Music And Mathemathics

[sigurdV]

(Mahutma enters, takes a quick woof and d#appears."I must report this to the villain" he thinks")

 

Reading a book on Harmony theory I realized there is but three basic concepts: Scale,Chord and Tonality

 

Why the nunber three, I wondered... A quick check shows they imply each other:

 

Begin with "the natural" seven notes of the scale (as names I use numbers):1234567

 

(They are said to arise as overtones of a single note but that story is seldom covered in an ordinary Harmony book)

 

Apply the basic operation and the result is the chord: 1357246 (the ordinary chord consists of the first three members, the full chord is called a thirteen chord ...at least in swedish)

 

Apply it again and the result is the tonality: 1526374 (Which is the part of the circle of fifths containing the chords used in tonality 1... dont worry if you dont get the meaning here)

 

Apply it again and youre back with the scale: 1234567

 

So THATS why! I thought: the three concepts form a group based on 7!

 

Other basenumbers should give other amounts of concepts,

Will they differ in semantical content?

Isnt twelve tones the optimal number a la Schoenberg?

 

Actually ive forgotten most of the questions entering my mind some thirty years ago while i painstakingly calculated the values received on the first sixteen bases...

 

Hmmm this is very tiresome... isnt there whiskey in the jar... if i were a...well whatever they call themselves... hick...slurp...

Hey you guys! Do a poor composer a favor and figure some values out so the hidden secrets of music might get visible ...

I distinctly remember doing the list, but where did I put it and what function can give such values ... and (snore).

[SaxonViolence]

Did anyone ever see, "Children of a Lesser God" ?

 

The congenitally deaf woman asked her hearing husband (or boyfriend) to explain Music to her.

 

Remember the Spectrum?

 

ROY G BIV

 

Red Orange Yellow Green Blue Indigo Violet.

 

Seven, just like in an Octave.

 

Some folks feel that Indigo isn't a true color, and Newton was straining to get to lucky #7. That is a topic in itself.

 

But Indigo may be a little too close to Violet for our purposes. Our "Indigo" will be pure White.

 

C D E F G A B C

 

I think that if I remember correctly, there are eleven octaves on a piano.

 

Take some colored diodes, or lights.

 

C= Red; D=Orange; E=Yellow; F=Green; G=Blue; H=Indigo(White); B=Violet; A=Red.

 

Put a vertical row of twenty-five or thirty lights to represent every note (11*8=88)

 

It would look something like:

 

R O Y G B I V R O Y G . . . .

R O Y G B I V R O Y G . . . .

R O Y G B I V R O Y G. . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

 

Okay, set up a frequency detector that can distinguish between each musical note. You have to play with sampling frequency a bit--between every .20 to 1.0 Seconds.

 

The louder a given note is, the higher its column lights up--once again, you will have to experiment with the thresholds.

 

What about Sharps and Flats?

 

Space your Diodes far enough apart that there's room for another row between them, and start your sharps from the top.

 

C C# D D# E F F# G G#

R R O O Y G G B I

R O O Y Y G B B B

R R O O Y G G B I

R O O Y Y G B B B

 

Anyway, C# Is also D Flat--so they start from the top, the more decibels, the further Down the column lights--and C#/D flat alternate Red and Orange Lights. I think that this could give a deaf person at least the idea of what music does.....

 

Fact is, the human Brain is great at doing much World building with a minimal of data. Someone who watched the light show enough might be able to hear the music, or even compose using the light show.

 

But anyway, that's a relatively complex idea, and I hope that I got it across clearly.

 

Now my point is--assign mathematical values to all the lights--in some kind of reasonable way. Pitch can go on one axis,volume of discrete frequencies another.....

 

And put it on a time axis--or perhaps you could use Parametric equations.....

 

But I like the idea of time as a third spacial axis. That means that if we select our dimensions wisely, we can project a 3-D Model that's an excellent representation of a work of music as a static multi-colored object.

 

Also, you can look for patterns both numerically and visually (Intuitive and Rational--Side by Each).

 

Saxon Violence

[sigurdV]

With pleasure I greet newcomer saxon :)

 

My first critique of his work is a potzible spoiling error: "Octive" = "Octave"

I suppose it will be edited out eventually. His catching idea of "1234567=roebvir" adds a new dimension to our origin...

One cannot help woedering how many other interesting identifications the ingenious hypographicians will introduce!

