Review of "The Mathematics of Music" By John O'Sullivan

Hi all. John gave me a copy of his book and asked me to post a review of it to the Tuning List, so I'm obliging him. I don't expect the content of my review will be of much interest to most here, as you're all already familiar with his work and have your own opinions. You know me as well, and I'm sure most of you could guess at what I think of his book. This review is thus mostly for John's benefit, and the benefit of the lurkers.

To start off, I have to say John deserves a lot of credit for his modesty. Throughout the book he makes plain that his work is based on educated guesses and his own psychoacoustic preferences, and he acknowledges he is probably not the first to look into these ideas and that some or most of his conclusions may turn out to be mistaken. This modesty and honesty is rare among those filled with the eager joy of breaking the confines of the 12-TET paradigm.

That said, the title of the book is quite the misnomer, and very much at odds with John's attitude throughout the book. If he'd called it "My Search for a Better 12-Tone Scale" or something like that, I would approve whole-heartedly, as that's *really* what this book is. However, calling it "The Mathematics of Music" makes it sound much more definitive and authoritative than it really is, especially considering John is not a mathematician and is apparently ignorant of all the mathematical models and theories of music and consonance--and most importantly, temperament--that have been proposed throughout history (and especially on the Yahoo Tuning List).

Reading this book, it is clear that John did hardly any research into the history of tuning and temperament, aside from reading Doty's "The Just Intonation Primer". That lack of research is a massive detriment to his writing. For instance, in his search for a musical formula to quantify the consonance of a ratio, he settles on 1/x+1/y (i.e. 1/numerator+1/denominator) and says he considered but did not actually try 1/xy. However, if he had done any research, he would have found that the product xy is known as "Tenney Height" and is used as a standard measure of dissonance for ratios below a certain complexity threshold, so 1/xy would have given him a better (although still not problem-free) model. If he had done a little more research, he would have discovered the work of Paul Erlich and the theory of harmonic entropy, which would allow him to quantify the discordance of *any* dyad, even if it is irrational, and this model works considerably better than his 1/x+1/y formulation.

More importantly, if he had done his music history homework, he would have discovered not just that he is not the first person to undertake the task of temperament, but that he is only one of many people throughout history to seek the very same goal of a maximally-consonant 12-tone octave-repeating scale. He would have discovered the different varieties of Meantone tunings, and more importantly, the various "well-temperaments" that circulated (no pun intended) prior to the adoption of 12-TET. He would greatly have benefited from study at least the works of Andreas Werckmeister, and a comparison between his Blue Temperament and the "best" well-temperaments that have been historically documented would have lent much greater strength to his conclusions about his temperament's superiority.

Lack of research aside, John also makes the same fallacious assumption of the vast majority of tuning theorists: the assumption that a greater quality of consonance means an objectively better scale. Let me state this as plainly as possible: there is no such thing as a universally "best" scale. A scale is only good or bad relative to the wishes of a composer, and it _should_not_be_assumed_ that all composers would be happier with an unequally-tempered scale where a handful of keys approximate simple Just harmonies while a few keys are unacceptable. To assume this ignores all the reasons that 12-TET came to be adopted in the first place: namely, that composers began to desire that all 12 keys should work equivalently so that they could modulate freely. 12-TET conquered the musical world due to the demands of composers, not because it is the "best sounding" tuning but because of the sort of compositions its structure enables. Neither of John's tunings would likely be acceptable to a composer who shares compositional goals with the likes of Schonberg, Stravinsky, or perhaps even Beethoven. And though John mentions alternative EDOs like 19, 22, and 31 off-hand at one point, he does not mention the fact that any of these tunings do a much better job of approximating the ratios that he likes than does 12-TET, but without sacrificing key equivalence. Any of those EDOs would be a viable alternative to his Blue Temperament in terms of the quality of consonance they offer, and I suspect John did not delve further into them only because they would have weakened his case for his own temperament (or else out of a fear of more than 12 notes).

To continue this point, I should note that neither of John's tunings fit my own compositional desires, either. Like many musicians with an interest in microtonality, I sought escape from 12-TET not because I thought it sounded "out of tune" and wanted something that sounded "better", but because I sought alternative tonal frameworks, new scales, new harmonies--ways of making music that is totally impossible in 12-TET and capable of expressing emotions and concepts that 12-TET only hints at. I have found far and away the most success in this endeavor using tunings that deviate significantly from ideals of consonance and/or rely heavily on nontraditional consonances. I am admittedly somewhat of a maverick in this regard, but nevertheless I am a great example of a composer whose "best" tuning would be nowhere near either of John's tunings.

