Music and the Fibonacci Sequence and Phi

Musical scales are related to Fibonacci numbers.

Piano keyboard showing that even music is based on the Fibonacci series The Fibonacci series appears in the foundation of aspects of art, beauty and life. Even music has a foundation in the series, as:

  • There are 13 notes in the span of any note through its octave.
  • A scale is composed of 8 notes, of which the
  • 5th and 3rd notes create the basic foundation of all chords, and
  • are based on a tone which are combination of 2 steps and 1 step from the root tone, that is the 1st note of the scale.

Note too how the piano keyboard scale of C to C above of 13 keys has 8 white keys and 5 black keys, split into groups of 3 and 2.While some might “note” that there are only 12 “notes” in the scale, if you don’t have a root and octave, a start and an end, you have no means of calculating the gradations in between, so this 13th note as the octave is essential to computing the frequencies of the other notes.  The word “octave” comes from the Latin word for 8, referring to the eight tones of the complete musical scale, which in the key of C are C-D-E-F-G-A-B-C.

In a scale, the dominant note is the 5th note of the major scale, which is also the 8th note of all 13 notes that comprise the octave.  This provides an added instance of Fibonacci numbers in key musical relationships.  Interestingly, 8/13 is .61538, which approximates phi.  What’s more, the typical three chord song in the key of A is made up of A, its Fibonacci & phi partner E, and D, to which A bears the same relationship as E does to A. This is analogous to the “A is to B as B is to C” basis for the golden section, or in this case “D is to A as A is to E.”


Musical frequencies are based on Fibonacci ratios

Notes in the scale of western music are based on natural harmonics that are created by ratios of frequencies.  Ratios found in the first seven numbers of the Fibonacci series ( 0, 1, 1, 2, 3, 5, 8 ) are related to key frequencies of musical notes.

Fibonacci
Ratio
Calculated
Frequency
Tempered
Frequency
Note in
Scale
Musical
Relationship
When
A=432 *
Octave
below
Octave
above
1/1440440.00ARoot432216864
2/1880880.00AOctave8644321728
2/3293.33293.66DFourth288144576
2/5176174.62FAug Fifth172.886.4345.6
3/2660659.26EFifth6483241296
3/5264261.63CMinor Third259.2129.6518.4
3/8165164.82EFifth162 (Phi)81324
5/21,100.001,108.72C#Third10805402160
5/3733.33740.00F#Sixth7203601440
5/8275277.18C#Third270135540
8/31,173.331,174.64DFourth11525762304
8/5704698.46FAug. Fifth691.2345.61382.4

 

The calculated frequency above starts with A440 and applies the Fibonacci relationships.  In practice, pianos are tuned to a “tempered” frequency, a man-made adaptation devised to provide improved tonality when playing in various keys.  Pluck a string on a guitar, however, and search for the harmonics by lightly touching the string without making it touch the frets and you will find pure Fibonacci relationships.

* A440 is an arbitrary standard.  The American Federation of Musicians accepted the A440 as standard pitch in 1917.  It was then accepted by the U.S. government its standard in 1920 and it was not until 1939 that this pitch was accepted internationally.  Before recent times a variety of tunings were used.  It has been suggested by James Furia and others that A432 be the standard.  A432 was often used by classical composers and results in a tuning of the whole number frequencies that are connected to numbers used in the construction of a variety of ancient works and sacred sites, such as the Great Pyramid of Egypt.  The controversy over tuning still rages, with proponents of A432 or C256 as being more natural tunings than the current standard.


Musical compositions often reflect Fibonacci numbers and phi

Fibonacci and phi relationships are often found in the timing of musical compositions.  As an example, the climax of songs is often found at roughly the phi point (61.8%) of the song, as opposed to the middle or end of the song.  In a 32 bar song, this would occur in the 20th bar.


 Musical instrument design is often based on phi, the golden ratio

 

Stradivarius violin design using phi, the golden ratio or golden section, in its design

Fibonacci and phi are used in the design of violins and even in the design of high quality speaker wire.

Insight on Fibonacci relationship to dominant 5th in major scale contributed by Sheila Yurick.

Do you know of other examples of the golden ratio in music?  Submit them below.

