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.
Music and Mathematics
(Picture from http://cc.wsd3.org/media/images/piano_keys.jpg)
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. Music brings joy to those who play, write, or listen. Ask them if it is connected to mathematics, and many people will be puzzled. However, once the shock recedes and they think about it, they realize that there are striking similarities. Not only the obvious ones, like beats in a measure, but the wavelengths and ratios between notes. Studies have shown that babies who listen the classical music can grasp mathematical concepts quicker than those who did not. In essence, math is creative and beautiful, which is why such elegant connections are made between them. This paper will attempt to discuss the simple and complex connections between the two studies as well as attempt to uncover new ideas in the fields of music, such as a wind instrument with a piano’s range. (Background from http://www.math.niu.edu/~rusin/uses-math/music/ and http://www.woodpecker.com/writing/essays/math+music.html)
Music is made up of beats. Beats are pulses in which time is marked. The most common measure has four beats in it, which means no matter the combination of notes, they must add up to four beats. For example
(image from http://www.uoregon.edu/~kford/picturegallery/intromusic_files/image005.gif)
ü Level one is called a whole note, meaning that this note will get four beats. 1note @ 4 beats
ü Level two is composed of two half notes, which are two beats each. 1 note @ 2 beats
ü Level three is 4 quarter notes representing one beat each. 1 note @ 1 beat.
ü Level four is 8 8th notes, which equal one half a beat each. 1 note @ .5 beats
ü Level five shows 16 16th notes, which would each receive one quarter of a beat. 1 note @ .25 beats.
The same holds true for rests: one whole rest equals two half rests equals four quarter rests, and so on and so forth. The following picture shows the relationships between rests and beats.
(picture from http://www.enchantedlearning.com/music/label/notesrests/answers.GIF)
(This and similar pictures from http://www.dkimages.com/discover/previews/756/223171.JPG)
Basics first. Before attempting to play music, any musician will look at the time signature. The top number tells a musician how many beats are in one measure and the bottom one reveals which note will get one beat. In the above example, there will be four beats in one measure, and a quarter note (1/4 a measure) will get one beat. But this does not mean that only quarter notes can take be in that measure. Take a look at the first measure above
(From http://www.dkimages.com/discover/previews/756/223171.JPG, edited by myself)
Another simple aspect of music is the tempo. Tempo is recorded in beats per minute and tells the conductor and musicians how fast a piece of music should go. This is marked at the beginning of a piece above the staff (see above). A tempo of 60 is very slow at one beat per second while a tempo of 180 is very fast at three beats per second. The formula to figure out how long a song will be is:
60S=MB
T
S equals time in seconds, m equals the number of measures, b equals the beats in one and t equals the tempo
The key signature tells the musician which keys to play in. Different keys are made up of different notes even though the ratios to the first note remain the same.
(image from http://www.ericweisstein.com/encyclopedias/music/wimg3.gif)
Take any note, and the transition to the note directly to the left or right of it is called a half step. The note two notes to the left or right is called a whole step. For example, take “F”. An “F#” and an “E” are considered half steps, while “G” and “D#” are considered whole steps. To create a major key we used this pattern. Start with any note and then:
+1 +1 +½ +1 +1 +1 +½
which will give you the key of that note. This will also give you a major scale. According to www.answer.com, majors scales are defined as: “an ascending or descending collection of pitches proceeding by a specified scheme of intervals”. This interval is: x+1+1+½+1+1+1+½, with x representing any note, which was discussed earlier. There are twelve major scales that are distinct, C, C#/Db, D, Eb, E, F, F#/Gb, G, Ab, A, Bb, B/Cb. ( # represent sharp, b represents flat). Any scales after these would be defined as octaves, one whole pattern above the others. To find octaves we use the formula (with x representing the first note of any scale):
x=x+11
Take the formula for the major scale again:
x +1 +1 +½ +1 +1 +1 +½
Apply this to any note and every individual answer you get along the way makes up the key. For example, start on Bb.
