# Harmonic series and God



## winklepicker

Pythagoras believed that the universe had its own music, the music of the spheres. But sadly, when he applied maths to making notes, he found that things didn't work out so neatly. This is called the Pythagorean comma. 

Wikipedia:

"In music, when ascending from an initial (low) pitch by a cycle of justly tuned perfect fifths (ratio 3:2), leapfrogging twelve times, one eventually reaches a pitch approximately seven whole octaves above the starting pitch. If this pitch is then lowered precisely seven octaves, it will be discovered that the resulting pitch is 23.46 cents (a very small amount) higher than the initial pitch. "

If you were setting out to design a universe, wouldn't you make the physics work out a bit better than this? So is the Pythagorean comma evidence for the non-existence of God?

Full Wikipedia articles here, here and here.


----------



## .   1

I prefer the Einsteinium theorum;
The absence of miracles proves the existence of God because the world is so perfect that it follows comprehensible laws with no requirement for Divine Intervention.

.,,


----------



## Brioche

Which God is it who does not exist according to the Pythagorean comma?

Is it Vishnu, Brahma, Shiva, Zeus, Jupiter, Elohim, YHWH, or Allah or ?


----------



## winklepicker

Erm... whichever one designed the universe.


----------



## cuchuflete

Sounds like a trombone player invented the perfect fifth.  Proof of the existence of a god will
come when there is a perfectly tuned trombone player, if not sooner, or later.


----------



## Kajjo

winklepicker said:


> Pythagoras believed that the universe had its own music, the music of the spheres. But sadly, when he applied maths to making notes, he found that things didn't work out so neatly.


First at all, the "music of the spheres" is an esoteric and alchimistic approach without  any fundament in science or nature. It was just a wrong, superstitious idea of Pythagoras and has been re-awakened by medieval alchimists along with concepts of quintessentia and the philosopher's stone. Personally, I do not see any point to discuss these ideas with regards to God, mathematics or music.



> "In music, when ascending from an initial (low) pitch by a cycle of justly tuned perfect fifths (ratio 3:2), leapfrogging twelve times, one eventually reaches a pitch approximately seven whole octaves above the starting pitch. If this pitch is then lowered precisely seven octaves, it will be discovered that the resulting pitch is 23.46 cents (a very small amount) higher than the initial pitch."


The whole concept of this explanation is misleading. The truth is that (3/2)^12 is not equal 2^7, but fairly similar. This is neither a miracle nor to be expected mathematically. Further, the exponents 12 and 7 are arbitrarily choosen. Pythagoras' observation is valid, but meaningless and has nothing at all to do with an imperfect universe.

Since both the ratios 2/1 and 3/2 are very important in music, tuning of instruments in a way that both potencies match up, would have a lot of practical advantages -- so many in fact that an equal-tempered instrument with tiny deviations is preferred above perfectly tuned fifths and octaves. Musicians are pragmatic here!



> If you were setting out to design a universe, wouldn't you make the physics work out a bit better than this? So is the Pythagorean comma evidence for the non-existence of God?


Again, this has nothing to do with physics, but only with straight-forward mathematics. Mathematics itself are flawless and I believe there is only exactly one system of mathematics that does and could work. This, however, would be another topic.

You can find an unlimited number of fractions whose arbitrarily choosen potencies are approximately similar, but not exactly identical. Just choose mutual exclusive prime numbers and natural numbers as potencies. The importance of 3/2 and 2/1 in music makes the Pythagorean example somewhat outstanding, but mathematically this example is just one of an infinity number of possible examples!

Kajjo


----------



## maxiogee

What has the way our ears perceive noise got to do with the design of the universe. Will all music sound the same, and be in the same proportions on all possible planets?


----------



## danielfranco

It could also be argued, rather wanly, that the proof of God's existence is the fact that He also created hairy morose creatures with the intellectual capability of understanding the reality of such a maddeningly precise mathematical musical scale in their brain even though their bodies can never perceive it.


----------



## Qcumber

Why is it important to use mathematics or any other intellectual tool to prove or disprove God's existence? Whether God exists or not, and whatever its name, what does it change for a human being?


----------



## Kajjo

Qcumber said:


> Whether God exists or not, and whatever its name, what does it change for a human being?


This is getting highly off-topic, I am afraid: For example, if people knew there is no God, they could decide for themselves and would not have to surrender their personal liberty to the rules of religions. Many people behave in certain ways and refrain from enjoyable actions just because they believe God forbids it. They would be liberated to a high degree.

Kajjo


----------



## Joca

winklepicker said:


> Pythagoras believed that the universe had its own music, the music of the spheres. But sadly, when he applied maths to making notes, he found that things didn't work out so neatly. This is called the Pythagorean comma.
> 
> Wikipedia:
> 
> "In music, when ascending from an initial (low) pitch by a cycle of justly tunedperfect fifths (ratio 3:2), leapfrogging twelve times, one eventually reaches a pitch approximately seven whole octaves above the starting pitch. If this pitch is then lowered precisely seven octaves, it will be discovered that the resulting pitch is 23.46 cents (a very small amount) higher than the initial pitch. "
> 
> If you were setting out to design a universe, wouldn't you make the physics work out a bit better than this? So is the Pythagorean comma evidence for the non-existence of God?
> 
> Full Wikipedia articles here, here and here.


 
I am no expert on Physics or Mathematics, but I am not sure whether you shouldn't be talking about Mathematics here instead of Physics. Anyway, that the Physics isn't working perfectly here doesn't prove anything against or in favour of God, in my opinion. Rather, I would say that it is perhaps a sign that the universe was designed to come to (or must have) an end.


----------



## LouisaB

winklepicker said:


> Pythagoras believed that the universe had its own music, the music of the spheres. But sadly, when he applied maths to making notes, he found that things didn't work out so neatly. This is called the Pythagorean comma.....
> 
> If you were setting out to design a universe, wouldn't you make the physics work out a bit better than this? So is the Pythagorean comma evidence for the non-existence of God?


 
What kind of person would argue 'God cannot exist because he does not fit into a theorem I have created for him?'

What kind of person believes human knowledge ends with what he himself knows?

A humble and intelligent person faced with a problem in the universe says only 'I do not understand this yet'. An arrogant person cannot accept the possibility of being in any way fallible, and says instead 'This does not make sense'.

Eventually, it may prove _not_ to make sense. It may indeed prove to be a design flaw (as defined by a race who think they're really clever to be able to make cars, and haven't yet noticed they can't create a flower or a human life). Not being either superhuman or God, I don't know whether it will or not. But I can't definitely say that it won't. For me to argue the existence or non-existence of _anything_ on the basis of one man's failed theory would surely be the act of a fool.

Louisa


----------



## Josh_

maxiogee said:


> What has the way our ears perceive noise got to do with the design of the universe. Will all music sound the same, and be in the same proportions on all possible planets?


Yes, interesting questions.  I know you meant them rhetorically, but I will venture a response for those who may be curious.  Sound waves need a medium with which to convey them.  Here on Earth we have our atmosphere composed of various gases and whatnot.  In outer space, or in a vacuum, you would not be able to hear anything as there is no medium to carry sound waves.  On other planets we probably would hear things differently as the consistency and composition of the atmosphere would most likely be different than Earth.

Maybe we could use a vague comparison to get an idea of these differences.  Most people know that helium, when inhaled, causes the voice to go really high.  This is so because it is a light gas -- lighter than air, anyway.  Well, argon, if inhaled*, causes the voice to go really low because it is a heavy gas.  So imagine an atmosphere whose composition is lighter than Earth's atmosphere.  We would hear things in higher frequencies than we do on Earth.  Everybody would sound squeaky, relative to Earth.  Likewise, in an atmosphere heavier than Earth's everyone would have a deep voice(again, relative to Earth).

