow minor thirds."--_David B. Doty: The Just Intonation Primer_



Che2007 said:


> Saying that is rooted in earlier history is trivial, nonspecific and, in the terms you are using, misleading.


You will have to quote me exactly.


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## millionrainbows

Che2007 said:


> No, many intervals are quite discrepant. For example, an Equal Temperament semitone is 100c and the pythagorean has two semitones: C-Db=113.7c or E-F=90.22c. That is quite a difference! How about the interval G#-Eb (the wolf 5th in pythagorean tuning)? That would be 678.5c in pythagorean compared to the 700c of ET.


That's all irrelevant to my point.



che2007 said:


> You should know, ET is actually a meantone tuning. Meantone tunings spread the diatonic comma out across the scale in different ways, ET does it in a completely flat profile. To my mind, that is pretty different from pythagorean tuning.


No, I do not see it that way. Meantone tunings were devised to get better maj thirds.

"In a sense, the rise of equal temperament can be seen as a partial resurgence of the old Pythagoran doctrine, since the Pythagoran tuning also produced good fifths (and fourths), wide major thirds, and narrow minor thirds."--_David B. Doty: The Just Intonation Primer_


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## millionrainbows

Che2007 said:


> Since this thread is ostensibly about Pythagoras (although it is clearly a not so guarded attack on yours truly)...


Sorry you're paranoid.


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## Guest

I drew a horsey once.


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## millionrainbows

That's nice, Victor. Be sure to put up your crayolas.

~


Che2007 said:


> Since this thread is ostensibly about Pythagoras (although it is clearly a not so guarded attack on yours truly) why have you veered off into talking about ET anyway? Are you not really interested in the Pythagorean tradition?


ET is our present system, and Pythagoran tuning, most especially its treatment of fifths, is very relevant to the development of ET.



Che2007 said:


> Back to topic: what do you guys think about Ptolemy's debunking of Aristoxenus' mathematical irregularities? He shows that Aristoxenus' proofing method of his postulate that a 4th amounts to 2 and a half tones is incorrect mathematically. This draws out a pretty stark distinction between the two theorists: On the one hand, Aristoxenus is concerned with concordance as a rough estimate of an interval, his method seeming to require a level of compromise to succeed. On the other hand, Ptolemy is concerned that the mathematics to underpin such a claim be accurate.


Ptolemy wanted tunings in which both ear and ratio are in agreement.



Che2007 said:


> To me, both theorists have something to add. Aristoxenus' intuitive modelling of relationships has a very 'real-world' feel to it, while Ptolemy has a particular clarity and elegance. I think this debate is much the same as those still alive in theory today, between pragmatism and formalism. Thoughts?


To me, ratios are as real as the vibratory relationships they model. Sound is physics, not strictly mathematical, so ratios are simply the expression of real phenomena, not just formalisms.

According to my source, _Tuning and Temperament_ by Barbour, which I paraphrase as follows:

Aristoxenus opposed the Pythagorans because he felt that the ear was superior to mathematics in determining intervals and octave divisions, and spoke in terms of "parts" of an octave.

One of Aristoxenus' scales was composed of equal tones and equal halves of tones, so he was hailed by 16th century theorists as the inventor of equal temperament. However, he may have intended this for application to the Pythagoran tuning, for most of the other scales he has expressed in this unusual way correspond closely to the tunings of his contemporaries, so it may be that his protest was not against current practice, but against the rigidity of mathematical theories.(Barbour)

"Temperaments" are adjustments to tunings. "Tunings" are systems, like the Pythagoran and just, in which all intervals may be expressed as the ratio of two integers. Thus for any tuning, it is possible to obtain a monochord in which every string-length is an integer (Barbour). A "temperament" is a modification of a tuning, and needs radical numbers to express the ratios of some or all of its intervals (Barbour). Therefore, in monochords for temperaments, the numbers given for certain (or all) string lengths are only approximations, carried out to a particular degree of accuracy. (Barbour)

I think Bach, as a tuner as well as performer/composer, had plenty of real-world experience with tuning by ear. See Bradley Lehman's Bach tuning, and video demonstrations of it, at his Larips website. In my opinion, I like Bach in this tuning, because it is not "true" ET, and yet sounds good in all 12 major and minor keys. Each key has minute differences between certain intervals, which to me gives each one a unique sound.


