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 Forum index » How-tos » Micro Tuning
Microtonal keyboard - circle of 5ths
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Octahedra



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PostPosted: Wed Dec 15, 2010 12:45 pm    Post subject: Microtonal keyboard - circle of 5ths Reply with quote  Mark this post and the followings unread

It's amazing where the circle of 5ths turns up... (yes this is going to get into some heavy-ish music theory!)

The layout I used on my microtonal keyboard (31 notes per octave which I'll be using as an equal temperament) is arranged so when you move a given distance in a particular direction, you will always get the same change in pitch (it's isomorphic).


vertical = 1 step

right&up diagonal = 3 steps (diatonic semitone)

left&up diagonal = 2 steps (chromatic semitone)

along a row (e.g. from one white key to the next) = 5 steps (whole tone)


A lot of the other microtonal keyboards I've been influenced by have this feature too. But I've noticed something special about these designs that have the staggered rows of keys. (I don't know who was the first to spot this, but here we go anyway!)

The 1st diagram out of the four shows an octave of the keyboard with all 31 notes being used. The label (1,31) on the key means for example that C is the first of the 31 notes in the octave. The D key is physically exactly half way between the top and the bottom of the keyboard. I've written out a circle of 5ths to the left of the octave with D in the middle.

So to the point of this post:
It turns out that with these keyboards, the vertical position of each key corresponds with its position in the circle of 5ths. D is exactly on the centre-line, G is slightly higher, then C, and so on. The notes that would sound more exotic or dissonant if played against a nice safe white-key mode like A minor, are the ones further towards the top and bottom of the keyboard.

Onto this keyboard you can map scales with fewer notes per octave (see the mappings below of 29, 19 & 17 notes, which are some of the best numbers for tonal/diatonic music). While writing this I'm genuinely surprised to find that the 19 mapping seems to be isomorphic - I didn't expect any of them to be.

Anyway, the fact that the key positions fit the circle of 5ths has a useful consequence. To find a mapping for an odd number of notes simply leave out keys, starting with the very highest (Gbb) and lowest (A##), and working your way in towards the middle until you only have left the number of keys that you need. To get an even number of notes you have to assign Ab and G# to the same pitch, which is useable if not perfect.

Having D in the middle (and Ab and G# ennharmonic in an even-number scale) goes with the fact that the octave is a symmetrical sequence (palindrome) around D - in terms of both the colour and the vertical position of each key. And I'm fairly sure this is true for any scale you map to this keyboard by the rules above.

Right, that's my brain properly toasted for one evening! Smile

Gordon


keyboard mappings.png
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Keyboard of 31 notes per octave, with scales of 31, 29, 19 & 17 mapped onto it
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seraph
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PostPosted: Wed Dec 15, 2010 2:30 pm    Post subject: Re: Microtonal keyboard - circle of 5ths Reply with quote  Mark this post and the followings unread

Octahedra wrote:

Right, that's my brain properly toasted for one evening! Smile

Gordon,
that's probably the first of a long list of epiphanies you'll get while working with microtones!
Excellent reasoning Very Happy

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Octahedra



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PostPosted: Thu Dec 16, 2010 12:23 am    Post subject: Re: Microtonal keyboard - circle of 5ths Reply with quote  Mark this post and the followings unread

seraph wrote:
that's probably the first of a long list of epiphanies you'll get while working with microtones


I hope so, although at the moment I'm just glad that I haven't got myself too badly stuck in a corner by designing the keyboard for just one favourite scale.

When I started this, I thought that by optimising the keyboard for one scale I'd be spoiling the others quite badly. This kind of keyboard of course gives you less freedom than a huge grid of identical hexagons! But I realised early on that there were decent mappings for some other scales. I only spotted the circle of 5ths explanation after building the thing, and I'm a lot more confident now about dividing the octave into lots of different numbers...

Have you settled on a 'definitive' Carlos Gamma mapping on your Chameleon yet, or are you still experimenting?

Gordon
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seraph
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PostPosted: Thu Dec 16, 2010 1:01 am    Post subject: Re: Microtonal keyboard - circle of 5ths Reply with quote  Mark this post and the followings unread

Octahedra wrote:

Have you settled on a 'definitive' Carlos Gamma mapping on your Chameleon yet, or are you still experimenting?



this is my latest note layout for Carlos Gamma Very Happy

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