Mathematical Questions

[AwakeInADream]

I thought it would be a good idea to start a thread about mathematical questions,
because whatever I happen to be researching, it always leads back to numbers.
And from numbers to infinity and ultimately back to the mind of God.

I am hoping this thread will attract any mathematician's out there with a desire to teach,
and also others like myself who desire to learn or re-learn a subject that was sadly
made so dull at school and yet remains so vitally important to every field of knowledge.

It is becoming increasingly obvious that the establishment cannot be trusted to teach us
anything of value, particularly when it comes to things like statistics.
This is why I seek to test things out for myself, in order to gain a practical understanding,
and this attitude applies just as much to spirituality as it does to the science's.
I feel it makes no sense at all to believe anything blindly.

I think this thread fit's in pretty well with Project Avalon's main theme, and I think there is a
real need for the spiritually minded to have a strong foundation in mathematics.
I will be eternally grateful to anyone who can help with this.

My first question is this and has to do with ratio's and music:

QUESTION: The 12 note equal temperament system of western music is represented by 2 1/12.
What does this ratio mean? 2 1/12?

(*Note ET stands for equal temperament throughout)

I was lucky enough to find a decimal value online in order to work out the frequencies for
a 12 note equal temperament scale which is 1.0594631.
You take the start note 440Hz(A) and times it by 1.0594631 to get the 2nd note 466.16Hz(A#) and so on until you reach the octave 880Hz(A).

How does this number (1.0594631) relate to the ratio 2 1/12?

The reason I needed to know was because I wanted to work out how to calculate other equal temperaments like 22ET and 14ET. 22 equal temperament is expressed as 2 1/22 and 14ET is 2 1/14.

I have now figured out all of the decimal values needed to get any equal temperament from 2-26, which I did
in a spreadsheet through trial and error, changing the values slightly until the octave was right. This process took many unnecessary hours.

This is what I got for 22ET: 1.03200827973421 but how does it relate to 2 1/22?
And 14ET: 1.05075663865322 to 2 1/14?

This process would have been much easier if I had known more about ratio's and math's.
I'm sure it could have been done in no time at all with a much more elegant solution.

Does this have something to do with logarithms(I don't understand them at all)?

The only relationship I have found between these decimal values is that
the square root of the 12 equal temperament number gives you the number needed for a 24ET. And if you square the 12ET number you get the 6ET number...

If you're wondering how all this relates to spirituality, it's because I wanted first to be able
to represent the 22 letters of the holy Hebrew alphabet by musical notes, and secondly I am interested in how music made with various different ET's might affect consciousness.

[araucaria]

Posted by AwakeInADream (here)
I thought it would be a good idea to start a thread about mathematical questions,
because whatever I happen to be researching, it always leads back to numbers.
And from numbers to infinity and ultimately back to the mind of God.

I am hoping this thread will attract any mathematician's out there with a desire to teach,
and also others like myself who desire to learn or re-learn a subject that was sadly
made so dull at school and yet remains so vitally important to every field of knowledge.

It is becoming increasingly obvious that the establishment cannot be trusted to teach us
anything of value, particularly when it comes to things like statistics.
This is why I seek to test things out for myself, in order to gain a practical understanding,
and this attitude applies just as much to spirituality as it does to the science's.
I feel it makes no sense at all to believe anything blindly.

I think this thread fit's in pretty well with Project Avalon's main theme, and I think there is a
real need for the spiritually minded to have a strong foundation in mathematics.
I will be eternally grateful to anyone who can help with this.

My first question is this and has to do with ratio's and music:

QUESTION: The 12 note equal temperament system of western music is represented by 2 1/12.
What does this ratio mean? 2 1/12?

(*Note ET stands for equal temperament throughout)

I was lucky enough to find a decimal value online in order to work out the frequencies for
a 12 note equal temperament scale which is 1.0594631.
You take the start note 440Hz(A) and times it by 1.0594631 to get the 2nd note 466.16Hz(A#) and so on until you reach the octave 880Hz(A).

How does this number (1.0594631) relate to the ratio 2 1/12?

The reason I needed to know was because I wanted to work out how to calculate other equal temperaments like 22ET and 14ET. 22 equal temperament is expressed as 2 1/22 and 14ET is 2 1/14.