*Please don't try this at home!


----------



## xrayspex

The question for me is why there is a 12-tone system in the first place.   Not all cultures use it, but most (?) western ones do.  It sounds OK, sometimes, but it definitely has problems, as anybody who has every tried to intonate a guitar knows.


----------



## ireney

Don't get me wrong, Pythagoras was a great mind and a pioneer in mathematics and all and we still learn the Pythagorean theorem in Greece in ancient Greece (which is thankfuly in this case extremely close to modern Greek and we make no effort to reproduce the "right" pronunciation)  but then  Thales was also great but had some obviously wrong ideas 

Mind you he didn't go as far as Pythagoras who created a whole mystical, secret cult of his own in his effort to shoehorn universe in his own theorems I'll give him that.

In other words, Pythagoras was a great mind that went a wee bit too far into the astral planes. It is very interesting if you ask me that he noted the connection/relation of music to mathematics and this is one more reason for simple me to applaud his genious but his "religious" wa/onderings were a bit off. He committed the deadly sin of science: He tried to fit the world into his preconceived model you see.


----------



## Qcumber

Kajjo said:


> For example, if people knew there is no God, they could decide for themselves and would not have to surrender their personal liberty to the rules of religions. Many people behave in certain ways and refrain from enjoyable actions just because they believe God forbids it. They would be liberated to a high degree.


Of course, you are kidding. Look at all the crimes committed by some so-called believers.


----------



## Fernando

What is really strange is that Maths FITS with the real world. Most of the times you can apply a mathematical model to the world. And it works!

This morning I went to work, driving the 20 km distance at 60 km/h. It lasted 20 minutes!!!!!

As Pythagoras could have said to you, the World does not exactly fits the Maths because Maths (God?) are perfect, World is not. Paradise is not in this world.

Now for serious Pythagoras mystical thoughts are (basically) crap.

1+2+3+4 = 10
C2+c2 = H2


----------



## Fernando

Qcumber said:


> Of course, you are kidding. Look at all the crimes committed by some so-called believers.



Just imagine they would not believe in any upper ethical code. 

Wait!, better no imagination needed. Recall instead Hitler and Stalin.


----------



## Qcumber

Fernando said:


> What is really strange is that Maths FITS with the real world. Most of the times you can apply a mathematical model to the world. And it works! This morning I went to work, driving the 20 km distance at 60 km/h. It lasted 20 minutes!!!!!


This is not the real world, but the scientific images numbers give of the world. It is fairly easy to apply maths to physics applied in their turn to technology because it makes it possible for man to create an artificial world ... quite different from the real world.
Maths cannot prove the existence of God, but the existence of a mathematical idol some mathematicians will call "God".


----------



## Qcumber

Fernando said:


> Just imagine they would not believe in any upper ethical code.


The majority of people do not seem to believe in anything that could be called "upper ethical code". They just fear the law, and if they can get away scotfree after they have committed a crime, they just think they are very smart.


----------



## alexacohen

winklepicker said:


> If you were setting out to design a universe, wouldn't you make the physics work out a bit better than this? So is the Pythagorean comma evidence for the non-existence of God?


I don't know anything about mathematical reasoning, but what has Maths to do with beliefs?
Alexa


----------



## Macunaíma

Kajjo said:


> This is getting highly off-topic, I am afraid: For example, if people knew there is no God, they could decide for themselves and would not have to surrender their personal liberty to the rules of religions. Many people behave in certain ways and refrain from enjoyable actions just because they believe God forbids it. They would be liberated to a high degree.
> 
> Kajjo


 
It's not all that off-topic, and I must say I absolutely agree with Qcumber.

If people knew there is a God... If people knew there is no God... But as they will never know, what difference does it make in the end? 

And besides, there are those who don't believe in a personal God, but do believe in a Creator or something, and who are not restricted by religious beliefs.

And last but not least, wouldn't it be so nice if people 'loved their neighbours' witnout being told to do so, and not for fear of punishment? I highly doubt religions are a necessity of the human beings.

Uhm... perhaps a little off-topic, alright. But still on the topic of God...
Feel free to delete it if I've strayed too far.


----------



## Lorixnt2

ireney said:


> Don't get me wrong, Pythagoras was a great mind and a pioneer in mathematics and all and we still learn the Pythagorean theorem in Greece in ancient Greece (which is thankfuly in this case extremely close to modern Greek and we make no effort to reproduce the "right" pronunciation)  but then  Thales was also great but had some obviously wrong ideas
> 
> Mind you he didn't go as far as Pythagoras who created a whole mystical, secret cult of his own in his effort to shoehorn universe in his own theorems I'll give him that.
> 
> In other words, Pythagoras was a great mind that went a wee bit too far into the astral planes. It is very interesting if you ask me that he noted the connection/relation of music to mathematics and this is one more reason for simple me to applaud his genious but his "religious" wa/onderings were a bit off. He committed the deadly sin of science: He tried to fit the world into his preconceived model you see.





Ahem ireney what's the modern Greek for "universe"?  I know nothing of Neo-Greek


----------



## Fernando

Qcumber said:


> This is not the real world, but the scientific images numbers give of the world. It is fairly easy to apply maths to physics applied in their turn to technology because it makes it possible for man to create an artificial world ... quite different from the real world.



The scientific image numbers give of the world has taken me to the office. My office is quite real to me.


----------



## alexacohen

. said:


> I prefer the Einsteinium theorum;
> The absence of miracles proves the existence of God because the world is so perfect that it follows comprehensible laws with no requirement for Divine Intervention.
> 
> .,,


Yes... any theory can be used to demonstrate the existence or non existence of God. Believers will use it to demonstrate His existence, non-believers to prove He doesn't exist.
If we are to apply theories to demonstrate His existence, or the existence of life after death, or whatever belief that cannot be proved, I'd rather apply Murphy's Law:
"The light at the end of the tunnel is the headlamp of an oncoming train".
Alexa


----------



## .   1

cuchuflete said:


> Sounds like a trombone player invented the perfect fifth. Proof of the existence of a god will
> come when there is a perfectly tuned trombone player, if not sooner, or later.


How will God tune the trombone player?
Where will He insert the adjustment key or will it just be with a tap with Maxwell's Silver Hammer?

.,,


----------



## Joca

alexacohen said:


> Yes... any theory can be used to demonstrate the existence or non existence of God. Believers will use it to demonstrate His existence, non-believers to prove He doesn't exist.
> If we are to apply theories to demonstrate His existence, or the existence of life after death, or whatever belief that cannot be proved, I'd rather apply Murphy's Law:
> "The light at the end of the tunnel is the headlamp of an oncoming train".
> Alexa


 
Alexa:

I know this is off-topic, but I have to say that I believe in miracles. Every time a thing happens for the very first time, that's a miracle. Like the first time you fell in love.  

JC


----------



## alexacohen

Joca said:


> Alexa:
> 
> I know this is off-topic, but I have to say that I believe in miracles. Every time a thing happens for the very first time, that's a miracle. Like the first time you fell in love.
> 
> JC


Or the first time your baby smiles at you...
Alexa


----------



## Athaulf

Qcumber said:


> Whether God exists or not, and whatever its name, what does it change for a human being?



Not much in practice. Various substitutes that play the same role are easily found.


----------



## .   1

Fernando said:


> Wait!, better no imagination needed. Recall instead Hitler and Stalin.


Hitler and Stalin were both believers.

