|Musical Mathematics — Cris Forster|
This is the book on tuning we have been waiting for. It was just
released in July 2010 after more than a decade in preparation.
Cris Forster is an instrument designer, builder and composer who
lives out in California. He’s originally from Brazil and has lived in
Germany as well. Cris even spent time restoring the old Harry Partch
instruments for the Harry Partch Foundation, and once packed them up
and shipped them to Germany and back for a concert.
Cris is a very serious and conscientious guy who always does the
best possible work. He’s a perfectionist and a rare craftsman in this
regard. His instruments are each gorgeous works of art.
Cris has always been a microtonalist but for 12 of the last 20 years he
has been researching tunings very intensively for the purpose of a
book he was working on. It took ten years to research and write, and
then it took another two years to format it, create all the hundreds
of necessary diagrams and graphs, index it, footnote it, and get it
published. That publication was just a few weeks ago, by Chronicle
Books in San Francisco who has made a well made book physically, very
well bound with sewn folios and high quality paper.
The book clearly reflects 12 years of work and is the most
comprehensive book on tuning I have seen. For many years people have
asked me if I know of a single excellent reference that will orient
them in all this tuning stuff. There hasn’t before really been a
single book that contained a real breadth of coverage, and much of the
most interesting material, particularly regarding ethnomusicology, was
buried in obscure out of print publications or extremely expensive
references. Finally I have a single book I can unequivocally
recommend. Whew. Thank goodness. At last.
It’s not just about tunings; the book begins with many chapters
extensively discussing acoustic musical instrument physics, providing
the solid foundation needed not just to build instruments, but to
understand where scales came from and why we hear things the way we
It then moves through 200 pages on western tunings, with a
comprehensive and extensively cited coverage of Ancient Greek music
before heading into the history of Western European tunings that most
of us are somewhat familiar with.
It finishes up with 400 pages of coverage of ethnic tunings
covering four different major traditions: Chinese, Indian,
Middle-Eastern (distinguishing between Arabic, Turkish and Persian),
and Indonesian tuning systems, each which have unique approaches.
There is coverage of Cris’ own instruments, a brief biography at the
end, and also directions for building your own small harmonic canon,
which will be helpful in understanding tunings in a direct and sensory
way. The canon is referred to occasionally in the text with suggested
experiments relevant to the tuning discussion.
The book is giant. Originally nearly 2000 pages long, the format of
the book was reworked so it would fit into 1000 pages. The result is
pretty dense text with long scan lines in a large and heavy book. If
there is one thing that could have been done differently it would be
to break this wonderful monster into a three volume set as reading it
can be backbreaking work due to the sheer weight. But that would have
increased the cost surely. Reading it pretty much needs to be done
lying on the floor as holding it up will hurt your wrists after a
while. Just letting you know that you get your money’s worth
with this book; it has so much useful information.
One tip is to bookmark the footnotes at the end of each chapter and
read them in tandem. In addition to the usual citation information
which comprises most of the footnotes, the notes also have quite a bit
of secret and fascinating information in them that probably would have
distracted from the flow of the text if placed in line. Reading these
is like going down the rabbit hole though. I quickly was led off into
other references and found I was spending days and days following up
on interesting things I didn’t know about just to advance a single
page in the book.
Cris’s style is dry. There’s not a lot of blatant humor here or
hilarious anecdotes. If you crave that sort of stuff there are other
books that can be used as a supplement. But it’s also not useless or
obscure and impenetrable academic information like we see in so many
of those journal articles and university theses about scales that come
out nowadays which try to make tuning into some sort of incredibly
abstract thought experiment. In contrast to that, the math in here is
necessary and describes real things. There are formulas or numbers on
almost every page, but much of this is because he doesn’t skip
steps but tries to explain everything in detail. The math is junior
high or high school level depending on where you went to school.
Possibly elementary level if you went to school outside the US. You
need to know fractions, multiplication, division, and a little about
logarithms. No calculus. It’s not pure math formulas for page
after page either, I don’t want to give the wrong idea. There is
a great deal of historical information as well that is clearly
relevant to the history and use of tuning. In fact, the math serves
these parts rather than the other way around.
