Fifteen Note Equal Temperment

with examples

...........what in the world??


Introduction

    In the dim ages of the past, I wrote my Master's Thesis in Music Composition. At the time (1985), making music on a personal computer was still very much in its infancy, midi was barely getting started to be used, and the glitziest gassiest machine available was the Commodore Amiga, which I had one of.

    Now, to prove that I had actually thought about stuff to the graduate committee, I actually had to think about stuff for a while. And what I came up with was (and, believe it or not, this is the condensed version), this:

    If one is going to go to all the trouble to write music for computers that is to be performed, one should go to all the trouble to write music that computers would be good at. That is, one should exploit the things that computers can do that humans cannot. Or at the very least, have a hard time with.  At the time, there seemed to be 3 things that computers could do better than people:

    1) Pick up an instrument, play a note, put it down, pick up another instrument, and play a note. All within a millisecond. My thesis piece involved an orchestration that did that with some success - the software I used had an limited additive synthesis engine (only 5 harmonics of the 1st 32 overtones could be used on any instrument and only 16 instruments available to choose from) with  constant rapid successive change in the instrument used.
    2) Play a note at any time. To some degree, my thesis piece did this - I used very difficult meter changes (15/16, 7/8, 11/16, etc) at very fast tempos. There were, though, timing resolution limitations that could not be broken.
    3) Play any pitch whatsoever. I only had moderate success with this. The software I used has limitations of only 4 note polyphony, and 12-note equal temperment.

    The piece did not go far in demonstrating any true prowess on my part toward the full realization of these ideals, but I had the advantage of this all being new, so I got away with it all its deficiencies. Maybe someday I'll post it here.

Introduction, part 2

    The 3rd Principle from above is something I have since thought about for a long time, and came to the obvious conclusion that a hallmark of computer music should involve microtuning, as the computer can realize all 4000+ microtuning systems that come with Scala with (relative) ease.  What I found was that there were oodles of the scales that only changed part of  the character of the piece. What would happen was that different chords would sound better in some temperments/tunings, worse in others. But the different temperment/tunings were difficult to recognize as being significant to the musical ideas: they just changed the rendition. At best, the traditional alternative tunings (meantone, Pythagorean, just) didn't increase the possibilities - they decreased them. I found myself shying away from complex chord progressions because they had bad elements in the alternative tunings that were glossed over (and thereby usable) by 12-note equal temperment.

    Enter Easley Blackwood.

    In 1980 Easley Blackwood released the Microtonal Etudes. They were a set of pieces he wrote when he got an endowment to explore the possibilities of extended equal temperments. Instead of 12 notes per octave, the Etudes comprised of pieces made with 13 thru 24 equally-tempered notes per octave. What Blackwood found was astounding: instead of all being a terrible mish-mash of unlistenable trash, there were parts of the extended equal temperment systems that had musical sense in them. For sure, there was no shortage of note-combinations that are generally agreed to sound terrible. What was amazing was the number of acceptable-to-good chords.

    Of all the things Blackwood tried, he indicated that 15-note equal temperment was the likely candidate for the next-big-thing. Of course, that hasn't panned out yet. The reason he thought this was that harmonically speaking, 15-note presented the composer with some acceptable triads to resolve to, and chord progressions that simply aren't available in 12-note equal temperment, and a variety of modes with which to experiment. It sounds alien and often out-of-tune, for sure. Useless for the existing 12-note repertoire. But it gives us something we haven't had for a long time: New Harmonic Material.

    The main reason it hasn't panned out is that it's hard. Hard to think about, hard to play, hard to get one's head around it at all. Blackwood wrote some stuff to try to enlighten us lesser mortals, and the one I'm interested in today is:
 Blackwood, Easley. "Modes and Chord Progressions in Equal Tunings,"  Perspectives of New Music, Vol. 39, No. 2, 1992, pp. 167-200. 
They'll sock you up for $30 for a reprint here:  http://www.perspectivesofnewmusic.org/backissues.html   I found it at the local college library. Sections 17-26 deal with 15-note equal temperment. This page provides some renderings of the examples. I can't of course, repost the section of the article without permission. But I can paraphrase. Badly, too - and get away with it.