All of these issues notwithstanding, I do have to admit that John succeeded in meeting his own goals for the tuning. As far as 12-note temperaments that seek to approximate as many simple-ratio consonances as possible go, John's is definitely up there. It is a slight improvement on 1/4-comma Meantone, and though not as versatile as, say, Werckmeister III in terms of modulational flexibility, in its "good" keys it sounds a bit better. So despite his lack of research and rather informal methodology, he did in fact arrive at a "good" scale for his stated purposes. Whether other composers will find his scale preferable to any of the number of Meantones and well-temperaments that once flourished in the Western musical community, I cannot say, but John's Blue Temperament is definitely worthy competition.

In conclusion, I can't say it's a "bad" book; it is well-written, honest, succinct, and clear. But I cannot recommend it, because it does not include any of the thoroughly-researched and tested (and much more general and elegant) mathematical-musical theories of the modern age (harmonic entropy, the regular temperament paradigm, Tenney-Optimal tuning optimization, etc.); nor does it include any of the vast and relevant history of temperament. Also, the failure to include any competitive tunings, such as various well-temperaments, alternative EDOs, TOP temperaments such as TOP Meantone or TOP Pajara (both of these latter can produce 12-note scales, too) makes for a very shallow book, and their absence is very conspicuous to those who know better. This book is not an authoritative source on the relationship between music and mathematics, it is only a documentation of one man's individual quest for a certain type of harmony. It is my sincere hope that John will take my advice and immerse himself in some research so that he can come to understand both the historical evolution of temperament and the modern advances in temperament-math that have arisen from the Yahoo Tuning List in the last few decades, and that he actually explores some of the alternatives to his Blue Temperament and comes to understand their relative strengths and weaknesses. Perhaps then he might be prepared to write a book *worthy* of the title "The Mathematics of Music".

-Igs

Thanks Igs,

you promised me a fair and unbiased review of my book and that's exactly what you gave and I am very happy with it. I am also very grateful that you took the time to read and review my book.

One point: you said "he considered but did not actually try 1/xy".
On the top of page 18 I say "In the end, just to be sure, I had a look at the 1/xy formula and found one clear inconsistency and so ruled it out".

Thanks again and I know I have a lot more to learn.

John.

