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Category Archives: Music Theory

To be fair, the new plugin is the same as the old one but with different numbers.  I’m planning on making a Sibelius plugin factory at some point, to allow custom plugins.  For now, though, enjoy the ability to write in 17-tone equal temperament!

Though “temperament” is a bit of a misnomer, because while 17-TET certainly sounds like it has quite a temper, it doesn’t actually temper out anything useful.  Here are the names of the 17 notes of 17-TET:

C Db C# D Eb D# E F Gb F# G Ab G# A Bb A# B (C)

If you’ve read my previous post on the mathematics of diatonic tunings (and actually understood my ramblings), you might remember (or not) that our diatonic 12-tone system depends on two kinds of half steps, the chromatic, from C to C# where the note name doesn’t change, and the diatonic, from C to Db where the note name does change.  The chromatic half step is seven fifths forward in the circle of fifths, and the diatonic half step is five fifths backward in the circle of fifths.  In 12-TET, there are only 12 notes, so those two half steps meet at the same notes, and they’re enharmonic.  C# is the same as Db, just spelled differently.  In other diatonic tunings, though, the chromatic and diatonic half steps have different sizes!  In n-TET, the circle of fifths has n notes.  Each diatonic half step steps backwards 5 fifths, and each chromatic one steps forward 7 fifths, so if your chromatic and diatonic half steps meet after c chromatic ones and d diatonic ones, there are n = 7c + 5d notes in the circle of fifths.  Heh, like that derivation?  Anyway, one of the values of n is 17, where equivalence is reached in 1 chromatic and two diatonic half steps because c = 1 and d = 2.  1 chromatic half step from B is B#; 1 diatonic half step is C and two is Db.  In 17-TET, B# = Db.  Kinda weird, isn’t it?  c and d are also the sizes of the intervals in 17-TET steps.  The chromatic half step is 1 17-TET step and the diatonic is two.  This explains the weird order evident in the note names.  I excluded double flats and double sharps (as well as Cb, E#, Fb, B#), but it’s easy to see where they go.

Problem number 1 with 17-TET: the major triad.  Let’s look at the cent values: the major third is 423.5 cents and the fifth is 705.9 cents.  12-TET has 400 and 700, and JI has 386.3 and 702.0.  The fifth is a little sharp — not much sharper than the 12-TET fifth is flat.  You barely notice it if at all.  But the third is 23.5 cents sharper than 12-TET and a ridiculous 37 cents sharper than JI!  It resembles a major third but it’s REALLY out of tune.  19-TET has a pretty close fifth as well, but its major third is 378.9 cents, closer to JI than 12-TET.  When you hear a major chord in 19-TET, it’s very consonant; in 17-TET, it’s an elementary school class of violinists.

So, you have to wonder: how would the second movement of Beethoven’s Sonata No. 8 “Pathetique” sound in 17-TET?  Well, it’s fairly odd that you were wondering that, but that’s OK because it’s right here:

Sonata No. 8 – II (12-TET)

Sonata No. 8 – II (17-TET)

Sonata No. 8 – II (19-TET)

These recordings were made in Sibelius; the 17-TET and 19-TET recordings using my plugins for the respective purposes.  It’s really easy to retune: select all, apply the plugin, done.  (So long as you wrote the music with one note in a staff at a time, anyway; I also had to transcribe the E major section as being in Fb major to keep spellings correct.)  Hopefully they give a good sense of how 17-TET works.

And that is that it doesn’t.

17-TET is AWFUL for tonal music.  Most of the piece is beautiful major chords, and if there’s one thing that sounds awful in 17-TET, it’s major chords.  Minor chords aren’t as bad.  17-TET does have a neutral third that sounds good, the D# (which is the same as Ed, E half-flat).  However, 12-TET does not have a neutral third, so Beethoven wasn’t gonna use it!

