Music Unraveling the Mystery of Music Intervals

[Stephanus]

Dear PF forum,
Do anyone know, why music interval is always
\sqrt[12]{2}?

 

[Greg Bernhardt]

Did you read through this wiki?
https://en.wikipedia.org/wiki/Interval_(music)

 

[Stephanus]

Did you read through this wiki?
https://en.wikipedia.org/wiki/Interval_(music)

Hi Mr. Greg Bernhardt,
It's an honor to be answered by you.
Okay, I'm reading https://en.wikipedia.org/wiki/Interval_(music)#Latin_nomenclature
and this https://en.wikipedia.org/wiki/Interval_(music)
and this https://en.wikipedia.org/wiki/List_of_pitch_intervals
https://en.wikipedia.org/wiki/Diatonic_scale
All of these are based on 2^\frac{1}{12}
Or Exp(log(2)/12), I have to use this formula, Latex is difficult to make bold type font
But, why 12? Several music scale, diatonic, pentatonic, they are all based on 12. All those link refer to 12.
The 2, is easly understood Exp(log(2)/12), it has something to do with harmony.
Take a child(boy or girl) and an adult male. When they sing together, the adult will take 1 octave below the child (except if you mention Robert Plant or Ian Gillian). It's remarkable that a non trained adult can automatically switch to half frequency below the child. It's human nature.
But I am wondering about the 12.
Every music instrument (except violin and drums) such as guitar, piano, whistle, flute, they are all using 12 interval. Okay,.. they are modern music. Perhaps there were some consensus back then, such as, Latin for animal names, Arabians for numbers, Greek letter for mathematic equations.
Any old cultures on earth, most of them use 12 interval, even tough there's no historical connections between them. It predates alphabet and of course silk road.
See: https://en.wikipedia.org/wiki/Diatonic_scale#Prehistory
Is it human nature?
If anybody remember Spielberg's "Close Encounter of the Third Kind". We human communicate with the aliens with music.
Okay, it's understandable. We can't easly mimic dog or cat voice with our own vocal chord, or horse, tiger, etc. And our ear are not trained to their voice. The ears of 300 of 400 millions world population are trained to hear English. But only 200 millions people are trained to hear Indonesian language, non native speakers are difficult to hear Indonesian language. So, why we consider we are trained to hear Alien language if we make contact.
So in the movie, human use music, and the alien, too. So there's the famous John William's Five Note,
http://www.ars-nova.com/Theory Q&A/Q35.html
Again, it's based on 12 interval.
Okayy, that was fiction.
But, is 12 interval human nature or universal?

 

[Stephanus]

Additional information:

https://en.wikipedia.org/wiki/Diatonic_scale#Prehistory
Prehistory
... There is evidence that the Sumerians and Babylonians used some version of the diatonic scale...

So this \sqrt[12]{2} predates alphabet, much less Newton G, Avogadro constant, Planck constant, or even Hubble constant.
If the discovery is made in modern times, it's understandable. But if it was found in ancient times, that raises a question. Is it human nature?

 

[DrGreg]

The point is that the 12-equal-interval scale (or equal temperament) contains within it good approximations for ratios such as 2:1 (octave), 3:2 (perfect fifth), 4:3 (perfect fourth), 5:4 (major third), 6:5 (minor third). These ratios sound good to the human ear because the notes share some of the same harmonics.

 

[Stephanus]

Does 12 has something to do with 4!

 

[Mark44]

Does 12 has something to do with 4!

Not as far as I know, and besides 4! = 24. If you look at a piano keyboard, there are twelve keys between successive C keys. A note that is an octave higher (12 halftones) has twice the frequency.

 

[Stephanus]

Thanks Mark44 for your answer.

Not as far as I know, and besides 4! = 24.

