Review: John O'Sullivan's _The Mathematics of Music_

This is cross-posted to the Tuning and Making Microtonal Music lists.

Before I start, since I'm not a known commodity on either list, it may
help to know my background and perspective.

I was educated as an engineer, and I like knowing how things work. I
don't love working out the actual math problems, though, and am
perfectly happy to have someone else do that. In other words, I like
understanding equations more than working them.

I'm not a musician. I played some classical guitar when I was a teen,
but now I just play chords to Irish drinking songs for my kids. I
learned some music theory in high school (traditional harmony and Bach
chorale kinds of things) and stuck with composition enough to create
some little ditties. You can see the vast majority of my meager output
here: http://www.freivald.org/~jake/music.html

My tastes are considered eclectic, but they're probably less so than
many on this list: Rush, Peter Gabriel, and Ultravox in popular music;
Romantic and Modern classical music, including Mahler, Ives,
Stravinsky, and (sometimes) Schoenberg; bebop; old blues; and even
country that's not too pop-inflected. I can only rarely tolerate Top
40.

I've understood basic concepts of harmonics for a long time, but only
recently did I realize that there's a discrepancy between 12-EDO and
perfectly just tuning. Learning this led me to the Tuning list, the
Xenharmonic wiki, and a ton of other resources that I'm barely
starting to grasp. Although I'm better educated about music than many
non-musicians, and although I'm reading and listening a lot these
days, I'm a complete novice at this stuff.

It was in this context that I received John O'Sullivan's _The
Mathematics of Music_.

==========

What it is
----------
A story of John O'Sullivan's quest to create a better scale than the
12-equal-division-of-the-octave (12-EDO) scale that is the mainstay of
modern Western music.

Who would like it
-----------------
* Tinkerers, who would try out his methods
* Composers who want to see what his goals were, whether his scales
met those goals, and whether those goals turn out to be useful in
real-world composition
* Mathematically inclined non-composers ONLY IF they want to hear a
story about developing a scale.

Length
------
71 pages + front matter & back matter

Availability
------------
Amazon.com: http://www.amazon.com/Mathematics-Music-Temperament-Explained-Alternative/dp/0956649203

John's Website
--------------
http://www.johnsmusic7.com

==========

Let me start by saying that reading John's book has made me respect
him a great deal. His book details his efforts to improve on the
12-EDO scale, which has reigned for several hundred years as the basis
of most Western music, and he makes his goals, methods, and results
admirably clear. I will use the word "amateur" quite a bit in this
review, but that's not a pejorative -- it was an amateur who proved
the "professionals" wrong and proved that the Troy of the Iliad
actually existed, after all.

I should say up front, though, that this isn't anything like a text
book. It's the story of a dedicated amateur's quest to create better
scales. (The scales that result are called "Blue Just" and "Blue
Temperament".) Tinkerers will like it and will want to try out his
processes for themselves. Composers may want to see what his goals
were, and to try out his scales to see if he met them and if his goals
match their compositional goals. Mathematically inclined non-composers
who just want to know how music is structured will only like it if
they're interested in how he gets to his scales. (That describes me,
and I liked the book.)

As I understand it, people historically developed scales
experimentally, almost viscerally. They stretched wires or blew
through tubes or banged on surfaces, and tuned them by tightening or
cutting or boring or otherwise physically manipulating their systems.
They relied only on their ears -- what else was there? -- to make
these materials resonate in ways that seemed more-or-less harmonious,
and over time developed an understanding of some of the mathematical
relations in the scale. The 9/8 step didn't come from abstract math;
it's the gap that naturally comes from tuning fifths up from a unison
and down from an octave.

John's book feels similarly experimental and visceral, but comes at
things from the opposite direction. He pushes around frequencies and
oscillations instead of wires and tubes. He starts with ratios and
ends with frets and guitar strings.

To his credit, John shows that he's making educated guesses. He
describes the guesses he made, prior guesses that he discarded, and
where he thinks refinements are probably necessary.