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> Hi all. John gave me a copy of his book and asked me to post a review of it to the Tuning List, so I'm obliging him. I don't expect the content of my review will be of much interest to most here, as you're all already familiar with his work and have your own opinions. You know me as well, and I'm sure most of you could guess at what I think of his book. This review is thus mostly for John's benefit, and the benefit of the lurkers.
>
> To start off, I have to say John deserves a lot of credit for his modesty. Throughout the book he makes plain that his work is based on educated guesses and his own psychoacoustic preferences, and he acknowledges he is probably not the first to look into these ideas and that some or most of his conclusions may turn out to be mistaken. This modesty and honesty is rare among those filled with the eager joy of breaking the confines of the 12-TET paradigm.
>
> That said, the title of the book is quite the misnomer, and very much at odds with John's attitude throughout the book. If he'd called it "My Search for a Better 12-Tone Scale" or something like that, I would approve whole-heartedly, as that's *really* what this book is. However, calling it "The Mathematics of Music" makes it sound much more definitive and authoritative than it really is, especially considering John is not a mathematician and is apparently ignorant of all the mathematical models and theories of music and consonance--and most importantly, temperament--that have been proposed throughout history (and especially on the Yahoo Tuning List).
>
> Reading this book, it is clear that John did hardly any research into the history of tuning and temperament, aside from reading Doty's "The Just Intonation Primer". That lack of research is a massive detriment to his writing. For instance, in his search for a musical formula to quantify the consonance of a ratio, he settles on 1/x+1/y (i.e. 1/numerator+1/denominator) and says he considered but did not actually try 1/xy. However, if he had done any research, he would have found that the product xy is known as "Tenney Height" and is used as a standard measure of dissonance for ratios below a certain complexity threshold, so 1/xy would have given him a better (although still not problem-free) model. If he had done a little more research, he would have discovered the work of Paul Erlich and the theory of harmonic entropy, which would allow him to quantify the discordance of *any* dyad, even if it is irrational, and this model works considerably better than his 1/x+1/y formulation.
>
> More importantly, if he had done his music history homework, he would have discovered not just that he is not the first person to undertake the task of temperament, but that he is only one of many people throughout history to seek the very same goal of a maximally-consonant 12-tone octave-repeating scale. He would have discovered the different varieties of Meantone tunings, and more importantly, the various "well-temperaments" that circulated (no pun intended) prior to the adoption of 12-TET. He would greatly have benefited from study at least the works of Andreas Werckmeister, and a comparison between his Blue Temperament and the "best" well-temperaments that have been historically documented would have lent much greater strength to his conclusions about his temperament's superiority.
>
> Lack of research aside, John also makes the same fallacious assumption of the vast majority of tuning theorists: the assumption that a greater quality of consonance means an objectively better scale. Let me state this as plainly as possible: there is no such thing as a universally "best" scale. A scale is only good or bad relative to the wishes of a composer, and it _should_not_be_assumed_ that all composers would be happier with an unequally-tempered scale where a handful of keys approximate simple Just harmonies while a few keys are unacceptable. To assume this ignores all the reasons that 12-TET came to be adopted in the first place: namely, that composers began to desire that all 12 keys should work equivalently so that they could modulate freely. 12-TET conquered the musical world due to the demands of composers, not because it is the "best sounding" tuning but because of the sort of compositions its structure enables. Neither of John's tunings would likely be acceptable to a composer who shares compositional goals with the likes of Schonberg, Stravinsky, or perhaps even Beethoven. And though John mentions alternative EDOs like 19, 22, and 31 off-hand at one point, he does not mention the fact that any of these tunings do a much better job of approximating the ratios that he likes than does 12-TET, but without sacrificing key equivalence. Any of those EDOs would be a viable alternative to his Blue Temperament in terms of the quality of consonance they offer, and I suspect John did not delve further into them only because they would have weakened his case for his own temperament (or else out of a fear of more than 12 notes).
>
> To continue this point, I should note that neither of John's tunings fit my own compositional desires, either. Like many musicians with an interest in microtonality, I sought escape from 12-TET not because I thought it sounded "out of tune" and wanted something that sounded "better", but because I sought alternative tonal frameworks, new scales, new harmonies--ways of making music that is totally impossible in 12-TET and capable of expressing emotions and concepts that 12-TET only hints at. I have found far and away the most success in this endeavor using tunings that deviate significantly from ideals of consonance and/or rely heavily on nontraditional consonances. I am admittedly somewhat of a maverick in this regard, but nevertheless I am a great example of a composer whose "best" tuning would be nowhere near either of John's tunings.
>
> All of these issues notwithstanding, I do have to admit that John succeeded in meeting his own goals for the tuning. As far as 12-note temperaments that seek to approximate as many simple-ratio consonances as possible go, John's is definitely up there. It is a slight improvement on 1/4-comma Meantone, and though not as versatile as, say, Werckmeister III in terms of modulational flexibility, in its "good" keys it sounds a bit better. So despite his lack of research and rather informal methodology, he did in fact arrive at a "good" scale for his stated purposes. Whether other composers will find his scale preferable to any of the number of Meantones and well-temperaments that once flourished in the Western musical community, I cannot say, but John's Blue Temperament is definitely worthy competition.
>
> In conclusion, I can't say it's a "bad" book; it is well-written, honest, succinct, and clear. But I cannot recommend it, because it does not include any of the thoroughly-researched and tested (and much more general and elegant) mathematical-musical theories of the modern age (harmonic entropy, the regular temperament paradigm, Tenney-Optimal tuning optimization, etc.); nor does it include any of the vast and relevant history of temperament. Also, the failure to include any competitive tunings, such as various well-temperaments, alternative EDOs, TOP temperaments such as TOP Meantone or TOP Pajara (both of these latter can produce 12-note scales, too) makes for a very shallow book, and their absence is very conspicuous to those who know better. This book is not an authoritative source on the relationship between music and mathematics, it is only a documentation of one man's individual quest for a certain type of harmony. It is my sincere hope that John will take my advice and immerse himself in some research so that he can come to understand both the historical evolution of temperament and the modern advances in temperament-math that have arisen from the Yahoo Tuning List in the last few decades, and that he actually explores some of the alternatives to his Blue Temperament and comes to understand their relative strengths and weaknesses. Perhaps then he might be prepared to write a book *worthy* of the title "The Mathematics of Music".
>
> -Igs
>

Igs>"However, if he had done any research, he would have found that the
product xy is known as "Tenney Height" and is used as a standard
measure of dissonance for ratios below a certain complexity threshold,
so 1/xy would have given him a better (although still not problem-free)
model."

   Tenney height is a model I am certainly NOT a big fan of.  Numerator times denominator, wow...any fifth grader could have easily come across it.
   What exactly proves Tenney Height as so superior in the first place?  For example, it fails miserably for dyads with a complexity over about 70...at best, it's a good "very low limit dyad-only" model.
  Tenney Height throws a whole lot of 11-limit and a fair deal of 9-limit straight out the window, and leaves so much of what makes Middle Eastern and Eastern music work ignored.  It basically points of right back at Western theory for the most part.

>"He would greatly have benefited from study at least the works of
Andreas Werckmeister, and a comparison between his Blue Temperament and
the "best" well-temperaments"

    IMVHO, if you're serious about that point, this warrants a dyad-by-dyad analysis of the best well temperaments vs. John's temperaments (since John's work is based on dyadic analysis).   Actually (on the side), since my "Dimension" scale is really a cross between Mohajira and Meantone, I would not mind a bit if people challenged by scale against those two.

