Notating and Keyboarding

This page is strongly influenced by, and in many ways a response to, Erv Wilson's paper On Linear Notations and the Bosanquet Keyboard from Xenharmonikôn 3, among others. Go here for the index. I'll still try and make this page self explanatory for those of you who can't handle a load of bitmaps.

I've also found this page which has stuff on Bosanquet keyboards.

This (my) page isn't finished. However, it's been not finished for a few weeks now so I may as well put it up in its imperfect state.

the basics of generalized keyboards
duodecimally oriented scales
septimally oriented scales
generalizing generalized keyboards
case study: 3+4 scales
generalizing staff notation
piano roll type notation

So, on with the fun!

The Basics of Generalized Keyboards

In abstract terms, a generalized keyboard is a way of arranging a scale in two dimentsional space. Here's an example:

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

See the Wilson Archive for better pictures of these. This keyboard would be suitable for meantone scales. A related concept is that of 7+5 scales so let's say this is a way of keyboarding 7+5 scales. That is, the octave is made up of 7 large semitones, here called s, and 5 small semitones, here called r. A step up and to the right is r. A step down and to the left is s. So, any interval defined in terms of r and s is also defined on this keyboard (or one like it with more keys).

The philosophy behind generalized keyboards is that any given scale or chord can be transposed without changing its shape. Real keyboards can be made like this. Unfortunately, they're too expensive for most of us at the moment.

A similar layout for 5+7 scales would look like this:

C/    D/
   C#/   Eb/   F/    G/
C     D     E/    F#/   GA/   Bb/   C/
   C#    Eb    F     G     A/    B/   
C\    D\    E     F#    GA    Bb    C 
   C#\   Eb\   F\    G\    A     B    
            E\    F#\   GA\   Bb\   C\
                           A\    B\   

That uses my the positive notation I mentioned in my schismic scales page. If you understood the last diagram but not this one, don't worry, hopefully all will become clear by the end of the page. It's equivalent to Wilson's positive notation of the keyboard.

Duodecimally Oriented Scales

The second layout above is for what Wilson calls "duodecimally positive" scales. "Duodecimal" means 12, so duodecimal scales are based around 12 notes. The "positive" means these are positive linear temperaments in Bosanqet's terminology, or 5+7 scales in mine.

7+5 scales are duodecimally negative. They can be written with the same kind of mapping as this, only the layout will be reversed top to bottom. Sort of.

10+2 scales are also twelve based. However, they can't be written with this kind of keyboard layout. They are, in general, double-positive, or duodecimally double-positive.

Septimally Oriented Scales

This layout, which you can have again

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

Is based around 7 symbols, and so is septimally oriented. (Note that this has nothing to do with septimal harmony.) As such it is a useful for 5+2 scales. The same scale could be written in duodecimal form, as evidenced by the fact that it falls into 12 columns. For a meantone, B# would be lower than C, so C\, hence meantones are typically duodecimally negative.

However, this section is on septimal scales, so let's call it septimally positive. To make the septimal nature clear, I'll magically reconfigure my abstract keyboard to have 7 rows to the octave.

C#  D#  E#                    
C   D   E   F#  G#  A#  B#    
Cb  Db  Eb  F   G   A   B   C#
            Fb  Gb  Ab  Bb  C 

The positive-ness means the sharps go up the keyboard. A scale where C# is flatter then C is septimally negative, or a 2+5 scale.

Generalizing Generalized Keyboards

Well, the simple general case is that any m+n scale can be mapped to a keyboard with m+n rows. or m rows or n rows or 2m+n rows or whatever.

It's this positive/negative notation that gets me, though. I'll abbreviate it so that "12-" stands for "duodecimally negative", "7+" stands for "septimally positive" and "12++ stands for "duodecimally double-positive".

A scale is positive or negative depending on whether its fifth is sharper or flatter than that of the equal tempered scale it's based on. So, a 7+ scale will have a fifth sharper of that in 7-equal. The fifth here is defined as the best approximation to that in the basic scale.