Because he's telling you the story of his quest, he sometimes gives
details that seem out of place. For instance, early on he says, "For
years I wrote musical ratios in the form x/y where x was less than or
equal to y (e.g., 2/3). This goes against the norm and musical ratios,
also called intervals, are usually written with x equal to or greater
than y (e.g., 3/2). I have since adopted this approach." I really
didn't need to know that: More important to me is the convention that
most people use.

As an amateur, he tells us what he's interested in rather than
surveying the scope of relevant literature. He doesn't discuss MOSs,
non-12 equal temperaments (though he mentions them once), dwarves or
hobbits or kleismic scales, the creation of scales by stacking ratios
(a la La Monte Young's _Well-Tuned Piano_), or many other things that
xenharmonicists seem to talk about. He doesn't discuss the difference
between regular temperaments and others. He never mentions the word
"comma". He also limits himself to 12-note scales. There's nothing
wrong with any of this -- he gives his reasons for limiting himself to
12-note scales, for instance -- but as someone who was starting to
look at lots of microtonal information, I know that there's a lot more
to scale development than what he covered.

His writing style is pretty straightforward. For instance, on the
bottom of page one he says, "For example the 4/5 interval is 13.7
cents out of tune." I already knew what cents were, but I thought that
someone who didn't know might be confused; however, at the top of page
two he says, "A cent is one hundredth of a standard semitone [etc.]".
When you need information or definitions, he usually has it just in
the nick of time.

What follows is a walk through the book. I won't try to touch on
everything John does, but I'll hit the highlights roughly in the order
you'd see them if you read it.

In the introduction, John says he wants to "develop a better tuning
system", but doesn't explicitly say the ways in which it would be
better. For people who know something about tuning, it's clear that
he's looking for a Just Intonation system; people who don't know what
that means would be unlikely to know that JI is only one type of
tuning system, with its own pros and cons. By the time I hit the Blue
Temperament chapter, I really came to think that that's the most
important issue in the book. More on that later.

He also says, "I suspect that if my system is correct, it will make
redundant many of the rules of standard music theory." This was a
warning flag, making me believe that John's view was too narrow.
Regardless of the scales he produced, there would still need to be
concepts of chord progressions, voice leading, and so on. He might
replace or modify some rules of music theory, or provide a surer basis
for them, but make them redundant? Not likely.

When he gives historical background he shows, I think, the narrowness
of the amateur. I think it would have been useful to talk about how
tuning systems evolved; with terms like "Pythagorean" being bandied
about on the Web (though not in his book), it would be nice to know
what Pythagoras had to do with anything. Also, this would be a good
place to tip the hat to non-Western tunings; those tunings don't
necessarily agree with his principles, and in some ways they compete
with 12-EDO the same way Blue Just and Blue Temperament do. (John
noted in a private email exchange that he's not specifically looking
at Western music, but is just following the paths down which his math
takes him.)

In the second chapter, he makes a few guesses at a mathematical
formula for consonance and promises to get back to his reasons for the
correct one (for interval x/y, where x and y are frequencies, he
claims it's 1/x + 1/y), promising to tell us in chapter 5 why that
seems to fit. He does: In essence, it's because it's simple and
because it seems to work for melodic movement.

When discussing the harmonic series, he gives a straightforward
description of a vibrating string, and then relates it to how
different notes sound. As I said earlier, this is an empirical book;
while I more-or-less trust John's ear (partly because I like the scale
it produced), when he talks about "resolution" and "similarity" and
"sweetness" I get the feeling that I'd have to walk through all of the
same experiments he did before I could understand what he meant. This
is one of those points where I think this book is the perfect gift for
a musical tinkerer. I also wonder if there isn't room for bias and
cultural conditioning in his method: Mightn't people who are used to
other scales find certain intervals to be sweet just because they are
used to them?

He brings up "periodicity", which he defines as finding "repeat
points" where the all wave components have zero crossings at the same
point. He doesn't say what a periodicity value should be, though, or
what its importance is. It seems important, but remains mostly
unexplained.

