h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Entry: Mathematics in Music - A2135882
Author: vanamonde - U546428

this is my paper that connects everything from pythagorus to Fibbanacci through music.

Well done Vanamonde. This is a fascinating topic and you have brought up some interesting facts and figures. I covered a small part of the topic, with a different emphasis, in A1339076 "Sol-Fa - The Key to Temperament".

The figures are most important (naturally) so let me say first: your explanation of concord and discord leaves some loopholes.

In your "list of notes, frequencies of these notes, and ratios of said notes to Middle C"
C 261.6
D 293.7 (ratio 9:8)
E 329.6 (ratio 5:4)
F 349.2 (ratio 4:3)
G 392.0 (ratio 3:2)
A 440.0 (ratio 5:3)
B 493.9 (ratio 17:9)

The reason that B:C comes out at 17:9 is because of temperament. If B:G is 5:4 (and it is, the same as E:C) and G:C is 3:2 then B:C should be 15:8, which is a bit more harmonious than 17:9. It is still discordant enough to show how B doesn't fit in a chord with C; but two things are getting mixed up here. You have taken equal-tempered frequencies and simplified their ratios. Unfortunately a more distant interval (a fifth plus a third) gets simplified to a relatively meaningless ratio, where the meaningful one would make its point perfectly well. What you have done is not wrong, but it is (possibly) confusing.

By the way you need to put "note - frequency - aproximate ratio" into the table format so that your headings line up. Have a look at the GuideML Clinic.

The explanation of Hz is also a little confusing. You say that "Frequency is the measure of the number of times a single wave passes a given point. Therefore, a note is determined by the length of the wave produced." -- this "therefore" assumes the truth that sound waves travel at a regular speed; which makes it hard to swallow "When two or three notes are played together, their waves are traveling at different rates." Careful choice of words can clear this up.

The other bit I find hardest to follow is under Beethoven: "it was found that the famous opening "motto" appears not only in the first and last bars (bar 601 before the Coda) but also exactly at the golden mean point, 0.618 of the way through the symphony (bar 372) and also at the start of the recapitulation which is phi, or 0.382, of the way through the piece."

Now you are talking about the first movement, not the whole symphony. The famous theme is only in the first movement. And of the first movement, is bar 601 the last bar, or the last bar before the coda? (The coda is an integral and very large part of the movement, and leaving it out of the count would look suspicious.) In my score the first movement has only 502 bars, and bar 372 is in fact the start of the coda. The recap starts in bar 248 (about halfway through). Did you mean that phi comes at the start of the development (bar 125)? This just needs clearing up. The four-note theme is hardly absent anywhere in the movement, so it is not particularly enlightening to say that it "appears not only in the first and last bars (bar 601 before the Coda) but also exactly at the golden mean point, 0.618 of the way through the symphony (bar 372) and also at the start of the recapitulation". It appears all over the joint!

Please check spelling of Pythagoras, Fibonacci, prodigious, innate, supplemented, dichotomy, referenced; and you might reconsider systemized/systematized, many if not most/most, their conribution . . . several of these/much of this, esthetically (typo), its own genre (no ') and two or three dubious claims: "Consequently the world is made of math" (it may be, but not consequently on your previous statement); "Humans have included music in their lives since the dawn of civilization" (must have been long before that); and "as we have progressed as a race" which is simply dubious

x2
I must confess that despite my (somewhat unique) ability to understand such things as cultural change, grammatical structure, and dynastical inheritance (see A2048690), I am completely inept both in the fields of mathematics and music. I understand barely anything beyond trigonometry, and don't get things like fifths and thirds. Thus I present myself for the ultimate guinea pig for the comprehensability of your entry to the outside world. I have so managed to bog through the first paragraph. I will let you know when I get further on.
-MotDoc

As an engineer and musician, this is pretty plain sailing to me, and a damn interesting entry to boot. Well done that man.