I have now figured out all of the decimal values needed to get any equal temperament from 2-26, which I did
in a spreadsheet through trial and error, changing the values slightly until the octave was right. This process took many unnecessary hours.

This is what I got for 22ET: 1.03200827973421 but how does it relate to 2 1/22?
And 14ET: 1.05075663865322 to 2 1/14?

This process would have been much easier if I had known more about ratio's and math's.
I'm sure it could have been done in no time at all with a much more elegant solution.

Does this have something to do with logarithms(I don't understand them at all)?

The only relationship I have found between these decimal values is that
the square root of the 12 equal temperament number gives you the number needed for a 24ET. And if you square the 12ET number you get the 6ET number...

If you're wondering how all this relates to spirituality, it's because I wanted first to be able
to represent the 22 letters of the holy Hebrew alphabet by musical notes, and secondly I am interested in how music made with various different ET's might affect consciousness.

I think it may be a mistake using decimals here rather than vulgar fractions. Just as studying ancient monuments with a decimal value for pi will lead to a dead endbut 22/7 will not.

[AwakeInADream]

That's what I mean. I don't understand how to use fractions and ratio's.

I want to know what 2 1/12 mean's so I can use it practically...In layman's terms...

So 22/7 is pi then, interesting, 2 holy numbers. Thank's

[lightpotential]

Dear Awake,

I can indeed answer your question regarding the value 1.0594631, as you cite. This is a ratio that is found within the celestial realm, and is concerned with the transformation of the earth-moon system. The more accurate value you are looking for is 1.059531131.

Current Earth Tropical Year (days) = 365.242184. Ideal year = 360.

365.242184 / 360 = 1.014561622

1.014561622^4 (raised to the 4th power) = 1.059531131

If this value is then reduced to the 3rd root, one gets = 1.019462465 (1100/1079)

This is the ratio of change of increase for the mean distance between the moon and the earth (as accompanied the noted change of its year). Also, in units of 6000 feet - a special unit called an ideal geographical mile, the very mean distance between the moon and the earth is 210194.62465 IGM.

If you set aside the number 2, you will see the ratio of increase embedded within the distance itself.

If you go to my website Ancient World Mysteries, you can see the math in greater detail. The link is below.

[araucaria]

Posted by lightpotential (here)
Dear Awake,

I can indeed answer your question regarding the value 1.0594631, as you cite. This is a ratio that is found within the celestial realm, and is concerned with the transformation of the earth-moon system. The more accurate value you are looking for is 1.059531131.

Current Earth Tropical Year (days) = 365.242184. Ideal year = 360.

365.242184 / 360 = 1.014561622

1.014561622^4 (raised to the 4th power) = 1.059531131

If this value is then reduced to the 3rd root, one gets = 1.019462465 (1100/1079)

This is the ratio of change of increase for the mean distance between the moon and the earth (as accompanied the noted change of its year). Also, in units of 6000 feet - a special unit called an ideal geographical mile, the very mean distance between the moon and the earth is 210194.62465 IGM.

If you set aside the number 2, you will see the ratio of increase embedded within the distance itself.

If you go to my website Ancient World Mysteries, you can see the math in greater detail. The link is below.

If Awake wasn't completely flummoxed already, he sure is now I reckon

Edit: what have the fluctuations of the Moon to do with music I wonder??

[AwakeInADream]

That's amazing Lightpotential!

The 12 note western system of music is related to the dance of the Earth and the Moon!

Pythagoras really did hear 'The Music of the Sphere's'!

EDIT: I still could do with someone to explain all these things in layman's terms though...

EDIT2: Great Website you've got BTW Lightpotential!

[ghostrider]

looks like it's just a version of yin/yang when you boil it all down. IMO

[indigopete]

Posted by AwakeInADream (here)
QUESTION: The 12 note equal temperament system of western music is represented by 2 1/12.
What does this ratio mean? 2 1/12?

Hi Awake

If you can grasp the difference between a "linear" relationship (between 2 values) and a "logarithmic" one then you're in there.


