Relationship(s) between Music and Math: References please?
Kharim Hogan
I am looking for any references, bibliographies, pointers to anything
that discusses or otherwise deals with the relationship(s) between
music and math. Does anyone have any such references, know of anyone
doing work/research along these lines or have pointers to where I might
find anything along these lines? Specific or general is fine.
Many thank in advance,
Kharim Hogan
Public Cluster Macintosh
<kha...@cs.indiana.edu> wrote:
Tuning theory is quite mathematical. If you have something like this in
mind, please post and I'll give you a bunch of references. I'm doing
research in this area myself.
--
This article/message was posted using NewsWatcher from a Macintosh in an Academic Computing Services public cluster at Yale University. Yale University accepts no responsibility for the identity of the author or the content of this message.
Jeffrey Dods
A book you may want to look at is by (the late) Sir James Jeans.
It is called "Science and Music"
Jeans says his book relys much on the work of the great physisist
Hermann Helmholtz.
The good news is that both these books can be ordered from Dover.
In looking through the Dover Math and Science catalog, I can see
that there may be lots more titles interesting to you.
Jeff.
jd...@chem.utoronto.ca
gr...@netcom.com
: Hi,
You might delve into "Bach, Esher, and Goedel" by Hostetler. He has lots
of interesting insights into Bach's music and mathematics.
grip
--
Gene Ward Smith
> I am looking for any references, bibliographies, pointers to anything
> that discusses or otherwise deals with the relationship(s) between
> music and math.
I used to do stuff along these lines, but got discouraged when the
Computer Music Journal turned down a paper for being "too mathematical".
I doubt the stuff I did (on scales, groups of transformations, etc.) is
what interests you. I will leave you to ponder the following formula:
if T is a "large" local maximum of |zeta(1/2 + it)|, where zeta is the
Riemann zeta function, and "large" can be made precise in terms of
the usual conjectures about the rate of growth of such maxima, then
I = 2400 pi / (ln(2) T) is, in cents, a "good" value for an equal-step
musical scale--if approximating rational numbers is "good".
I leave you to ponder the significance, if any.
--
Gene Ward Smith/Brahms Gang/University of Toledo
gsm...@math.utoledo.edu
Jon Claerbout
at your local city library.
--
Jon Claerbout, j...@sep.stanford.edu, Stanford Exploration Project
<a> HREF="http://sepwww.stanford.edu/sep/jon"> Jon Claerbout </a>
Gerry Myerson
scales and the generalized circle of fifths, Amer. Math. Monthly 93
(1986) 695--701.
G. Myerson
Public Cluster Macintosh
<gsm...@math.utoledo.edu> wrote:
> On Thu, 1 Dec 1994, Kharim Hogan wrote:
>
> > I am looking for any references, bibliographies, pointers to anything
> > that discusses or otherwise deals with the relationship(s) between
> > music and math.
>
> I used to do stuff along these lines, but got discouraged when the
> Computer Music Journal turned down a paper for being "too mathematical".
> I doubt the stuff I did (on scales, groups of transformations, etc.) is
> what interests you. I will leave you to ponder the following formula:
> if T is a "large" local maximum of |zeta(1/2 + it)|, where zeta is the
> Riemann zeta function, and "large" can be made precise in terms of
> the usual conjectures about the rate of growth of such maxima, then
> I = 2400 pi / (ln(2) T) is, in cents, a "good" value for an equal-step
> musical scale--if approximating rational numbers is "good".
I work in this area and would love to see what results you get with this
function. Please post. This sounds like the sort of thing I've been
pondering but my physics degree leaves me lacking in the mathematical
knowledge.
Gene Ward Smith
Gene Ward Smith/Brahms Gang/University of Toledo
gsm...@math.utoledo.edu
On Mon, 5 Dec 1994, Dave Rusin wrote:
> OK, I'll bite. I had decided that the number of steps in an equal step
> scale ought to occur in the continued fraction approximation of
> ln(3)/ln(2). What's your connection with zeta?
Zeta(s+it) = 1 + 2^(-s)(cos(t) + isin(t)) + ...
when s > 1, so if you chose a place like you suggest, it makes both
the "2" and the "3" term large outside the critical strip--and even
inside, by the Riemann-Siegel formula, etc.
Really, though, we want to do better than just approximate thirds and
fourths, which is what your method aims at.
Mark Chrisman
Most of the responses to this thread have dealt with things like tuning and
scales. Anyone heard of a mathematical analysis of (for example) harmony and
voice leading in Baroque music?
--------------------------------------------------------
Mark Chrisman (if your reply is of general interest
please post it, don't email it.)
