Music for Conworlders: Polyphony

Polyphony is the texture in which you have melodies simultaneously. The interest is created by their rhythmic and melodic independence.

Several styles of polyphony exist in western music. Out of these, some are fairly obscure local folk styles (e.g. Outer Hebrides choral singing, various folk song styles in the Balkans, etc), but there is one that is not all that obscure – in fact, fairly prominently known. It has the additional benefit of being well formalized, well studied, and material on it is easily available - I am of course speaking of baroque counterpoint.

Baroque polyphony developed out of renaissance polyphony, which developed out of medieval polyphony. Each of these nodes in the family tree of western counterpoint has branches that did not contribute much to the descendants.

The formalizations that exist may be a bit too strict in the sense that they do in fact forbid things that were common. In part, it seems these formalizations might have been more like guidelines or even good things to practice, because doing so made one more aware of the possibilities and problems that various things brought with them.

The rules of counterpoint operate on several levels: there's rules that govern how melodies are to be written. In part, these seem to favour for single melodies that they be identified as "a single voice". There is probably some form of biological reason why we tend to hear things with very similar pitches as coming from the same source if they have the same timbre, but large differences in pitch even on the same timbre is easily parsed as multiple separate sources of sound. This, in fact, is an easy way of "faking" polyphony: make multiple big jumps. The brain will parse distant melodical snippets as part of different melodies, and those that are close together in pitch, but separated by intervening notes in other registers will still be heard as something akin to a melody.

For pairs (or larger sets) of melodies, the rules seem to favour designing them in ways that keep them distinct, so that they not merge into single "supertone melodies" – i.e. if you do a lot of, say, parallel movement, the brain will soon merge the separate melodies that are moving in parallel.

However, only when melodies are very consonant does the risk for that seem to be very high - and this is why parallel fifths and octaves were quite vigorously avoided in baroque music (except as a timbral effect!). Thirds and sixths are slightly less consonant, so some amount of parallel movement on those is not forbidden. With a different tuning or different timbre, the consonance profiles of our different intervals might make the ban on parallel movement less important, or maybe even more important - parallel movement in just intonation might make consonant intervals almost vanish into each other, conversely, 19 tone equal temperament might make parallel fifths unbearably badly tuned?

So, counterpoint favours melodies that move in different directions a lot. Of course, as soon as you have more than two melodies, you'll have to compromise on that at least some of the time. (If you have more than three, it's even impossible not to compromise on it). You have to end up deciding which voices to move similarly or parallelly, etc.

It pays off to read up on the rules of counterpoint, and consider their role in the musical context, and then consider to what extent they can be applied in your conculture's music.

In western music, an important aspect has been development. Development is reusing melodical elements in new combinations with other material - either material you already presented earlier in the piece, or new material. Development seems rather uncommon globally – variations are a much more popular thing worldwide. A variation is simply a melody with some elements changed – a sequence turns down instead of going up, a tone is substituted for another, the order of some tones is reversed, etc.

Microtonal Theorizing: A roadmap to my ideas

So, while I am writing these posts I am also kind of developing my own take on the ideas I present – this is why it takes the better part of forever to make new posts. In large parts, this is stuff based on things other xenharmonists and theorists have worked out, but it probably bears some undeniable traces of my own mind as well: I might have misunderstood other theorists' ideas, or maybe I have even introduced some ideas of my own.

So, a roadmap:

• overtone series and JI to approximations of finite subsets thereof in an equal temperament
• subsets of the overtone series, esp. partially overlapping ones given different starting points in the equal temperament
•  from temperaments and JI to chords as subsets of an ET
• from temperaments and JI to MOSes and MODMOSes as subsets of an ET
• chords as subsets of MOSes and MODMOSes
• movements by restricted paths /by step-size or step-type combinations) within and between group-structures of chords within a temperament; the subset of movements that is within a MOS/MODMOS is of special interest
• movements by restricted paths from MOS/MODMOS to rotation of the same MOS/MODMOS as a group operation. movements from one type of MOS/MODMOS to another type as a group operation.
• OTHER thing altogether: Comma pumps. Making comma pumps and the above come together. Using the inter-MOS/MODMOS modulation or extra-MOS/MODMOS modulation to create compelling chord progressions
• embedding MOSes/MODMOSes in larger MOSes/MODMOSes and relating this to the previous set of ideas (i.e. melodies moving in chords moving in scales^{moving in scalesmoving in scales...})
• Question: is there anything analogous to the comma pump in higher-level modulation?
• Question: does any other weird phenomenon happen in modulation that involves multiple levels of modulation, where weird things start happening?