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## Che2007

millionrainbows said:


> In my opinion, I like Bach in this tuning, because it is not "true" ET, and yet sounds good in all 12 major and minor keys. Each key has minute differences between certain intervals, which to me gives each one a unique sound.


I agree well-temperaments, like Lehman's suggestion, sound good playing that repertoire. The overlap between well-temperaments and ET is an interesting area for research. If you are interested, here is a website where you can explore these tunings and ideas: http://www.sim.spk-berlin.de/bach:_wtc_973.html


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## drfaustus

> Why, then, is this remainder of the circle of fifths called the "Pythagoran comma" if he did not divide the octave into 12 parts?


If the circle of fifths isn't a circle but a spiral. If we divide the octave into 12 parts, as a result the actual tuning of the electric pianos. Really, They are the most out of tune instruments, because no interval is correct. 1=2, 2/3, 3/4, etc. all are broken.


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## millionrainbows

drfaustus said:


> If the circle of fifths isn't a circle but a spiral. If we divide the octave into 12 parts, as a result the actual tuning of the electric pianos. Really, They are the most out of tune instruments, because no interval is correct. 1=2, 2/3, 3/4, etc. all are broken.


...But fifths are the most accurate. Western music is based on fifths. The fifth is the most prominent harmonic besides the octave. When perfect 2:3 fifths are stacked, an almost perfect circle is created which yields all 12 notes. The small error is spread out over all 12 fifths, so in ET, fifths are only 2 cents flat (a cent is 1/100th of a semitone).

Anybody approaching Pythagoran methods should acknowledge this.

There are more definite, substantive connections between the Pythagoran scale than there are relevant differences. Equal temperament is closely similar to the Pythagoran scale because of the preference given to fifths.

Mean-tone temperaments were in search of better major thirds, for harmonic reasons.

Therefore, a good case can be made that Western tonality is not actually based on harmonic principles, but on fifths which have been adjusted slightly. You can call this method "harmonic" if you wish, but the reality is that it is based on fifths and that this led to the 12-note chromatic scale, and to equal temperament.


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## drfaustus

millionrainbows said:


> ...But fifths are the most accurate. Western music is based on fifths. The fifth is the most prominent harmonic besides the octave. When perfect 2:3 fifths are stacked, an almost perfect circle is created which yields all 12 notes. The small error is spread out over all 12 fifths, so in ET, fifths are only 2 cents flat (a cent is 1/100th of a semitone).


Look out! Because, the thirds keep on out of tuning. An human fine ear can be appreciated 3 cents. In ET only exists two types of tuning: Pythagoric (fifths right) and Just, (fifths and thirds right). The Temperamenting is conveniently out of tune a series of consonances to come up practicable scales.


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## millionrainbows

drfaustus said:


> Look out! Because, the thirds keep on out of tuning. An human fine ear can be appreciated 3 cents. In ET only exists two types of tuning: Pythagoric (fifths right) and Just, (fifths and thirds right). The Temperamenting is conveniently out of tune a series of consonances to come up practicable scales.


In the end, equal temperament favors fifths.

The way you are using the term 'temperament' is inaccurate. In tuning terms, scales are created with ratios. These ratios are then "tempered" or adjusted. Mean-tone temperaments were developed to try to get better thirds, but only worked well within a certain range of keys.

By the way, I took a test, along with another subject who was a punk rocker in a band called "The Big Boys." We were both able to perceive a pitch difference of one cent.


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## Robert Eckert

This is probably not the case but, I recently heard that Pythagorous Tuned with [email protected] 432. This, I was told was more in tune with 
the inner harmony of people's physical being...whatever that might mean. Anyone know?


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## KenOC

Robert Eckert said:


> This is probably not the case but, I recently heard that Pythagorous Tuned with [email protected] 432.


One wonders what ancient scroll yielded this information...


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## millionrainbows

Robert Eckert said:


> This is probably not the case but, I recently heard that Pythagorous Tuned with [email protected] 432. This, I was told was more in tune with
> the inner harmony of people's physical being...whatever that might mean. Anyone know?


If anything, you are speaking of a ratio, not a quantity. If A=433 resonates with anything, it is because of a relationship, not a fixed quantity such as 432 cycles per second. Besides, humanity could not measure absolute frequencies until the early 20th century. Up until then, they used tuning forks and counted "beats."

The human brain operates at 8 to 13 Hz (formerly called cycles per second). You do the math.


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