But his selection of elements is not ideal since a standard identification is "1234567=cdefgab" which makes two elements carry two meanings :(

And since ""meaning"= 1" is possibly to be the future case, care should be taken when introducing

new possible isomorphies. Ideally no two bases contain the same element!

Theres a lot of work in keeping our Basement Clean, not to mention the cellar...So a word of warning: Nah! i need more... here we go:

[Turtle]

Reading a book on Harmony theory I realized there is but three basic concepts: Scale,Chord and Tonality

 

Why the nunber three, I wondered... A quick check shows they imply each other:

 

Begin with "the natural" seven notes of the scale (as names I use numbers):1234567

 

(They are said to arise as overtones of a single note but that story is seldom covered in an ordinary Harmony book)

 

Apply the basic operation and the result is the chord: 1357246 (the ordinary chord consists of the first three members, the full chord is called a thirteen chord ...at least in swedish)

 

Apply it again and the result is the tonality: 1526374 (Which is the part of the circle of fifths containing the chords used in tonality 1... dont worry if you dont get the meaning here)

 

Apply it again and youre back with the scale: 1234567

 

So THATS why! I thought: the three concepts form a group based on 7!

 

Other basenumbers should give other amounts of concepts,

...

 

i'm a musical dunce but somewhat handy with different bases. could you sing out what "the basic operation" is as i can't quite see and/or hear it. if i get that in my notebook i might be able to join the chorus. :note2: :note: :singer:

[sigurdV]

i'm a musical dunce but somewhat handy with different bases. could you sing out what "the basic operation" is as i can't quite see and/or hear it. if i get that in my notebook i might be able to join the chorus. :note2: :note: :singer:

OF COURSE! How stupid can I be?????????????????????????????????????????????????????????????? :doh:

 

Its simply to select the first element in the base, jump over next, select next, jump over next etc etc until theres nothing left. :)

 

So op(xyz)=xzy.

[sigurdV]

Theres been much ado about nothing (base 0?) so instead of repeating my words in here i suggest readers go to: Universe from nothing #42

[Turtle]

OF COURSE! How stupid can I be?????????????????????????????????????????????????????????????? :doh:

 

Its simply to select the first element in the base, jump over next select next jump over next etc etc until theres nothing left. :)

 

So op(xyz)=xzy.

 

ok. so you said then before

Other basenumbers should give other amounts of concepts,

Will they differ in semantical content?

Isnt twelve tones the optimal number a la Schoenberg?

 

Actually ive forgotten most of the questions entering my mind some thirty years ago while i painstakingly calculated the values received on the first sixteen bases...

 

so taking 11 for example:

 

1 2 3 4 5 6 7 8 9 10 11

1 3 5 7 9 11 2 4 6 8 10

1 5 9 2 6 10 3 7 11 4 8

1 9 6 3 11 8 5 2 10 7 4

1 6 11 5 10 4 9 3 8 2 7

1 11 10 9 8 7 6 5 4 3 2

1 10 8 6 4 2 11 9 7 5 3

1 8 4 11 7 3 10 6 2 9 5

1 4 7 10 2 5 8 11 3 6 9

1 7 2 8 3 9 4 10 5 11 6

1 2 3 4 5 6 7 8 9 10 11

 

is this want you want to construct?

[sigurdV]

YES! but it also should be presented as the results of a function from numbers to numbers...Like:

f0=

f1=

etc up to ...say f16

 

Then questions arise (Its a bloody shame this breakthrough comes so long afterwords , i doubt if i can resurrect many of my earlier hunches.)

 

 

ok. so you said then before

 

so taking 11 for example:

 

1 2 3 4 5 6 7 8 9 10 11

1 3 5 7 9 11 2 4 6 8 10

1 5 9 2 6 10 3 7 11 4 8

1 9 6 3 11 8 5 2 10 7 4

1 6 11 5 10 4 9 3 8 2 7

1 11 10 9 8 7 6 5 4 3 2

1 10 8 6 4 2 11 9 7 5 3

1 8 4 11 7 3 10 6 2 9 5

1 4 7 10 2 5 8 11 3 6 9

1 7 2 8 3 9 4 10 5 11 6

1 2 3 4 5 6 7 8 9 10 11

 

is this want you want to construct?

 

God bless them Mathematicians

 

Im thinking : Perhaps this merits composing a "symphony of turtles"?