Dean Schulze
__The Science of Fractal Images__. It has been quite some time since
I looked at this book, however.
Dean Schulze
TITLE The Science of fractal images.
OTHER AUTH Peitgen, Heinz-Otto, 1945-
Saupe, Dietmar, 1954-
Barnsley, M. F. (Michael Fielding), 1946-
PUBLISHER New York : Springer-Verlag, c1988.
SUBJECTS Fractals.
NOTE Based on notes for the course Fractals--introduction, basics, and
perspectives given by Michael F. Barnsley, and others, as part
of the SIGGRAPH '87 (Anaheim, Calif.) course program.
Dean Schulze
(I mean this in the most casual way - this is just my observation.) It
seems that many scientists and mathematicians had a musical background
in their youth. Has anyone else noticed this?
Dean Schulze
Dennis J. Ciplickas
Check out this book:
@book(weigend-gershenfeld-92
,author="Weigend, Andreas S. and Gershenfeld, Neil A."
,title="Time series prediction: forecasting the future
and understanding the past: proceedings of the
NATO advanced Research Workshop on Comparative
Time Series Analysis"
,publisher="Addison-Wesley"
,address="Reading, MA"
,year="1992"
)
There was an interesting article describing the application of time
series analysis to (one of?) Bach's Unfinished Fugue(s). I seem to
recall references at the end of the chapter.
-Dennis
William C. Snyder
>
>In article <Pine.SOL.3.91.941203202711.1374C-100000@math>, Gene Ward Smith
><gsm...@math.utoledo.edu> wrote:
>
>> On Thu, 1 Dec 1994, Kharim Hogan wrote:
>>
>> > I am looking for any references, bibliographies, pointers to anything
>> > that discusses or otherwise deals with the relationship(s) between
>> > music and math.
>>
>> if T is a "large" local maximum of |zeta(1/2 + it)|, where zeta is the
>> Riemann zeta function, and "large" can be made precise in terms of
>> the usual conjectures about the rate of growth of such maxima, then
>> I = 2400 pi / (ln(2) T) is, in cents, a "good" value for an equal-step
>> musical scale--if approximating rational numbers is "good".
>
>I work in this area and would love to see what results you get with this
>function. Please post. This sounds like the sort of thing I've been
>pondering but my physics degree leaves me lacking in the mathematical
>knowledge.
I don't know how this relates to the aforementioned method for obtaining a
musical scale, but here is an interesting experiment. I'd like to hear if
others have seen this:
Postulate just two requirements for a musical scale.
1. Ability to transpose.
In other words, the scale should have equal geometric steps
in the octave so that a piece could be moved up or down
without changing the relation between notes.
step = r*f, r=step ratio, f=frequency
number of steps = n,
k^n = 2 (octave), --> nlogk = 2
2. Existance of many "good" intervals.
The scale should provide many or all of the small integer
ratios of frequecies. ( 2:1, 3:2, 4:3, ... ). These ratios
should be accurate enough to sound good to the human ear (1%).
I plotted the results and found that 12 steps is "unusually" efficient
in providing the above requirements. You have to go several more steps
to get any improvement in the number of intervals supported. There is also
a peak at seven steps, which I believe is the number for certain eastern
scales.
--
William C. Snyder
University of California Santa Barbara
CRSEO: Center for Remote Sensing and Environmental Optics
Ed Stokes
Rauscher, F.H., G.L. Shaw, and K.N. Ky, "Music and spatial task performance",
Nature, 365, 611 (1993).
"...We performed an experiment in which students were each given three
sets of standard IQ spatial reasoning tasks; each task was preceded
by 10 minutes of (1) listening to Mozart's sonata for two pianos
in D major; (2) listening to a relaxation tape; or (3) silence.
Performance was improved for those tasks immediately following the
first condition compared to the second two."
Public Cluster Macintosh
Author: Lindley, Mark, 1937-
Title: Mathematical models of musical scales : a new approach / Mark
Lindley, Ronald Turner-Smith.
Published: Bonn : Verlag fîur Systematische Musikwissenschaft, 1993.
Description: 308 p. : ill. ; 25 cm.
Author: Panthaleon van Eck, C. L.van.
Title: J.S. Bach's critique of pure music / C.L. van Panthaleon van
Eck.
Published: Netherlands : C.L. van Panthaleon van Eck, c1981.
Description: 170 p. : music ; 26 cm.
Author: Mann, Chester D.
Title: Analytic study of harmonic intervals / by Chester D. Mann.
Published: Tustin, Calif. : C.D. Mann, 1990.