.,,


----------



## alexacohen

Well, to go back to the first question... trying to prove or disprove the existence of God using a Mathematical theory, whatever it be, is impossible.
Let the Philosopher's Stone belong to where it belongs: to the realm of a dream...
Alexa


----------



## ireney

Lorixnt2 said:


> Ahem ireney what's the modern Greek for "universe"?  I know nothing of Neo-Greek



It's σύμπαν or κόσμος. What is more interesting for this discussion is that both words (symban and cosmos/kosmos) are the same ones used in ancient Greek. Pythagoras used the second one, "cosmos" which means a) universe b) ordered "arrangement" c) adornment. The first one, "symban" means simply "everything" (to be exact, it means "plus everything".


----------



## Lorixnt2

ireney said:


> It's σύμπαν or κόσμος. What is more interesting for this discussion is that both words (symban and cosmos/kosmos) are the same ones used in ancient Greek. Pythagoras used the second one, "cosmos" which means a) universe b) ordered "arrangement" c) adornment. The first one, "symban" means simply "everything" (to be exact, it means "plus everything".



Thank you ireney ,
Of course I beg your pardon. There's a word in Italian that could define very well my English and of course I would'nt know very well how to translate . This word is "accrocchio" . Anyway my problem when I read the first  winklepicker's statement  

_Pythagoras believed that the *universe* had its own music_

was merely linguistic.

I've thought this sentence attributed to an ancient Greek a category of thought that is tipically latin cause the word "universe" stems from _universus_ (or _universum_) that means simply 

_versus unum _ that is to say  concerning the unity or, keeping better the adversative sense, face to the unity.

According to you do σύμπαν or κόσμος mean exactly the same thing?


----------



## ireney

Well, "sympan" means "everything together" whereas, as I explained, "cosmos" means (at least I am willing to bet this is the meaning Pythagoras had in mind) "an arrangement that is orderly" (can't think of a better word that arrangement).

Sympan and universe have so similar meaning as to be considered synonyms really.

Kosmos on the other hand does not refer to a unity really. It refers to something that is orderly, that the world is arranged in a tidy way so to speak 

Since accrochio seems to be pertinent to the question I googled it and found the following results

http://forum.wordreference.com/showthread.php?t=26826

http://www.cs.unibo.it/~fabbria/accrocc.html

Does this help at all?


----------



## Lorixnt2

ireney said:


> Well, "sympan" means "everything together" whereas, as I explained, "cosmos" means (at least I am willing to bet this is the meaning Pythagoras had in mind) "an arrangement that is orderly" (can't think of a better word that arrangement).
> 
> Sympan and universe have so similar meaning as to be considered synonyms really.
> 
> Kosmos on the other hand does not refer to a unity really. It refers to something that is orderly, that the world is arranged in a tidy way so to speak
> 
> Since accrochio seems to be pertinent to the question I googled it and found the following results
> 
> http://forum.wordreference.com/showthread.php?t=26826
> 
> http://www.cs.unibo.it/~fabbria/accrocc.html
> 
> Does this help at all?




Piece of crap and crappy stuff help a lot for accrocchio thank you ireney. I'm glad  to see you have thought to "orderly arrangement" cause I had thought in the same way trying to figure out the opposite forms.
In the case of universe I had thought something as pluriverse or multiverse
but in the case of kosmos I couldn't imagine anything else than 

_chaos

_So what did reallyPythagoras  mean?

Did he want to describe the uncertain statute  of this "universe"  or simply settle an order?


----------



## .   1

ireney said:


> Well, "sympan" means "everything together" whereas, as I explained, "cosmos" means (at least I am willing to bet this is the meaning Pythagoras had in mind) "an arrangement that is orderly" (can't think of a better word that arrangement).
> 
> Sympan and universe have so similar meaning as to be considered synonyms really.
> 
> Kosmos on the other hand does not refer to a unity really. It refers to something that is orderly, that the world is arranged in a tidy way so to speak.


This makes the whole thing sound less like a Pythagorean Comma and more like a Pythagorean Pun.

.,,


----------



## alexacohen

Well, the only way Pythagoras can explain and demonstrate his theory is by returning from the dead. And by returning from the dead he will demonstrate, without a doubt, the existence of God.
Alexa


----------



## Lorixnt2

alexacohen said:


> Well, the only way Pythagoras can explain and demonstrate his theory is by returning from the dead. And by returning from the dead he will demonstrate, without a doubt, the existence of God.
> Alexa




Tell me more Alexa.  Will you? I will read you tomorrow.


----------



## Lorixnt2

. said:


> This makes the whole thing sound less like a Pythagorean Comma and more like a Pythagorean Pun.
> 
> .,,




Alexa is right since we actually don't know what Pythagoras said and a lot of the things we know come from Proklos who lived some 7-800 years after him.
But assuming  hypothetically Pythagoras had used the word kosmos in the sense of "order" and trying to figure out where this pun could be I've 
rephrased the original post just as a passtime

_Pythagoras believed that an arrangement that is orderly had its own music, the music of the spheres. But sadly, when he applied maths to making notes, he found that things didn't work out so neatly. This is called the Pythagorean comma. [...]
If you were setting out to design an arrangement that is orderly, wouldn't you make the physics work out a bit better than this? So is the Pythagorean comma evidence for the non-existence of God?_

Seen this way I'd say the Pythagorean comma demonstrates only the need of different physics but I suspect the pun could be  in the fact that while kosmos as opposed to chaos has an univocal meaning  the meaning of both universe and sympan seems to me to oscillate.


----------



## .   1

What word would Pythagoras have used for 'comma'?
Is there a pun involved there as well?

.,,


----------



## Lorixnt2

. said:


> What word would Pythagoras have used for 'comma'?
> Is there a pun involved there as well?
> 
> .,,



Interestingly for me I had never considered before the quite logical possibility of using  in English the word comma as pause, separation. My dictionary which is rather old gave me only the punctuation mark meaning. Thinking to the possible metaphorical usage of the Italian "virgola" non cambia una virgola or non si è spostato di una virgola it is possible Italian gives to it a more spatial connotation and English a more temporal one. I don't know how Pythagoras
could have said it in Greek. Maybe ireney knows.


----------



## Chaska Ñawi

Moderator note:  Winklepicker's original question was:



> If you were setting out to design a universe, wouldn't you make the physics work out a bit better than this? So is the Pythagorean comma evidence for the non-existence of God?


----------



## Lorixnt2

Chaska Ñawi said:


> Moderator note:  Winklepicker's original question was:




And my original answer was 

1) this question, ascribed to Pythagoras's thought, could be anachronistic from a linguistic point of view

but, if this wasn't the case,

2) you could maybe teach me how to design a winklegarden but, in the case of a universe, how could I get down to work?


----------



## alexacohen

Pythagoras philosophy was intended to explain the world as he knew it. Even so, I doubt that any mathematical theory could be used then, or now, to demonstrate the existence of God. Or his non existence.
Beliefs, dreams, illusions, faiths simply don't belong to the world of maths.
And for designing an entire universe according to a mathematical theory, I will quote Mrs. Murphy's words of wisdom:
If you follow all the rules you'll miss all the fun.
Alexa


----------



## ireney

May I add that, the fact that the world may not fit in a mathematician's  model and -according to him - therefore there is no God implies that God is a mathematician too   If there is an omnipotent God that has indeed created all the laws the govern the universe I cannot see why He cannot make for example a disc world on the back of four elephants themselves on the back of a giant astro-turtle . He can if He wants to and He wouldn't care if this goes against any laws He made.  He can unmade them or make any number of exceptions to the rules. I realise that may not sit well with those of my fellow members who are too much into hard science but there you go 

As to what Pythagoras called it, I have absolutely no idea. I'll have to look up his works on-line (don't have them around) and see what he calls it if you are really interested. Will post it here if it has something to do with the overall discussion, will be glad to PM it to whomever is interested if it's not (or if I cannot see the relevance which has happened before since I am way to far from omnipotent, omniscient, omniwhatever including omnibus)


----------



## .   1

Lorixnt2 said:


> And my original answer was
> 
> 1) this question, ascribed to Pythagoras's thought, could be anachronistic from a linguistic point of view


Yup.  That's original!



