The Connections Between Music and Mathematics: Revised and Better Than Ever

When the word music comes to mind, many people think of their favorite hip-hop artists, rappers, and guitarists. Many people have at one point or another played an instrument in a band, learned piano, or have sang. Notes elegantly drawn across the page can calm and sooth people, or get them in the zone for sporting events.

207.65

G#5/Ab5

830.61

G#7/Ab7

3322.44

A1

55

A3

220

A5

880

A7

3520

A#1/Bb1

58.27

A#3/Bb3

233.08

A#5/Bb5

932.33

A#7/Bb7

3729.31

B1

61.74

B3

246.94

B5

987.77

B7

3951.07

(A partial listing of all musical note frequencies.  From http://www.phy.mtu.edu/~suits/notefreqs.html, edited by Alex Donnelly)

In music, the lowest note possible to play is a low “C” at 16.35 hertz and G# is the highest note at 13289.75 hertz.  There are two constants involved with the frequency of notes.  The first constant is that the A above middle C (A4) is equal to 440Hz.  The second constant is the relationship between each consecutive note.  There is a number that each note is multiplied by to get each successive note.  This number is 2 to the power of 1/12 or 1.059463094.  To get each successive note, a formula similar to the population or interest formula is used.

Fnew=Fold ∙ (1.059463094)n

(Where F is equal to the frequency and n is equal to the number of half-steps of half-tones [the space between C and C# for example])

For instance, to find the frequency of the note 12 half-steps above A4, multiply 440Hz by 1.059463094 to the power of 12.  Because 1.059463094 is equal to 2 to the power of 1/12, this simplifies to be 2, therefore 440Hz times 2, to give an answer of 880Hz.

F= 440{A4} ∙ (21/12)12  =  880Hz{A5}

The difference between A4 and A5 is what as known as an octave.  The difference in frequency between this octave is 440Hz and 880Hz.  This octave is 2 times the first note.  The same is true for all octaves, which you can tell from the chart above.  Therefore, all octaves have a ratio between them of 1:2.

Note that these ratios can be applied to all notes.  The difference between two notes is always the same ratio..  For instance, middle C (C4) has a hertz of about 262 and the G above it (G4) is 392hz.  When you divide the two, you get the ratio between the notes: 3/2 (example number 7 below).  The difference between Cn and Gn will always remain in proportion to eachother in the way of 3:2.  This means that every 3 radians of C will match up with every 2 radians of G at the x-axis.  This is a relatively low ratio, which sounds good to our ears.  The ratio between C and C# (about 262Hz and 277Hz respectively) is 135:128.  This is a huge ratio, which sounds terrible to a listener.  (Research from http://en.wikipedia.org/wiki/Hertz, http://en.wikipedia.org/wiki/Pitch_%28music%29) (Picture below from http://en.wikipedia.org/wiki/Mathematics_and_music)

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68 Responses to “The Connections Between Music and Mathematics: Revised and Better Than Ever”
1. Emmel Says...

On May 30, 2008 at 11:33 am

I am sorry to have to conclude that you did not show any link between music and mathematics in your article. The problem is probably not the math, but the music.

You describe a lot of superficial elements of music, like notes, pitches, durations and instruments, but if that would be music, it would suffice to give a lot of one-year-olds a musical instrument and let them hammer away and the result would be music.
These superficial elements do indeed show connections with mathematics, but do not represent music. It is even possible to imagine music without any of these elements.

As mathematics is not about formulas, neither is music about notes. Mathematics is a formal language to describe logic and patterns. Music is a language to represent meaning and emotion.

Rather than an article on notes, it would be far more interesting to read an article that explains the mathematical properties behind why the musical logic of a song as Frere Jacques does not change when played in a different pitch or temperament.

2. Peter Griffin Says...

On May 30, 2008 at 11:37 am

Article, article, article, article, “This reminds me of the the time I tried to find an instrument with the same range as a piano.” article, article, article.