Also, he points out that he is using a "Sine Lead voice" on his
synthesizer to do the testing, noting that using "a voice with a full
harmonic series", which I would call a more complex timbre, "makes
testing the notes harder." It seems to me that timbre is critical in
understanding how things will sound together. For example, a perfect
fifth sounds hollow on a classical guitar, but rich and full on a
distorted electric guitar. For all that John is very pragmatic and
empirical in his approach, if his method doesn't take into account
different timbres then I wonder how practical the result will be. He
does address it in Chapter 6, but in a way that I don't agree with:
Timbres are different because they are composed of different
overtones, but John just breaks the world into "sine waves" and
"complete harmonic series". But whether this matters is a question for
composers and arrangers to deal with; I'm struggling to make any music
with any timbres, so for now I'll note the issue and leave it alone.

Sometimes, as John is casting about for the right equations, I feel
like he goes down paths he doesn't need to go. I'm most interested in
the relationships between the notes: For instance, as consonance goes
up, his formula for consonance should go up. I'm less worried about
what threshold it crosses as the formula reaches 1 -- there's enough
guesswork going on that I'll assume that everything could be
multiplied by a constant to get the "right" measure for a particular
purpose.

One thing to note is that John describes "Major" or "minor" as being
related to the strengths (as calculated by his formulae) of various
interval pairings instead of the character of the sound. I don't know
whether these calculations relate to the character of the sound or
not.

In chapter seven, John details his Blue Just Tuning. Everything else
has essentially been groundwork leading up to this point. Based on all
the math he has laid out before, he chooses the intervals that should
sound sweetest.

It's important to realize that this is a somewhat circular
mathematical description of what he hears: His math was made based on
what his ear told him was "sweetest", and now his math tells him which
intervals should sound sweet.

His Blue Just tuning is a familiar set of intervals: 15/14, 9/8, 6/5,
5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 9/5, 15/8, and 2/1.

It's a pretty good scale. Since I'm still trying to figure out how to
control non-12-EDO scales, I won't offer any of my compositions on it,
but Chris Vaisvil has composed a few items with it:
http://chrisvaisvil.com/?p=317
http://chrisvaisvil.com/?p=376
http://chrisvaisvil.com/?p=443

Now, from what I've seen, you could probably give Chris a fish and a
hand grenade and he'd make them sound musical. But I don't think he's
performing any special wizardry here; I think these clearly show that
Blue Just is a viable tuning for mainstream compositions at least, and
that it achieves John's goals of providing a sweet sound.

John briefly touches on the fact that this is a seven-limit tuning and
explains what that means, but doesn't much go into why that's
important, or if it has anything to do with the "periodicity" he had
mentioned before.

Upon seeing the scale, I recalled his comment about making much
traditional music theory redundant. I also recalled the fact that the
traditional ii-V-I cadence that I learned in high school isn't nearly
as simple as it looks, since it involves the "wolf" fifth (D-A) in
early ill-tempered systems. Sure enough, in Blue Just tuning, the
second scale degree has the following steps available for harmony:
112, 182, 294, 379, 498, 610, 680, 814, 884, 996, 1116, 1200. The
minor third there would be about 20 cents flat, and the 680-cent
"fifth" would be really flat. (The V-I cadence would sound perfect, of
course.) Thus this tuning has some of the same benefits and drawbacks
of untempered systems or "non-well" temperaments.

The next chapter essentially shows how much he's willing to temper a
note -- 256/255, or 6.775877 cents -- and shows us how he tempered the
Blue Just tuning. The reason for the 256/255 ratio is apparently
mystical, since he says, "powers of 2 such as 64, 128, 256, etc. are
magic numbers for me". (His programming experience probably affects
his view of magic -- mine does for me. :) ) The 6.8-cent limit doesn't
seem out of line, though, so I'll accept it. Microtonal composer Kyle
Gann uses five cents, I think. John ultimately gives us a temperament
that seems pretty good to my amateur eyes: In cents, Blue Temperament
is 121.6, 200.7, 313.5, 388.4, 501.2, 580.4, 702.0, 816.9, 889.4,
1012.5, 1085.1, and 1200.

