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Music interval

  1. May 23, 2015 #1
    Dear PF forum,
    Do anyone know, why music interval is always
    [itex]\sqrt[12]{2}[/itex]?
     
  2. jcsd
  3. May 23, 2015 #2
  4. May 23, 2015 #3
    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^[itex]\frac{1}{12}[/itex]
    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?
     
  5. May 23, 2015 #4
    Additional information:
    So this [itex]\sqrt[12]{2}[/itex] 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?
     
    Last edited: May 23, 2015
  6. May 23, 2015 #5

    DrGreg

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    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.
     
  7. May 23, 2015 #6
    Does 12 has something to do with 4!
     
  8. May 23, 2015 #7

    Mark44

    Staff: Mentor

    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.
     
  9. May 23, 2015 #8
    Thanks Mark44 for your answer.
    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.

    Of course Mark44, that goes without saying.
    [itex]\sqrt[12]{2}[/itex]
    An "octave" higher is always twice the frequency. And we can make any interval if we want. And an octave still twice the frequency.
    [itex]\sqrt[14]{2}[/itex]
    [itex]\sqrt[16]{2}[/itex]
    [itex]\sqrt[18]{2}[/itex]
    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.
     
  10. May 23, 2015 #9

    atyy

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    It's a deliberate error. An ideal perfect fifth should have a frequency ratio of 3/2 = 1.5, but if one uses [itex]\sqrt[12]{2}[/itex] 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
     
  11. May 23, 2015 #10

    atyy

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    But this guy can play the quarter tones!
     
  12. May 23, 2015 #11
    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.
     
  13. May 23, 2015 #12
    Wow, that's great and odd, too. But only in violin, right. Guitar, Piano, Harp, whistle, flute can't.
     
  14. May 23, 2015 #13

    atyy

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    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.
     
    Last edited: May 23, 2015
  15. May 23, 2015 #14

    atyy

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    Here is an example of a non-diatonic scale.



    But Arabic music also has the Western major scale.

     
    Last edited: May 23, 2015
  16. May 23, 2015 #15

    atyy

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    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.
     
  17. May 23, 2015 #16

    Mark44

    Staff: Mentor

    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.
     
  18. May 23, 2015 #17
    good electric keyboards have pitch knobs on the side.
     
  19. May 23, 2015 #18
    Yeah, you right :smile:
     
  20. May 23, 2015 #19
    Damn, should have read your answer before I answer to Mark44.:smile:
    Thanks, Thankz.
     
  21. May 23, 2015 #20
    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)
     
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