3. Bill S. Preston Esq. Says...

On May 30, 2008 at 11:39 am

Don’t mean to be a joyless pedant, but…I’m going to be anyway.

“There are 13 major scales that are distinct”

Only if you count E# and F as two different root notes. You’re not quite there with the connection between radians and Hertz, either. Or the connection between time signatures and fractions – 3/4 and 6/8 are not the same, for example.

4. sajori Says...

On May 30, 2008 at 11:41 am

i’ve always had this same idea… that once you’re good at music, youre good at math( to some point, can’t leave out the “not likey math” emotion) but i think hes right…

5. prock Says...

On May 30, 2008 at 11:42 am

Something worth reading related to this is Godel, Echer, Bach by Douglas Hofstadter.

6. Ross Handler Says...

On May 30, 2008 at 11:44 am

There are TWELVE major scales. E# and F are the same thing. You can actually build 15 from harmonic notes, but our ears can only distinguish 12 unique scales.

7. Bob White Says...

On May 30, 2008 at 11:44 am

The unit of Hertz is not radians per second! 1Hz = 1 oscillation/second. There are 2pi radians per oscillation so 1 Hertz in unit of radians/oscillation = (1 oscillation/second)*2pi (radians/oscillation) = 2pi radians/second. This is pretty important if you want to actually build something.

8. http://www.reginasuniverse.com Says...

On May 30, 2008 at 11:46 am

This is amazing and Godel, Escher, Bach is one of the greatest published works of all time.

9. misses some basic point... Says...

On May 30, 2008 at 12:04 pm

well how can you have a three-page dissertation about music theory and “math” and not even mention the difference between the Major and Minor keys, or even the Cycle of 5ths?

Basically, the minor key substitutes the 3rd and 6th notes of the major key for ones a half-step lower, given the key a much more “sad” or dramatic feel. You can still build chords the same way, but the 2nd note of the chord will be different ie.

Cmaj scale – C D E F G A B
Cmin scale – C D D# F G G# B

Cmaj chord – root 3rd 5th = C E G
Cmin chord – root 3rd 5th = C D# G

10. bob white is right Says...

On May 30, 2008 at 12:04 pm

2pi radians per second is one Hz.

11. Rob Says...

On May 30, 2008 at 12:08 pm

I am dumber for having read this. Bravo, sir. Brav.o.

12. rik Says...

On May 30, 2008 at 12:11 pm

yeah, better get that 2pi = 1hz thing correct. f = 2*pi*omega where omega is the angular frequency.

On May 30, 2008 at 12:14 pm

“Perhaps that is why musicians are so good at math.” You have got to be kidding me.

There are connections between music and math, but this article does nothing to show it. Your argument is weak and very poorly written. I hope they didn’t pay you for this.

14. James Says...

On May 30, 2008 at 12:17 pm

Great article. Very insightful, all amateur music makers should read this and absorb as much as possible.

15. mjc Says...

On May 30, 2008 at 12:33 pm

Even though I knew this, it was still good to read the part about how the different pitches line up in different frequencies. The graph helped to illustrate the point. Just fix the typos, check to see if the previous commenter’s criticisms about the math are correct and you have a good article.

16. Vuk Says...

On May 30, 2008 at 12:42 pm

There’s no E sharp. It would be called “F”.

I believe there are 12 distinct major scales, not thirteen, no?

17. mes Says...

On May 30, 2008 at 12:48 pm

duh i like music!!!

18. Jonny Says...

On May 30, 2008 at 12:54 pm

“No wind instrument has ever been created that can mimic the range of the piano.”

Um… the organ?

19. Pancakes Says...

On May 30, 2008 at 12:58 pm

“well how can you have a three-page dissertation about music theory and “math” and not even mention the difference between the Major and Minor keys, or even the Cycle of 5ths?