For a major scale: Bb +1 +1 +½ +1 +1 +1+½
Which corresponds to: Bb,C, D, Eb, F, G, A, Bb
Which gives us the key of Bb: Bb, C, D, Eb, F, G, A, Bb
There is also something called a minor scale, which involves lowering certain degrees in the major scale. This makes the scale sound sad, and is characteristic of many slow, sad songs as well as ominous sounding classical music. To get a minor scale, we lower the third, sixth and seventh note of the major scale by a half step
Where as the scale used to be: x +1 +1 +½ +1 +1 +1 +½
1 2 3 4 5 6 7 8
The minor scale becomes: x +1 +½ +½ +1 +½ +½ +½
So in the major scale of C: C, D, E, F, G, A, B, C
1 2 3 4 5 6 7 8
We get the C minor scale: C, D, Eb, F, G, Ab, Bb, C
Now take the Eb major scale Eb +1 +1 +½ +1 +1 +1+½
In order to get: Eb, F, G, Ab, Bb,C, D, Eb
Notice that the scales of Eb major and C minor have the same notes, but in a different order:
C, D, Eb, F, G, Ab, Bb, C
Eb, F, G, Ab, Bb, C, D, Eb
This is because C major and Eb minor are what is called relative keys. Eb is the relative minor to the key of C. This may seem confusing, but it is easily summarized with what musicians call the “Circle of Fifths” (shown below). Although they keys on a piano (the white ones) read; C, D, E, F, G, A, B; musicians memorize the notes according to their structure in the circle of fifths: F, C, G, D, A, E, B. The scales are organized this way because this is the way the number of sharps and flats that appear in the key are organized. Forget about the F for a moment:
Circle of Fifths: F, C, G, D, A, E, B
Scale(key) number 1, 2, 3, 4, 5, 6
7
Number of Sharps: 0, 1, 2, 3, 4, 5
6, 7
The first scale (key) we start with is C. C has no sharps (0). Next we go to G. G has one sharp (1). D has 2 and E has 3. In order to find out what these sharps are, we start with F and count the number. So G, with one sharp, gives us only F. This means that F is the only note that is sharp in the key of G. Take the key of E. E has 4 sharps. Starting with F, we get F, C, G, and D. This means that the four sharps in the key of E are F, C, G, and D.
However, this gets complicated when the circle goes on past B after reading clockwise. The notes F# and C# appear. These notes were not in the original sequence of letters. This is because the that sequence is read both forwards and backwards (which we’ll discuss later) and that F# and C# do not fit when it is read backwards. To help the reader understand this, imagine there are two separate sequences, one for the sharp notes, and one for the flat notes. So in order to include F# and C#, the sharp sequence should look like this:
F, C, G, D, A, E, B, F#, C#.
So therefore, following the same method discussed above, A C# scale would have seven sharps (remember to start counting at C and not at F). These seven sharps are F, C, G, D, A, E, and B. Which gives us the scale and key of C sharp major: C#, D#, E# (played as F), F#, G#, A#, B# (played as C).
Now the circle of fifths also works for the flat scales. Just drop the F# and C#, and reverse it and add an F to the beginning:
F, B, E, A, D, G, C, F
1, 2, 3, 4, 5, 6, 7
This is the “Circle of Fifth” for the flat scales. F is an anomaly here because it is the only major scale to use flat notes without itself being flat. Take a look:
F is the first flat scale. Because F is the first flat scale, it gets one flat. To figure out what particular note is flat, start counting from the beginning, always skipping the first F. Therefore the one and only flat in the key of F major is a Bb. Let see:
F +1 +1 +½ +1 +1 +1 +½
F G A Bb C D E F
Here it is easy to see that the B note is indeed the only flat note in the F major scale. As another example, the key of Gb. G is the sixth note in the circle and therefore gets six flats. Starting from the beginning and leaving off the first F, we get that the six flats are Bb, Eb, Ab, Db, Gb, and Cb, meaning that the B, E, A, D, G, and C all become flat in the key of Gb:
Gb, Ab, Bb, Cb (played as B) Db, Eb, F, Gb
The inner circle of the “Circle of Fifths” also described what the “relative minor” is. “C” on the inner circle corresponds to “Eb” on the outer circle. This means that C minor uses the same notes as they key of Eb, except the scale starts with C and not with Eb, as discussed earlier. This holds true for all notes. Therefore, “A minor” uses the same notes as “C” and so on and so forth. Just as patterns are found with numbers, so too can they be found in music. The circle of fifths is a table that summarizes the answers to the major and minor formula.
Now this pattern is the same for all notes and scales. To transpose a scale or change it to a different key, you use a method similar to a shift cipher. For instance; take the notes C, D, E, F, and G in the key of C (no sharps of flats). If one wants to transpose these notes, one first attaches numerical values to the notes based on their scale. C is the first note of the scale so it gets a value of one. D is the third note so it gets a value of 3:
C D E F G
1 3 5 6 8
Then one takes a new key, say G(one sharp only which is an F). Due to the fact that G is first note in a the G scale, it equals the note value of one given to C in the previous example. A is the third note in the G scale which matches up with the third note in the C scale: D.