Comments

  1. Ryan says

    The whole overtone series is a series of golden ratios. If you divide an octave by a perfect fifth, (13/20), you get the golden ratio. If you divide a perfect fifth by an octave, (8/13), you get the golden ratio. If you divide a perfect fourth by a major sixth, (6/10), you get the golden ratio. And if you divide a major third by a perfect fifth, (5/8), you get the golden ratio. The overtone series is a natural order of notes that is played by horn instruments and found in other instances in music.

    • Becky says

      I’m sorry but I don’t understand. I’ve done research and it says that the golden ratio 1.61803… and so on, but 13/20 = 0.65, 8/13 = 0.615… , 6/10 = 0.6, and 5/8 = 0.625. Sorry for not understanding, could you please elaborate?

      • says

        One of the properties of Phi, 1.618, is that its reciprocal is 1 less than itself, or 0.618. You can either compare the numbers you listed to 0.618 or just flip the numerator and denominator to get 21/13 = 1.615, 13/8 = 1.625, 10/6 = 1.666 and 8/5 = 1.6000. Note that you should use 21 though, not 20. You can use 1.618 and 0.618 interchangeably by doing multiplication or division.

    • Kenny says

      Well… if you’re using the overtone series as your natural guidepost, which I fully advocate, it doesn’t make much sense to then switch to equally tempered half-steps. Those half-steps aren’t really a direct result of the overtone series, they result from the 12th-root of 2 (2^(1/12)), the 2 representing the octave overtone and the 12 being the number of half-steps needed to fit in one octave… so the equally tempered half-steps you’re treating as similarly natural to the overtone series are actually produced by an equation designed by humans. I suspect you may be onto something with the relationship of phi to ratios within the harmonic series, but a proof of such a phenomenon would require much more than you’ve provided. Also, the perfect fourth and major sixth (12th overtone isn’t quite major) don’t appear naturally in the overtone series — at least, not until the partials are very close. If you could explain the existence of the Pythagorean comma by way of phi, then you’d really have something going.

      • Kenny says

        And in his Theory of Harmony, Schoenberg gives a much simpler explanation of the phenomenon of the I, IV, and V chords based on the overtone series, with no “approximations” required.

  2. Anthony says

    Interesting they pointed out this applies to western-style music, where the smallest increment in a mode is a helf-step. I’m going to have to research how it applies to eastern-style modes and scales where the smallest increment is a quarter-step.

  3. Alan Ingram says

    Fascinating! I’m wondering how Fibonnaci applies to the sort of chord sequences that I love working with which are based on jazz and bossa nova. I wonder if rhythms work within the same principle but based on the human heart beat?

  4. Eric says

    Also, if you continue the series, you arrive at 55, which is of course a lower octave of A=440hz. If you multiply 13, the 7th number of the fibonacci sequence, in order to find its octaves, you arrive at 416hz, which is approximately 1/2 step below 440hz, with an error of +.70hz.

  5. says

    In agreement with Becky and perplexed by Gary’s reply, I suggest a simple equation; 440 Hz multiplied by 1.618 is 711.92 Hz. This is not a tone in the western diatonic based on A=440 Hz. The frequency 711.92 Hz lies between F4 and F#4.

    @Gary, how is it that 21/13 = 1.615, 13/8 = 1.625, 10/6 = 1.666 and 8/5 = 1.6000 have anything to do with 1.618 ?

    I did some research in microtonal relationships for a job a couple years ago and found that Phi does not lead to anything the western ear recognizes in either harmony or rhythm. It does have some basis in eastern and northern African music though.

    • says

      2, 3, 5, 8 and 13 and 21 are numbers in the Fibonacci sequence. The relationship to Phi is that that ratio of each one to the one before it converges on Phi as you go further in the sequence. See the Fibonacci Series page for more information. You’re correct that in music the fundamental relationships between notes in Western scales are much more closely related to the numbers of the Fibonacci series rather than Phi itself. The simplest harmonics, such as those heard when plucking a guitar string, are based on the string being divided into 2, 3, 4, 5 or more sections. Western music sales use a scale of 8 tones, while oriental scales use 5 tones.

    • says

      Hi Robert,
      I’d be interested to hear more about your research, particularly anything pertaining to the significance of the golden section in north African music.
      Best,
      Alex

  6. Alena Rose says

    Hi so I’m not great musician but I do love math and music. I find myself somewhat confused though on part of your first defense of the Fibonacci sequence in music. The part I speak of is “5th and 3rd notes create the basic foundation of all chords, and are based on whole tone which is 2 steps from the root tone, that is the 1st note of the scale.” I was wondering if you could explain this in simpler terms for me as I find this quite fascinating.