Lorixnt2 said:


> but, if this wasn't the case,
> 
> 2) you could maybe teach me how to design a winklegarden but, in the case of a universe, how could I get down to work?


You would wear winkle-pickers  riding a horse wearing blinkers (another work for winker)  eating periwinkles  dumped from a Winnebago  into Lake Winnebago  by the decendents of Arnold von Winkelreid  from Winnipeg. 

.,, 

.,,


----------



## alexacohen

The only thing that can be proved with a mathematical precision is that if you hear bark, bark, then it must follow that there is a Dog.
Alexa


----------



## .   1

alexacohen said:


> The only thing that can be proved with a mathematical precision is that if you hear bark, bark, then it must follow that there is a Dog.


Nope, you're wrong!
Gee that felt cheeky.  This is my first opportunity to give it to you for not thinking about your answer fully.
Crude bosses bark orders to servile employees.
You could be referring to a kid falling off a bike and barking both knees and one elbow.
Seals bark at seal lions that bark at walrusses that bark at each other.
*To bark up the wrong tree *is (Collins dictionary) _informal _to misdirect one's attention, efforts, etcetera; to be mistaken.

I reckon that you have bark in your teeth. 

.,,


----------



## Lorixnt2

. said:


> You would wear winkle-pickers  riding a horse wearing blinkers (another work for winker)  eating periwinkles  dumped from a Winnebago  into Lake Winnebago  by the decendents of Arnold von Winkelreid  from Winnipeg.
> 
> .,,
> 
> .,,




All this stuff to put together in a unity!!  But it's a heavy task my dear triple punctuation  mark . I'm not  _pantokrator_ and I wouldn't like to lose something along the way.
But maybe there's an even more fatiguing engagement when you try to see 
a commitment to a unity as something that you could see with your eyes or even touch. If a commitment became a physical thing what about its shape, its dimensions, its _nature_? To see there a shape I should embrace the "universe" from an "outside" and in this case I wouldn't have the "one" anymore but, at least a "two" or even better a "three" cause I'd need a third thing to keep them separated: me? God? an ideal wall? And what if I wondered _when_ this "universe" began? And don't even try to say :"Change pusher!" now .  It might be a good drug for everyone if after many many years we go on losing our head behind it . In any possible sense: metaphorical or not. Have a good day and take care of your head! Mi raccomando!


----------



## Chaska Ñawi

We now interrupt this broadcast for a brief moderator note.

"Ahem."

Thank you.  We now return to our scheduled programming.


----------



## .   1

Lorixnt2 said:


> All this stuff to put together in a unity!!  But it's a heavy task my dear triple punctuation mark . I'm not _pantokrator_ and I wouldn't like to lose something along the way.


It's just a string of words surrounding 'winkle' in the dictionary.

I think that you just gave me a very clever dissertation on existentialism that I can only just vaguely comprehend but I love the sound of the words as they drift past in the same ether that is the stuff of the universe.

I am sure that I know what you said but when the immovable object that is my brain met the irresistable force of your logic my synapses went into temporary meltdown.

See ya tomorrow mate,

.,,
Sorry cross post 



> "Ahem."


----------



## winklepicker

Thank you all for your wonderful responses. I am now in a position to award minims.

Best sensible answer to a silly question: LouisaB ddddd (five minims)
Wittiest answer: cuchu dddd 
Best Murphy's Law: alexacohen dddd
Best mention of babies: alexacohen dddd
Best answer from someone in denial: kajjo ddd
Best phantasmagoria: ,,. ddd _(can I have a puff after you?)_
Most cerebral answer: Lorixnt2 ddd

And the special award for self-control: Nun-Translator ddddd.

Wisest remark:

_'A humble and intelligent person faced with a problem in the universe says only 'I do not understand this yet'. '_

Thanks guys.


----------



## gaer

winklepicker said:


> If you were setting out to design a universe, wouldn't you make the physics work out a bit better than this? So is the Pythagorean comma evidence for the non-existence of God?


I assume that you realize our modern system of equal-temperament is all about our attempt to force the natural harmonic series into a very artificial 12-note system.

The fact that 2*1.5^12/2^7=2.027287 simply says that we have almost pounded a slightly deformed peg into a round hole. 

By the way, trombone is one of the few instruments that can be tuned very well to any tuning system, unlike valve instruments.

How all this relates to God totally escapes me, by the way. 

Gaer

PS : an error of 23.46 cents is not at all a small error to a musician. If the octaves on a piano were tuned that far off, you would not be able to bear listening to it!


----------



## Fernando

There was a more important "comma" problem involving "irrational" numbers. It is believed that the demostration that sq. root(2) is irrational was kept as a secret, because it was someway "dangerous".



. said:


> Hitler and Stalin were both believers.
> 
> .,,



Simply, no.


----------



## .   1

Fernando said:


> Simply, no.


Absolutely yes.
I will actually contribute rather than just saying , 'You're wrong.'  I gave that tactic up when I reached double digits in age.

Read the context.
Both Hitler and Stalin believed in an upper moral code and that they were the final word on what was moral.
Hitler was the Grand High Priest of Natzism and Stalin was the Grand High Priest of some ism or other.

Thanks for your reasoned response.

.,,


----------



## Fernando

. said:


> Thanks for your reasoned response..,,



Not at all.


----------



## Kelly B

winklepicker said:


> [...]
> So is the Pythagorean comma evidence for the non-existence of God?
> 
> Full Wikipedia articles here, here and here.


Does the Easter bunny visit your house? (please don't actually answer that. I'm trying to evoke a metaphor....)

I think God is a proud parent with a sense of humor, and he is purely delighted when we learn enough to notice the little eggs he has hidden here and there for us to find.


----------



## .   1

Fernando said:


> Not at all.


I fully agree.


----------



## .   1

Kelly B said:


> Does the Easter bunny visit your house? (please don't actually answer that. I'm trying to evoke a metaphor....)
> 
> I think God is a proud parent with a sense of humor, and he is purely delighted when we learn enough to notice the little eggs he has hidden here and there for us to find.


 
Little Eggs
From this scattered jumble
of past places ideas wander
to a history of our nature
an alternate look at live​ 
We all seek a different piper
see a seperate stately message
as we wend our puzzled passage
through this changling maze called time​ 
As we think and taste and touch it
take a nervous nibble quickly
we must follow what we did before
to get to where we go​ 
Every portion is a gamble
to find marvel all should ramble
with some luck and balanced planning
live to taste the lotus bloom​ 
As we seek evasive answers
look about and try to open
all the tiny perfect puzzles
of this planet we infest​ 
Please consider logic's lesson
to step light upon existence
careless leaching users vanish
when they kill their only host​ 
.,,​


----------



## TimLA

WP, I congratulate you on an excellent question (as usual)!!
But let me throw some sand into your transmission, just for fun D).

Later in the text from Wiki they say:


> Put more succinctly, twelve perfect fifths are not exactly equal to seven perfect octaves, and the Pythagorean comma is the amount of the discrepancy.


Which speaks to the fundamental issue - the relationship between fifths and octaves - both of which are not mathematical concepts, but rather derived from human biology. They can be *described* mathematically, but they were derived from humans 10,000 years ago (just a guess) who thought..."Hmmm, if I pluck this string over here (A), and that string over there (B), they sound similar, but B is "higher" than A. Somewhere around 300 AD someone said...hmmm I think I'll call that an "octave". Later they became more technical and developed "black keys" "white keys" )) "fifths", and all the other desciptors of how our ear and brain process these funny movements of air that hit our eardrum.