As to whether you should get this book or not there is no question:
you should get it. You’re here on this site, you have an
interest in tuning, you probably have written or performed microtonal
music, well oh yes oh my certainly you definitely should be getting
this one without a doubt. It’s a brilliant work of scholarship
by someone who really knows what he is talking about. We have really
needed someone to take this approach and spend the time needed to
assemble all this information in just such a giant book and present it
in a straightforward, conscientious and accurate manner. Eventually
you are going to buy it and if you wait too long it could go out of
print since this is definitely specialist fare and not for the general
reading public. I recall years ago I bought the other giant red book
of tuning, Owen Jorgensen’s amazing manual of bearing plans and
common practice tuning history called “Tuning”. It was
about the same price, just under $70. Couldn’t afford it at the
time as I was a starving student, but I bought it anyway. Nowadays
“Tuning” runs $250-$700 for copies in poor to middling
condition assuming you can find it at all. So, you should get it right
now. And you should not be surprised if it takes you 10 years to read
if you do it like I am doing, one page at a time, trying to absorb
I’ve been going through the Greek and non-western sections and the
coverage is excellent and just the sort of specific and accurate
detail I have been craving for so long. I am so glad this book has at
last been published and I think you will be as well.
Here is Cris’ web site which has several sample chapters as
well as more information about his work: Chrysalis Foundation
[Musical Mathematics, On the Art and Science of Acoustic Instruments at Amazon]
|Genesis of a Music — Harry Partch|
Harry Partch built all his own instruments, invented his own scales, and is one of the
most important figures in american music, and is the most important historical figure
in modern microtonality. This book covers his philosophies, his rantings, the history
of tuning starting with the ancient greeks and chinese, technical descriptions of the
instruments he invented, and performance notes on his operas.
Some of the theory, particularly his ideas about psychoacoustics,
have been superseded by more contemporary research. But that
doesn’t matter at all because the ideas he had all proceeded
from his ear and not his theorizing and thus remain completely
This book, particularly the historical and ranting sections, is a
blast to read. Lots of fun. You’ll be highlighting or
underlining every page, laughing with Harry, and trying to remember
the marvelously erudite ways in which he says things.
It doesn’t matter what your field of interest is. You need
this book. This is a book by a microtonalist which I can recommend to
everyone and anyone with no hesitation.
[Genesis of a Music at Amazon]
|Tuning, Timbre, Spectrum, Scale — William Sethares|
What makes a set of notes sound dissonant or consonant when played
together? It depends strongly not just on the tuning used but the timbre as
Over the years there has been a lot of academic music theory papers
about consonance and dissonance. Most of them are nonsensical, dense,
obscure, wrong, and can be disregarded.
The works and research of Bill Sethares though can not be disregarded
because they are the first time someone understood everything and explained
Tuning, Timbre, Spectrum, Scale is not only the most important book
about tuning written to date, but it is the most important book
about music theory written in human history.
An understanding of the relationship between timbre and tuning is a
prerequisite for really understanding music. Since this book was published,
the bar has been raised. If you are not familiar with this material, you
don’t yet understand the theory of music.
Yes, this book is expensive, it is full of math, and it is challenging
to read. However, it is not complex for its own sake or to intimidate, but
to explain the truth as precisely as possible.
There is a great bibliography and discography included, as well as a CD
of musical examples.
Rather than get into great detail about it, you just need to read it and
understand it. If you don’t want to spend the cash, most of the key
insights are available in articles freely available on the author’s site.
[Tuning, Timbre, Spectrum, Scale at Amazon]
|Harmonograph — Anthony Ashton|
This fine little book is currently in print and is rather inexpensive.
Essentially this book is about ways in which to visualize musical dyads.
Ashton notes that this not the original Harmonograph book from the 1800s,
but he found that old book and reprinted its diagrams with his own notes. The diagrams
were made over 100 years ago using a mechanical device called a harmonograph,
which is a contraption with pendulums that is able to draw waveforms and patterns.
The book is not full of deep tuning insights, but it is delightful and worthwhile
nonetheless. There is information about a few basic just intonation intervals,
and a table of basic intervals in the appendix.
A fun vacation project might be to make one of the old mechanical harmonographs.
Directions on how to do so are included in the book, and also an overview of
how to make a Kaleidophone, and how to make Chladni Patterns by bowing a metal plate
covered with sand.
If you have a Mac, I have a program called IntervalCalc
that will create one of the types of harmonographs discussed in the book.
[Harmonograph at Amazon]
|Harmonic Experience — W. A. Mathieu|
This book is in textbook form and is nicely put together, with plenty of diagrams and discussion.
