## Easely Blackwood, Twelve Microtonal Etudesfor Electronic Music Media, opus 28

For each of Blackwood's twelve microtonal etudes, I've explored various ways of using
color, geometry, and motion to depict their use of intervals, scales, harmony, and tonality.

#### Approach

The music I was most familiar with before encountering Blackwood's microtonal etudes
is based on a system with twelve pitch classes (C, C#, D, ... A, B-flat, B), seven-note scales
selected from those pitches, and three-note triads that included intervals with frequency
ratios 2:3, 4:5, and 5:6 (aka the perfect fifth, major third, and minor third).

In this system, it's useful to look at pitches, harmonies, and scales/tonalities as being related
by the interval of a perfect fifth, and it's possible to make a chain of all twelve pitches
according to this ordering (C, G, D, ... B-flat, F, C), known as the circle of fifths.

So, that was the lens I was looking through when I was trying to make sense of these etudes.
I was asking questions like: "which sets of pitches are used together in sections of the piece?"
and "which notes are combined into chords?"

For each etude, I began by choosing a mapping between pitch class and color. Though the
etudes use nonstandard divisions of the octave, they do something that conventional tonal
music does: modulate between various tonalities—that is, they use a certain collection of
pitches for a while, then switch to a different one. So, I tried to find colors that would make
these modulations obvious.

Although an interval close to a perfect fifth plays an important role in all the etudes I studied,
it's not always the same role. In some etudes, there is a circle of fifths that works same way as
in a 12-note system, but in others there is no single circle of fifths that goes through all the
pitches, but there are instead several interlaced circles.

In addition to depicting sequences of fifths in a circle, I've also developed a display in which
the sequence is displayed horizontally and slides to show modulations (sliding fifth row).

To show triads (and other relationships between thirds and fifths), I've used a hexagonal pitch
lattice
in which the three axes correspond to three intervals.

And, finally, for each etude, I've made one or more scrolling scores. They are in this playlist.

#### Index

In the following index, the etudes are listed in the order given in the score, which is also
the order Blackwood prescribes when they're played as a set, and their order in his original
recording and my YouTube playlist, but for ease of navigation, in the text on this page,
they're in order according to the number of notes in the scale.

Items included in the YouTube playlist are marked (p), and ones used as examples in this
page are listed in italics.

N.b. Videos with both a pitch lattice and a scrolling score are listed in both columns.

 description PDF CDtrack colorwheel affinitymap (*) fifthsmap pitchmap slidingfifths row pitchlattice scrollingscore 16 NOTES 16 2 jpg, jpg, jpg, jpg jpg jpg video (p), video, video, video video (p), video, video, video 18 NOTES 18 3 video jpg jpg video (p) 21 NOTES 21 4 jpg, video jpg jpg video video (p) video (p) 23 NOTES 23 5 jpg jpg video video (p) 13 NOTES 13 6 jpg, jpg, jpg, jpg jpg, jpg, jpg, jpg, jpg jpg jpg, jpg, jpg, jpg, jpg, jpg jpg, video, video video, video, video (p) 15 NOTES 15 7 jpg, jpg, video, video (grid) jpg jpg video, video jpg, jpg video, video, video (p), video, video, video, video, video video (p), video, video, video, video, video, video, video 17 NOTES 17 8 video, video jpg jpg video (p), video, video, video video video 22 NOTES 22 9 jpg, video jpg jpg video video, video (p) 24 NOTES 24 10 jpg jpg jpg video (p) 14 NOTES 14 11 jpg, video jpg jpg jpg video video video, video (p) 20 NOTES 20 12 jpg, video jpg jpg jpg video (p), video 19 NOTES 19 13 video, video jpg jpg video video (p), video (slowed)

#### Acknowledgements

I am grateful to Matthew Sheeran for supporting this project and providing much of the score data I used, and to
him, Stephen Weigel, and Carl Lumma for their insights, suggestions, criticisms, corrections, and encouragement.