Description: vii, 196 p. : ill. ; 28 cm.
Author: Barbour, James Murray, 1897-
Title: Tuning and temperament; a historical survey, by J. Murray
Barbour.
Published: New York, Da Capo Press, 1972 [c1951]
Description: xii, 228 p. 22 cm.
Author: Pierce, John Robinson, 1910-
Title: The science of musical sound / John R. Pierce.
Edition: Rev. ed.
Published: New York : Freeman, c1992.
Description: xi, 270 p. : ill.; 24 cm.
If you're interested, I'll post some journal references.
Gerhard Niklasch
In article <3c27t5$g...@ucsbuxb.ucsb.edu>, wi...@crseo.ucsb.edu
(William C. Snyder) writes:
[...]
|> I don't know how this relates to the aforementioned method for obtaining a
|> musical scale, but here is an interesting experiment. I'd like to hear if
|> others have seen this:
|>
|> Postulate just two requirements for a musical scale.
|>
|> 1. Ability to transpose.
|> In other words, the scale should have equal geometric steps
|> in the octave so that a piece could be moved up or down
|> without changing the relation between notes.
|> step = r*f, r=step ratio, f=frequency
|> number of steps = n,
|> k^n = 2 (octave), --> nlogk = 2
|>
|> 2. Existance of many "good" intervals.
|> The scale should provide many or all of the small integer
|> ratios of frequecies. ( 2:1, 3:2, 4:3, ... ). These ratios
|> should be accurate enough to sound good to the human ear (1%).
|>
|>
|> I plotted the results and found that 12 steps is "unusually" efficient
|> in providing the above requirements. You have to go several more steps
|> to get any improvement in the number of intervals supported. There is also
|> a peak at seven steps, which I believe is the number for certain eastern
|> scales.
Viggo Brun (the same who showed that \sum 1/p extended over twin primes
is finite) once wrote an article on this. Try the collected works, or
ask again and I could search for the reference.
With respect to the above postulates, you should really go one step
further: Ability to transpose doesn't require the octave to be
represented _exactly_ . You could have scales that produce more
accurate fifths than octaves... perhaps also approximating the 4:1 ratio
well, or 8:1, but not necessarily 2:1 ...
Regards, Gerhard
--
+------------------------------------+----------------------------------------+
| Gerhard Niklasch | All opinions are mine --- I even doubt |
| <ni...@mathematik.tu-muenchen.de> | whether this Institute HAS opinions:-] |
+------------------------------------+----------------------------------------+
Andrew Mullhaupt
: These books are good and all contain much interesting mathematics:
In the usual tradition of shamelessly advertising my own work:
Douthett, Entringer and Mullhaupt, "Musical Scale Construction: The Continued
Fraction Compromise", Utilitas Mathematica, v. 42 (1992).
This paper is about how different notions of best approximation apply to
the construction of "fifth harmonious" equal tempered systems. The most
interesting part mathematically, is that an actual new theorem.
Later,
Andrew Mullhaupt
Daniel Kornhauser
<On Thu, 1 Dec 1994, Kharim Hogan wrote:
<
<< that discusses or otherwise deals with the relationship(s) between
<< music and math.
<
<Computer Music Journal turned down a paper for being "too mathematical".
<I doubt the stuff I did (on scales, groups of transformations, etc.) is
<if T is a "large" local maximum of |zeta(1/2 + it)|, where zeta is the
<Riemann zeta function, and "large" can be made precise in terms of
<the usual conjectures about the rate of growth of such maxima, then
<I = 2400 pi / (ln(2) T) is, in cents, a "good" value for an equal-step
<musical scale--if approximating rational numbers is "good".
<
<--
This sounds intriguing to me. Could you please post a reference for this
result, and others that you have obtained? (Or, e-mail me.)
-- Dan Kornhauser
korn...@oasys.dt.navy.mil
Gerry Myerson
=> [I've edited this a lot]
=> Postulate just two requirements for a musical scale.
=>
=> 1. Ability to transpose.
=>
=> 2. Existence of many "good" intervals.
=>
=> I plotted the results and found ...
Something resembling the diagram on p. 45 of Steinhaus, Mathematical
Snapshots, 3rd American edition, Oxford 1969?
For other editions, look up "scale" in the index.