Basically, the minor key substitutes the 3rd and 6th notes of the major key for ones a half-step lower, given the key a much more “sad” or dramatic feel. You can still build chords the same way, but the 2nd note of the chord will be different ie.

Cmaj scale – C D E F G A B
Cmin scale – C D D# F G G# B

Cmaj chord – root 3rd 5th = C E G
Cmin chord – root 3rd 5th = C D# G”

You’re naming the notes wrong in the C minor scale. In every diatonic scale you only use each note name once, so you can’t have a D and a D# in the same diatonic scale.

It would be:
Cmin scale – C D Eb F G Ab B

This of course makes no difference on a piano where D# and Eb are the same notes, but a skilled musician will play those notes differently on a non-fretted instrument such as a violin.

20. Andy Says...

On May 30, 2008 at 1:14 pm

I have to admit I was pretty disappointed by this article. The title was promising, but it just fell flat…

I was expecting to see math relating to harmony. There is concrete math that explains all of this… like an octave is 2:1 frequency ratio, and a perfect 5th is a 3:2 frequency ratio. Also, naturally-produced sounds include overtones, which are also mathematically related, which is why a major third interval sounds “major” and a perfect fifth interval is so harmonious — because the root note contains overtones that match the upper note’s fundamental. The human mind is capable of subconsciously identifying these patterns through physiological/neurological things that I don’t understand so well.

I suppose this will be of interest to some people, but most of the people that saw the headline and clicked it in excitement will already be a hundred levels above this.

This stuff is covered (more clearly and in greater depth usually) in the first pages of any beginners music book. (Just trying to give some constructive criticism.) Better luck next time.

21. Andy Says...

On May 30, 2008 at 1:19 pm

Oops, I did not notice there were 2 more pages until I had pasted that last comment… you did hit on the harmonics a little bit, and gave the more elementary explanation for why harmonies sound so… but look up overtones and work that into the article. C, E, and G sound so nicely because C by itself contains E and G already in its overtones!

BTW, this was in my iGoogle news feed from Doggdot.us.

22. Sarah Says...

On May 30, 2008 at 1:21 pm

23. sciency guy Says...

On May 30, 2008 at 2:20 pm

24. Poemind Says...

On May 30, 2008 at 2:32 pm

As a person with a music composition degree and a computer science degree, I was naturally drawn to this article for the title. But the moment I saw the word “reseed” and then saw that you placed 13 pitches in the chromatic scale, I had to write it off as a waste.

x=x+13 is simply wrong!!

There are many connections between mathematics and music but this article does little to elucidate them. Try again, or not…

25. GW Says...

On May 30, 2008 at 2:42 pm

I didn’t see much relating to mathematics here, and the little bit that did is incorrect and confusing. Your equation for Hz is wrong, and in this case, frequency has nothing to do with an angular measurement like radians. The introduction to music theory is nice but doesn’t really fit the title. Ditto for the comparison of reeds and strings. I was expecting a clearer article with a little more depth and understanding.

Also, I’m not a spelling nazi, but a quick proof-read wouldn’t hurt.

On May 30, 2008 at 3:04 pm

Horrible article… 13 major scales? No, only 12. And since when is D the third note in the C major scale?

27. gh0st Says...

On May 30, 2008 at 3:39 pm

“emmel” could not be more wrong.

Now, I know the freebird hippie guitar hero wannabe in you does not want to hear it, but from someone who is a musician (18+ years guitar, 6+ years sax, 2+ years piano, I can make some organized noise on a harmonica too) as well as someone familiar with mathematics (computer science) I feel confident in proclaiming: the freebird hippie guitar hero wannabe might be wrong.

Music is impossible without math. While the poetic minded emmel would make the point a chord is a chord becuase it is pleasant sounding, I can prove to you that a chord is a chord because of math. The chord is a chord because of sevreal mathematical principles. No, Bach did not understand and appreciate how a 440 MHz and a 820 MHz waves would interact with each other, but his ignorance and the ignorance of others does not change the fact that it works not because of “beauty” and “love” or the “universal language that is music” but because of math and physics.