#### 13 NOTES

In a 13-note system, there are six orthogonal ways of mapping the color wheel to the pitches: by taking
1, 2, 3, 4, 5, or 6 steps in one circle for each step in the other (the other six step sizes are just those six,
but going in the opposite direction). The question is: which one of these mappings is the most useful?
It depends what you want the color to help you notice. Here's an overview of the piece with those mappings:

1 step 6 steps

2 steps 3 steps

4 steps 5 steps

I find it's most useful for the colors to show which sets of pitches are being used, and where it changes
from one set to another. As you can see, if one step in color corresponds to one step in pitch, the score
looks like a uniform rainbow throughout the piece. The mapping which makes the sections of the
piece look most different turns out to be the one with 5 steps. This surprised me a bit, since the 5-step
interval (which lies between a major third and a perfect fourth) doesn't play a conventional role in
the piece. Here's what it looks like closer up (and with lines marking the sections):

Given that, the pitch/color wheel looks like this (the second one is the one used in the animation, because
I chose to call the 9th scale degree "tonic" and I usually make that blue):

Because the 3-step and 4-step intervals figure prominently in the piece, the most effective layout for the
pitch lattice turned out to be one with the horizontal and diagonal steps corresponding to those:

Here's a video with the chart:

#### 14 NOTES

To get a sense of which pitches tended to be used together (that is, harmonically) in this piece,
I ran it through my affinity map tool. This showed pretty clearly that Blackwood used the odd-
numbered pitches together a lot, with the even-numbered pitches used together (though less
often). Because the inverval closest to a perfect fifth was rather flat, I disregarded it at first, but
Matthew Sheeran tipped me off, and I quickly found a pitch/color map that made sense, with
the pitches segregated into two groups, each ordered by the 8-step (close to perfect fifth) interval:

Here's an overview of the piece with that coloring, with the pitches ordered by circle-of-fifth position:

And here's what it looks like with pitches ordered by pitch height (that is, from low to high, as in a normal score):

I think the display that shows the tonal motion best is the circular one ...

Here's a view with a pitch lattice ...

#### 15 NOTES

In an equal-tempered 15-note system, chains of intervals of size {1, 2, 4, 7, 8, 11, 13, 14} traverse all fifteen
pitches, but those of size {3, 5, 6, 9, 10, 12} do not. The result is that of all the intervals used in conventional
triads (major/minor thirds, perfect fifth, and their invesions), the only one that has a close match in this system
andcan be used to construct a "circle-of-______" (the analogue of the "circle-of-fifths" in traditional Western
harmony) that includes all the pitches, is the minor third (or, counter-clockwise, its invesion, the major sixth).

Applying equally-spaced hues from the color wheel to pitches arranged in a circle-of-minor-thirds didn't yield a
useful result, though, since chords related by recognizable pitch relationships (root movement by fourths/fifths)
had very different sets of colors, while harmonically distant chords (e.g. ones a tritone apart) were similar in color.

So, I decided to abandon the constraint of having a completely symmetrical system (one in which all steps within
the color wheel corresponded to the same musical interval), and instead focused on the five (interlocking) sets of
augmented triads. Starting from the home/tonic pitch/color (zero/blue), the first two steps are by (approximately)
a major third (0→5→10); this is followed by an (approximately) perfect fifth (0→5→10→4).

However, while repeating this pattern through 5 cycles (0,5,10 → 4,9,14 → 8,13,3 → 12,2,7 → 1,6,11) does take
you through all fifteen pitches, when you reach the end, another perfect fifth step doesn't take you back to the
beginning (since a perfect fifth step from 11 takes you to 5, not zero). So, there has to be a break somewhere.
I decided to put it at the opposite side of the circle from the tonic. To do that most comfortably, the tonic pitch had
to be in the middle of the center 3-note group, like this: (2,7,12 → 6,11,1 → 10,0,5 → 14,4,9 → 3,8,13) ...

Because the tonic of this etude is the third scale degree (aka pitch 2), the numbers I've used are offset by 2 from that:

A useful way to show pitch relationships is to arrange the pitch classes in a repeating lattice, with the primary
common intervals (thirds, fifths, and their inverses) adjacent. Because the computer display is wider than it is
high, and because a chain of 15 minor thirds (each 4 steps wide) traverses all the pitch classes, that's the first
thing I tried, with the minor thirds (and major sixths) ordered horizontally and the other intervals diagonally:

Here's the video using that system:

Another way to organize pitches in the lattice is to have the horizontal order correspond to the interval closest to a
perfect fifth (in this case, the 9-step interval). That results in this:

Here's the video using that:

#### 16 NOTES

In an equal-tempered 16-note system there are three equal-step sequences that take you through all the pitches: ones of 3, 5, or 9 steps.
There are therefore three possible circle-of-______s that are symmetrical:

The 9-step interval is the one that's closest to a perfect fifth , and it's an interval that Blackwood uses a lot in this piece,
so even though it's noticeably out-of-tune compared to a pure fifth (flat by about 27 cents), it seemed like the best choice.