From the perspective of the physics of the universe there is nothing fundamental about a "fifth" or an "octave" - these are human-derived terms to describe a complex system of eardrum, malleous, incus, stapes, labrynth, endolymph, the eight cranial nerve, and probably the most important part (in my opinion) the auditory cortex. It is the sum of these individual parts that defines what we hear as a "fifth" or an "octave". And an argument can be made that even within humans, it is possible to interpret sound differently - Eastern music and their tetracords, Rast scales, etc - can sound foreign to a Mozart-trained ear.

The differences between individual notes are "the twelfth root of 2". 
I would wonder if on Vulcan, Spock might have a different set of organs that sense the movements of air,
and if his cortex might interpret "notes" based on "the twelfth root of 3"))...

So I wonder if we should use the foibles of human biology to prove or disprove the existence of a higher being...

Superb question, and I thank you.


----------



## Kajjo

TimLA said:


> Which speaks to the fundamental issue - the relationship between fifths and octaves - both of which are not mathematical concepts, but rather derived from human biology.


That is not exactly correct. We have to clearly distinguish _physical/mathematical properties_ from _biological/sensoric issues_. 

It is a _physical phenomenon_ that overtones, e.g. of vibrating strings, are natural factors (1, 2, 3, ...) of the fundamental frequency, these _harmonics_ are a physical reality. This is entirely independent from the human ear and applies to the whole universe, not restricted to musical applications. Typical muscial instruments produce not only the designated fundamental, but also the harmonics in varying proportions, depending on the specific instrument -- these observations are again independent from the human ear. The human ear is used to receive no clear, singled-out frequencies, but sets of harmonics. The fact that octaves (1:2 ratios) are considered perfect consonances to the human ears is very solidly based on physics and reality, not on definition or culture. Viewing the resulting superimpositions of different frequencies, it is easy to comprehend why simple frequency ratios like 2:3 and 3:4 sound more consonant than odd ratios.

To return to the title question, the introduction of exactly 12 equal notes in one octave is a purely cultural decision. Back again to mathematics one last time: Having accepted the physical harmonic reality of an octave having a ratio of 1:2, and (b) having arbitrarily decided for 12 steps, and (c) having decided that adjacent notes of equal steps shall have the same frequency ratio, we have only one possibility: the ratio of two adjacent note is 2^(1/12). The same would have been possible with _any _number _n _of notes in one octave, just using the frequency ratio 2^(1/_n_).

The musical problem described in the title question occurs only, because human musician would like to be able to play the 3:2 ratio with one of the 12 steps mentioned in the previous paragraph. However, this is mathematically impossible. If _k_ were the correct step (out of our 12 steps), the equation 2^(k/n) = 3:2 would need to have a natural solution, but _k = (ln 3  - ln 2) / ln 2_ is an irrational number, thus cannot be expressed as a fraction, no matter how many steps you would use! In summary, it has nothing to do with our cultural decision of 12 notes in once octave, it's just a mathematical phenomenon that 1:2 and 3:2 cannot be built with a number of constant ratios!



> Somewhere around 300 AD someone said...hmmm I think I'll call that an "octave". Later they became more technical and developed "black keys" "white keys", "fifths", and all the other descriptors of how our ear and brain process these funny movements of air that hit our eardrum.


Well, have a look on the superimpositions and naturally occuring overtones, and you will learn that probably the contrary effect took place: The brain and ear adapted to what was to being heard, to what sounds reached the ear, rather than humans arbitrarily describing random ratios!



> From the perspective of the physics of the universe there is nothing fundamental about a "fifth" or an "octave"


To the contrary, it is! See above and compare all frequencies resulting from big bang to today, they usually exhibit overtones. 



> the most important part (in my opinion) the auditory cortex. It is the sum of these individual parts that defines what we hear as a "fifth" or an "octave". And an argument can be made that even within humans, it is possible to interpret sound differently - Eastern music and their tetracords, Rast scales, etc - can sound foreign to a Mozart-trained ear.


Well, it is true that complex musical arrangements and systems of notes are cultural and arbitrary. However, this does not apply to octaves and so on, but only to the system of "valid notes" actually used by a society or individual musician.



> The differences between individual notes are "the twelfth root of 2". I would wonder if on Vulcan, Spock might have a different set of organs that sense the movements of air, and if his cortex might interpret "notes" based on "the twelfth root of 3"


No, it is highly unlikely, boundering on the impossible -- given the same natural laws apply on Vulcan.

Kajjo


----------



## winklepicker

Take a taut string. Make it vibrate. It wobbles along its whole length. That wobbling gives you a note. We call this the fundamental.

But it also wobbles along HALF its length. The two halves wobble independently at double the speed the whole length wobbles. That double speed, half-length wobble gives a note double the frequency of the full length - which we call an octave above.

Not content with these two wobbles, our string also wobbles along a quarter of its length. This wobble produces a note related to the fundamental by an interval we call a fifth. In the key of C it would be a G. 

You can go on halving the wobble length, and each halving produces a new note. Generally speaking, the higher you go, the quieter the note, but the relative volumes of these higher vibrations are what makes an oboe sound different from a violin.

The first 12 halvings produce 12 notes that make up a scale. They're not precisely evenly divided, but they're near enough. 

So the 12 note thing is not entirely arbitrary. Bach's great breakthrough was to tune the 12 notes to equal spacings, smoothing out the slight variations to produce a doctored, unnatural scale that the human ear could live with, but which in terms of maths and physics is a cheat.

But when you get to the 13th, it all goes horribly wrong. The 13th note should be a Bb. And it nearly is. But enough NOT nearly that it sets up a throbbing sound when played with a Bb, a sound that indicates that the two notes are not resonating harmonically.

The Chinese also worked out this 12 note thing - it's not just a Pythagorean cock-up.


----------



## gaer

winklepicker said:


> Take a taut string. Make it vibrate. It wobbles along its whole length. That wobbling gives you a note. We call this the fundamental.
> 
> But it also wobbles along HALF its length. The two halves wobble independently at double the speed the whole length wobbles. That double speed, half-length wobble gives a note double the frequency of the full length - which we call an octave above.


So far this is correct.


> Not content with these two wobbles, our string also wobbles along a quarter of its length. This wobble produces a note related to the fundamental by an interval we call a fifth. In the key of C it would be a G.


No. The string vibrates in thirds. If A=440, then 3*440=1320, which is an octave and a 5th higher than the fundamental. To get a fifth, we divide by 2. This is why a fifth is shown as 3/2 * the fundamental.


> The first 12 halvings produce 12 notes that make up a scale. They're not precisely evenly divided, but they're near enough.


Again, this is wrong. If you go on halving, you get octaves. The string vibrates in thirds, fourths, fifths, etc. Vibrating in 5ths produces the next "new note", because 2, 4, 8, 16 are just octaves. We would call this "C sharp".

I don't want to say more, because most people will not be interested, but in every octave twice as many notes appear. This means that valveless horns can play a full scale (diatonic, not chromatic) in the high register. The notes are very out of tune to our ears, but French horn players used to tune these notes by moving their hands in the bells to make the pitch rise or fall, and this is still done when such "period instruments" are used in modern recordings of music from the Baroque and Classical periods.


> So the 12 note thing is not entirely arbitrary. Bach's great breakthrough was to tune the 12 notes to equal spacings, smoothing out the slight variations to produce a doctored, unnatural scale that the human ear could live with, but which in terms of maths and physics is a cheat.


Every scale that we use that does not directly correspond to the natural harmonic series could just as easily be called doctored and unnatural. "Equal temperament" is simply one of many possible ways of tuning, although it is the dominant one in the West.


> But when you get to the 13th, it all goes horribly wrong. The 13th note should be a Bb. And it nearly is. But enough NOT nearly that it sets up a throbbing sound when played with a Bb, a sound that indicates that the two notes are not resonating harmonically.