C D E F G
1 3 5 6 8
G A B C D
The transposition is complete. The spaces between CDEFG will be the same as GABCD. This is exactly the same as transformations on a graph. Under a transformation, the image looks the same, with the same distance between points. This is the same as transposing; the piece sounds the same, just with different points.
Above, whole steps, half steps and keys were discussed. The reason artists use keys is so that notes in piece sound pleasant to a listener. It may seem difficult to makes people’s taste connect to math. However, there is a way to determine what will make notes sound sweet to one’s ear. It all starts with a Hertz (Hz).
A Hertz is a measure of frequency. Hertz is a ratio to radians per second. The more radians per second in a sound wave, the higher the pitch or frequency. One radian per second is equal to:
1radian/second= 1 Hz
2
(http://en.wikipedia.org/wiki/Radian_per_second)
Pythagoras, one of the most influential mathematicians of all time, was one of the first mathematicians to make connections between music and math. He believed that everything in life could be expressed as a fraction. He took this approach towards music as well. Pythagoras lived during Ancient Greek times. During this era, music was much simpler than it is today. The octave had only five notes. Pythagoras noted that each note was a fraction of a string. Take for example the double bass, a string instrument. The last string on a double bass, when bowed or plucked, vibrates at 196 Hz (which corresponds to the note “G”). If one places their finger about ¾ of the way down the string and presses down, the note becomes a C, which is approximately 4/3 (the reciprocal) of the frequency, at 262 Hz.
This “reciprocal rule of music” applies to all notes of all non-electric instruments. When the no valves of a trumpet are pressed, a “C” is heard, (corresponding to C4 on the chart below). When the first valve is pressed, a “G” is heard, (corresponding to G4 on the chart below). “C” has a frequency of 261.63 and “G” has a frequency of 393.
393__ is approximately equal to: _3_
261.63 2
Being that trumpets make sound by air vibrations, trumpets change notes by changing the amount of space the air has to vibrate. By pressing down the first valve, the air space is cut by 2/3, the reciprocal. The ratios between the frequencies of notes always equal the reciprocal of the fraction of space the note can vibrate in. The ratio between C and G is always 3 to 2, and therefore to play a G, one must take 2/3 of the space used to create the sound, whether it be 2/3 of the string, 2/3 of the air tube, 2/3 the area of the block (on a xylophone).
|
Note |
Frequency (Hz) |
Note |
Frequency (Hz) |
Note |
Frequency (Hz) |
Note |
Frequency (Hz) |
|
C0 |
16.35 |
C2 |
65.41 |
C4 |
261.63 |
C6 |
1046.5 |
|
C#0/Db0 |
17.32 |
C#2/Db2 |
69.3 |
C#4/Db4 |
277.18 |
C#6/Db6 |
1108.73 |
|
D0 |
18.35 |
D2 |
73.42 |
D4 |
293.66 |
D6 |
1174.66 |
|
D#0/Eb0 |
19.45 |
D#2/Eb2 |
77.78 |
D#4/Eb4 |
311.13 |
D#6/Eb6 |
1244.51 |
|
E0 |
20.6 |
E2 |
82.41 |
E4 |
329.63 |
E6 |
1318.51 |
|
F0 |
21.83 |
F2 |
87.31 |
F4 |
349.23 |
F6 |
1396.91 |
|
F#0/Gb0 |
23.12 |
F#2/Gb2 |
92.5 |
F#4/Gb4 |
369.99 |
F#6/Gb6 |
1479.98 |
|
G0 |
24.5 |
G2 |
98 |
G4 |
392 |
G6 |
1567.98 |
|
G#0/Ab0 |
25.96 |
G#2/Ab2 |
103.83 |
G#4/Ab4 |
415.3 |
G#6/Ab6 |
1661.22 |
|
A0 |
27.5 |
A2 |
110 |
A4 |
440 |
A6 |
1760 |
|
A#0/Bb0 |
29.14 |
A#2/Bb2 |
116.54 |
A#4/Bb4 |
466.16 |
A#6/Bb6 |
1864.66 |
|
B0 |
30.87 |
B2 |
123.47 |
B4 |
493.88 |
B6 |
1975.53 |
|
C1 |
32.7 |
C3 |
130.81 |
C5 |
523.25 |
C7 |
2093 |
|
C#1/Db1 |
34.65 |
C#3/Db3 |
138.59 |
C#5/Db5 |
554.37 |
C#7/Db7 |
2217.46 |
|
D1 |
36.71 |
D3 |
146.83 |
D5 |
587.33 |
D7 |
2349.32 |
|
D#1/Eb1 |
38.89 |
D#3/Eb3 |
155.56 |
D#5/Eb5 |
622.25 |
D#7/Eb7 |
2489.02 |
|
E1 |
41.2 |
E3 |
164.81 |
E5 |
659.26 |
E7 |
2637.02 |
|
F1 |
43.65 |
F3 |
174.61 |
F5 |
698.46 |
F7 |
2793.83 |
|