Likewise, the 5-step interval is fairly close just-tuned major third (off by 11 cents, same as in 12-note equal temperament),
so a pitch lattice based on fifths and thirds was as good a fit as any.

Here's a video based on those choices ...

From that, it's easy to see that much of the music is built upon minor seventh chords (the chord which begins and ends the piece),
with diminished seventh chords (diagonal stripes in the lattice) featuring prominently, and "bells" adding more harmonic spice.

#### 17 NOTES

The 17-note system is similar to the 19-note one in that it has an interval that's close to a perfect fifth, so the
considerations for that etude (q.v.) mostly apply here. The main difference is that the thirds (major and minor)
are not so well in tune; Blackwood softens them by almost always combining them with fifths.

The coloring based on the circle of fifths works well for this etude; the changes of tonality are very clear:

Another way to look at this is to arrange the circle of fifths vertically (repeated so as not to break the circle):

Here's what that looks like in real time (with pitch classes in order on a circle) ...

... and here's an alternate version that spreads out the circle of fifths horizontally and shows the treble and bass
staves and the motion of the notes (the horizontal shifts happen when the key signature changes):

#### 18 NOTES

The 18-note system doesn't have any interval that's close to a perfect fifth, so I chose to divide the 18 pitches into
six interlocking augmented triads, and give each triad its own color. Here's a video based on that coloring ...

... and here's a static chart video that shows the six groups (and positions within the group):

Here's an overview of the whole piece using that coloring ...

#### 19 NOTES

Of the microtonal tunings used in the Blackwood etudes, the one with 19 notes is the most similar to the
conventional 12-note system, because they both contain intervals that are good approximations of those used in
perfectly-in-tune triads (which use frequency ratios 3:2, 4:3, 5:4, and 6:5), and both contain a "circle of fifths"
(sequence of pitches separated by an interval close to a perfect fifth) that includes all the pitches. For 12-pitch music,
I often map colors from the "color wheel" to pitches on this circle (see here for a description of this system), and
the same system can be easily extended to 19 pitches.

What's different in a 19-pitch system is that the circle of fifths has approximately three sets of diatonic pitches
as opposed to the 12-pitch system which has approximately two.

So, whereas the 12-pitch system calls G♯ and A♭ the same note ...
A♭— E♭— B♭
F — C — G — D — A — E — B
F♯— C♯— G♯
... a 19-pitch system goes further around the circle ...
C♭— G♭— D♭— A♭— E♭— B♭
F — C — G — D — A — E — B
F♯— C♯— G♯— D♯— A♯— E♯
... and calls B♯/C♭ and/or E♯/F♭ the same note.

To familiarize yourself with the pitches and colors, here's a video showing the pitch/color circle ...

... here's the horizontal-sliding version of that (like the one for 17 notes) ...

... and here's the scrolling score video using those colors ...

... an overview of the changes in tonality (circle-of-fifths view) ...

... and the pitch view of that ...

#### 20 NOTES

The most stable possible harmonies in this tuning are ones based on dividing the octave into five equal parts. There are
four ways of doing this (0/4/8/12/16, and the same pattern transposed up 1, 2, or 3 steps), and in this piece, Blackwood
spends most of his time in one of those sets, as can be seen in this pitch map:

Because there are therefore, in effect, four "circles-of-fifths," I divided the color wheel into four regions, and
mapped each set of five pitches to a region:

Here's an animation showing the music's movement through those regions:

#### 21 NOTES

The 21-note system is interesting in that it has intervals that can function as thirds and fifths (7=M3, 12=P5),
but it doesn't have an orthogonal circle of fifths that includes all 21 pitches. If you follow the chain of fifths, you
return to starting point after seven steps, and there are three of these chains, interleaved. In the coloring of the
pitches, I decided to preserve the integrity of each of these chains, and put them one after another, starting with
{9,0,12,3,15,6,18}, then up one scale degree to {10,1,13,4,16,7,19}, finishing with {8,20,11,2,14,5,17} ...