I have for a long time wished that more people would be able to grasp this notion. I think in it are several possibilities to get kids more excited about math.

28. math&music Says...

On May 30, 2008 at 4:30 pm

Alex, Alex, Alex …. how could you do this! What a waste of people’s time. The idea is good. After all, it’s been written on ad nauseum. And the presence of mathn in music is real interesting!! But so many of your basic “facts” are wrong, wrong, wrong, as has been pointed out by many of the comments.

Maybe you should’ve stuck with your brine shrimp article, where you decided that brine shrimp hatch best in the ocean.

Duh!

29. wikibuddha Says...

On May 30, 2008 at 6:07 pm

Johnny said “‘No wind instrument has ever been created that can mimic the range of the piano.’ Um… the organ?”

Wouldn’t it be more accurate that the piano was created to mimic the organ?

I couldn’t find proof, but I think a theremin may well exceed the range of a pianer.

30. wikibuddha Says...

On May 30, 2008 at 6:10 pm

Yes, I confirmed that a theremin has a range of 12 octaves while pianos only push 8 octaves.

31. wikibuddha Says...

On May 30, 2008 at 6:11 pm

But it’s not a wind instrument

On May 31, 2008 at 12:10 am

As an educated musician (meaning I have a degree in music), I’m convinced that both the author of this article and those posting comments don’t have the background knowledge to talk about this subject. For example, there is such a thing as an E#. We use it in the F# major scale. Or you could play an E# major scale if you wanted to. Yes, it will sound the same as an F major scale.

D is not the third note in the C major scale. If, however, it resides in a tone row using C as pitch class 0, you would call it 3.

Basically, if you don’t have the background knowledge for this subject, don’t try and act like you know what you’re talking about. Music isn’t learning to play the guitar from tabulature pages or quickly reading wikipedia articles online. There’s a LOT more to the discipline.

If you all want to have some real fun with music and math, look up “serialism” or “twelve-tone music”. Arnold Shoenberg, Anton Webern, and others championed it in the early half of the 20th century. It’s good stuff.

33. comet Says...

On May 31, 2008 at 12:52 am

music is as mathmatismic as formulatin a heart break.

music is expressionism. express yourself. knockin on heavens door by bob dylan is three chords and yet can ruin your life forever , even before flight of the bumble bee or anything else formulated or not. music should be about expressing ones self no matter what. math is science and it should be! give us the answers right? music should raise qwestions, and shed light in a way that is accepted as expression from another soul. dont give me harmony or dissonance or verse or stanza or definition, give me expression and not commercialism or capitalism. art is wrongly critiqued as is music.. all you people on the poorch, give us what you got like you people on the bully pulpit, we want it all!
love will keep us together, dont worry about it!

ryanbear

34. Clistina Says...

On May 31, 2008 at 10:51 am

I think I’v got something which I didn’t know before.Although it’s not very clear,thank you all the same.

35. Alex Donnelly Says...

On June 1, 2008 at 7:28 pm

Alright first let me start off by saying that I did not mean to publish this as the be-all-and-end-all of the connections between music and mathematics.

I am only a tenth grade math student who was required to write a ten page research papers on something math related. When I heard about this website I decided to post it just for fun.

I apologize, when writing it the night before, yes most of the connections were weak and spelling, well, non-existent.

I appreciate all of your comments, and now that I realize how important this actually is, I plan on reworking it after I take my finals at the end of June.

Thnak you to all who read my article and I hope you will continue to read my work

36. Lori Says...

On June 1, 2008 at 8:48 pm

Lol… that’s funny Alex! 10th grade project! Perhaps a brush up is in order! http://tinyurl.com/4gnhkk

37. G Says...

On June 1, 2008 at 9:25 pm

Hey, don’t write things if you can’t write them correctly!