A. BACKGROUND

To illustrate, a linear relationship is where one value changes in proportion to the other. For example:

BRICKS and HEIGHT
I have a brick that's 10 cm thick. If I build 2 bricks high, the wall will be 20cm. If I build 3 bricks high, it will be 30cm. i.e. if I double the number of bricks, the height also doubles. If I treble the number of bricks, the height also trebles.

TABLE A - LINEAR RELATIONSHIP
1st brick layer = bricks are 10cm thick
2nd brick layer = bricks are 10cm thick
3rd brick layer = bricks are 10cm thick

This is called a linear relationship because if you draw a graph of the wall height against the number of stacked bricks (brick count) it will be a straight line.

Now lets look at a LOGARITHMIC relationship. Do do that, you have to imagine a wall where, in each successive 10cm brick layer, the bricks are only a tenth of the thickness of those in the preceding one below it. So, each time I added another brick level, the height of the wall increases by a proportionately less amount. For example:

TABLE B - LOGARITHMIC RELATIONSHIP
1st brick layer = bricks are 10cm thick
2nd brick layer = bricks are 1 cm thick
3rd brick layer = bricks are 0.1 cm thick

... and so on. So by the time I'm at the 3rd layer of bricks, I need to add 100 more bricks just to get the 10cm increase I had with the first one. If we plot the number of bricks at each layer to get back to our 30 cm height of TABLE A, then it looks like this:

TABLE C - BRICK COUNT (LINEAR SCALE)
1st brick layer = 1 brick (We've reached 10cm height)
2nd brick layer = 10 bricks (We've reached 20cm height)
3rd brick layer = 100 bricks (We've reached 30cm height)

This is annoying, because the increment between the number of bricks at each layer is different (9 in the first, 90 in the second) so it's a headache to calculate things.

MATHEMATICAL TRICK !!!
Even mathematical boffins such as Einstein experience headaches with anything other than linear relationships, so they use a useful tool: We can get the graph back to a straight line if we plot the wall height against the LOGARITHM of the brick count instead of just the brick count itself. In this case TABLE C looks like this (try it yourself on your calculator. Use the Base 10 logs, not the natural (e) ones):

TABLE C - BRICK COUNT (LOGARITHMIC SCALE)
1st brick layer = log(1) = 0
2nd brick layer = log(10) = 1
3rd brick layer = log(100) = 2

Suddenly, were back in business ! We have the same increment between each brick layer (1) instead of a varying increment, which is easier on our tender brains.

To be specific, the Log is just the number of zeroes in each brick count. (So say the log of a million (in base 10 logs) would be 6, because it's got 6 zeroes after it).

B. HOW DOES THIS RELATE TO YOUR QUESTION

In your example, the pitch is the height of the wall (brick layer Nº) and the frequency is the number of bricks. Each time we go up 1 note, we need a much bigger increment in frequency (more bricks to achieve the same height increase). As with the example above, however, we can go up in equal increments if we use the logarithm of the frequency instead of the frequency itself.

The only difference, is that instead of Base 10 logarithms, we use base 2. That's where the 2 1/12 comes from. (It's actually 2 to the power 1/12, which is written as the digit 2 with 1/12 as a superscript. The superscript part is the LOGARITHM).

C. HOW DOES THIS NUMBER (1.0594631) RELATE TO THE RATIO 2 1/12

Try it for yourself ! On your calculator, type 2, then the power button (usually marked Y with an x superscript), then [open bracket] 1 / 12 [close bracket] [equals]. You should get the result 1.0594631. (That's just a decimal representation of 2 to the power of a twelfth). This can also be expressed as the twelfth root of 2.

Now lets continue to completion. If we plot each note against the base frequency expressed as powers of 2 (i.e. logarithms in base 2) we get this table:

TABLE D - PITCH and FREQUENCY

1st Note = Base Frequency x [2 to the power 0/12]
2nd Note = Base Frequency x [2 to the power 1/12]
3rd Note = Base Frequency x [2 to the power 2/12]
4th Note = Base Frequency x [2 to the power 3/12]
5th Note = Base Frequency x [2 to the power 4/12]

Do you see the numbers on the right hand side going up in equal increments ? 1/12, 2/12, 3/12 etc ? Those are the logarithms (in base 2) of the frequency instead of the frequency itself. Our ear agrees with the maths because our ear is listening to the logarithm of the frequency and hearing it go up in equal increments - not the frequency itself.