    • says

      Take a C chord as an example. It consists of the C, E and G notes. E and G are the 3rd and 5th notes of the C scale. E is 2 whole tones from C. The 13 note scale breaks down into key notes defined by 8, 5, 3, 2 and 1, as does the Fibonacci series.

    • says

      Per the composer: “What Phi (the golden ratio) Sounds Like” is a musical interpretation of the mathematical constant Phi. The formula I use to translate the digits of Phi into music is as follows: 1=C, 2=D , 3=E , 4=F , 5=G, 6=A, 7=B, 8=C octave, 9=D octave, 0=no note is played. The melodies that you hear throughout this piece are taken directly from the first 39 digits of Phi. The tempo is set at 161.8 BPM.” A very creative, and well executed, interpretation. It would be interesting to create and hear other interpretations of phi in which the frequencies of the notes themselves are based on golden ratio proportions.

        • says

          Flats and sharps are just half steps down or up between whole tones, so they could still be half of the way between whole tones in a golden ratio scale or … perhaps there should be two dividing points between whole tones, one at 38.2% and one at 61.8%, using 1/phi and 1/phi squared.

          • Darren says

            I have a question, if you wanted to express a Pythagorean 3,4,5 triangle musically what 3 keys would you need to strike to get the same relationship?

  7. Derial says

    I don’t know a lot about music…but I heard an interesting conversation earlier this week that came to mind when I saw this site a few minutes ago. Does a Fibonacci sequence have anything to do with the so-called “Devil’s Interval or “Flatted Fifth” chord?

    • says

      The flatted fifth doesn’t seem to be related to the Fibonacci series or phi. A flatted fifth is six half tones up from the root, with six more half tones up to the octave. The ratio of its frequency to the root is the square root of two, and the ratio of the octave to the flatted fifth is also the square root of two. As an example, starting at A440, the flatted fifth E flat, has a frequency of 622.254, which is 440 times root two (1.414214), with the octave at 880.

  8. Glen Kennedy says

    Thats great Gary really good information regarding equal temp and overtone series, it would be quite something to link it with the pythagoras comma. I have been using math and playing with ideas re composition for some time. Interesting idea to experiment with Phi and frequencies etc. I will have a listen to that piece soon.
    @ Darren, might not be quite what you are after, here’s some ideas,you could try calculating the frequencies of notes and squaring them and then try to find 3 frequencies that match the Pythagorean triangle, or find another series of pythagorean triplets as they are known eg 5^2+12^2=13^2 or a larger one 99^2+4900^2=4901^2. I’d be surprised if any of them matched the equal temp tuning.
    Or in the overtone series, the 3rd harmonic is G, 4th is C and 5th is E, so a C chord or squared , 9+16=25. The 9th overtone is D, 16th is C and the 25 th is a G of sorts -27 cents.
    I’d be interested to see if anyone else has some ideas in this regard.

  9. Billy says

    Hey Gary. This may be off topic, but I was wondering if you could apply colors to the phi ratio and to the corresponding notes in the scale? Also, if there was a 3d representation of these sounds or of the phi ratio itself? Thanks so much.

    • says

      You can apply the golden ratio / phi to colors. Musical scales though are based on octaves, with each octave being based on a doubling of the frequency, and with each note in a 12 tone scale being a increase of the 12th root of 2 from the one before it. That fact would probably override the exact application of the golden ratio. Anyone else have any creative thoughts?

