A Composition, and a concept

This piece is basically the finished version of the little draft I previously posted; some structure was added, some stuff removed, some stuff adjusted. Basically I am somewhat happy with it, and I think I am going to leave it well enough alone now for a while.

The name - lambda calculus - refers to the scale it is in, and it is this scale I want to discuss now. But first, let me refresh you on regular scales. I will express this in a roundabout manner, just to illustrate the kind of challenge microtonal music sometimes forces on composers.

The regular major scale, natural minor, mixolydian etc - the white keys on the piano - consist of a stack of fifths. You basically start on F, and you go (...Eb → Bb → )F → C → G → D → A → E → B (→ F# → C# →  ...). So we basically have picked six notes that are adjacent by the cycle of fifths. 

Let us imagine we want to know which chords are major and which are minor in the scale C D EF G A Bc. We know the major third is wider than the minor third, and we know the diminished fifth is less wide than the regular fifth, so we take a paper with grids, and place the notes so that an empty grid represents a non-scale tone, and one filled with a note name represents, obviously, that note. We take another paper and cut out a small stencil, where we leave an opening for C, E and G. We continue the scale so we have C D EF G A Bc d ef. Now, we can slide our stencil along the scale, and see which tone is the root of a major chord, just by looking at when the openings in the stencil have a named note or not in them. We can make a similar stencil for minor chords, diminished chords, seventh chords, etc. But this is tedious and inefficient.

Turns out one reason we don't do this is that musicians learn these things by heart or at least to really intuitively calculate it. Ask me in the middle of the night which chords in F# major are minor chords, and I will tell you. However, let us figure out a reasonable way of doing this. So, we notice that a major chord will retain the same shape also along the cycle of fifths:
F → C → G → D → A → E → B The bolded tones form the F major chord.
F → C → G → D → A → E → B The bolded tones form the C major chord.

Basically, we are back at cutting stencils, in a manner of speaking. However, let us figure out something more useful than the stencil - distance! We notice that the major third and the fifth both reside in the same direction from the root, so if there is a major third, there is also a fifth. We thus ignore the fifth for now, and focus on
F → ... → ... → ... → ...A
So basically, if there are four fifths to the right of our root, we have a major chord. We know our entire thing is constructed by stacking fifths, so we ask how many such we have stacked, and it turns out that there are seven. Thus, we can move this structure twice to the right before the third falls out of the permitted range.

We can do the same with regards to the minor chord:
F → C → G → D → A → E → B (minor chord, note that I marked the chord root with red)

We can tell this cannot be shifted to the left without having to go outside of the key of C (since then we'd have to go to the left of F for the minor third), but we notice we can shift it up to twice to the right.

This obviously doesn't work if we order the tones by size:
C → D → E → F → G → A → B → c
C → D → E → F → G → A → B → c
The chord shape's been shifted once, but since the step sizes are not uniform, this has changed the type of the chord.

Now, these are things any musician who's done a fair share of chord practice and theory does without reflecting on it, and I doubt the above methods are used in the mental computations most musicians do. I am pretty sure I do this by thinking of chord shapes and knowing which fret on which string produces what tone, for instance, and I bet pianists are more likely to do it by reference to the keyboard somehow.

So, we know now that F, C and G produce major chords, and D, A and E produce minor chords. Thus, if we reorder these along the usual scale we get the familiar, downright old C major, D minor, E minor, F major, G major, A minor thing. B dim also exists, obviously, but I will not go into that at the time being.

So, this might seem to lack utility. But, what if we suddenly build a different scale - but base our new scale on similar principles? Let us imagine we stack nine fifths:
F →C → G → D → A → E → B → F# → C# → G#

And let's say we want to know how many minadd9 chords we get. If we map it out in something along the lines of a normal order we get C, C#, D, E, F, F#, G, G#,A, B, c. Just looking at that set of notes doesn't tell me much. So, we look at the structure we're looking for in the segment of the cycle of fifths we're using:
F →C → G → D → A → E → B → F# → C# → G#
This structure can be pushed all the way to
F →C → G → D → A → E → B → F# → C# → G#
So we know now that D, A, E, B and F# all can be roots of that chord-shape without going outside of the scale.