Bosanqet's definition of nthly positive scales is that the Pythagorean comma is n steps. This can be generalized for other kinds of positivity: a scale is 7++ if 7 fifths are two steps sharper than 4 octaves.

An analogy can be made to m+n scales as follows:


The left hand column is my way of writing Bosanqet's terminology as extended by Wilson. The middle column is my terminology with some symbols given to the intervals. The higher letter in the alphabet is always the larger interval. The right hand column is a twist on my terminology, where it doesn't matter which is the larger interval. I'm using & instead of + to make this distinction clear. I can't be bothered to write & everytime, although the HTML guides tell me to do so.

There isn't a one to one mapping between the Bosanquet/Wilson terminology and mine. For example, 50-equal is a 7+5 scale (s=5, r=3) but it's 12--, not 12-. That's because its q is two steps. For all these scale types, the left hand column matches the right hand one for those cases where the right most letter stands for an interval of one step. That is q=1 for duodecimal scales, r=1 for septimal scales and s=1 for quintal scales.

I contend that the concept of 7+5 scales is more useful than that of 12- scales. All 12- scales are also 7+5 scales, but not all 7+5 scales are 12-. The same keyboard mapping can be used for all 7+5 scales. Although all equal temperaments can be uniquely assigned a degree of positivity or negativity, It is meaningless to say whether a 7+5 scale that isn't an equal temperament is really 12- or 12--. Although the closest match is obviously with 12-.

Case Study: 3+4 Scales

As all these new temperament classes won't mean much to you, I'll discuss the case of 3+4 scales. These are analagous to 7++ scales, but not the same as I explained just above.

To write down 7++ scales, I'll use the normal note names and re-assign them. Here's what I'll make the standard in terms of t and s, where t>=s:

 s t s t s t s 

It can then be mapped to a 7 column keyboard as follows:

   D#    F#    A#    C#
C     E     G     B    
   D     F     A     C 
Cb    Eb    Gb    Bb   

The interval C-G can be associated with a fifth. As E splits the fifth equally, it is a neutral third. So, this is a type of temperament based around fifths and neutral thirds. As a neutral third is very close to 11/9, we would expect these scales to provide good approximations to the prime numbers 11 and 3.

The set of 7++ scales, all of which are also 3+4 scales, is 3, 10, 17, 24, 31, 38, 45, 52, ... and some more 3+4 scales are 6, 13, 20, 27, 34, 41, 48, 55, ... The second set is actually that of 7--- scales, which is equivalent to the mythical 7++++. There are indeed some good 11-limit scales here, and how refreshing to find them by putting 11 and 3 ahead of 5 and 7.

After working all this stuff out however long ago it was, I've got quite involved in these scales. I hope to put some more documentation and some MIDI files up as evidence of this.

Generalizing Staff Notation

Conventional staff notation works pretty well as a way of writing septimal scales, positive double positive or whatever. Wilson suggests using a modified # symbol for shifts of the equivalent of a chromatic semitone. This is the amount by which 7 fifths exceed 4 0ctaves. I suggest using # for raising a note by the interval I call r in the scales above. That means C# is always between C and D. I think this makes sense, but it's all a matter of taste.

I suppose it all depends on how familiar you are with conventional notation. If the staff is fixed in your mind as a way of writing down scales with 12 notes to the octave, it may be difficult to break out of this. However, there's nothing about the way it's all written to even suggest there should be 7 notes to the octave. I'm sufficiently familiar with it, though, that I would have difficulty if this were not the case.

If you understand any form of notation for 12-equal, you can bend it to notate any duodecimal scale!

Piano Roll Type Notation

This is something you always get with sequencers. Although some microtonalists object to systems of notation based explicitly around 12 notes to the octave, I welcome as a way of writing meantone and schismic scales. All you need to do is define the modifiers and the standard scale. I would like to hotwire a sequencer to produce such a microtonal piano roll view.

The other thing is that it's pretty easy to adapt the piano roll to scales that aren't based around 12 notes to the octave. Because you can make the 12-ness clear, you can also make 11-ness clear when the need arises.

I'm also planning to add a section on the Bosanquet keyboard, explaining which scale types fit it well, and which do not.

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