On second thought: Nah! Zenon beat me to it :)

[Turtle]

Theres been much ado about nothing (base 0?) so instead of repeating my words i suggest readers go to: Universe from nothing #42

 

the question mark is apropos; there is no mathematical meaning to base 0 or base 1 for that matter.

[Turtle]

YES! but it should be presented as a function from numbers to numbers...Like:

f0=

f1=

etc up to ...say f16

 

not sure what you mean. the instructions "select the first element in the base, jump over next, select next, jump over next etc etc until theres nothing left" constitute an algorithmic function. i would replace the term "base" with "sequence" or "set" however.

 

so you would want to get at that for a 7-term sequence it takes 4 steps -inclusive of the last matching the original- to get back to start and for an 11-term sequence it takes 11 steps. oui/no?

[Turtle]

ok. try this from 2 terms through 13 terms. f=number of steps to repeat inclusive.

 

f=1

1 2

1 2

 

f=2

1 2 3

1 3 2

1 2 3

 

f=2

1 2 3 4

1 3 2 4

1 2 3 4

 

f=4

1 2 3 4 5

1 3 5 2 4

1 5 4 3 2

1 4 2 5 3

1 2 3 4 5

 

f=4

1 2 3 4 5 6

1 3 5 2 4 6

1 5 4 3 2 6

1 4 2 5 3 6

1 2 3 4 5 6

 

f=3

1 2 3 4 5 6 7

1 3 5 7 2 4 6

1 5 2 6 3 7 4

1 2 3 4 5 6 7

 

f=3

1 2 3 4 5 6 7 8

1 3 5 7 2 4 6 8

1 5 2 6 3 7 4 8

1 2 3 4 5 6 7 8

 

f=6

1 2 3 4 5 6 7 8 9

1 3 5 7 9 2 4 6 8

1 5 9 4 8 3 7 2 6

1 9 8 7 6 5 4 3 2

1 8 6 4 2 9 7 5 3

1 6 2 7 3 8 4 9 5

1 2 3 4 5 6 7 8 9

 

f=6

1 2 3 4 5 6 7 8 9 10

1 3 5 7 9 2 4 6 8 10

1 5 9 4 8 3 7 2 6 10

1 9 8 7 6 5 4 3 2 10

1 8 6 4 2 9 7 5 3 10

1 6 2 7 3 8 4 9 5 10

1 2 3 4 5 6 7 8 9 10

 

f=10

1 2 3 4 5 6 7 8 9 10 11

1 3 5 7 9 11 2 4 6 8 10

1 5 9 2 6 10 3 7 11 4 8

1 9 6 3 11 8 5 2 10 7 4

1 6 11 5 10 4 9 3 8 2 7

1 11 10 9 8 7 6 5 4 3 2

1 10 8 6 4 2 11 9 7 5 3

1 8 4 11 7 3 10 6 2 9 5

1 4 7 10 2 5 8 11 3 6 9

1 7 2 8 3 9 4 10 5 11 6

1 2 3 4 5 6 7 8 9 10 11

 

f=10

1 2 3 4 5 6 7 8 9 10 11 12

1 3 5 7 9 11 2 4 6 8 10 12

1 5 9 2 6 10 3 7 11 4 8 12

1 9 6 3 11 8 5 2 10 7 4 12

1 6 11 5 10 4 9 3 8 2 7 12

1 11 10 9 8 7 6 5 4 3 2 12

1 10 8 6 4 2 11 9 7 5 3 12

1 8 4 11 7 3 10 6 2 9 5 12

1 4 7 10 2 5 8 11 3 6 9 12

1 7 2 8 3 9 4 10 5 11 6 12

1 2 3 4 5 6 7 8 9 10 11 12

 

f=12

1 2 3 4 5 6 7 8 9 10 11 12 13

1 3 5 7 9 11 13 2 4 6 8 10 12

1 5 9 13 4 8 12 3 7 11 2 6 10

1 9 4 12 7 2 10 5 13 8 3 11 6

1 4 7 10 13 3 6 9 12 2 5 8 11

1 7 13 6 12 5 11 4 10 3 9 2 8

1 13 12 11 10 9 8 7 6 5 4 3 2

1 12 10 8 6 4 2 13 11 9 7 5 3

1 10 6 2 11 7 3 12 8 4 13 9 5

1 6 11 3 8 13 5 10 2 7 12 4 9

1 11 8 5 2 12 9 6 3 13 10 7 4

1 8 2 9 3 10 4 11 5 12 6 13 7

1 2 3 4 5 6 7 8 9 10 11 12 13

[sigurdV]

not sure what you mean. the instructions "select the first element in the base, jump over next, select next, jump over next etc etc until theres nothing left" constitute an algorithmic function. i would replace the term "base" with "sequence" or "set" however.