Tuning Theory for Conworlders: On With 11 - Chords

Now, we've previously considered the intervals of 12-tone equal temperament as approximations of a specific set of just intonation intervals, viz. something along the lines of the following:

1/1 16/15 9/8 6/5 5/4 4/3 45/32 3/2 8/5 5/3 16/9 15/8 just intonation major scale intervals bolded, other intervals of some kind of chromatic just intonation scale intervals inbetween

For the record, 16/15 ≃ 17/16, 8/5 ≃ 25/16, 6/5 ≃ 64/54, 16/9 ≃ 9/5, etc.
What just intonation intervals do we approximate now?

We have previously noted how an interval in a temperament is an approximation of another interval - such as four steps in 12-edo approximating 5/4. However, a thing I never said explicitly is that the tempered interval really approximates multiple just intonation intervals. C-E thus approximates 5/4, 81/64, 65536/50625, etc. Some of these approximations may have really big errors, however - and the error might grow as we stack them - even if C-E approximates 5/4, this doesn't mean C-G# (which is C→E→G#, that is, we repeat the same interval twice) is a very good approximation of (5/4)². Now, one thing we might notice is that 11-edo's minor second is pretty close to that of 12-edo, so they probably approximate the very same intervals - we can probably claim all of 16/15, 17/16, 18/17 and even maybe 15/14 as intervals it approximates. Of course, we also have intervals further up the overtone series we could claim - 181/170 very probably could be included, for one. (Try calculating the cents value for these intervals and compare to the cent value obtained for 11-edo above.) (In fact, a simple way of showing that any interval can approximate any number of intervals is to decide on a permissible error, then find a ratio within that error boundary, increase the denominator and enumerator by some factor, then reduce or increase one of them just a little towards the 'center' of the permitted range. We can repeat this procedure an infinite number of times.)

When looking at the major second, we get a much wider gap. We're somewhere between 9/8 and 8/7. So we just go ahead and sort of claim both as being represented by the second.

The minor third is worse. Is it a rather wide 6/5? A really narrow 11/9? Is it 17/14? Note - multiple options may all work for us.

The major third is somewhat close to 9/7, but also to 14/11 and 22/17. This is a good time to note certain things: if we want to build chords, we may want to find reasonably simple series in this. Major chords, as noted, consist of the very elegant triplet 4:5:6. We won't find something that simple in 11-edo, because our 3/2 (6:4) is totally missing. However, to make it easier to find series, we might want to try and find a) things with similar denominators, or b) things with similar enumerators. If we find a/a, b/a, c/a, however, we'll also find c/c. c/b, c/a. So we only need to look for either the same denominators or the same enumerators.

We go on: the perfect fourth is pretty close to 11/8, but also 15/11. The "perfect fifth", of course, is 16/11 (inverse of 11/8), or 22/15 (inverse of 15/11).

Minor sixth - 11/7, 14/9, 17/11.
Major sixth - 5/3, 18/11, 28/17
Minor seventh - 9/5, 16/9, 14/8, 30/17
Major seventh - 15/8, 17/9, 28/15, 32/17

So, given in the form above, we get something like (and note that each number may be replaced by other options):

1/1 17/16 8/7 17/14 14/11 11/8 16/11 11/7 18/11 14/8 17/9

 I won't arrange them as a table, however, arranging them by tables such that a column represents a shared denominator is a good idea. Recall that e.g. 7/4 = 14/8, so you may want to reproduce any number with relatively simple denominators in terms of prime factorization in other columns, so .e.g 7/4 also appears as 21/12, 14/8, etc. Anyways, some series I have found in this are:

8:11:14:(17) (so, starting at 8/8 we add 3 to the enumerator at each step!)
7:9:11 (13/7 and 13/9 are reasonably well approximated as well, so that might be an option - 13/11 is badly approximated, though - but of course, the precision you may want is up to you)
17:22:28 (17+5, 22+6)

I like having a common constant added at each step: a/a, (a+b)/a, (a+2b)/a, (a+3b)/a ..., but this is not strictly speaking a necessity. 8:9:12 chords sound fairly good and are popular in some western music, as are 6:8:9 chords.