Under the 'mechanics' aspect, it might be worth mentioning the mathematical reason why we use the 12-note scale in the first place.

Successive octaves are successive powers of two. And we use the 3/2 ratios to fill the notes in between. It would make more sense for these notes to be evenly spaced, so later musicians extended Pythagoras' seven-note scale by adding notes between (the black notes as we now know them ). They used twelve because (3/2)^12 is almsot exactly 7 octaves [(3/2)^7=129.7 2^7=128].

Also, if you want to stretch yourself under the range of your title, you could mention campanologists. Bell-ringers consider themselves mathematical musicians, working as they do with permuatation and combinations.

B

Very interesting entry. I seem to remember there was a connection between Mozart and birdsong, as well.

There are quite a few typos, so it would be worthwhile putting this entry through a spell checker. Or, if you like, I can go through (or someone else can) and point them up for you.

I agree with everyone on the spelling. There are more misspellings than I would want to deal with; Zarquon's Singing Fish is more merciful and kind than most. Recumbentman has cued you into a lot of them, if you don't have a spellchecker.

I think Recumbentman's entries are already in the Edited Guide. If they are, it would be cool to link to them.

In post 4, there's the phrase 'it would make more sense'. It would make more sense to whom? Even temperament is an acquired taste like any other, and consorts and polyphony aren't obvious as a be-all or end-all of music. This needs a small qualification of some sort if it is included in the entry.

In the entry the phrase 'considered to be consonant' also deserves a little qualification. The author already acknowledges Middle Eastern music, but there are also Gamelan, Jegog (these two Indonesian), Raga, and Chhandra (these two from India) tunings. Also, back in the days of monophony, chant, and early motets, a second was viewed as perfectly consonant, and today in western music, this is not the case. I am perhaps more persnickety than this than most, because for some reason my ears and brain don't seem to instinctually prefer western tunings over others.

Being mathematically inclined, and knowing nothing about music, I found this interesting ... and baffling.

It mentions assonance and dissonance, but then describes consonants - which I thought were non-vowels. It mentions simple frequency ratios, but octaves are more or less assumed. The maths left me cold - for me 5:4 then 5:4 gives 25:16, not 3:2. If the approximate ratios can be that crude, why does tuning have to be so precise?

As a former electrical engineer, I really understand stuff like fundamentals and harmonics, which I think the music fraternity call overtones, but these are never mentioned. I also thought they were the main reason for the importance of octaves. Am I wrong on this?

I would be interested to know the frequency ratios of some of the non-Western music traditions, which are not stated here, and why they too are harmonious in their own way. And why it is that when we divide our octave, we use a bizarre tone tone semi-tone tone tone tone semi-tone arrangement, rather than equal steps.

In summary, I did not find in the entry that which the title led me to believe might be there. However, I am sure it will improve. Well done for your first effort.

i have just taken the first step to revision and run everything through spell check.

there is certainly more that i could say about the subject, and plan on going into it. i have actually gone over some of the things mentioned (including overtones and a few others mentioned), so including those will not be hard.

thank you, everyone for your comments so far, and keep them coming

" 'it would make more sense'. It would make more sense to whom?"

Well, to me, obviously.

B

I've read through this entry and it is very interesting. I can see a place for it in the Edited Guide. Well done! I have got serious reservations about some of it, though.

1. Most importantly, I'd agree with Recumnbentman about Beethoven's 5th. The "motto" is repeated hundreds of times throughout the first movement, and does not occur anywhere else, so it is meaningless to cite this as an example of the golden ratio. You'd be much better to move the section on Fibonacci numbers to the contemporary exmaples section, where it belongs. Leave out Beethoven altogether unless you can come up with something more believable.