 

so you would want to get at that for a 7-term sequence it takes 4 steps -inclusive of the last matching the original- to get back to start and for an 11-term sequence it takes 11 steps. oui/no?

 

I think you follow the scent like a blood- ...eh...-turtle

 

I will immedeately abandon my selection of words to convey meanings when offered standard alternatives! I always was seclusional and had to use whatever was offered in my mind.

 

Its four steps but theres THREE steppingstones and them stones are gold nuggets:

 

SCALE=1234567, CHORD=1357246 and TONALITY=15...(being lazy)

 

The numbers carry a semantic content you see and semantics...eh... well...Now i think i get what I was after in the first place:

 

How does the semantic content vary with bases?

 

Also i searched for a...mathemathical function to replace the algorithmic function, and at first sight i thought no single function would do... the function must be composed of at least two what do I call them...Arithmethic functions? Well exactly there is the problem that stopped my thinking.(stop)

[Turtle]

I will immedeately abandon my selection of words to convey meanings when offered standard alternatives! I always was seclusional and had to use whatever was offered in my mind.

 

Its four steps but theres THREE steppingstones and them stones are gold nuggets:

 

SCALE=1234567, CHORD=1357246 and TONALITY=15...(being lazy)

 

no worries. i'm in that habit myself. (google katabatak ;))

 

so, subtract 1 from my f's in the above list. i'll have 14, 15, & 16 presently. so far i see no pattern relating the number of elements to the f. while you wait, have a look at this. >> Katabatak Powers

[Turtle]

14, 15, & 16 terms. f= # of steps.

 

f=12

1 2 3 4 5 6 7 8 9 10 11 12 13 14

1 3 5 7 9 11 13 2 4 6 8 10 12 14

1 5 9 13 4 8 12 3 7 11 2 6 10 14

1 9 4 12 7 2 10 5 13 8 3 11 6 14

1 4 7 10 13 3 6 9 12 2 5 8 11 14

1 7 13 6 12 5 11 4 10 3 9 2 8 14

1 13 12 11 10 9 8 7 6 5 4 3 2 14

1 12 10 8 6 4 2 13 11 9 7 5 3 14

1 10 6 2 11 7 3 12 8 4 13 9 5 14

1 6 11 3 8 13 5 10 2 7 12 4 9 14

1 11 8 5 2 12 9 6 3 13 10 7 4 14

1 8 2 9 3 10 4 11 5 12 6 13 7 14

1 2 3 4 5 6 7 8 9 10 11 12 13 14

 

f=4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1 3 5 7 9 11 13 15 2 4 6 8 10 12 14

1 5 9 13 2 6 10 14 3 7 11 15 4 8 12

1 9 2 10 3 11 4 12 5 13 6 14 7 15 8

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

 

f=4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1 3 5 7 9 11 13 15 2 4 6 8 10 12 14 16

1 5 9 13 2 6 10 14 3 7 11 15 4 8 12 16

1 9 2 10 3 11 4 12 5 13 6 14 7 15 8 16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

[sigurdV]

!

ok. try this from 2 terms through 13 terms. f=number of steps to repeat inclusive.

 

f=2

1 2

1 2

 

f=3

1 2 3

1 3 2

1 2 3

 

f=3

1 2 3 4

1 3 2 4

1 2 3 4

 

f=5

1 2 3 4 5

1 3 5 2 4

1 5 4 3 2

1 4 2 5 3

1 2 3 4 5

 

f=5

1 2 3 4 5 6

1 3 5 2 4 6

1 5 4 3 2 6

1 4 2 5 3 6

1 2 3 4 5 6

 

f=

1 2 3 4 5 6 7

1 3 5 7 2 4 6

1 5 2 6 3 7 4

1 2 3 4 5 6 7

 

f=4

1 2 3 4 5 6 7 8

1 3 5 7 2 4 6 8

1 5 2 6 3 7 4 8

1 2 3 4 5 6 7 8

 