So, some kind of use might be obvious from this, but let's go onwards even further. What if we're using a scale that not only is different from the major scale, but uses a different generator. In 12-tet, we've only got two generators that can produce scales with the kind of "irregular" structure that the major and natural minor scales have: the fifth and the fourth; since fifths and fourths are each other's inverses, they basically build the same scales but from the opposite ends. The reason only the fifth and fourth really can be used for this is that the only numbers under twelve that do not share any factors with 12 are 1, 5, 7 and 11. 1 and 11 generate fairly boring scales though: C → C# → D  → D# → ...

One number that has no lower numbers sharing any factors with it is 13. There has been some recent development in the xenharmonic community regarding the use of 13-tone equal temperament, and composers are increasingly learning how to use it to good effect. However, I've gotten interested in another 13-based temperament, viz. Bohlen-Pierce. Bohlen-Pierce divides the octave's fifth - the ratio 3/1 - into 13 equal steps - instead of using the octave, we use the perfect twelfth as a pseudooctave.


Now, for some reason this scale depicted I've come to use a scale that uses 10 or 3 as the size of the 'generator', so basically a cycle of thirds (or tenths) - either a charming sharp third (9/7, for those who have any idea about just intonation things), or a low flat tenth (7/3).
The full chromatic scale is thirteen tones, and the scale I use is called lambda.

If we do like the piano keyboard, and make scale member notes white keys, and chromatic notes between them black keys, we get a layout like this (I was going to make a css thing to generate keyboard layout illustrations using divs for this post, but alas, it needs some debugging, will be incorporated later):

C ^C# D E F ^F# G H ^H# J A ^A# B C

The notes have been named C DEF GH JA Bc (the spaces are distributed like the tones of the temperament that are not in the scale), where for now, C=C3, c=G4. To reduce confusion, I've decided to use bold italic for bohlen-pierce. Now, applying chord-intuitions is immediately somewhat less trivial! Obviously, the chords available in such a system will be different from the usual chords; we have certain chords in Bohlen-pierce that sound relatively nice, due to mathematical reasons. These include sets of tones such as CGA, CFA, CFH and CEH. How do we find which roots other than C give us these chords, without having chord member tones going into the chromatic zone? (It is not a big problem if they do, but knowing that that is happening might be useful.)

We rearrange the scale as a chain of thirds:
F H A C E G J B D
We spot the structures we're looking for:
F H A C E G J B D
F H A C E G J B D
F H A C E G J B D
F H A C E G J B D

And we can now easily tell how far we can move these shapes around until they drop off the edge of the scale. In addition, it turns out some musicians have found that it pays off to change D to D# on occasion to get progressions to work out well - this is fairly analogous to how the melodic minor permits a lot of wonky stuff in the upper notes. Unlike the melodic minor, though, such a change actually is the same as transposing the scale by a third. We add that note to the scale, but mark it with parentheses:
(Db) F H A C E G J B D
There is a genuine problem with this approach, that may pop up if the chord structure we are looking for is wider than the full set of omitted tones, i.e. if we are looking for the interval corresponding to F-B, we will find that the same interval appears stretched over the range of omitted tones.

F H A C E G J B D F# H# A# C# E#
E# = F in this scale, so we've finished a full rotation, and we substitute that in and go
F H A C E G J B D F# H# A# C# F H A C E G J B D...
We see that the interval F-B - which we'll call a major sixth (essentially the same as our dominant seventh, btw) also appears for H-D. but for A,C,E,G it does not. The interval starts making an appearance with J, however - J-F,B-H,D-A. However, for these it corresponds to another number of steps in the scale, and calling it a sixth would therefore be wrong. (This is like how the tritonus is a diminished fifth when it's B → f or C→Gb, but an augmented fourth when it is F → B or C# → G. This type of overlap between interval classes is more common in Lambda.)

The point I am going for, really, is that working with unusual scales - especially scales where some of the predictable mathematical properties of 12tet break down, properties we all are so familiar with that we seldom even lend them a second thought - forces learning some interesting approaches to things. Many musicians learn these chord-scale relationships over many years of practice, sight-reading, listening and so on. Relearning this for other kinds of structures without getting confused is a challenge, and relearning it for several in a short time is far from trivial.