Gerry Myerson (ge...@mpce.mq.edu.au)
Centre for Number Theory Research (E7A)
Macquarie University, NSW 2109, Australia
Kevin M. Johnson
: In article <ph-051294...@conn-hall-kstar-node.net.yale.edu>, p...@directory.yale.edu (Public Cluster Macintosh) says:
: >
: >In article <Pine.SOL.3.91.941203202711.1374C-100000@math>, Gene Ward Smith
: ><gsm...@math.utoledo.edu> wrote:
: >
: >> On Thu, 1 Dec 1994, Kharim Hogan wrote:
: >>
: >> > I am looking for any references, bibliographies, pointers to anything
: >> > that discusses or otherwise deals with the relationship(s) between
: >> > music and math.
: >>
: I plotted the results and found that 12 steps is "unusually" efficient
: in providing the above requirements. You have to go several more steps
: to get any improvement in the number of intervals supported. There is also
: a peak at seven steps, which I believe is the number for certain eastern
: scales.
A German composer, Schonberg I think, (early 20th century?) wrote several
12-tone compositions and performed them. I think I listened to one, and
it sounded discordant. I don't think audiences at the time appreciated
them. You might check into that.
--
Kevin Johnson
k...@bilbo.baylor.edu
"There are two kinds of geniuses. An ordinary genius is a fellow that
you and I would be just as good as, if we were only many times better.
The other kind are magicians. Even after we understand what they have
done, the process by which they have done it is completely dark."
-- Mark Kac, Mathematician
Public Cluster Macintosh
> In article <3c27t5$g...@ucsbuxb.ucsb.edu>, wi...@crseo.ucsb.edu
> (William C. Snyder) writes:
> [...]
> |> I don't know how this relates to the aforementioned method for obtaining a
> |> musical scale, but here is an interesting experiment. I'd like to hear if
> |> others have seen this:
> |>
> |> Postulate just two requirements for a musical scale.
> |>
> |> 1. Ability to transpose.
> |> In other words, the scale should have equal geometric steps
> |> in the octave so that a piece could be moved up or down
> |> without changing the relation between notes.
> |> step = r*f, r=step ratio, f=frequency
> |> number of steps = n,
> |> k^n = 2 (octave), --> nlogk = 2
> |>
> |> 2. Existance of many "good" intervals.
> |> The scale should provide many or all of the small integer
> |> ratios of frequecies. ( 2:1, 3:2, 4:3, ... ). These ratios
> |> should be accurate enough to sound good to the human ear (1%).
> |>
> |>
> |> I plotted the results and found that 12 steps is "unusually" efficient
> |> in providing the above requirements. You have to go several more steps
> |> to get any improvement in the number of intervals supported. There is also
> |> a peak at seven steps, which I believe is the number for certain eastern
> |> scales.
>
I calculated this using a log-normal distribution with a s.d. of 1% for
each interval, and then taking the geometric mean. The intervals are
weighted in inverse proportion to the total number with the same "limit"
classification, i.e., in the calculation for the 5-limit, 3:1 is weighted
twice as much as 5:1 and 5:3. Notice that the exactness of the octave
takes care of all other intervals automatically.
notes/octave 3-limit 5-limit 7-limit
1 0 0 0
2 2.11E-16 0 0
3 3.19E-15 3.14E-09 5.74E-25
4 2.11E-16 8.08E-10 8.13E-07
5 0.3228273 3.54E-08 2.70E-07
6 3.19E-15 3.14E-09 2.01E-06
7 0.4001807 3.51E-02 5.69E-05
8 3.30E-04 4.76E-04 6.25E-04
9 1.32E-02 1.19E-02 1.30E-02
10 0.3228273 0.0315652 9.37E-02
11 4.08E-04 3.49E-03 6.14E-03
12 0.9868176 0.6038773 0.1965739
13 9.78E-03 2.69E-02 2.86E-02
14 0.4001807 3.96E-02 0.1068103
15 0.3228273 0.5402809 0.3450564
16 8.02E-02 0.2802013 0.2298534
17 0.9478536 7.51E-02 0.1596261
18 0.0327528 0.1793055 0.2021525
19 0.8345147 0.8841485 0.5710635
20 0.3228273 0.2319016 0.2539463
21 0.4001807 0.2517376 0.3790156
22 0.8379369 0.7874842 0.6610211
23 0.1423669 0.3095289 0.3896932
24 0.9868176 0.6038773 0.4497839
25 0.3228273 0.4219834 0.4679604
26 0.7238607 0.6008141 0.6124499
27 0.7474462 0.7134926 0.7959956
28 0.4001807 0.5549609 0.5009584
29 0.9922869 0.606263 0.5349888
30 0.3228273 0.5402809 0.5327793
31 0.9110161 0.9296547 0.9522942
32 0.6797012 0.571204 0.6125636
33 0.6546545 0.5977407 0.6419944
34 0.9478536 0.9735731 0.62182
Much Eastern and African music is based on the 3-limit, so it is not
surprising that 5-, 7-, and 12-note equal temperaments are used there.