F#1/Gb1 |
46.25 |
F#3/Gb3 |
185 |
F#5/Gb5 |
739.99 |
F#7/Gb7 |
2959.96 |
|
G1 |
49 |
G3 |
196 |
G5 |
783.99 |
G7 |
3135.96 |
|
G#1/Ab1 |
51.91 |
G#3/Ab3 |
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)
|
Note |
Ratio |
|
|
0 |
1:1 |
|
|
1 |
135:128 |
major chroma or minor second |
|
2 |
9:8 |
|
|
3 |
||
|
4 |
||
|
5 |
||
|
6 |
45:32 |
diatonic tritone |
|
7 |
||
|
8 |
8:5 |
|
|
9 |
27:16 |
Pythagorean major sixth |
|
10 |
9:5 |
|
|
11 |
15:8 |
|
|
12 |
2:1 |
|
|
|
Notice how every third C (green) matches up with every second G (red), making the notes sound nice. (picture above and below taken from http://www.musicmasterworks.com/WhereMathMeetsMusic.html)
Here, F sharp is red, which matches up with green C. These notes will only match up every 45th time for C and 32nd time for F#, making them sound unpleasant. The above figures are sine curves, starting at the origin, which are used to graph function is math, but also to map out sound waves in music, one of the deeper connections thus far. ( Idea from Jessica Seminelli in Eureka!, “The Symphony of Sine Curves”. Research compiled from http://www.musicmasterworks.com/WhereMathMeetsMusic.html and http://thinkzone.wlonk.com/Music/12Tone.htm)
Harmonics are also a little on the complicated side. Harmonics the vibrations and are what makes notes on a flute and a violin sound different. These two distinct instruments make vibrations in very different methods. The strings vibrate when bowed in string instruments. The density, length, and tension of the string impact the vibrations made. In a flute, the air vibrates when a player blows across a narrow opening, similar to blowing across a water bottle. Brass instruments make vibrations from one’s lips. Reeds instruments make vibrations between the reed an the mouth piece, similar to a kazoo. Finally, harmonicas make sound by the little strips of metal vibrating quickly. The less area there is where the vibrations are made determine pitch. A flute, which is a very small instrument, has very little are to vibrate, creating notes with very high pitch. A tuba a an extremely large are in which the air can vibrate, creating notes with very low pitch. One can alter the vibrations by elevating of lowering the space the vibrations have to expand. A violin player moves his or her fingers up or down the strings to make them smaller or larger, affecting pitch. A brass player presses and releases valves, which allow the air to flow into larger of smaller chambers, affecting pitch. Reed players open or close holes in their instrument, allowing air to escape earlier or later than normal. The physical properties of each instrument limit their range, because no instrument out on the market can have accommodations for every note. Keep in mind that players do not drill openings in their instrument, but rather they are already there, making a definitive range for that instrument. Therefore, the instrument with the largest range is a piano. (http://en.wikipedia.org/wiki/Mathematics_and_music)
No wind instrument has ever been created that can mimic the range of the piano. This is limiting because pianos are expensive and are not easily transported. They also have to have their inner workings inspected after every move, and once a year to insure they are pitch perfect. I will try to solve this problem by creating a wind instrument that can cover the range of a piano.
To start with, I had to look at which instrument have the extreme low and high ranges, because there are not many instruments that can do this. Now there are trombones that exist that have a trigger, allowing them to cover more notes towards the lower reaches of the piano. Taking form those advancement, a trigger could b e used, located near the mouthpiece, to switch from a higher end of an instrument the lower end. They also have to be of the same type, say two brass instruments, because they need to share a mouthpiece.