These differ from the 4:5:6 triplet in a few interesting ways:
8:11:14:17 starts with a wide fourth, on top of which there's a major third; these add up to a minor seventh (unlike the major sixth we'd get in 12-edo by stacking the intervals that most closely correspond over there). The chord thus covers the octave quite differently from our major triplet: root-fourth-seventh, rather than root-third-fifth. Even if we compared the most similar inversions, we get a slight difference: 8:11:14 -> root-fourth-seventh, 4:5:6 -> root-fourth-sixth.

Let's try some rather unrestricted chord progressions using chords of these kinds. My samples here are not in 11-tet, but rather use (almost) just intonation and lots of inversions and "smooth voice leading", i.e. I try to move voices by small steps and preferrably as few changes when going from one chord to the next:

Now, the chords given above are almost perfectly in tune compared to the overtone series; in 11-edo, this is a luxury we don't have. On the other hand, we gain another benefit: we won't keep drifting by tiny bits of a semitone for (almost) every chord change we do.

Another thing, though - we have not yet considered how these chords fit into any scale - the above progressions are just unstructured movements around pitch space, which can be all nice and dandy, but it's hard to compose with such an open-ended pitch-world. Let's try building scales with this chord as our building block. Our chord consists of the tones at indexes {0, 5, 9}. Let us build an identical chord from 9: {9, 3, 7}. We now have {0,3,5,7,9} for our scale. A reasonable pentatonic scale. It also gives us two instances of the "minor" version of our chord - {5,9,3}, {7,0,5}. The scale we now have obtained goes by the name machine[5] in xenharmonic circles. Usually, it is constructed as a stack of major seconds - starting at the tone we now indexed '3'. We could repeat this operation, by stacking a new chord on top of 7: {7, 1, 5}. We're in luck - we only add one tone to the entire scale, giving us {0,1,3,5,7,9}, which is machine[6], starting at index 1. Much like each major scale (meantone[7]) contains three pentatonic major scales (meantone[5]) (CDEFGABC contains CDEGA, FGACD and GABCD), machine[6] contains two machine[5] - {1,3,5,7,9,11} contains both {0,3,5,7,9} and {9,1,3,5,7}. This can be of some melodic interest, possibly, or of some use for transposition and the like?

We could try stacking our chords in a different manner. {0,5,9}, and a new identical stack on top of 5: {5,10,3}, and maybe even once more on top of 10: {10, 4, 8}. We now have {0,3,4,5,8,9,10}. This does not look all that great - lots of huge stacks of minor seconds, coupled with large empty swathes.

How about one chord on top of both member tones of the original chord:
{0,5,9}, [5,10,3}, {9,3,7}. We get {0,3,5,7,10} - a slightly uneven pentatonic scale. Might be useful though.

What about the utonal chord? {0,4,9}. Repeated stacking on top of the topmost note gives {0,4,9}, {9,2,7}, {7,0,5} which gives {0,2,4,5,7,9} which is a rotation of the {0, 1, 3, 5 ,7, 9} scale we already looked at once.

Music Theory for Conworlders: An Example: 11-tone Equal Temperament

Looking at something that is very close to 12-tone equal temperament in one sense (very similar size for the semitone), yet very different in other senses (complete lack of perfect fifths and fourths, very sharp major thirds and minor thirds, ...) might be useful.

11-tone equal temperament's lack of fifths might seem to be a rather challenging thing, given the prevalence of fifths in the music we're used to. Let us compare the intervals of 11 and 12. In the cents given, I will underline repeating sequences. 12-edo to the left, 11-edo to the right.

minor second: 100 cents, 109.09... cents
major seconds 200 cents, 218.18... cents
minor third: 300 cents, 327.27... cents
major third: 400 cents, 436.36... cents
perfect fourth: 500 cents, 545.45... cents

Here already we're seeing a pretty large divergence already - the perfect fourth is already almost as close to the tritone as it is to the perfect fourth. We compare:

tritone: 600 cents; 545.45... cents? 654.54... cents?

Therefore, we'll consider the next half of the octave to be flat versions of intervals - rather than sharp versions as we did in the first half:

perfect fifth: 700 cents, 654.54... cents
minor sixth: 800 cents, 763.63... cents
major sixth: 900 cents, 872.72... cents
minor seventh: 1000 cents, 981.81... cents
major seventh: 1100 cents, 1090.90...

So, our main "weird" place is the fourths, fifths, and tritones - none of them seem to have very good intervals to correspond to. Of course the thirds as well are pretty far off - only the two kinds of seconds really do rather well (but even then, they're somewhat off). Here's our familiar composition (the mixolydian version) with all intervals replaced by their closest "neighbour" in the other tuning.