A C minor scale contains 3 flats, Bb, Eb, and Ab. This is a NATURAL MINOR SCALE. (C, D, Eb, F, G, Ab, Bb)

The Minor scales you are showing have a raised 7th (B), thus creating a HARMONIC MINOR scale. This scale does NOT occur naturally, since it has been altered from the original, which does occur mathmatically.

38. sean d Says...

On June 1, 2008 at 9:33 pm

alright G id like to see you do better

39. Moozik Says...

On June 1, 2008 at 9:35 pm

G and readplato are right … everyone else get a clue!! I especially liked keyboardologists post …. The idea of cutting a tube or string in half to get an actave higher was not discovered by Bach … try Pythagoras! Also, sharp and flat usage is defined by the diatonic scale, and not just the direction. If you are in D major, you don’t use a C# going up, and a Db going down … it’s always a C#! However, in 12 tone writing, or chromatic passages, this rule can apply.

People … don’t comment on things you don’t have any clue about. I agree whole heartedly with readplato.

40. sean d Says...

On June 1, 2008 at 9:45 pm

hey all you smart musicians try to name 40 instruments one by one ill start

trumpet

41. dikooo Says...

On June 1, 2008 at 9:46 pm

flute

42. vanet Says...

On June 1, 2008 at 9:48 pm

3 clarinet

43. jon Says...

On June 1, 2008 at 10:40 pm

cello

44. valli Says...

On June 2, 2008 at 1:00 am

Very interesting info!

45. Jon Says...

On June 2, 2008 at 6:22 am

Don’t keep knocking Alex guys. Im sure he tried very hard, and its understandable that he made a few mistakes. Its a 10th grade paper for goodness’s sake! Get off his back.

46. ugoo Says...

On June 3, 2008 at 2:18 pm

5. trombone

47. grentlow Says...

On June 3, 2008 at 2:19 pm

6. snare

48. sugby Says...

On June 3, 2008 at 10:16 pm

Did you know that the different instruments that you list are tuned differently? Piano tuning is designed to fulfil the translation symmetry, but fail in the harmonies. Violin tuning gives good harmonies but fails in the translation symmetry. You can see this in one of the comments earlier about whether you count E# and F as different notes. On pianos they are, but on violins they aren’t.

Working out these compromises between tuning systems might take you to some interesting stuff that you haven’t seen yet.

49. nikita_devil Says...

On June 9, 2008 at 9:06 am

i can say that it is good but it reallly doesnt show the connection..you are telling about the piano but not maths!!…its good that you have told about notes and stuff and indeed explained evertig well…but it is out of the topic!

50. bob Says...

On August 29, 2008 at 10:08 am

drum

51. HYR Says...

On August 29, 2008 at 10:09 am

8. piano

52. heqklkmb Says...

On September 1, 2008 at 12:14 pm

[URL=http://fowzlrck.com]egwbcboz[/URL] ehfxnmqa http://xkfjdogv.com ycnrettc qmfwimqq rceakezi

53. Annu Says...

On October 9, 2008 at 12:59 am

I expected more relations between math & music when i saw this site.But i’m sorry to say that i could find only few or in other words, nothing. sob
Can any1 tell a gud site with articles on the same topic?

54. Annu Says...

On October 9, 2008 at 1:01 am

its really urgent, plz help.

55. sally suck butt Says...

On October 21, 2008 at 10:13 am

this was lame

56. Skysurfer Says...

On December 2, 2008 at 10:07 am

Not very well executed I’m afraid…

For a fantastic read on music and it’s effects on the brain – including more in-depth information on the subjects touched on here, suggested reading is “This is Your Brain on Music” by Daniel Levitin.

57. yayayayayayayayaya Says...

On January 3, 2009 at 1:32 pm

58. yayayayayayayayaya Says...