What note are you starting with? Why note number 13? 

Gaer


----------



## TimLA

What an excellent discussion Kajjo! Thank you.
But now let’s go to the next harmonic! ) )



> That is not exactly correct. We have to clearly distinguish _physical/mathematical properties_ from _biological/sensoric issues_.


Exactly! We must clearly distinguish between the “perfect” physics of fluid movement (air on earth), and the less than perfect human biological aspects of sound, which include how we interpret sound.



> It is a _physical phenomenon_ that overtones, e.g. of vibrating strings, are natural factors (1, 2, 3, ...) of the fundamental frequency, these _harmonics_ are a physical reality. This is entirely independent from the human ear and applies to the whole universe, not restricted to musical applications.


Agreed, but how we define the original notes is “human” and thus less than perfect. Though Newton was born 40 years before Bach, he had no influence on Bach polyphony and fugues – Bach’s “ear” knew all it had to, without understanding frequencies.



> To return to the title question, the introduction of exactly 12 equal notes in one octave is a purely cultural decision (*EXACTLY!*). Back again to mathematics one last time: Having accepted the physical harmonic reality of an octave having a ratio of 1:2, and (b) having *arbitrarily* decided for 12 steps (*EXACTLY!*), and (c) having decided that adjacent notes of equal steps shall have the same frequency ratio, we have only one possibility:


We agree




> The musical problem described in the title question occurs only, because human musician would like to be able to play the 3:2 ratio with one of the 12 steps mentioned in the previous paragraph. However, this is mathematically impossible… In summary, it has nothing to do with our cultural decision of 12 notes in once octave, it's just a mathematical phenomenon that 1:2 and 3:2 cannot be built with a number of constant ratios!


*EXACTLY! *The physics that you describe is perfect, yet the human interpretation of fifths/octaves leads to the Pythagorean conundrum.



> Well, have a look on the superimpositions and naturally occuring overtones, and you will learn that probably the contrary effect took place: The brain and ear adapted to what was to being heard, to what sounds reached the ear, rather than humans arbitrarily describing random ratios!


This would require more thought and discussion on my part. I doubt the ear has “evolved” over the last 1,000 years, but I would have to think about the word “adapted”




> To the contrary, it is! See above and compare all frequencies resulting from big bang to today, they usually exhibit overtones.


The mathematical relationships of fifths thirds and octaves are/were clearly established – far after they were defined by our “ears”. Yet the notes themselves are arbitrary, as you have already stated.





> Well, it is true that complex musical arrangements and systems of notes are cultural and *arbitrary*. However, this does not apply to octaves and so on, but only to the system of "valid notes" actually used by a society or individual musician.


I agree



> No, it is highly unlikely, bounding on the impossible -- given the same natural laws apply on Vulcan.


We must diverge here. I just talked to Spock, and he said that their octaves (1:2 relations just like ours) have 37 notes each. So his “fifths” are definitely different from ours. You should hear what he says about Klingon music! D )
 
Excellent discussion.
We should write a book together!!!


----------



## Kajjo

TimLA said:


> This would require more thought and discussion on my part. I doubt the ear has “evolved” over the last 1,000 years, but I would have to think about the word “adapted”.


Overtones and other harmonics occured much earlier to mammal ears than just 1000 years ago! Wind through slits, all sounds made by vocal chords of animals and so on exhibit the same basic overtones that musical instrument produce. Even mankind probably started actively producing musical sounds several ten thousand years ago, most surely at least 5.000 to 7.000 years ago.



> The mathematical relationships of fifths thirds and octaves are/were clearly established – far after they were defined by our “ears”. Yet the notes themselves are arbitrary, as you have already stated.


Right. Frequencies and their ratios are physical, but notes and our musical system are cultural.


> We must diverge here. I just talked to Spock, and he said that their octaves (1:2 relations just like ours) have 37 notes each. So his “fifths” are definitely different from ours. You should hear what he says about Klingon music!


Wrong idea, I am afraid. For example, if we would divide our octaves in 53 notes, both fifths and fourths could be represented with minimum deviations, both less than the human ear can distinguish anyway -- it is just unpractical to do so. The problem of 3:2 not being compatible mathematically stays the same and a harmonic fifth will be exactly the same for Spock and his 37-tone music as our 12-tone system! Both will deviate somewhat and both will try to match (but fail) the physical correct fifth.

Kajjo


----------



## winklepicker

gaer said:


> No. The string vibrates in thirds. If A=440, then 3*440=1320, which is an octave and a 5th higher than the fundamental. To get a fifth, we divide by 2. This is why a fifth is shown as 3/2 * the fundamental.


Damn. The fellow's right. Again. Serve me right for over-simplifying.


----------



## gaer

winklepicker said:


> Damn. The fellow's right. Again. Serve me right for over-simplifying.


In fact, I've been tempted to write a great deal, but I'm afraid I'll lose people.

Let me try to write a little, and please tell me if it's pointless, because I don't want to try to explain something that interests no one.

It's true that harmonics or overtones are actually part of any natural sound. However, brass instruments illustrate how they work much better, because when we play such instruments, we use them. If, for instance, someone is playing a C trumpet, he can play:

Third space C in the bass clef (very hard to play though, sometimes called a "pedal tone").

Then C, G, C, E, G, *Bb, C, D, E, *F-F#, G *? *? *Bb, C

Note that the number of possible "notes" doubles in each octave.

This is going higher and higher, and the "F-F#" harmonic is actually so close to the middle of what those two notes are in the tempered scale that you would not be able to hear which it is closest two.

The "?" marks merely show that those notes, absolutely playable, are not at all close to the notes in our equalled temperament system. (These are pitches that ARE used to play a perfectly normal sounding scale on valveless horns, using the hand to raise or lower these pitches.)

But C, D, E and G ARE close. The D is a couple cents sharp, the E is about 12 cents flat (measured against equal temperament), and the G is less than 2 cents sharp (a ridiculously small discrepancy). These notes are all very high though, and you will hear them mostly in jazz. French horn works the same way (forget about the key), and French horn players use these notes all the time. When I say that the notes are high or low in cents, it is relative to the tempered scale. In nature, they are by definition right where they are supposed to be.

But what about the F-F# harmonic? It's RIGHT between two notes in a typical blues scale, and the Bb is also right for blues. It is no accident that people who have been labeled "primitive" often use pitches that are suspiciously close to those in the natural harmonic series.

All these harmonics that we can play on brass instruments are present in chimes (when one chime is struck), or in any other sound that is struck, plucked, etc.

There have been complicated replies about various tuning systems, but since you brought "God" into the discussion, perhaps you will realize why at one time only octaves and fifths were considered consonant sounds. Even a third was considered dissonant and inappropriate for the final "sound" at the end of a chant. (I'm oversimplifying, but you get the point.)

I'll continue this if anyone is interested. 

Gaer


----------



## Kajjo

gaer said:


> All these harmonics that we can play on brass instruments are present in chimes (when one chime is struck), or in any other sound that is struck, plucked, etc.


Thanks. This is a clear indication that harmonics are part of the natural world and not just human creations. The human response to sound and harmonics is hard-wired and music is a nice consequence of the ability of the human ear to analyse sounds. Our musical systems tried to organise and simplify, but of course has nothing to do with God. Why should it?

Kajjo


----------



## Forero

I think one reason violins and voices sound more in tune with our emotions than say pianos and trombones is that they, when used by experts, actually _are_ more in tune with the natural, non rounded-off, notes musicians have in their minds, both the melodious/harmonious ones and the complementary colorful ones, without having to round everything off to the available string lengths, tensions, and valve positions.