4! = 24, Ahh, foolish me of course. Perhaps 12 is the smallest number that have many factor. 2,3,4,6,12.
And of course it makes a good chord. Can't do harmonic with any number beside 12. But we can do that with 24 if we insist!
Is 12 universal? Do alien civilization use this number for their music? Well, it's an interesting idea isn't it. If not speculative.

A note that is an octave higher (12 halftones) has twice the frequency.

Of course Mark44, that goes without saying.
\sqrt[12]{2}
An "octave" higher is always twice the frequency. And we can make any interval if we want. And an octave still twice the frequency.
\sqrt[14]{2}
\sqrt[16]{2}
\sqrt[18]{2}
About "octave", even if you use pentatonic scale, we still call it an octave.

But why do re mi fa so la si do?
Why 1 1 ½ 1 1 1 ½? (Of course there many other scale.
Minor scale: 1 ½ 1 1 ½ 1 1
Minor melodic scale: 1 ½ 1 1 1 1 ½ up, 1 1 ½ 1 1 ½ 1, down
Minor harmonic scale: 1 ½ 1 1 ½ 1+½ ½
But why? Why not just 1 1 1 1 1 1?

And about natural scale or minor scale, not minor harmonic and minor melodic
In piano, we can play natural and minor scale with only the white parts.
It's strange isn't it, if you play any song on natural or minor scale in C, and once you hit one of the black key, any ordinary people will spot (or hear). "THERE!", there's is a flat/sharp key that you hit.

 

[atyy]

Dear PF forum,
Do anyone know, why music interval is always
\sqrt[12]{2}?

It's a deliberate error. An ideal perfect fifth should have a frequency ratio of 3/2 = 1.5, but if one uses \sqrt[12]{2} the perfect fifth is 1.498.

The reason is that in Western music, modulation from one key to another is important is an expressive tool. The most frequent modulation if one is starting in C major, is to G major. Ideally you want to go through the "circle of fifths" and return to C. If the fifth is perfect, this is impossible. So we make it wrong to fool the ear and make the circle of fifths possible.

But then why 12? It's the simplest, and after that, most musicians are too inaccurate - even great ones - Menuhin was not able to play the quarter tones in the Bartok sonata accurately, so Bartok wrote him an "easier version". However, there historically been attempts to use 41 and 53, as those are the next closest.

http://en.wikipedia.org/wiki/41_equal_temperament
http://en.wikipedia.org/wiki/53_equal_temperament

 

[atyy]

But this guy can play the quarter tones!

 

[Stephanus]

It's a deliberate error. An ideal perfect fifth should have a frequency ratio of 3/2 = 1.5, but if one uses \sqrt[12]{2} the perfect fifth is 1.498.

The reason is that in Western music, modulation from one key to another...

Thanks Atty for your idea.
But according to this,
http://en.wikipedia.org/wiki/Diatonic_scale#Prehistory
Prehistoric Babylonian and Sumerian also used diatonic scale. So it dates back thousands of years ago.

 

[Stephanus]

But this guy can play the quarter tones!

Wow, that's great and odd, too. But only in violin, right. Guitar, Piano, Harp, whistle, flute can't.

 

[atyy]

Thanks Atty for your idea.
But according to this,
http://en.wikipedia.org/wiki/Diatonic_scale#Prehistory
Prehistoric Babylonian and Sumerian also used diatonic scale. So it dates back thousands of years ago.

Yes, but in that case they did not necessarily use the division into 12. There they could use "true" intervals, ie. they could make their fifth exactly 1.5. If you don't change key a lot, then the division into 12 is not needed. To clarify, if you go to the bottom of the Wikipedia article on the diatonic scale, you will see that there are more intervals between successive notes of the scale than only semitones and tones.

 

[atyy]

Here is an example of a non-diatonic scale.



But Arabic music also has the Western major scale.