Definitions and Conventions.

    If one were to take an octave of 1200 cents, divide it by 12, you'd have 12) 100 cent divisions (duh).

    If one to take that same 1200 cent octave, divide it by 15, you'd have 15) 80 cent divisions. And hence the 1st problem. 3 extra notes to put one's fingers on.

    Blackwood took a traditional theorist's  approach to the 15-note scale and notation system by following circles of fifths, and he found that if he followed that idea far enough, the pitch E was enharmonically equivalent to the pitch F.  Now, if this wasn't confusing enough for us lesser mortals, in order to create a triad that was similar to one in 12-note equal temperment (of C-E-G), he was forced to reason that since E=F in 15-note, the  pitch that was closest the the middle note of the target triad that proved agreeable to listen to was in fact called E-Down, which was 80 cents lower than the E-F enharmonic pitch, which was not the E-F that it looks like on the page. Funny thing was, that this E-down that was derived from F that was only sort of like the 12-note pitch of F  was in fact the exact same pitch as 12-note equally tempered E. Confused yet?

    Blackwood laid out the notes with some enharmonic differences, and an extra symbol that indicated it was "up"[ ↑ ]  (or "down" [  ↓ ]) from the base pitch which may or may note be related to the similarly-named note in 12-note equal temperment, by 80 cents. He was trying to create a universal symbol that indicated that the pitch was microtonally adjusted. And by the time one got around the scale a few times, his enharmonic equivalents can make sense. It works for chord spelling, too. I found it hard to remember which pitches are up, which are down, and which ones are really not the ones they look like.

Positions 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
12-note pitch
C
C#
C#
D
D#
E
F
F#
F#
G
G#
A
A#
A#
B
C
Enharmonics

Db
Db

Eb


Gb
Gb

Ab

Bb
Bb



















12-note cents
0
100
100
200
300
400
500
600
600
700
800
900
1000
1000
1100
1200
15-note cents
0
80
160
240
320
400
480
560
640
720
800
880
960
1060
1120
1200
Cents off 0 -20 +60 +40 +20 0 -20 -40 +40 20 0 -20 -40 +60 +20 0

















Blackwood C C↑ D↓ D D↑ E↓ E F↑ F#↓ G G↑ A↓ A A↑ B↓ C
Enharmonics B
C#↓
Eb↑
F


Ab↑
Bb Bb↑
B















 

Myriad C C# Db D D# E F F# Gb G G# A A# Bb B
C
Enharmonics



Eb




Ab





Cents from Named
0
-20
+60
+40
+20
0
-20
-40
+40
+20
0
-20
-40
+60
+20
0

    As I am using Myriad's Melody Assistant to work with the scale, there are limitations to its notational capabilities (such as no custom symbols for pitch). And the way I've implemented the rules limits how to apply it to pitches - makes the Blackwood equivalences of C=B and E=F really really hard. I simply laid out 15 notes against 12 (17 when you count enharmonics), sorted out a couple enharmonic equivalences, adjusted pitches to force a nearest-fit and marked the differences from 12-note equal temperment in cents. So I've sacrificed some Blackwood's tonal sense for implementability, and created new modalities to have to think in. It'll work, it just looks funny.



Clicking on the graphic will convince you that yes, this is a different intonation scheme - some notes are noticibly out of tune with what we are used to, when played against what we're all used to- both lines will play together.

    Since the notation software I'm using won't support Blackwood's method anyway, I'll quit carrying on about it at this point and only show and use the the Myriad notation in the following examples. Were you to actually get your hands on the actual article, you'll have to refer to the table above to get it all to match (or, re-write it yourself!). The example numbers refer to the original article's example numbers. So we will start with Example 22, in Section 21. I've added some extra examples to illustrate more of what we're talking about.



Primary Triads: Section 21



Example 22 and 23


The closest-fit pitches that outline the major triads in 12-note equal temperment are in the ballpark in 15-note. It's out of tune, for sure, but not enough to make one storm from the room.