Mathieu believes that 12tET is a good tuning to use because it approximates many just intervals
and the ear is able to warp the mistuned degrees of 12 into actually hearing the just intervals
I don’t agree with Mathieu’s basic thesis there, but even so, this is a thoughtfully produced book,
a labor of love, and a lot of people really dig it.
Mathieu spends a lot of time teaching the reader to sing various intervals, and to breath and feel and experience the subtleties
of each interval. This is good training and a productive approach.
Music is fundamentally a spiritual, mystical science — something where I do agree
in total with Mathieu, who studied the teachings of the Sufi mystic Hazrat Inayat Khan.
Khan’s insights into music are gathered in a collection of essays titled “The Mysticism of Sound and Music”,
which I recommend to those interested in that subject.
I also feel an affinity towards Mathieu when he points out that infants and animals can not
distinguish octaves: that it is a learned skill. Yum to that!
The exercises are so good and the production values so high that I would say that this book
is worth getting, despite my disagreement with his central thesis. You will get maximum value from it if you plan to go through the exercises
and really try to feel and experience the harmonies on an intuitive, visceral level, an
approach I deeply respect.
[Harmonic Experience at Amazon]
|Tuning — Owen H. Jorgensen|
This ponderous tome is an encyclopedia of the art, practice and theory
of keyboard tuning for Western art music.
Meantones, well temperaments and equal temperaments are covered
here, including piano bearing plans for most of them. if you want to
know who first thought of a harpsichord or piano tuning, this book
will tell you that, along with all the advantages and controversies
surrounding tunings that were discussed back in the day. Lots of
references and historical data in here, backed by solid research.
Well-written, and an essential resource for anyone with any
interest in historical European tunings.
Unfortunately this has gone out of print and due to scarcity the
price on the used market can be high.
[Tuning at Amazon]
|Tuning and Temperament — J. Murray Barbour|
For those interested in an inexpensive technical history of modern
western tuning, the Dover reprint of Barbour’s Tuning and Temperament
is a reasonable choice. It’s well researched and has much concrete
information, such as tables of values in cents and ratios. Its historical focus
is quite specific and will be of most interest to those interested in
tunings of the Classical and Romantic periods. It does cover Greek
tunings briefly, and returns to Greek thought throughout the
text as appropriate. There is no coverage of non-western tunings.
Barbour’s university thesis that lead to this book was used as a
primary reference by Harry Partch for the historical tuning
information in Genesis of a Music, so it does make a useful complement
to go further.
[Tuning and Temperament at Amazon]
|Temperament — Stuart Isacoff|
Temperament is a social history of various events and drama in the
history of Western European tuning thought, culminating in the
adoption of 12 tone equal temperament.
Stuart Isacoff’s book garnered a rave review in the New York Times.
Because of this, this book may be the most well known, widely read and
popular book ever written on the subject of tuning. The book is
entertaining, with many lively anecdotes and gossip regarding famous
historical personages such as Leonardo da Vinci.
Despite these attributes, I can not recommend this book as a tuning
reference. Most of the book has nothing to do with tuning itself, but
is the titillating though irrelevant anecdotes.
On the few pages he dedicates to the topic of tuning, the
path he lays out for its history is one-dimensional and misleading or
even erroneous. As far as actual tuning material is concerned, he stays
close to the path of promoting his thesis that twelve tone equal
temperament is the triumph of centuries of battles between those who
moved towards the superiority of that tuning, and the backwards folks,
stubbornly holding on to tradition, who rejected its inevitability as the
supposed pinnacle of tuning technology and artistic expressiveness. The
stories chosen stick to the agenda, which unfortunately completely omits
the deep and illuminating richness of the true history of musical tunings.
The book is technically weak as well. It does not examine what
makes tuning unique or musically interesting. Throughout the book,
Isacoff talks to the readers as if even simple arithmetic is beyond
them. He avoids all issues of what intervals really are and keeps
everything at a do-re-mi technical level.
Those wishing to cut to the chase are advised to read the last chapter,
written after the book was finished. In it, he is hanging out with Philip
Glass and Michael Harrison and actually, for apparently the first time in
his life, hears a non-standard tuning (Harrison’s Revelation tuning)
in a composition. Liking the music, he is forced to admit that perhaps
everything he has said, and the conclusion of his book is simply wrong. At
least he speaks honestly to those who make it to the last page!
[Temperament at Amazon]
An older page with a short bibliography of articles and books, particularly about ethnic
music, and articles discussing nonoctave tunings, can be found at this link.