Because the chains of fifths don't contain a major third (or the version of the minor third that adds up to a fifth),
notes that make up close approximations of triads are sometimes not very close to each other, as can be seen in this
animated chart (using that ordering of pitches and colors) ...

... or in this horizontally-oriented circle-of-fifths animation ...

Still, looking at the whole piece with the notes arranged vertically in that order (and horizontally by time) ...

... it's clear that the use of pitches is very coherent. A pitch lattice (based on both P5 and M3) is a better tool for seeing that ...

#### 22 NOTES

After 12, 17, and 19, the next EDO tuning with acceptably in-tune thirds and fifths is 22.

Here's its color wheel / circle of fifths:

Viewing the piece in that layout ...

... it's easy to see that notes related by fourths and fifths are clustered together in one section of the circle,
and the thirds and sixths are in regions distant from that.

A striking feature of this piece is that the pitch centers of the 4th/5th notes and the 3rd/6th notes rotate one-half turn
through the circle in the course of the piece. This can also be seen in the fifths map:

The triadic structure is easy to see in the pitch lattice (lower-left-hand corner of this video):

#### 23 NOTES

Because the 23-note tuning doesn't map to normal diatonic/chromatic scales, Blackwood chose to instead use
pentatonic modes from Javanese/Balinese music (pelog and slendro). The affinity map showed which pitches
were used most prominently, and I used that as a starting point and shuffled the pitches around until the pitch
map for the piece was arranged in a small number of coherent clusters:

That's the ordering used here ...

(N.b. Unlike the other displays of this sort, here I've ordered the pitches in each tonality cluster according to
pitch—since they were not based on fifths; this made it a little easier to follow melodic motion.)

#### 24 NOTES

The 24-note can be looked at as two interlaced 12-notes scales and that's the tack Blackwood took:
alternating between one set and another. This can be seen in the affinity map for the piece ...

... in which all the even-numbered pitches are on one side of the circle, opposite the odd-numbered pitches.

For the color-wheel/pitch-class ordering, I took a tip from that map: many of the adjacent and nearby pitches were
minor thirds apart (0/6/12, 2/8/14/20, 16/22/4). The colors groups within each quarter-tone scale are matched to
the three diminished seventh chords (0/6/12/18, 2/8/14/20, 4/10/16/22, 1/7/13/19, 3/9/15/21, 5/11/17/23):

(I can't say that I found this coloring particularly useful.)

Here's an overview showing how the two interlaced quarter-tone scales are used:

The odd-numbered pitches are in the top half, the even-numbered pitches are in the bottom half.
In each of the eight repetitions of the passacaglia, the odd-numbered pitches are interspersed for
a few measures, then the even-numbered pitches are used alone for the end of the section.

In the scrolling score for this etude, the even-numbered pitches are ellipses,
and the odd-numbered ones are inverted ellipses ("stars"):

The passacaglia subject is indicated by size.

#### Related online resources

Easley Blackwood, text of booklet accompanying the CD of the microtonal etudes

Easley Blackwood, Modes and Chord Progressions in Equal Tunings

Mavila Temperament, score follow-alongs for etudes 13, 16, and 18 (Blackwood's original notation)

Matthew Sheeran, MIDI renditions of the etudes with piano roll score (from Stephen Weigel's Now&Xen podcast)

Bruce Duffie, conversation with composer/pianist Easley Blackwood

Stephen Weigel, Now&Xen podcast, discussion with Easely Blackwood (YouTube/audio, transcript/PDF)

Stephen Weigel, master's thesis (analysis of 13, 15, and 23 note etudes)

Stephen Weigel, Easley Blackwood Analysis Goals Experience (Gesundheit! Institute, UnTwelve 2019, 8/6/2019)

Stephen Weigel, arrangements of etudes for keyboard: 15-note, 16-note, 13-note, Patreon (for access to scores)

MonoNeon, playing along with the 16 note etude

Stephen Malinowski — April 9 2021 - May 16, 2022