Speaking of Spock's scale, that was already tried on Earth long before Star Trek.  Helmholtz?  The 37-equal-tone scale produces a much more natural fifth than our 12-equal-tone scale.  (22/37 of an octave is much closer to 2:3 than 7/12 of an octave).  But a piano built that way would have to have either very narrow keys or very wide octaves, or both.

Electronic computers, on the other hand, do not have such limitations when they emulate pianos, trombones, and such, except that the manufacturers of sound cards have chosen to "build in" the familiar limitations of "standard" musical instruments rather than to provide an easy way to go beyond them and produce the kind of music we have in our minds (like refined musicians have been talking about for centuries).

It is ironic to me that building lots of complex instrument sounds into a sound card is a real tour de force, but building in natural musical intervals would be very easy, if only they considered it marketable.

Another issue I have run into while attempting to create music by computer is that, though I can work out an "equivalent" midi file from listening lots of times to a wav file, programs designed to do this automatically invariably produce results fairly "understandable" to the ear but really weird as music.  My son the physics major says what is operating in this case is actually the same Uncertainty Principle we hear talked about by Quantum Theorists.


----------



## winklepicker

Forero said:


> It is ironic to me that building lots of complex instrument sounds into a sound card is a real tour de force, but building in natural musical intervals would be very easy, if only they considered it marketable.


Terry Riley and John Adams (among others) have used just intonation in their works. Adams' description of the problems he encountered in writing for orchestral musicians is illuminating!

Why we have 12 notes in our scale is discussed interestingly here. The problem of just intonation is that you may be stuck in one key unless you retune (trombones etc excluded!).

But it is (to me) a beautiful sound. There are some examples here.


----------



## Forero

Actually, just intonation is only part of what I'm talking about.  Just intonation is for consonances and those beautiful mathematical transitions between overlapping harmonies, including modulation between keys and borrowing from related keys.  But for wonderful "color" (chromatic passages, etc.), the dissonances need to be "tuned" off the scale of consonances for best artistic effect.

If sound cards were designed to allow you to specify any pitches you want at any given time, or from a given time to a given time, a musical artist could use the best hue for the context, without time delay and without getting out a wrench (or other heavy-duty tool).  I can also imagine software anyone could use that would make it easy to learn the art.

I think most people have a music sense tied in with their grammar sense, but unless we practice extensively at voice training or intoning our string passages, we are forever prevented from realizing what we hear in our minds because of not having an "obedient" instrument.

Of course, I think of music as a very human pursuit, but it does seem like converting files from one format to another could be automated.  Unlike "perfect" intonation though, the converting process is complicated indeed, because what a wave form "means" has to be in terms of the human intellect and human experience.

We are designed to understand sounds efficiently and don't really know how we do it.  That makes it difficult to program.

And then there's that Uncertainty Principle.


----------



## Kajjo

Forero said:


> If sound cards were designed to allow you to specify any pitches you want at any given time


I believe this is not a problem of sound cards, but of musical software. Programmers can make sound cards produce any pitch and frequency they desire.

Kajjo


----------



## ireney

Moderator's note: Please keep your very interesting (if somewhat unintelligible to me ) conversation within the topic. Definition of harmonics and such are OK but sound cards and musical software are not.


----------



## Forero

A few disorganized thoughts:

The mathematics of harmonies may be (somewhat) confusing, but it is not impossible.  It is not fair to say that it "doesn't work".  The difficulty is essentially this:

If you take a frequency and add 50% twice, you don't get 100% more than the original frequency but 125% more.  And no power of 2 is ever an odd number (except 2^0 = 1).

Addition has not failed and percentages are not, in relation to music anyway, a tool of the devil.

The familiar (tertian) harmonies of Western music are based on consonant perfect fifths (adding 50%) and consonant thirds (adding 20% or 25%).  Adding a smaller third (20%) and then a larger third (25%), or vice versa, is the same as adding a fifth (50%).  An octave is adding 100%, but you can't get there by thirds or fifths.  Note that this says nothing about God.

When Johannes Kepler discovered the actual relationships between the "spheres" (his 3 laws of planetary motion), he tried to apply facts about the Platonic solids as well as the "music of the spheres" idea.  Interesting forced coincidences with the solids, but the music was ugly!

The shape of the orbits, the changes in speed as the planets revolve, and the relation between the distance from the sun and how often they return whence they started are regular, but their relative frequencies that would determine their "harmony" are as arbitrary as the length of the day compared to the length of the year (think leap years).

I mentioned sound cards to point out that the "necessity" that Bach had to upset the serious musicians of his day no longer need apply.  Nothing is perfect, but it is now possible to create music that used to be impossible.  We can combine the best of Bach's modulations and symmetries with the best intonation on something that sounds like an organ, or whatever.

Comma in Pythagoras is komma, from koptein, to cut - just his way to say a small gap (discrepancy).  The punctuation mark of the same name represents a small gap in a sentence.

Galileo Galilei:  "La matematica e l'alfabeto nel quale Dio ha scrito l'universo."

Music is a language (don't know who said it first) that is both universal and culture dependent.  Conventions change from place to place and from time to time, but the language remains for the most part comprehensible to all.


----------



## gaer

Forero said:


> If you take a frequency and add 50% twice, you don't get 100% more than the original frequency but 125% more. And no power of 2 is ever an odd number (except 2^0 = 1).


I have no idea what you are getting at here. I assume you mean to start with an original pitch, then muliply by (3/2)^2, which gives you 9/4. This would give you a 9th. Or multiplying by 9/8 will give you the second note in a major scale if you wish to stick to harmonics and then use octaves to find pitches in the same scale. What you seem to be doing is taking the "fifth" of a note, then taking a "fifth of the resulting note", perhaps then moving down an octave.

This uses the 9th harmonic to get the second degree of a scale.


> I mentioned sound cards to point out that the "necessity" that Bach had to upset the serious musicians of his day no longer need apply.


Wait a minute. If you are saying that you don't need equal-temperament to play the 48 preludes and fugues, you may have a point, since although each prelude and fugue is in one of the 12 major or minor keys, the modulation may not be exotic enough to cause serious problems.

On the other hand, the moment you deal with the kind of chromaticism that is characteristic of composers such as Chopin, Liszt, Wagner, etc., then anything else but equal-temperament is going to create enormous problems and very unpleasant sounds. We still need equal-temperament. We just don't need it all the time. Not even close to all the time.

Consider this elementary chord movement:

C, E, Ab

You start out with C,E,G. Moving to E, you have B,E,G#. Then moving to Ab, you have C, Eb, Ab. Such chord progressions are very common in music of the 1800s and 1900s, right through to today. _*So the "necessity" that Bach had to deal with is just as present today in music that is highly chromatic in nature.*_


> Nothing is perfect, but it is now possible to create music that used to be impossible. We can combine the best of Bach's modulations and symmetries with the best intonation on something that sounds like an organ, or whatever.


This is true. You can, for instance, program things like the Two-Part Inventions and switch tuning systems at will to experience how this changes the sound of the music. It's unfortunate that the demand for software giving a great deal of freedom in this way is not large. The hardware exists.

Gaer


----------



## gaer

Forero said:


> The 37-equal-tone scale produces a much more natural fifth than our 12-equal-tone scale. (22/37 of an octave is much closer to 2:3 than 7/12 of an octave). But a piano built that way would have to have either very narrow keys or very wide octaves, or both.


Recheck your math. 

A natural fifth=3/2=1.5
A tempered fifth=2^(7/12)=1.4983071, which is less than 2 cents flat.

2^(22/37)=1.51005, almost 12 cents sharp.

In a 53 note system, the 31st note is a slight improvment, less than 1 cent flat, but I don't think you'd enjoy dealing with such a system! 

Gaer


----------



## Forero

gaer said:


> I have no idea what you are getting at here.



