Musical Scales with (n != 12) Notes

Musical Scales with (n != 12) Notes

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Just been listening to a fascinating programme on Radio 4 whilst travelling between offices, which coincidentally managed to follow on exactly from to something I was musing about last night. Western music has twelve notes in each octave, as is well known and understood by everyone from Bach to Julie Andrews in the Sound of Music (who just sang about the white notes!). But why twelve? Why not a different number, like eight? What would music sound like if it was played on a completely different scale with an octave subdivided differently?

Some other cultures often play music with additional notes in the scale: Indian instruments like the sitar are capable of playing quarter-notes (half way between two semitones), and the Japanese shakuhachi can play an almost unlimited range of frequencies. Others, too, have experimented with a different number of notes to the scale. Most interestingly, the harmonies that are most pleasing to the ear are those which have low number ratios (2:1, 3:2, 4:3 etc.) - it's exactly these ratios that form the common intervals on our musical scale (e.g. a fifth: A=440Hz, E=660Hz = 3:2). The twelve-note scale offers a very high number of these golden ratios with very few that don't match. These few are the ones that sound dissonant to the ear (e.g. a major seventh = 15:8).

It would be nice to explore this territory by building a small piece of software to let you select a scale containing an arbitrary number of intervals. Unfortunately most of the APIs in Windows at least don't offer that flexibility of control - DirectMusic (part of DirectX) only offers the diatonic MIDI scale, and the simple Beep() Win32 API call only accepts integral values, which greatly reduces the accuracy of any such computerised instrument. Something for further exploration, anyway. I found some interesting websites on the subject here and here (although I couldn't get the Java applet in the latter page to work - great pity).

Something for further exploration, anyway...

  • This reaction is a little late. I bumped into your blog some time ago. As I liked it very much I started reading backward. Being a musician myself with a little experience with a sitar (thanks for mentioning this great instument) I would like to add a little comment to this post. I am not an expert at all but I do see some interesting aspects to your proposal. To you as a piano/organ player ther are 12 half notes going from C tot C. When you play in a scale which contains F# you use the same black key as when you play a scale containing a G-flat. But these notes do not have the same pitch. A violin player, not guided by frets, will use a different position for both notes. We owe our piano tuning to Bach, his "well temperiertes klavier" led to a tuning which works for all scales and gives you the posibilty to play chords, multiple notes sounding together in nice intervals. When it comes to an instrument like the sitar things get even more compicated. Indian music does not have any chords, the only intervals really used are very basic not getting much more complicxated than 2:3. The pitch of notes on a sitar is even more complicated than quarter notes, almost every pitch is used. But not in a random way. The melody in Indian music is based on a raga, a selection of notes with a certain pitch some rules which notes to use when going up, which to use when going down and some idea how a melody should develop. The individual notes in a raga do not have an absolute pitch. The Sa (Do/C) of one musician can differ from that of another (not when playing together of course). And the Ma (Fa/F) of one raga can be of a different relative pitch then that of another raga. Besides that Indian music is very rich in the bending (meend) of a note. Where the different patterns of the bend (coming down / going up, pitch difference, timing) are a major form of expression. So Indian music doesn't really do much with harmony, but it can be very impressing nevetheless. Imho the things that bring music to life have mainly to do with timing. When a PC plays some music using directX or the like the fact that annoys me is that all notes are played exactly on the spot and all have the same intesinty and duration. When a real person(ality) plays the same music on a piano, which has the same pitch possibilities as DirectMusic, some notes will be played louder and longer, some will be played (a little) to late or early. The person will move me, for my PC to move it has to do something different. Thanks for your blog. Peter
  • I was searching for Indian scales so I could at least try playing along with a Sandip Burman CD I have on my guitar... the string instrument on my cd is a Sarod, which sounds similar to a sitar. Anyway, I found this site which is pretty informative: http://chandrakantha.com/articles/scales.html

    It mentions the raga that you mention.

    This is what I have to add to the discussion:
    As you mention, pianos are tuned to a temperament... this means that all intervals are not mathematically precise. The reason for this is that mathematically precise pitches "sound bad" when played together in certain arrangements. Octaves would, but other intervals would not. The reason that they "sound bad" is due to the dominant pitches present in the harmonics of the pitch. As you may know, all musical instruments produce sounds which are a combination of varying levels of many different frequencies. This is a related article: http://www.ams.org/new-in-math/cover/piano1.html It just so happens that these harmonics are not mathematically perfect when played on a piano. The reason for this is that different strings have different thicknesses and therefore different stiffnesses. This alters the harmonics. In addition, there is a circular cyclical movement that strings undergo in addition to the perpendicular vibrations that produce the primary tones. This circular movement also impact the harmonics. I have no degree in music, so I'm not giving you a complete story... this is just what I have learned on my own.
  • Most music systems that don't have twelve tones have completely different mathematical foundations for coming up with the correct pitches. The disagreement on tunings comes from the fact that if you use pitch ratios, then you can't make an instrument in which all notes are "equally" spaced apart within the octave. (n:n+1, is the basis for pythagorean pitch ratios i believe.) Indian music has 22 tones which are not evenly spaced, and not all 12 tones are relative. I think some mid eastern keyboards may have 14 keys, etc. the unevenness of other pitch systems seems to be the reason why they do not focus too much on harmony...there's only a few intervals that sound reasonable without the well temperment. of course, such systems do have richer possibilities for melody though.
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    The 12 tone system is defines the frequency to be basefreq * log2(n div 12) (i think that's right...not in the mood to compute it at the moment). This was done so that there is symmetry between all the notes, and it's a pretty good approximation of the modes you get when you define scales via pitch ratios. Our system is a synthetic approximation to "fix" natural scales so we can make practical instruments. So where do scales come from? pick any note in the 12 tone system, go up some number of fifths (in half tones, so ... n+7), consecutavely. Fourths will do the same thing, because going up a fifth and down a fourth are equivalent, disregarding octave. If you go up 5 times, you get a pentatonic scale, if you go up 7 times you get a diatonic scale. The note in these sets that you choose to be the "center" determines the mode that it is in.
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    I once created a MIDI windchime application that noodled over this flexible scale. the "center", "top" and "bottom" when moved around end up smoothly interpolating between all available modes from the pentatonic to diatonic scales. It gives you a sweet debussy-ish sort of sound. I could never come up with more than a partial explaination that incorporated minor scales with sharp 7ths, the ascending harmonic scales, the whole tone and diminished scales. I am sure that some academic has worked such a system out somewhere though.
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