2. You mention ragtime in a throw-away comment. Would you like to expand on this?

3. Your equation for the Golden Ratio is wrong: it should be (1-x)/x = x/1

4. Mentioning both Phi and phi as two separate numbers is confusing. You'd be better only mentioning one by name.

5. The section on Arabic music suggest that it uses only these "1.5 intervals", (which I call three-quarter-tones), and not the whole tones and semitones of western music. In fact Arabic scales are made from combinations of semitones, whole tones and three-quarter-tones.

6. You describe the Blues scale as pentatonic, but you've given six intervals in the scale, not five. That would make it hexatonic.

Don't be disheartened by all this criticism. I like the entry and want to help you to make it as good as possible before it goes into the Guide.

Oh, and I don't think you should go into any more detail on the frequencies of notes than you do at the moment, because this information is already in the guide in Recumbentman's entry.

Old Hairy --

"5:4 then 5:4 gives 25:16, not 3:2" is true, but Vanamonde never said it would give 3:2.

5:4 is the relation of E to C. 3:2 is the realtion of G to C.

25:16 is the relation of G# to C; because G# stands to E (5:4) as E stands to C (5:4).

G:E is smaller ("minor third"--6:5)than E;C ("major third"--5:4). To add the thirds G:E and E:C you must multiply up those ratios 6:5 and 5:4 and eureka you get 30:20 viz 3:2.

Oh crikey: I'm amongst people who know what they're talking about

Anyway, here's my vanamonde . Like MotDoc, I'm no mathematician and a very amateur musician, so another perfect guineapig...

This reads a bit oddly and could perhaps make more sense as:

Why is it ? Is there great benefit in recognising the links? I love music, but don't feel that understanding the maths connection is necessary to enhance that enjoyment.

Could there be a footnote to explain and please? I have *no* idea what they are...

Presumably only a middle C???

I'm not even going to try to decipher that sentence!!

Modes - ooooh. I am awfully confused about this lot. I got totally lost in this section I'm afraid. I don't understand the difference / relation between modes, scales, numbers, cords, thirds and sixths, intervals Is there any way you can spell it out more clearly for non-mathmatical and relatively non-musical readers? It may not be possible of course, but I do get my head around most stuff and I simply hit a brick wall of confusing terminology here.

And then I gave up, because the discussion of Mozart and Beethoven and the proportions had me stumped...

*Pathetic Frenchbean!* Sorry vanamonde: you lost me.


F/b

I mean -

F/b

Recumbentman.

I quote the entry:-
"For example, the frequencies of the notes C, E, and G match up very well because the ratio of C to E is 5:4, the ratio of E to G is also 5:4, and the ratio of C to G is 3:2"

So, by my arithmetic, C to G is 25:16, and not 3:2. That's why I'm baffled.

Old H--Why, you are so right! I took the meaning he should have had and expected him to have got it right.

Vanamonde --you need to change that. You haven't given the relation of G to E so either give it (6:5) or don't mention it, just that G goes with C and E goes with C and (incidentally) they also go with each other. The figures you give are still close enough (1.189 close enough to 1.2) to say that G:E is 6:5.

You could avoid this particular quicksand by stating the fact of consonance and dissonance, and linking to A1339076 "Sol-Fa - The Key to Temperament" for figures.

The explanation of consonance, that waves fall into step, can be illustrated by the phenomenon of seeing railings through railings; when the distance between is right, each third railing in the distant lot will line up with one close up, for instance. Or regular tree plantations; they line up in different collections as you pass.

Dissonance can be defined as "too complex a relation for the ear to latch onto". This is obviously a matter of taste to some extent, and different cultures have differnt tolerances.

OK Got dissonance and consonance now, thanks Recumbantman. What about ? ?

Could there be a footnotes in the text to explain these terms please Vanamonde?


F/b

Assonance and consonance are the same thing. I could say that assonance is the more fundamental term, but that would be in poor taste

i am working on this some more today, and many of the revision should be finished. thanks again to everyone, and again keep them coming

and for the record, i'm a girl.

Single?

(sorry sorry sorry sorry sorry)

B