 

[atyy]

But why do re mi fa so la si do?
Why 1 1 ½ 1 1 1 ½? (Of course there many other scale.
Minor scale: 1 ½ 1 1 ½ 1 1
Minor melodic scale: 1 ½ 1 1 1 1 ½ up, 1 1 ½ 1 1 ½ 1, down
Minor harmonic scale: 1 ½ 1 1 ½ 1+½ ½
But why? Why not just 1 1 1 1 1 1?

The general idea against having just 1 1 1 1 1 1 is that to help a certain note be special or the tonic, there should be some asymmetry to help the ear know where it is. Of course this is not absolute, as Debussy did write for the whole tone scale.



One way of constructing the diatonic major is to first construct the triad. In C major, the triad on C would be C-E-G. Then one constructs the triad on the fifth above C, ie. G-B-D, then the triad on the fifth below C, ie F-A-C. This is why Western music in the diatonic major can be harmonised with just 3 chords. It is not too wrong to say that in Western music all other chords used to harmonise music written with the diatonic major are variations of these 3 basic chords.

 

[Mark44]

Wow, that's great and odd, too. But only in violin, right. Guitar, Piano, Harp, whistle, flute can't.

Although guitars have frets, unlike violins, you can still get intermediate sounds by "bending" the string. You can't do this with piano or harp.

 

[thankz]

good electric keyboards have pitch knobs on the side.

 

[Stephanus]

Although guitars have frets, unlike violins, you can still get intermediate sounds by "bending" the string. You can't do this with piano or harp.

Yeah, you right

 

[Stephanus]

good electric keyboards have pitch knobs on the side.

Damn, should have read your answer before I answer to Mark44.
Thanks, Thankz.

 

[Stephanus]

Here is an example of a non-diatonic scale.

But Arabic music also has the Western major scale.

Wow, that woman from the above video, she is great. Can sing in that tone. Must be hard practice.
"Western major scale".
Perhaps we should know that Arabic music predates Western music.
Many knowledge come from Arab World because of the crusade war.
Alchemy, chemistry, astronomy, algebra.
Even our number, except 0 (zero), come from Arab World.
1..9 come from Arab, 0 comes from Hindu.
Chess, originally from India, were introduced to western world by Arabs. (Now chess championship returns to India, Viswanathan Anand)

 

[Mark44]

Chess, originally from India, were introduced to western world by Arabs.

The word for chess in Russian is Шахматы (transliterates roughly to shakh matiy, "the shah is dead"). The Russian word comes from Persian, which is the origin of the English word "checkmate." (See http://en.wikipedia.org/wiki/Checkmate.)

 

[Stephanus]

The word for chess in Russian is Шахматы (transliterates roughly to shakh matiy, "the shah is dead"). The Russian word comes from Persian, which is the origin of the English word "checkmate." (See http://en.wikipedia.org/wiki/Checkmate.)

Yes, "Shah" is arabic, I think.
In Indonesian language, "Shah" means King, ruler, leader.
Mat in Indonesian, "mati", is dead.

 

[Mark44]

Yes, "Shah" is arabic, I think.

I'm pretty sure "shah" is Persian. Shah Reza Pahlavi was the ruler of Iran before the Revolution in 1979. I'm not aware of any Arab-speaking country whose ruler was a "shah."

In Indonesian language, "Shah" means King, ruler, leader.
Mat in Indonesian, "mati", is dead.

 

[Stephanus]

I'm pretty sure "shah" is Persian. Shah Reza Pahlavi was the ruler of Iran before the Revolution in 1979. I'm not aware of any Arab-speaking country whose ruler was a "shah."

You're right. I think I'm confusing the word "shah" in my country.
I think some "islamic" vocabularies that we use here, don't come from Arab, like Shah. Should learn my own language further.
Speaking of language.
Do you think music is a kind of language?
You put two people from different civilization (or two different alien for that matter).
A uses perhaps Latin letter
B uses perhaps Japanese kanji.
I mean before modern era, before silk road era, where the western world haven't discover Japan, yet. And before Japanese learn "latino".
They can't communicate, but their "music" interval is most likely $\sqrt[12]{2}$.