Again, I'll lean on Chris Vaisvil to provide an example of the scale in use:
http://chrisvaisvil.com/?p=411

When John finishes this section, he says, "Blue Temperament may or may
not be the best possible 12 key _temperament_ but it's pretty good,
viable, and to me is a clear improvement on 12 Tone Equal
Temperament." And clearly, composers who are looking for a
predominantly five-limit, 12-note scale could do much worse, IF
they're willing to give up the simplicity of 12-EDO.

And it was in thinking about these ideas -- Blue Temperament being a
"clear improvement" on 12-EDO, and the idea of people giving up 12-EDO
-- that I came to believe "Improvement in what sense?" is the most
important question of the entire endeavor.

It's important to remember that 12-EDO isn't exactly "flawed". It's
not like aliens came down to Earth and imposed it on us, and we have
to liberate ourselves from the 12-EDO hegemons. It's something that
evolved out of prior tunings and temperaments. And its flaws never
stopped, say, Mahler's symphonies from being sublime and beautiful.
For a new tuning to be "better" than 12-EDO -- in the sense that it's
something people would want to tune pianos to, create clarinets for,
play guitars with, and learn to control compositionally -- it would
have to be "better" than meantone, Werckmeister, Prinz, and other
temperaments that existed before the 12-EDO system was widely adopted.

Moreover, those who are most likely to adopt a new tuning systems are
the same people who are more likely to seek out something different
from their scales: more 7-limit, 11-limit, or 13-limit harmony;
smaller steps between notes; non-octave scales; ethnic and unusual
harmonies.

Don't get me wrong, there are things within Blue Temperament that
could be spicy and unusual, too. For example, if you choose the mode
starting on the third note of Blue Temperament, you have a pretty good
approximation of 7/6, 5/4, 4/3, 7/5, 3/2, 5/3, and 13/7. But that
wasn't how the scale was devised, and it seems likely that if I wanted
13-limit harmony then I could choose other scales that would be
better. In short, it seems unlikely that most non-microtonal composers
will retune their pianos to Blue Temperament anytime soon.

Near the end of the book, revisiting some of the principles from prior
chapters, John works out a variety of intervals that should make good
chords. While I don't disagree with his method for figuring out
chords, I don't understand his approach to chord *progressions*. He
makes it look like two chords should progress effectively if his math
works out, regardless of direction, but whenever I work out chord
progressions the one that comes first makes a difference; e.g., C
Major moving to E minor is a pretty weak progression, while E minor to
C Major is pretty strong.

Some nitpicks:
* I think it would be useful to call intervals by their traditional
names -- just major third, just minor third, harmonic seventh,
whatever. I'm not yet at the point that I think in ratios, so "just
major third" means a lot more to me than "5/4" or "386 cents".

* Sometimes John chooses his examples a bit arbitrarily. One of his
examples starts, "A has a frequency of 110Hz." Well, yes, and because
I know that the A below Middle C is 440 Hz, I know that he's talking
about the A two octaves below middle C, but why did he pick that A?
Also, because I know that the range of human hearing is about
20-20,000 Hz, I know that when he says, "if the frequencies of the
chosen notes are, say, 12, 13, 15, 16, 18, 19, 22 and 24," he's
talking about notes I mostly can't hear. The math would work just as
cleanly if they were 1200, 1300, 1500, 1600, 1800, 1900, 2200, and
2400 -- so why use such unrealistic frequencies at all?

* John has his quirks, and one of them is carrying things out to
unnecessary levels of precision. If he thinks a difference of 6.775877
cents is essentially unnoticeable, why would 0.075877 cents (the
difference between 6.775877 cents and 6.7 cents) be noticeable? He
could have rounded this number to 6.8 cents or 7 cents, or even
truncated it to 6 cents, just as easily.

In short, I think it's a good book for people who are interested in
developing scales, and perhaps for composers who are interested in
getting a maximally sweet 12-note non-EDO scale. It is not suitable
for people who are trying to dig into the widely varying opinions that
involve MOS, different EDOs, commas, and so on.

Respectfully submitted,
Jake Freivald