I started with finding instruments with far reaches in terms of pitch. The following picture is the range of many instruments. (image below from http://upload.wikimedia.org/wikipedia/en/timeline/d509b142febba1b37c8544b544ff1ac5.png)
The octocontrabass clarinet’s range can not be covered by the chart, as the instrument can reach the A below the last part of the chart. So if a octocontrabrass clarinet and a soprano clarinet were combines, they would almost cover the entire range of a piano. Notice how the high reach of the octocontrabass clarinet and the low reach of the soprano clarinet math up, so it would be a seemless connection. The shortcoming is, the combined range is still 6 notes short of a piano’s range, however it can reach 4 notes lower than a piano can. I believe that this is close enough.
octocontrabass clarinet (on left) (image from http://www.jayeaston.com/images/octocontra-Clarinets-Leblan.jpg)
Soprano Clarinet ( image from http://www.interstatemusic.com/wcsstore/InterstateMusic/ims/ipf/105773.jpg)
When combined, these two instruments would be connected at the mouth piece, with the smaller of the two sticking out at about a 45 degree angle to the ground.
In conclusion, there is a very elegant connection between math and music. Whether is be simple, such as beats in a measure. Or more complex, sine curves and hertz ratios, it is clear why they are so similar. The different measure connect to the patterns learned in early math, 1-2-3-4, 1-2-3-4,1-2-3-4. Just like mathematical sequences have patterns so does music. The different number of beats a note gets connects to the fractions in math. 1/1 equals 4/4. Simplification. No matter what, each measure will always have four beats in it, just like 4/4 and 1/1 are equal, the answer always being one. The ratios between frequencies, are also similar to word problems in early math where one objects would complete a task every two minutes, and another every three, and a student is asked when they will meet up. This is an early concept of the least common denominator. When attempting to add fractions without the same denominator, a student must find the LCD, just as the frequencies, make music more pleasant to people. The connection between transposing and translating were shown, as if the mold was picked up and moved somewhere else to create almost identical music. And finally, an pretty successful attempt was made to design a wind instrument that could cover the range of a piano. All in all, the connections between math and music are deep. Perhaps that is why musicians are so good at math.
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65 Responses to “The Connections Between Music and Mathematics: Revised and Better Than Ever”
On May 29, 2008 at 8:45 am
alex how do people look at this
On May 30, 2008 at 11:21 am
“once the shock reseeds and they thing about it…”
Well said.
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.
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.
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.
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…
On May 30, 2008 at 11:42 am
Something worth reading related to this is Godel, Echer, Bach by Douglas Hofstadter.
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.
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.
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.
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
On May 30, 2008 at 12:04 pm
2pi radians per second is one Hz.
On May 30, 2008 at 12:08 pm
I am dumber for having read this. Bravo, sir. Brav.o.
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.
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.
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.
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?
On May 30, 2008 at 12:48 pm
duh i like music!!!
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?
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.
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.
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.
On May 30, 2008 at 1:21 pm
Maybe I’m just another pedant to be dismissed, but this article is a good example of a lot that is wrong with blogging.
On May 30, 2008 at 2:20 pm
this article is so, so bad in so many ways. bad math. bad explanation of graphs. bad music. Just bad. Which is bad because there ARE so many cool relationships in music that are mathematical. And not just simple math, really hardcore math. Like have you ever wondered why you hear a note in one scale as flat but another it is sharp? Well, matrix math can explain that (a recent article in Nature, I think). Also cool? notes that go together are orthogonal to one another.
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…
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?
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.
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!
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.
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.
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.
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
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.
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
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
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.
On June 1, 2008 at 9:33 pm
alright G id like to see you do better
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.
On June 1, 2008 at 9:45 pm
hey all you smart musicians try to name 40 instruments one by one ill start
trumpet
On June 1, 2008 at 9:46 pm
flute
On June 1, 2008 at 9:48 pm
3 clarinet
On June 1, 2008 at 10:40 pm
cello
On June 2, 2008 at 1:00 am
Very interesting info!
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.
On June 3, 2008 at 2:18 pm
5. trombone
On June 3, 2008 at 2:19 pm
6. snare
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.
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!
On August 29, 2008 at 10:08 am
drum
On August 29, 2008 at 10:09 am
8. piano
On September 1, 2008 at 12:14 pm
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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?
On October 9, 2008 at 1:01 am
its really urgent, plz help.
On October 21, 2008 at 10:13 am
this was lame
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.
On January 3, 2009 at 1:32 pm
there was nothing to say about this DISGRACE
On January 3, 2009 at 1:34 pm
DISGRACE!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
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.
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.
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!
On August 23, 2009 at 4:17 am
it is the worst site i have visited!!!
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.
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