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Music and Mathematics

randulo

Playing saxophone 21 months - 2.4% of my life
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I stumbled across this web page quite by accident, and found myself captivated by the topic such that I couldn't stop reading it to the end, I thought I would share it with my friends at the Cafe. Music and Mathematics
Thanks for sharing!

Joseph Schillinger's Mathematic Basis of the Arts (1943) is a look at music and art at the atomic level.
Full text of "The Mathematical Basis Of The Arts ( Joseph Schillinger, 1943)"

For example,
"For a number of centuries philosophers have suspected that there are unconscious mathematical procedures behind conscious musical intentions. Music becomes "the mathematics_of-the-soul.'' he raw material of the mathematics of music begins with atomic structure and the life of living cells. It is quite simple to solve all the problems of musical creation with the mathematical equipment we possess at present. All musical procedures are only special cases of the general scheme of
pattern-making."

The bold sentence in the quote seems counter-intuitive as no one has heard a Schillinger compations played as far as I know. Famous composers and musician, such as George Gershwin, studied with Schillinger, though, so he wasn't "all wet", either.

The digitised version isn't perfect, so it's not easy reading. Re-issue copies are available but quite pricey ($40 to $100+).
I think there are many interesting thought-provoking things in this book, but there are also personal observations that are, uh, personal and some science that has been surpassed. Band in a Box would have blown Schillinger away, since he states something close to "machines can't create art".

Schillinger's work lead to Nicolas Slonimsky's Thesaurus of Scales and Melodic Patterns:


The Thesaurus takes the math elucidated by Schillinger and derives and elaborates patterns. It's a book of nothing but lines of standard notation, all in concert C. You will recognize many that have been used in jazz and classical music for decades. If you're a decent sight reader, this book would be a great thing if you're looking for new sounds on any instrument.
 

Halfers

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I'm very grateful that all these clever people used their maffs to work out all this stuff, so I can just sit back and stain their glorious legacy with some terrible rendition of something I just heard on the radio :)
 

randulo

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There's nothing wrong with that, as long as you realize that there's nothing wrong with examining and contemplating what's behind these things. In fact, that's all I can do, since I do not read music.
 

Halfers

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There's nothing wrong with that, as long as you realize that there's nothing wrong with examining and contemplating what's behind these things. In fact, that's all I can do, since I do not read music.
Absolutely. I wasn't criticising the Mathematical approach to understanding and examining Music, if that's what you picked up from my comment.
 

Halfers

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On the Mathematical front, what I find most interesting is the beautiful Geometric patterns that can be derived from the Mathematical connections of the musical notes - multi faced shapes created by linking notes from the Circle of Fifths etc.
 

Hipparion

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Having a quite strong background in mathematics I must admit that you've got me curious: if anyone has references (books/papers) relating music and maths to recommend, I am interested !
Bonus points if it involves group theory...
 

randulo

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@Hipparion check out either of the books above. The one is apparently public domain and available free, albeit in an imperfect format.
 

Dibbs

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Having a quite strong background in mathematics I must admit that you've got me curious: if anyone has references (books/papers) relating music and maths to recommend, I am interested !
Bonus points if it involves group theory...

There's no shortage of group theory in modern academic music theory.

Try neo-riemannian theory. For a gentle(ish) introduction Audacious Euphony: Chromatic Harmony and the Triad's Second Nature (Oxford Studies in Music Theory):Amazon.co.uk:Books

This is a little more challenging but very interesting and covers a lot more ground: https:://www.amazon.co.uk/Geometry-Music-Counterpoint-Extended-Practice/dp/0195336674/ref=pd_bxgy_14_img_2/262-4310014-5542751?_encoding=UTF8&pd_rd_i=0195336674&pd_rd_r=b72aa5b8-42c8-461d-a9bb-bebd905bec0a&pd_rd_w=BjuXq&pd_rd_wg=n5Nfx&pf_rd_p=7a9d3b22-47b7-4932-be38-57f4219c3325&pf_rd_r=85EAAQWX52ZC1QQ21BAN&psc=1&refRID=85EAAQWX52ZC1QQ21BAN

Here's an academic paper by the same guy that's more "mathsy" than his book. I struggle at this level. You might enjoy it.. http://dmitri.mycpanel.princeton.edu/voiceleading.pdf

How about this:

View: https://www.youtube.com/watch?v=krH0muWvQXY
 
OP
jbtsax

jbtsax

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It warms my heart to know I'm not the only "geek" here who is interested in this "stuff". Here is a quote from the article:

But he too [Issac Newton] was obsessed with musical ratios, devising a ‘palindromic scale’ in which the intervals were the same whether you went up or down: 9:8, 16:15, 10:9, 9:8, 10:9, 16:15, 9:8. He compared it to the seven rainbow colours of the spectrum.
I'm hoping that someone smarter than I am can figure out this "palindromic scale" and write it down on a staff. The common "palindromic" scales that are the same going up or down of course are the chromatic, and whole tone. But these are made up of equal intervals. Extra bonus points if they can "show their work" as to how they came up with the notes/pitches involved.
 

Dibbs

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It warms my heart to know I'm not the only "geek" here who is interested in this "stuff". Here is a quote from the article:



I'm hoping that someone smarter than I am can figure out this "palindromic scale" and write it down on a staff. The common "palindromic" scales that are the same going up or down of course are the chromatic, and whole tone. But these are made up of equal intervals. Extra bonus points if they can "show their work" as to how they came up with the notes/pitches involved.

You can't write it down on a staff because it's a specific just tuning. Normal notation isn't specific enough..

The nearest would be a dorian mode since it goes TSTTTST but there are two different sizes of whole tone; 10:9 and 9:8. Dorian is palindromic in equal temperament too, so you could call Newton's scale dorian in a form of just intonation.
 

Hipparion

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I don't know yet how to explain Newton's ratios, but it seems to me that he was trying to define a temperament, so his palindromic scale cannot be described with the pitches we currently use. It could be described using frequencies though (a tool he didn't have).

Using our current pitches, I don't think it is difficult to build palindromic scales. One just has to set the number of pitches he wants in said scale.

E.g., there are only 6 palindromic scales (starting on C) containing 3 pitches :
C-Db-B-C
C-D-Bb-C
C-Eb-A-C
C-E-Ab-C
C-F-A-C
C-Gb-Ab-C

But maybe I didn't understand correctly the question ?
 

Dibbs

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I don't know yet how to explain Newton's ratios, but it seems to me that he was trying to define a temperament,
...

Yes, they refer to frequency ratios and quite straightforward if you're not scared of a little maths.

2:1 is an octave
3:2 is a perfect 5th
5:4 is a just major third.

In Newton's scale the 9/8's and 10/9's are two different sizes of whole tone and the 16/15 is his semitone.

You can multiply intervals together as fractions to see how pure the thirds and fifths are. e.g. 9/8 x 10/9 = 5/4 - a just major third.

This is a really good introduction. Just Intonation Explained
 

nigeld

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Here's my attempt to compare Newton's palindromic scale with Pythagorean and equal temperament.
If we start on an A, then the notes in Newton's scale are approximately: A, B, C, D, E, F#, G
I have constructed a chart showing the frequencies of the notes in the three scales:
 

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