Western (renaissance and tonal) music is based on the 5-limit, thus
12-equal is used, with proposals for 19 and 31, based on their greater
purity of tuning, usually rejected because of their greater complexity. 22
and 34, though acheiving high ratings above, do not allow for all the
chords in a diatonic scale to be in tune, or for smooth modulations
between keys.
At 35, things get hairy since there are two ways to approximate 3:1 with a
1% error.
-Paul Erlich
Public Cluster Macintosh
> scales. Anyone heard of a mathematical analysis of (for example) harmony and
> voice leading in Baroque music?
>
Prem Sobel
Kornhauser) writes:
>In sci.math, Gene Ward Smith <gsm...@math.utoledo.edu< writes:
><On Thu, 1 Dec 1994, Kharim Hogan wrote:
><I used to do stuff along these lines, but got discouraged when the
><Computer Music Journal turned down a paper for being "too
mathematical".
><I doubt the stuff I did (on scales, groups of transformations, etc.)
is
><what interests you. I will leave you to ponder the following formula:
><if T is a "large" local maximum of |zeta(1/2 + it)|, where zeta is the
><Riemann zeta function, and "large" can be made precise in terms of
><the usual conjectures about the rate of growth of such maxima, then
><I = 2400 pi / (ln(2) T) is, in cents, a "good" value for an equal-step
><musical scale--if approximating rational numbers is "good".
> This sounds intriguing to me. Could you please post a reference
for this result, and others that you have obtained? (Or, e-mail me.)
Why bbother approximating the rationals when you can have an infinite
dimensional just scale consisting only of rational muiscal intervals.
In actual practice after studying the scales which people use, 99%
need not go beyond the 4 dimensional rational scale where all intervals
are of the form:
int = 2^n2 * 3^n3 * 5^n5 * 7^n7
1<=int<=2, integer exponents
Since it is not practical to allow the exponents to grow in magnitude
without bound, I have designed (on paper) a keyboard which makes chords
easy and gives the effect of unbounded magnitude by using a special key
to mean exponentially shift selected key to center of range. I have
sythesised some music in this scale and it is quite nice.
Prem
Randall C. Poe
|> In article <1994Dec1.1...@news.cs.indiana.edu>, "Kharim Hogan"
|> <kha...@cs.indiana.edu> wrote:
|>
|> > Hi,
|> >
|> > I am looking for any references, bibliographies, pointers to anything
|> > that discusses or otherwise deals with the relationship(s) between
|> > music and math. Does anyone have any such references, know of anyone
|> > doing work/research along these lines or have pointers to where I might
|> > find anything along these lines? Specific or general is fine.
|> >
|> > Many thank in advance,
|> > Kharim Hogan
|>
|> Tuning theory is quite mathematical. If you have something like this in
|> mind, please post and I'll give you a bunch of references. I'm doing
|> research in this area myself.
|>
Isn't the original mathematical theory of "perfect" intervals (thirds,
fifths, and octaves) from Pythagoras? I seem to recall the term "Pythagorean
tuning". I know that the shift to modern temperament in Bach's time involved
compromising on the perfect intervals so that all 12 key signatures were
usable.
I suspect there's much more involved and interesting theory in what makes
for a "pleasing" sound. I have seen Sci.Am. articles on sound in piano and
violin (why do Stradivari's violins sound so good?) which were quite
interesting.
Another mathematical aspect is in composition, under the rules of fugue
construction and counterpoint. See "Godel, Escher, Bach" for the astonishing
facility Bach showed with this form in not only solving the complex
multivariable problems involved, but in making beautiful music at the same
time.
Public Cluster Macintosh
> Why bbother approximating the rationals when you can have an infinite
> dimensional just scale consisting only of rational muiscal intervals.
> In actual practice after studying the scales which people use, 99%
> need not go beyond the 4 dimensional rational scale where all intervals
> are of the form:
>
> int = 2^n2 * 3^n3 * 5^n5 * 7^n7
> 1<=int<=2, integer exponents
>
> Since it is not practical to allow the exponents to grow in magnitude
> without bound, I have designed (on paper) a keyboard which makes chords
> easy and gives the effect of unbounded magnitude by using a special key
> to mean exponentially shift selected key to center of range. I have
> sythesised some music in this scale and it is quite nice.
>
> Prem
Just intonation has many problems: scales sound awkward, there is always
one out-of-tune chord in a diatonic scale, modulations often require
ugly-sounding microtonal adjustments, etc. How do you deal?
-Paul