I just didn't see why, in the original post, the math was being perceived as somehow perverse or something (though virtually the same math may be used by a usurer to the consternation of the borrower).  I was thinking someone might be confused about the difference between the linear space and the logarithmic space.  Late at night, especially, I am confused too. 



> On the other hand, the moment you deal with the kind of chromaticism that is characteristic of composers such as Chopin, Liszt, Wagner, etc., then anything else but equal-temperament is going to create enormous problems and very unpleasant sounds. We still need equal-temperament. We just don't need it all the time. Not even close to all the time.
> 
> Consider this elementary chord movement:
> 
> C, E, Ab



Can you give an example of a piece that uses this progression?  I am curious where it goes from there.

I don't know what to do with, say, Dorian mode, and I am pretty sure that 12-tone-row stuff may get "very unpleasant", but the Chopin and Liszt pieces I'm familiar with (which is not saying much actually) seem (to me) to be understandable in terms of natural consonances.



> You can, for instance, program things like the Two-Part Inventions and switch tuning systems at will to experience how this changes the sound of the music. It's unfortunate that the demand for software giving a great deal of freedom in this way is not large. The hardware exists.
> 
> Gaer



In my computer's case, it's as much the hardware as the software, but that's off topic.  Someday, I may figure out a way to hear what I am imagining.



gaer said:


> Recheck your math.
> 
> A natural fifth=3/2=1.5
> 
> A tempered fifth=2^(7/12)=1.4983071, which is less than 2 cents flat.
> 
> 2^(22/37)=1.51005, almost 12 cents sharp.
> 
> In a 53 note system, the 31st note is a slight improvment, less than 1 cent flat, but I don't think you'd enjoy dealing with such a system!
> 
> Gaer



Thanks.  I probably shouldn't do math online in the middle of the night.

At this point, no telling what I might enjoy.   I surprise myself nearly every day.


----------



## gaer

Forero said:


> [C, E, Ab]
> 
> Can you give an example of a piece that uses this progression? I am curious where it goes from there.


Certainly. Liszt's "Liebestraum No. 3", the very famous one, starts in the key of Ab, four flats. A page or so later it is in B major, 5 sharps, then it moves to C major. A few measures later it is in E major, four sharps. This is in another universe from tuning systems that stick to the harmonic series, "unadjusted", since the idea is that you can instantly "modulate" to any of the 12 keys (15 if you count enharmonic key signatures). The music demands this kind of "flexibility".

The first movement of Beethoven's "Moonlight Sonata" starts in C# minor, which of course is related to E major, but by the 2nd page of so it is temporarily in the key of C major.

These kinds of modulations are all over the place starting with 18th century Romantic music and so are by no means limited to piano music.

Note that I'm not saying that one tuning system is better than another, merely that non-tempered tuning does not work well at all for music written by people who composed music with tempered tuning in mind.

The important thing, in my mind, is that you start with the harmonic series for at least some notes in almost any system, then "tweak" them in various ways. 

Gaer


----------



## Forero

gaer said:


> Certainly. Liszt's "Liebestraum No. 3", the very famous one, starts in the key of Ab, four flats. A page or so later it is in B major, 5 sharps, then it moves to C major. A few measures later it is in E major, four sharps. This is in another universe from tuning systems that stick to the harmonic series, "unadjusted", since the idea is that you can instantly "modulate" to any of the 12 keys (15 if you count enharmonic key signatures). The music demands this kind of "flexibility".
> 
> Gaer



Thank you, Gaer.

Those are two of my favorites!  My grandmother used to play those and gave me her book.  Another one is Rachmaninoff's Prélude in C# minor.

My whole point is that we need not stick to one harmonic series or tuning system because we can at any time choose any note from any key, but we try to keep things reasonable.  Expert violin players know how to "entone" to the natural harmonies where appropriate as well as how to "round off" to the nearest 12th root of 2, whatever.

What I would do if I could play any notes whatever on the piano is to first analyze the piece in terms of tonalities and consonances.  That is, ignoring the notes used to create tension, and looking only at the notes meant to harmonize at any given point, determine which tones are intended to be held at the same frequency, which are meant to harmonize with those, and which tones - at the really interesting parts - will need to go down or up a comma or however much.

For example, in going from justly intoned key of C major to justly intoned key of G major, we need to alter two tones:  F -> F#, and A that goes with F and C -> A that goes with D and F#.  If the progression is from some inversion of a consonant F major to some inversion of a consonant D major, the A has to change just as the F changes to F#.  The A also has to change similarly from A minor to D major.

That's the case where the same written note represents a different natural note.  The opposite happens too.  When Liszt goes from 4 flats to 5 sharps, he writes a B major followed by a D#7.  I believe he intended this D#7 to be formed from the same frequencies as the Eb7 earlier on.  Working back from there, I deduce that what is written B major (for readability) could have been written as Cb major using the original notation with flats, and Cb-Eb-Gb is what he "really" meant.

Well anyway, when a choir is trying to sound their heavenly best, they will find the natural harmonies, including the comma shifts and such.  But if a comma-shifted voice "forgets" to come back to the original key, the rest of the choir tends to follow them and end up singing everything a comma too high, or more often, too low.  After several interesting parts, they may end up quite a bit flat (or sometimes sharp) from where they ought to be.  I say "ought" because composers, even the fancy modern ones, generally come back to the tonalities they start with, for symmetry.

I need to turn in now, but I'll look at Liesbestram some more this week, and see if I find any part that doesn't "work".


----------



## gaer

Forero said:


> Thank you, Gaer.
> 
> Those are two of my favorites! My grandmother used to play those and gave me her book. Another one is Rachmaninoff's Prélude in C# minor.


I hope I helped. I don't want to continue any kind of sophisticated musical discussion in this thread because I think we are leaving behind the subject of harmonics, which is complicated enough for most of the people who are reading this thread. 

Gaer


----------



## alexacohen

gaer said:


> I hope I helped. I don't want to continue any kind of sophisticated musical discussion in this thread because I think we are leaving behind the subject of harmonics, which is complicated enough for most of the people who are reading this thread.
> 
> Gaer


Maddening, Gaer...
Now please, I'd like to hear Träumerei n 15... for me, at least, music makes sense because it is beautiful... 
Alexa.


----------



## Forero

gaer said:


> I hope I helped. I don't want to continue any kind of sophisticated musical discussion in this thread because I think we are leaving behind the subject of harmonics, which is complicated enough for most of the people who are reading this thread.
> Gaer



I have enjoyed all the excursions we all have made in this thread, and  I think it's all pertinent to the original question.



winklepicker said:


> Pythagoras believed that the universe had its own music, the music of the spheres. But sadly, when he applied maths to making notes, he found that things didn't work out so neatly. This is called the Pythagorean comma.
> 
> ... [1.5)^12 is not exactly 2^7.]
> 
> If you were setting out to design a universe, wouldn't you make the physics work out a bit better than this? So is the Pythagorean comma evidence for the non-existence of God?



My current view of the "big picture":

Music that is all perfect fifths doesn't meet our Western standard anyway.  And even what we imagine we hear is not always exactly the way things happen - just what they "mean" to us.

Basically music, quantum theory, and other arts are all simultaneously simple and complex.  Finding meaning is what we do.  A lot of people have done a lot of work to continually simplify what we experience into manageable theories and pleasing melodies, harmonies, etc.

I think perhaps the universe, including ourselves, is designed to be understandable enough that we can create and use our theories and at the same time complex enough to keep us busy and keep us from thinking we can do without each other. 

There is a lot out there to learn, and we are learning it - together.  I don't know what "post-modern" means, but let's shun the dark ages.

I personally enjoy discussing theories, knowing that none is the end-all but that there is always more we can learn and do peaceful things with, like this forum - and like music.


----------