>"the assumption that a greater quality of consonance means an objectively better scale"

   Well, considering just about every person I've asked outside the list says microtonal music sounds too dissonance...I'll at least say increasing consonance vis-a-vis most existing microtonal scales would be an improvement in most people's ears.  Of course, that's not 100% across the board...but...close enough.

>"he does not mention the fact that any of these tunings do a much better
job of approximating the ratios that he likes than does 12-TET, but
without sacrificing key equivalence."

   But, Igs, key equivalence is no more a universal truth than the idea more consonance is always better.  If I'm reading this correctly, you are saying anything that fails to make the EDO "perfect transposition" standard is ignorant garbage...and if that's not blatant subjective opinion I don't know what is.

>"I have found far and away the most success in this endeavor using
tunings that deviate significantly from ideals of consonance and/or
rely heavily on nontraditional consonances. I am admittedly somewhat of
a maverick in this regard, but nevertheless I am a great example of a
composer whose "best" tuning would be nowhere near either of John's
tunings."

    But who said there was an effort made to write something that would work for 100% everyone?  I don't see it...but I do hear you being cocky.  A more appropriate way to say it "not to say it wouldn't work for other people but...it's simply not my cup of tea...to me half the fun is non-traditional consonances, something John does not appear to focus on".

>"But I cannot recommend it, because it does not include any of the
thoroughly-researched and tested (and much more general and elegant)
mathematical-musical theories of the modern age (harmonic entropy, the
regular temperament paradigm, Tenney-Optimal tuning optimization,
etc.);"

   This is the equivalent of saying "I don't care about how scales sound or what emotions they contain...but I care much more about the lack of current/less-complex mathematical models they are based on". 
   That's bass ackwards and elitist as heck (and elitist, with no goal except insulting those who 'aren't', IMVHO...is you're going to attack something, attack how certain chords/dyads/progressions...in the scale sound...and then go about proving how a "better" mathematical model would yield a better answer. 
 

On Sun, Feb 13, 2011 at 8:50 PM, Michael <djtrancendance@...> wrote:
>
> Igs>"However, if he had done any research, he would have found that the product xy is known as "Tenney Height" and is used as a standard measure of dissonance for ratios below a certain complexity threshold, so 1/xy would have given him a better (although still not problem-free) model."
>
>    Tenney height is a model I am certainly NOT a big fan of.  Numerator times denominator, wow...any fifth grader could have easily come across it.
>    What exactly proves Tenney Height as so superior in the first place?  For example, it fails miserably for dyads with a complexity over about 70...at best, it's a good "very low limit dyad-only" model.
>   Tenney Height throws a whole lot of 11-limit and a fair deal of 9-limit straight out the window, and leaves so much of what makes Middle Eastern and Eastern music work ignored.  It basically points of right back at Western theory for the most part.

That's where something like Harmonic Entropy comes more in handy. Or,
if what you're mainly concerned with is beating, Sethares' dissonance
model. I've wanted to expand on Sethares' model for quite some time
now to incorporate some of the periodicity buzz stuff we were delving
into last month.

-Mike

Okay, this is very late in coming, since John O'Sullivan sent me his book
ca. a month ago, but it has taken me this long to reach a point at which I
feel comfortable evaluating John's methods.

About me: I'm 30 y.o., a computer engineer by training, and I have an
abiding interest in music, listening, and related theory. I'm no expert in
these matters, however, although I have been involved with them practically
my entire life and at certain points was even pretty good at some of them.

The reviewers who have already responded to John's request have said a lot
already, of course. Jake F's review was excellent. Mine will be more brief.

John's book is meant for the enthusiast, and he does a good job carrying his
enthusiasm forward. His chapters are very manageable, and this book is brief
and skimmable. The motivations for his particular choices are not always
clear. He does, however, give a fairly clear description of what he is doing
at each step. He does explain his reasoning, which is great; when parsing
through ideas, statements of motivation can be like landmarks. What is not
crystal, however, is why these choices are theoretically significant.

I don't have the book with me right now, but, as I recall, the first 16
pages are a basic introduction to JI, and the most significant chapters are
6-10, where John lays out his evaluation methods for chords. Chapter 13 is
one page and is titled The Music Matrix; it wasn't entirely clear to me why
the method given there would only produce "sweet" chords.

I trust John's ear and think he has some good notions. Hopefully he will
continue to learn here how to elaborate and to refine his musical interests
theoretically in a community of experts and amateurs. I'm very glad to have
this book. I agree it need not be referenced onlist.

Dan N