Note that 19-TET sounds considerably better.  The intervals that most diverge from their 12-TET counterparts are the half steps, both diatonic and chromatic (12-TET: 100 cents, 19-TET diatonic: 126 cents, 19-TET chromatic: 63 cents, 17-TET diatonic: 71 cents, 17-TET chromatic: 141 cents).  The sonata movement has a few chromatic passages, and it’s really obvious.  17-TET deals pretty well with leading tones, actually, which 19-TET doesn’t, but 17-TET makes up for it by totally destroying the major triad.  19-TET has better major triads than 12-TET!

Of course, you can write great music in 17-TET.  You can; I haven’t yet.  I will post when (and if) I do.  One thing that is painfully obvious upon hearing retunings of 12-TET music is that music conceived in 12-TET probably sounds better in 12-TET, and other ideas need to be worked out for other tuning systems.  With the Offtonic 17-TET plugin for Sibelius, you can play as you will!

A while ago (a long while ago, I guess — hey, I have a job, so I’m busy), I posted about how 19-tone equal temperament is a diatonic tuning.  But is it the only one?  Other than 12-TET, that is?

Obviously not.  Let’s take a look at how it works!

Under our harmonic system, we base everything on the fifth.  A perfect fifth is a frequency ratio of exactly 1.5, which translates to about 701.955 cents.  Our 12-TET scale only uses multiples of 100 cents, so it approximates the perfect fifth quite well at 700 cents.  This is far from the only possible value, however!  There’s a small range where we can arbitrarily choose our fifth, and that’s what we’re about to explore.

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I have just recently written a Sibelius plugin that allows playback in 19-tone equal temperament, Offtonic 19-TET.  The 19-tone scale has also been available on the Offtonic Microtonal Synthesizer since its creation.  So what is it?  What is it about?  What is the deal with 19-tone equal temperament, or 19-TET?

To begin with, there are many ways to generate an EDO, an Equal Division of the Octave.  The most obvious way is to simply divide the octave into n equal parts, and voilà, n-TET.  You can think of this as starting from note 0 and adding 1 each time, and when you get to n, that’s really the same as note 0.  For example, in 12-TET, I can start at C:

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Unlike the overtone series of the previous post, there is nothing physical or natural about the undertone series, despite its very similar construction.  The overtone series consists of notes at frequencies f, 2f, 3f, 4f, and so on; the undertone series, on the other hand, consists of notes at frequencies f, f/2, f/3, f/4, and so on.  The undertone series is the mirror image of the overtone series.  If you start at one particular note and go up for the overtone series, you’d go down by the same amount for the undertone series.  It’s available in the Offtonic Microtonal Synthesizer as Undertone (C8).

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I’ve added a new scale to the microtonal synthesizer for your enjoyment.  The preset is called Harmonic (C1), and it’s the first 127 notes of the harmonic scale beginning at C1.  I think it’s pretty cool!

Before they invented valves, trumpets were limited to only playing certain notes.  A trumpet in C could maybe get a low C3 out (it sounds kind of terrible), but it could play its first few harmonics fairly easily.  Middle C (C4), G4, C5, E5 (a bit flat), G5, Bb5 (very flat), C6, D6, E6, Ft6 (that’s F half-sharp), G6, Ab6, A6, Bb6, B6, C7, …  Actually, it’s REALLY not easy to play that high, but you’ll notice the notes are closer together as you go up.  What’s going on?

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The Microtonal Synthesizer has been updated.  I used it to explain to someone how to listen to beats and realized that I couldn’t set the cents of notes.  Turned out I was forgetting some factors of 100 in one of the functions handling the text fields, and “50” was magically changing to “0.05”.  This is now fixed.  (Update: as I was writing this post, I noticed that changing the frequency of a note wasn’t working.  It should be now.)

So I might as well explain how beats work.  Let’s say you have a note at some constant frequency of f.  This means that you have some sort of wave that repeats at a frequency of f; if f = 440 Hz, then the wave repeats 440 times per second.  It’s not a simple sine wave; the shape of the wave is what gives instruments their characteristic sounds.  A sine wave sounds more like an “ooh”, while a brighter sound like “aah” will be a more complex shape.  For now, though, let’s assume we’re talking about simple sine waves.

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