On January 3, 2009 at 1:34 pm

DISGRACE!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

59. richard e Says...

On January 3, 2009 at 5:07 pm

well yayayayayayayayaya, i don’t see how this is very constructive… Alex i thought this article was very well written, with occasional typo’s, i just wish you went more into harmonics and the math, as well as physics, behind that.

60. rosiy Says...

On June 2, 2009 at 7:36 pm

Does it really matter if the detais are not correct. It is a valid point to make if you truely believe in it, and wish to share it with others.

61. Jer Says...

On June 11, 2009 at 10:38 am

If the details aren’t correct how do you learn?
I’ve spent the last two weeks researching all about the connection between these two topics. Starting with absolutely no musical knowledge. When i came to this site and after reading this i had started questioning everything i have previously learned. Thanks to all the commentors who quickly relieved me of that worry.

On the other hand if anyone has any sites that can show ways to teach Math dependently with music on these connections that would be greatly appreciated. I know of the benifits of independent music study to math, but i would like to find some when they are used together. Thanks!

62. aakash Says...

On August 23, 2009 at 4:17 am

it is the worst site i have visited!!!

63. newEnglander Says...

On August 27, 2009 at 12:35 pm

this is a decent article for any musician wishing to gain a deeper understanding of the underlying mechanics of the art form. as in any art form, such as sculpting, painting, dance, ect, a knowledge of the mathematical roots help to build a mental structure for the artist to enhance their performance. kudos to all the intellectuals, both mathematicians and virtuosos musicians, but there are many folks who would find this information, both enlightening and inspirational. as a lifelong professional musician and industrial artist, i find this article to be a valuable tool for both the novice and journeyman musician. i am grateful to Alex Donnelly for his, well spent. time and effort in the composition of this informative piece. i’ll be sure to direct any musician interested in deepening their understanding of their artform to this page.

64. Konishjit Singh Bedi Says...

On May 31, 2010 at 1:04 am

Amazing information, didn’t expect music and maths to have such deep connection. VERY INTERESTING INDEED!!

65. diva Says...

On September 28, 2010 at 11:00 am

itz pathetic i cnt m paper presentation wd dis ((

66. Soren Tornkvist Says...

On January 2, 2012 at 5:33 am

Mathematics can be used to describe the tonal structure of music. But it is in no way prescriptive. Music lives its own life in a space-time with many more (and different) dimensions than mathematics.

And it depends much on the cultural context: Africans divide the octave into 5 och 7 equal steps in order to create a “scale”. And the Arabs prefer to build tetra-chords including what we westerns would call 3/4 notes. Not to mention the Indian micro-intervals!

Yes, mathematicians and musicians both work with abstract entities. And some mathematicians do play music. But I have come across many of them who are “idiot savant” (boy geniuses) for whom music is an abstract toy without any relations to human emotions. Making music together with them is chilling. Their notion of beauty seems to differ from mine.

67. Curtis Stotlar Says...

On May 16, 2012 at 3:01 pm

I’ve never heard of a c minor scale with a D# or an A#. It has an Eb and a Bb plus an Ab. Also, yes there are 12 major scales and 12 minor scales. The major scale is in the Ionian mode and the minor scale is in the Aeolian mode. These are the modes familiar to us today. Five more modes also exist.

68. George DeMarse Says...

On July 21, 2012 at 2:16 pm

I see little connection between musical ability and mathematical ability beyond elementary principles like rhythm, metre, tempo, etc. Musicians, at least in the psychology literature, are generally right brained. Their most potent cognitive abilities lie with creativity, big picture thinking, visual concepts, thinking in auditory “musical patterns,” that type of thing.
The mathematicians and logicians like directions, like to “dissect” wholes into parts, pride themselves on following “chains of reasoning” which right brainers would find boring and pointless. Namely, they would say “I’ve already arrived at step 10–I don’t care about steps 2-9.”

The Sage of Wake Forest

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