f=7

1 2 3 4 5 6 7 8 9

1 3 5 7 9 2 4 6 8

1 5 9 4 8 3 7 2 6

1 9 8 7 6 5 4 3 2

1 8 6 4 2 9 7 5 3

1 6 2 7 3 8 4 9 5

1 2 3 4 5 6 7 8 9

 

f=7

1 2 3 4 5 6 7 8 9 10

1 3 5 7 9 2 4 6 8 10

1 5 9 4 8 3 7 2 6 10

1 9 8 7 6 5 4 3 2 10

1 8 6 4 2 9 7 5 3 10

1 6 2 7 3 8 4 9 5 10

1 2 3 4 5 6 7 8 9 10

 

f=11

1 2 3 4 5 6 7 8 9 10 11

1 3 5 7 9 11 2 4 6 8 10

1 5 9 2 6 10 3 7 11 4 8

1 9 6 3 11 8 5 2 10 7 4

1 6 11 5 10 4 9 3 8 2 7

1 11 10 9 8 7 6 5 4 3 2

1 10 8 6 4 2 11 9 7 5 3

1 8 4 11 7 3 10 6 2 9 5

1 4 7 10 2 5 8 11 3 6 9

1 7 2 8 3 9 4 10 5 11 6

1 2 3 4 5 6 7 8 9 10 11

 

f=11

1 2 3 4 5 6 7 8 9 10 11 12

1 3 5 7 9 11 2 4 6 8 10 12

1 5 9 2 6 10 3 7 11 4 8 12

1 9 6 3 11 8 5 2 10 7 4 12

1 6 11 5 10 4 9 3 8 2 7 12

1 11 10 9 8 7 6 5 4 3 2 12

1 10 8 6 4 2 11 9 7 5 3 12

1 8 4 11 7 3 10 6 2 9 5 12

1 4 7 10 2 5 8 11 3 6 9 12

1 7 2 8 3 9 4 10 5 11 6 12

1 2 3 4 5 6 7 8 9 10 11 12

 

f=13

1 2 3 4 5 6 7 8 9 10 11 12 13

1 3 5 7 9 11 13 2 4 6 8 10 12

1 5 9 13 4 8 12 3 7 11 2 6 10

1 9 4 12 7 2 10 5 13 8 3 11 6

1 4 7 10 13 3 6 9 12 2 5 8 11

1 7 13 6 12 5 11 4 10 3 9 2 8

1 13 12 11 10 9 8 7 6 5 4 3 2

1 12 10 8 6 4 2 13 11 9 7 5 3

1 10 6 2 11 7 3 12 8 4 13 9 5

1 6 11 3 8 13 5 10 2 7 12 4 9

1 11 8 5 2 12 9 6 3 13 10 7 4

1 8 2 9 3 10 4 11 5 12 6 13 7

1 2 3 4 5 6 7 8 9 10 11 12 13

 

And here are some of the votes from the jury: Note that f4=f3

 

f0=?

f1=?

f2=2

f3=3

f4=3

f5=5

ops i started correcting yor list until i realized i was doing nothing of that sort

 

I want to expose the number of steppingstones, f7 gives three steppingstones...the function I was after should tell me how many steppingstones there are in say: f2012 or f-272

 

So: I though at the time that exposing the number of steppingstones for each and every (((interesting)))number and researching each steppingstone for its semantical content might give me some surprises :)

[Turtle]

!

 

 

And here are the votes from the jury:

 

f0=?

f1=?

f2=2

f3=3

f4=3

f5=5

ops i started correcting yor list until i realized i was doing nothing of that sort

 

What I want is to expose the number of steppingstones, f7 gives three steppingstones...the function I was after should tell me how many steppingstones there are in say: f2012 or f-272

 

no worry on the correction. i'll edit my f's 4 u. :) based on what i see so far i posit there exists no function/expression/equation to give/predict the f for some arbitrary n-terms.

[sigurdV]

:lol:

no worry on the correction. i'll edit my f's 4 u. :) based on what i see so far i posit there exists no function/expression/equation to give/predict the f for some arbitrary n-terms.

 

I thought so, but notice there are functions that gives SOME of the values: f16 for instance!

 

Perhaps all values are given by a composed function: C= f+g+...? (cearching for C)(Originally i named the searched for function Q)

 

Can C be made continious as well? Can we then see what f1 and f0 should be?

 

I cant help observing that some turtles work with lightning speed:)