This is not particularly beautiful, is it? We will come back to this composition with some reworkings later on. For now, we'll try and consider scales, chords and chord-scale relationships. When we come back to the piece a bit further down, we'll have "better" chords, and maybe have figured out what particular rotation of what particular scale works best (or at least better) for the composition.

The scale I was using in that piece was, in cents, like this:

0, 218, 436, 545, 655, 872, 981

or in terms of step size:

LLssLsL

or in terms of the distribution of step-sizes between the notes: C _L͡  D _L͡  E_sF_sG _L͡  A_sB _L͡  C. L have ties on them to illustrate that they are 'bigger' than s. If we were to use piano keyboard design to illustrate this scale, it would be like this:

The keys at both ends are C. 

We find that the scale we're now using is not a MOS. (It is a MODMOS, however - a scale that is a 'distortion' of a MOS, given certain restrictions on what kinds of distortions we may use. Many MODMOSes are very useful - and some MOSes might not be! However, let's try and see if a MOS is a good tool for this right now.). By changing one note, we do get a MOS, the results being here:

Now the scale is LsLsLsL. Slightly more of a 'minor' tonality there. We alter the one tone we need altered (viz. we lower one note - basically, our E becomes an Eb). We're now playing on this somewhat more "evenly distributed" keyboard:

Notice again, how the keys at the edge are C:s.

In terms of cents, this is 0, 218, 327, 545, 655, 872, 981. The scale we're now using has a name, the history of which I don't quite know. It's orgone temperament.

Anyways, we may notice the chords are not particularly pretty there - it sounds somewhat off all the way through. We shall go and attempt to see if we could find something even more off - that does sound somewhat good or at least has its own logic, or something else along those lines in the next post. For this, we'll need to look at several different things - chords as approximations of parts of the overtone series, smooth voice movement from chord to chord, and how chords relate to the scales in which they reside.

A New Song

I had an idea and made it into a song pretty quickly. It did develop in some ways I had not expected.

This is still a bit of a work in progress, as I am not entirely happy with it. You might notice there's still some rough edges, and I'd guess some work on the structure might be good too.

It starts out in 11 edo/orgone, in this particular scale structure:

A

A#

B

C

C#

D

D#

E

F

F#

G

Slightly misleading in a sense, of course, since we're working with eleven tones per octave here - D# is about halfway between 'regular' D# and E (with regards to A), and F# is really closer to 'regular' G than the G given above is.

The funkier bit is in 11 edo/machine, in this particular mode:

A

A#

B

C

C#

D

D#

E

F

F#

G

We can compare the two, obviously, by setting them next to each other:

A

A#

B

C

C#

D

D#

E

F

F#

G

A

A#

B

C

C#

D

D#

E

F

F#

G

The notes I've taken to calling C# and G, which are slight misnomers, are lacking in both.

Music Theory for Conworlders: Other MOS structures

Okay, so I've established a handful of MOS structures this far:

ssLsL (pentatonic major, s = major second, L = minor third)
LLsLLLs (diatonic major, L = major second, s = minor second)
ssLsssL ('anti-diatonic' mavila, s = slightly wide minor second, L = slightly wide major second)

I am not sure if I gave any 'generally' applicable MOS patterns, but there's one pretty simple one. Examples of it will appear in any set:

AA*B 

where * is the kleene star. This reads out 'one or more A before one B'. Let A, B ∈ {L,s} and A≠B. Since we permit 'rotating' the set, M = A*BA*, where |M| > 2 also is a reasonable description - viz. any string that is at least two symbols long and has at most one B, and an arbitrary number of As before and/or after the B.

This gives us things such as LLLLs, ssssL, LLLLLs, sssssssssL, ... the reason why these are all MOSes should be fairly obvious - all seconds except one will be of the form A (and one of the form B), all thirds will be AA, except two - one will be AB, one will be BA, but the length of AB and BA is the same so we can consider them both to have the length |A|+|B|; for fourths, we get ABA, BAA and AAB as well as AAA, etc.

But these are only a subset of possible MOSes. We can look at an example that is in some use in Bohlen-Pierce:

LssLsLsLs

Or at the 11 edo version of a temperament by the name 'orgone':

LsLsLsL

and another one which I have not found a name for:

ssLssLsL

The plot thickens. Apparently, (AB)*A or (AB)*A(AB)* or somesuch qualifies - Any number of ABs, where A, B ∈ {L,s} and A≠B, and one A at the end - again, permitting rotation of these as well.
The last example also indicates that maybe things like (AB^N)*AB^N-1? I am not actually entirely sure on whether all scales generated by that form will qualify, although all the previous forms given here do qualify.