SUMMARY

[1] - The value 2 to the power one twelfth (2 1/12) is the ratio between each successive frequency in the western scale
[2] - This number can be represented as a decimal as approximately 1.0594631
[3] - If we represent each frequency as a ratio of the base frequency using powers of 2 (the logarithm of the frequency in base 2 instead of the frequency itself), then the increments are equal between each note

ONE FINAL THING

The number 2 to the power 1 twelfth is an IRRATIONAL NUMBER. These are quite fascinating phenomena in themselves because they represent a kind of "gap" or "black hole" in the number line. It is a number which does not exist and cannot ever be represented as a decimal. You can only represent exact numbers either side of it but not the number itself. You can, however, represent it with full precision as a ratio (i.e. an implied value which uses two other values to represent it, for example, the ratio of the circumference to the diameter of a circle (normally known as "Pi").

Hope that helps.

Pete

[AwakeInADream]

Wow! That was amazingly lucid, not only did I understand every word, but also I've just done logarithm's for the first time in my entire life, and understood them!

I did the calculations for 12ET using buttons on my calculator that I've never pressed before and it worked. I've also tested it for 2 1/22 and 2 1/14 and that worked too.
And just for fun 2 1/1200 = 1.00057779. That's amazing! Thank you so much for this, I was half expecting this thread to be buried before anyone super smart noticed.

Irrational numbers too, I've never been sure before, I guess the decimals just flow on till infinity, the mind of God. Cool.

Thank you so much Pete! I hope you'll pop back in to this thread from time to time, to help anyone else who might have questions.

[Rocky_Shorz]

25/12 is Christmas...

how does it relate to music?

let's go Caroling...

[AwakeInADream]

Posted by Rocky_Shorz (here)
25/12 is Christmas...

how does it relate to music?

let's go Caroling...

I might just be crazy enough to test that musical scale out you know!

I once re-worked 'The Hymn to St. John the Baptist' using Solfeggio tones and it sounded awful!

[Rocky_Shorz]

your post is at 22:22 as in the day after the Mayan Calendar ends...

my birthday, I'm starting to think we were meant to bump into each other...

[AwakeInADream]

I told you those numbers are everywhere lately!

The day after the Mayan calendar ends! Wow! That's really hopeful!

I guess the world won't end then after all...

[Rocky_Shorz]

naaa just the grumpy old crusty dudes and dudettes currently running it...

the looking glass doesn't show what is beyond the 21st...

the ones who depended on knowing the future before it happens have been cut off...


but you're right, I never miss a Birthday...

[leavesoftrees]

Posted by AwakeInADream (here)

If you're wondering how all this relates to spirituality, it's because I wanted first to be able
to represent the 22 letters of the holy Hebrew alphabet by musical notes.

See

http://www.joyousworld.com/qabalah/t...ributions.html

this lists notes assigned to each Hebrew letter

[Chester]

Posted by AwakeInADream (here)
I thought it would be a good idea to start a thread about mathematical questions,
because whatever I happen to be researching, it always leads back to numbers.
And from numbers to infinity and ultimately back to the mind of God.

I am hoping this thread will attract any mathematician's out there with a desire to teach,
and also others like myself who desire to learn or re-learn a subject that was sadly
made so dull at school and yet remains so vitally important to every field of knowledge.

It is becoming increasingly obvious that the establishment cannot be trusted to teach us
anything of value, particularly when it comes to things like statistics.
This is why I seek to test things out for myself, in order to gain a practical understanding,
and this attitude applies just as much to spirituality as it does to the science's.
I feel it makes no sense at all to believe anything blindly.

I think this thread fit's in pretty well with Project Avalon's main theme, and I think there is a
real need for the spiritually minded to have a strong foundation in mathematics.
I will be eternally grateful to anyone who can help with this.

My first question is this and has to do with ratio's and music:

QUESTION: The 12 note equal temperament system of western music is represented by 2 1/12.
What does this ratio mean? 2 1/12?

(*Note ET stands for equal temperament throughout)

I was lucky enough to find a decimal value online in order to work out the frequencies for
a 12 note equal temperament scale which is 1.0594631.
You take the start note 440Hz(A) and times it by 1.0594631 to get the 2nd note 466.16Hz(A#) and so on until you reach the octave 880Hz(A).