 

[atyy]

Wow, that woman from the above video, she is great. Can sing in that tone. Must be hard practice.
"Western major scale".
Perhaps we should know that Arabic music predates Western music.
Many knowledge come from Arab World because of the crusade war.
Alchemy, chemistry, astronomy, algebra.
Even our number, except 0 (zero), come from Arab World.
1..9 come from Arab, 0 comes from Hindu.
Chess, originally from India, were introduced to western world by Arabs. (Now chess championship returns to India, Viswanathan Anand)

To complete my using wrong terminology, let me demonstrate that they have the Western diatonic minor scale too.

 

[Stephanus]

What is the title above?
"Muwashah of Arab - Andalusia..."
Well, If I'm not mistaken, Andalusia is a region in southern Spain. Near Gibraltar strait, again "Gibraltar" comes from "Jeb Al Tarik", a general in Arab?

 

[atyy]

They can't communicate, but their "music" interval is most likely $\sqrt[12]{2}$.

The $\sqrt[12]{2}$ is very Western, and late. The Chinese had it, but it doesn't seem they used it extensively in their music. It was developed into Western "common practice harmony". In contrast, the expressive tools in Arabic music use a different kind of modulation, between melodic modes, and not so much harmony. According to some Arab music theorists, they use $\sqrt[24]{2}$ http://en.wikipedia.org/wiki/Arab_tone_system. The diatonic scale is much earlier, but it doesn't involve the $\sqrt[12]{2}$.

 

[jtbell]

But then why 12? It's the simplest, and after that, most musicians are too inaccurate - even great ones - Menuhin was not able to play the quarter tones in the Bartok sonata accurately, so Bartok wrote him an "easier version". However, there historically been attempts to use 41 and 53, as those are the next closest.

http://en.wikipedia.org/wiki/41_equal_temperament
http://en.wikipedia.org/wiki/53_equal_temperament

The American composer Easley Blackwood has experimented with writing music in equal-tempered scales with 13 through 24 notes to the octave:

http://www.cedillerecords.org/albums/microtonal

http://en.wikipedia.org/wiki/Easley_Blackwood,_Jr .

[added: I can't make the second link post properly. The URL is supposed to end in a period, but the forum software insists on putting the period outside the URL. When you follow the link, you'll get an error page from Wikipedia. If you then add the period by hand in your browser's address bar and hit "enter", you'll go to the correct page. Or just use Google to search for Easley Blackwood.]

 

[Stephanus]

Okay,... I think 12 is the "magic number" for music.
Not that the smallest number that have 4 factors. 2,3,4,6, but

The point is that the 12-equal-interval scale (or equal temperament) contains within it good approximations for ratios such as 2:1 (octave), 3:2 (perfect fifth), 4:3 (perfect fourth), 5:4 (major third), 6:5 (minor third). These ratios sound good to the human ear because the notes share some of the same harmonics.

Continuing your list Dr. Greg
5:3 (perfect sixth, La)
9:5 (perfect sixth, Si)
But 5:3 or 9:5 the divider and modifier distance is not 1, 5:3 is 2, whereas 9:5 is 4
I don't know if this is a good cause for 12 interval

 

[Mark44]

Speaking of language.
Do you think music is a kind of language?

Not really. Music can convey some emotions, maybe, but I don't see how it could be used to convey much more than that.

 

[Stephanus]

Not really. Music can convey some emotions, maybe, but I don't see how it could be used to convey much more than that.

Twenty six Alphabets + 10 numbers covers 3 octaves in diatonic scale.
In the movie "Close Encounter of the Third Kind", the alien and human use music to communicate.
Okay it's just a movie, but for diferent species with different vocal cords and trained ears just to hear their own vocal cord, perhaps music is a kind of communication tool.