And of course, I have previously mentioned the trick of pretending that a MOS is really an equal temperament, and then picking out notes within that MOS by numbering the notes as a sequence and then picking such notes by use of MOSes. I.e. Pentatonic-in-seven as MOSes within the diatonic scale, here given with the 'root' for each 'basic' rotation in large print. Keep in mind that one can further of course 'rotate' within the subset itself:

C	D	E	F	G	A	B
*	*	*		*	*
	*	*	*		*	*
*		*	*	*		*
*	*		*	*	*
	*	*		*	*	*
*		*	*		*	*
*	*		*	*		*

If we were to look at each of these scales in terms of 12-tone equal temperament intervals from the scale root, we'd get:

min2	maj2	min3	maj3	p4	aug4	p5	min6	maj6	min7	maj7
	*		*			*		*
	*	*				*		*
*		*				*	*
	*		*			*		*
	*		*			*		*
	*	*				*	*
*		*			*		*

And all of these have their own rotations as well, except of course the three ones I've marked with red, which have identical structures and thus we can ignore the rotations two of them contribute, and the same goes for the green pair. A short not very rigorous investigation suggests to me that the rotations. I probably should code a thing to generate and rotate these for me and then output them as html tables, but I am a bit too tired today.

Music Theory for Conworlders: Texture

I have this far developed theory that is suitable for harmony and counterpoint, but I have not really talked much about the concepts of harmony and counterpoint - in fact, I've remained quite silent as to why those would be of any interest whatsoever.

In many human cultures, the usual texture has been one of monophony.

Monophonic version of the familiar tune. The slight change in rhythm has nothing to do with the change in texture, but I felt like it added some slight interest. Note that the file has the wrong name.

Monophony simply consists of one pitch played at a time. The pitch can be played by multiple instruments - and in fact lots of middle east music basically consists of a whole ensemble playing the same melodies at the same pitch, and the combinations of instruments simply create different complex timbres. Timbral variation can be obtained by, say, having some instruments be silent in some parts. Whether octaves are to be considered separate pitches for this classification differs between authorities, but if the only simultaneous interval that is permitted is octaves, you probably won't go wrong by calling it monophony.

Examples of monophony include the various plainchants, Chinese art music and Arabic traditional music. Apparently also several African vocal traditions. You've also probably produced monophony yourself - singing unaccompanied is a form of monophony.

A slightly extended monophony can be achieved by including a drone - a tone that basically is sustained throughout the piece. Examples of this style can be found in many places - most bagpipe traditions, the hurdy gurdy and related instruments, and lots of Indian music. Continuing on this, we can again listen to the same tune:

Monophonic tune with C as its drone. 

The same tune can be turned into a somewhat different tune simply by changing the drone - and by that realization, we're getting quite close to homophony instead.

Same monophonic tune with A as its drone instead of C.

As a final comment on drones, they need of course not be bass notes - they could be higher in pitch than the melody stuff as well.

Homophony has more voices than just the melody - however, the other voices mainly serve to create a harmonic background, and not to provide all that much in ways of harmonic interest. Therefore, their rhythmic behaviour tends to be pretty similar. The versions of my tune with chords basically all have been pretty simple homophony.

One further type of texture is called heterophony.

A rather fantastic tradition of art music that is replete with heterophony is the Indonesian gamelan. It is well worth familiarizing yourself with it - there's hundreds of records, and of course youtube videos. Basically, heterophony consists of multiple voices playing the same melody - but they occasionally deviate in ways peculiar to their own style. Imagine that two people who have learned the same song, but slightly differently, come together and try singing it in unison - but occasionally, they deviate from each other's lines - their rhythm differs or their pitch selection differs, or both.

Finally, there's polyphony. Polyphony is famously present in Baroque music, but also appears in lots of African styles, Georgian folk music, various musics of the Balkans.  Polyphony.ge has a map over different forms of polyphony over the world, and a variety of interesting information in general.

Polyphony as a texture has a lot of variation. Generally, there's more than one melody simultaneously, and they both provide some interest. The interactions between the two or more melodies should be the central idea. Polyphony is quite complicated, however, and I will probably deal more with that in a separate post.