How does this number (1.0594631) relate to the ratio 2 1/12?

The reason I needed to know was because I wanted to work out how to calculate other equal temperaments like 22ET and 14ET. 22 equal temperament is expressed as 2 1/22 and 14ET is 2 1/14.

I have now figured out all of the decimal values needed to get any equal temperament from 2-26, which I did
in a spreadsheet through trial and error, changing the values slightly until the octave was right. This process took many unnecessary hours.

This is what I got for 22ET: 1.03200827973421 but how does it relate to 2 1/22?
And 14ET: 1.05075663865322 to 2 1/14?

This process would have been much easier if I had known more about ratio's and math's.
I'm sure it could have been done in no time at all with a much more elegant solution.

Does this have something to do with logarithms(I don't understand them at all)?

The only relationship I have found between these decimal values is that
the square root of the 12 equal temperament number gives you the number needed for a 24ET. And if you square the 12ET number you get the 6ET number...

If you're wondering how all this relates to spirituality, it's because I wanted first to be able
to represent the 22 letters of the holy Hebrew alphabet by musical notes, and secondly I am interested in how music made with various different ET's might affect consciousness.

If I recall correctly, the notes of the musical scale do not correspond in the way you mentioned as the scale is not based on a straight line (instead it is logarithmic). I hope that doesn't throw too much of a monkey wrench into your idea.

I also have heard that musical note correspondences have already been established (though I know not the authority nor the credibility of said authority) in relation to some alphabets and I seem to recall the Hebrew alphabet being one of them.

I am fascinated by your idea and I hope it leads you to some wonderful self discoveries... can't wait to hear about your adventures in this regard, Chester

EDIT: I broke one of my rules... read all the prior posts before posting as you may be unnecessarily redundant and I was! haha

[Chester]

Posted by lightpotential (here)
Dear Awake,

I can indeed answer your question regarding the value 1.0594631, as you cite. This is a ratio that is found within the celestial realm, and is concerned with the transformation of the earth-moon system. The more accurate value you are looking for is 1.059531131.

Current Earth Tropical Year (days) = 365.242184. Ideal year = 360.

365.242184 / 360 = 1.014561622

1.014561622^4 (raised to the 4th power) = 1.059531131

If this value is then reduced to the 3rd root, one gets = 1.019462465 (1100/1079)

This is the ratio of change of increase for the mean distance between the moon and the earth (as accompanied the noted change of its year). Also, in units of 6000 feet - a special unit called an ideal geographical mile, the very mean distance between the moon and the earth is 210194.62465 IGM.

If you set aside the number 2, you will see the ratio of increase embedded within the distance itself.

If you go to my website Ancient World Mysteries, you can see the math in greater detail. The link is below.

WoW! Are you friends with Christopher Knight and Alan Butler?

http://www.amazon.com/Built-Moon-Chr.../dp/190585711X

fascinating info - thanks, lightpotential

[AwakeInADream]

I've got a question I think might be easier for someone to answer.

If 22/7 represents Pi (thank's Araucaria!).

What fraction represents the golden ratio Phi?

[Chester]

Posted by AwakeInADream (here)
I've got a question I think might be easier for someone to answer.

If 22/7 represents Pi (thank's Araucaria!).

What fraction represents the golden ratio Phi?

http://www.jcu.edu/math/vignettes/continued.htm

Your answer might be found in this link... fascinating stuff. Cheers, justone

[AwakeInADream]

Posted by justoneman (here)

Posted by AwakeInADream (here)
I've got a question I think might be easier for someone to answer.

If 22/7 represents Pi (thank's Araucaria!).

What fraction represents the golden ratio Phi?

http://www.jcu.edu/math/vignettes/continued.htm

Your answer might be found in this link... fascinating stuff. Cheers, justone

OK! You're right, the answer is there, but all that stuff just looks like a foreign language to me.

It's more complicated than I thought it would be. Quadratic equations?

Thank's JustOneMan!

Can you explain this?

EDIT: I've got it now, I was just scared by the language. The square route of 5 plus 1 then divided by 2 is 1.618033989 the Golden Ratio. Cheers!