Here's all of the traditional diatonic-like triads, some of which aren't so good:



For something really bad, try this:



Even Blackwood admits: avoid this in any context.



Five Equal Parts: Section 22


Here Blackwood says that this root progression: C-F-A#-D-G-C, topped with any mix of major or minor triads is good:

example 24
(Example 24)
Funny thing: it has that very ii-V-I  progression we just got done saying was so bad.

Here it is all in minor:

example 24 in minor


Putting a major triad over a descending five-note equal scale gives a startling, alien, and appealing progression:


example 25
The five-note equal division is something that cannot be approximated on a piano, and using that with acceptable triads makes for remarkable progressions.


Section 23: More on 5-note equal division


A 12-note octave gives us divisions into three equal parts (an augmented triad), four parts (a diminished seventh chord) or six parts (a whole tone scale), but not five parts - for that, we need a 15-note scale.

The five-note equal scale, if we have to quantify it, sounds like something between a diminished seventh chord and a whole-tone scale.
As if that explanation helps - conceiving this in our little minds is utterly beyond human imagination (and those that can imagine it without having heard it can stop reading right now, and show us all a thing or two). Blackwood has conceded that it is a sound he can never quite get used to, and he's been listening to it longer than all the rest of us.


A straight run of the scale, ascending or descending, topped with major or minor triads works, but in ways Blackwood hasn't been able to find a theoretical explanation for: "piquantly discordant, but weakly dissonant subdominant" is as close as he could get, whatever that means. The bass note should be either the tonic or dominant note.

example 26
(Example 26)


Other good progressions involving subdominants:
:
Example 27
(Example 27)


Section 24: Ten-note Symmetric Mode


mode 1
(Example 28 Mode 1)


Mode 2
(Example 28 Mode 2)


mode 3
(Example 28 Mode 3)
Examples 22-27 could all be considered to be in Mode 3.


Here's 10 major triads successfully progressed:

example29
(Example 29)
Here it is in minor:
example 29 in minor
Alternating major and minor triads also supposedly work.


Section 25: 6-note Symmetric Mode



(Example 30)

The Six-note symmetric mode (alternating minor 3rd with minor 2nds)  in 12-note equal temperment is difficult to distinguish from the same mode in 15-note equal temperment - it's just a bit out of tune. It's not altogether useful as a singular mode - no major 2nds - as good melodic lines don't work in it. Cool, tho, huh.


  
(Example 31)

Section 26: Six-note and Ten-note Progressions, Combined!


example 32
(Example 32)
10-note mode can be established with 3 chords: 6-note mode can be established in 2. By picking a triad in 10-note mode, and equating it as closely as possible to 6-note, and then using that chord to connect into another 10-note mode, modal modulation opens up a large class of progressions with the alternation of 6-to-10-note model  Equal temperment requires symmetry or modality for cohesion, else it just wanders. Using multiple modes extend the intricacy one can use to extend pieces. Blackwood, tho, cautions against extensively mixing intricate modal modulations with diatonic progressions


Section 27: Other Cyclic Progressions


Maybe I'm just tired, but the text associated with this section generally boil down to "here's some other progressions".


another modulation
(Example 33)

example34
(Example 34)

example 35
(Example 35)


Conclusions, Afterwords, and Codas, oh my.......

    There is another section that discusses 15-note vs 16-note equal temperment. He points out that harmonically speaking, they work differently (imagine our surprise). It is difficult to spot the difference between the two scales when played as a monophonic melody, because of the difference of a mere 5 cents per interval. He goes on to expound on the value of 15-note progressions - it has beautiful, fascinating, and unusual harmonic forces at it's disposal, which could bring about an enrichment of musical repertoire. He advocated guitars as being practical instruments to build into these structure, and in fact, he did commission some to be built in later years.
    In later years, though, Blackwood gave up on evangelizing for his discoveries. Seems it was just a bit hard to get others to take up the call. Look, it took me a quarter century to get just this far.
    Obviously, computers help make this more reachable - no customized instrument building, just a powerful lot of thought.

..........for what it's worth.........
-bjc


p.s. here's all the examples, and the rules file for Melody Assistant