 

[Vanadium 50]

Continuing your list Dr. Greg
5:3 (perfect sixth, La)
9:5 (perfect sixth, Si)
But 5:3 or 9:5 the divider and modifier distance is not 1, 5:3 is 2, whereas 9:5 is 4
I don't know if this is a good cause for 12 interval

You really should read the article that Greg pointed you to.

There is no perfect sixth. There is a major sixth, and a minor sixth.

 

[Stephanus]

You really should read the article that Greg pointed you to.

There is no perfect sixth. There is a major sixth, and a minor sixth.

Ahh, you're right Vanadium.

 

[Pythagorean]

I think one issues in this thread is that 12-tone and equal-temperament (2^(1/12)) are not the same thing.

Before equal temperament, Pythagoras used perfect fifths to construct the 12 tone scale. As arty pointed out, thus made it difficult to modulate keys, because the errors in Pythagoras' method got larger as you got "further away" (distance measured on circle of fifths, probably) from the key in which the perfect fifths were constructed.

So there were lots of 12 tone scales, and methods of interval cobstruction and 2^(1/12) is really quite a recent one. It's advantage is that it's consistent - all the intervals are exactly the same from the first degree to the second regardless of key.

I think 12 tones (and 24) work so well because of the numerous factors you mentioned (2,3,4,6). I know there are other divisions - the most important thing is the octave (2:1), as it's relevant to how our brain processes audio signals audio signals. After that, the fifth (3:2) is the next most important, then it probably gets rather subjective, emergent, and cultural after that (Even the fifth isn't universally used like the octave is).

 

[Stephanus]

I think 12 tones (and 24) work so well because of the numerous factors you mentioned (2,3,4,6)

Thanks Pythagorean for your answer.
But, the more I think of it, the more I disagree with my previous statement. It's not that 12 can be divided by 2,3,4 or six.
It's that, as DrGreg before pointed out,

...2:1 (octave), 3:2 (perfect fifth), 4:3 (perfect fourth), 5:4 (major third), 6:5 (minor third). These ratios sound good to the human ear because the notes share some of the same harmonics.

Minor third: 2^{(1/12) x 3} ≈ 6:5
Major third: 2^{(1/12) x 4} ≈ 5:4
Perfect fourth: 2^{(1/12) x 5} ≈ 4:3
Perfect fifth: 2^{(1/12) x 7} ≈ 3:2
Octave: 2^{(1/12) x 12} is of course 2:1
Don't you think so Pythagorean?

I guess this is the answer of my curiosity for years. So simple
Okay..., one more question for anybody.
π, e, golden ratio, they are all, I think, universally accepted. I mean really universally. Any civilization even outside the Earth will use those constants. What about \sqrt[12]{2}, is it universally used?

Any idea?

 

[Pythagorean]

Thanks Pythagorean for your answer.
But, the more I think of it, the more I disagree with my previous statement. It's not that 12 can be divided by 2,3,4 or six.
It's that, as DrGreg before pointed out,Minor third: 2^{(1/12) x 3} ≈ 6:5
Major third: 2^{(1/12) x 4} ≈ 5:4
Perfect fourth: 2^{(1/12) x 5} ≈ 4:3
Perfect fifth: 2^{(1/12) x 7} ≈ 3:2
Octave: 2^{(1/12) x 12} is of course 2:1
Don't you think so Pythagorean?

I guess this is the answer of my curiosity for years. So simple
Okay..., one more question for anybody.
π, e, golden ratio, they are all, I think, universally accepted. I mean really universally. Any civilization even outside the Earth will use those constants. What about \sqrt[12]{2}, is it universally used?

Any idea?

My point was that these are two distinct issues. One is the issue of how many notes you subdivide the octave by, the other is how you distribute that subdivision. There's not one way to do it, the only reason that Dr Greg can talk about approximating these ratios in the first place is because these are the ratios that are already desired, and these ratios come from higher order subdivisions (dividing the string in half, thirds, and fourths).