      • Darwe says

        C-UTerus 16 32 64 128 256
        C# 17 34 68 136 272
        D- REign/region 18 36 72 144 288
        D# 19 38 76 152 304
        E-MIxtus orbis 20.25 40.5 81 162 324
        F -FAtum 21.5 43 86 172 344
        F# 22.75 45.5 91 182 364
        G-SOL/ar 24 48 96 192 384
        G# 25.5 51 102 204 408
        A-LActeus 27 54 108 216 432
        A# 28.5 57 114 228 456
        B-SIdereus orbis 30.5 61 122 244 488
        I found it interesting when i saw video of 440vs432 and on the “SOL” note you could see the sun(or multiple suns) (Cymatics experiment 432-440Hz by Holonmusic432Hz)
        So the “sol” frequencies are like the 24 hour cycle(cycles per second=hertz)
        60secondsX60minutesX24Hours a day=86400 (this is A=432X2) and so on.
        When i imput all the frequency in Excel i started subtracting the C# with C, D with C# and so on and i found this numbers that occur in Cycle computing For exsample 0.75 wiuld be 3/4 of a circle 1 =full circle 1.25=Circle and 1/4, 1.5=Circle and half..so on and so on.
        0.75 1.5 3 6 12 24 48 96 192 384 768
        1 2 4 8 16 32 64 128 256 512 1024
        1.25 2.5 5 10 20 40 80 160 320 640 1280
        1.5 3 6 12 24 48 96 192 384 768 1536
        2 4 8 16 32 64 128 256 512 1024 2048
        I find this much interesting. Computer RAM progression(geometric) and progresions for Screen resolutions (16:10 Phibonacci,its should be 16,2:10 roughly )
        13 is a magic number (star of david=male and female principle in the male,and another male and female in the female together they are 12 sided star of david and the point of manifestation is the new octave,”Do” dominion Point zero ,perfect state)
        Much to be said here but then again Im still learning. thank you for the post.

      • madhu says

        I haven’t had time to study the above page that you mentioned, but have been looking at the relationship of colors to musical tones.
        What I’ve found, is that C 41 octaves above middle C is probably 563 THz, equated by doubling 256 41 times, or 256X2e41. Rainbow color C is 566 THz, the range of Green being 540-610 THz.
        The other notes and colors are:
        D (288Hz) is 633Thz 41 octaves higher, equalling Rainbow Blue (638THz, blue ranging from 610-670THz)
        E (324 Hz) is 712 THz 41 octaves higher, equalling Rainbow Violet (714THz, violet ranging 670-750THz)
        F is 756THz, just above violet, G is 422 THz 40 octaves above middle C, rainbow red being 428THz, the range given for red 430-480 THz.
        A (432) is 474 THz 40 octaves higher, so on the very orange side of the red scale, B(486) is 534 THz 40 octaves higher, with is within the yellow range (510-540THz). I have more to add but need to do something else now.

  10. Carola says

    Hello,
    I´m writing my term paper for school about this topic, Does anyone know the author of this great article? =D
    I´ll be very thankful about an answer because I need the name for a citation.

  11. Chartsky says

    Hi. Thanks for your article. I came across it while doing generic research further applying Fibonacci to my day trading. I never before realized there are 13 notes in an entire octave. I’m often amazed at how this series is involved in so much around us.

    • says

      hi gary. thanks for the info.
      looking for 1 note that represent the phi. what note should it be ?
      you can suggest several notes ofcourse.
      thanks
      shai

      • says

        If I had to pick just one note, I suppose it would be the one with a frequency of 161.8 or 1618 cycles per second. The golden ratio though is about a ratio of two things though, so that would mean it would be better to pick a note and the one that is 1.618 or 0.618 times its frequency.

        • john cronen says

          hi , i have brain damage , so i keep it simple on 5ths and golden mean proportion ( which ratio stays the same despite the size used ) and for me its the model of perfect compresssion in nature ,..

          if starting at D 144 hz ( 432 @ 216 attune ) if starting at 144 to go up a golden ratio i would multiply it by 3 and divide it by 2 , like pythagoras did ,, all the heady music maths make me swoon ,, so :

          144 going up the scale is 144 x 3 then divides by 2 = 216 its perfect pythagoran fifth .. and very close to perfect phi golden mean compression tones .

          to go down just divide by 3 and multiply by 2 :
          144 / 3 x 2 = 96 G ( i also like 144 as its in fibonachi directly )

          so going up from D144 its A 216 , and down its g 96 ..

          g 96 to D 144 is perfect golden mean proportions ,, the music buffs seem to way over complicate simple fifths and pythagoran fifths both ..

          i see it as a shape of phases meeting a bit like a hurricane and implosion ..

  12. Hamda says

    I need some more vivid explanation of the relationship of music with Phi, if you can give me some more. Like, how is Phi found in the musical notes, or chords or other things relating to music.

    • says

      This is way too simplistic a reading of the way musical theory works and is basically a numerological approach of having a theory and trying to find facts to fit it.

      We *know* how the musical scale works: An octave is a doubling of frequency per octave and then whole tone ratios (not the golden series either) give the harmonic series, which in one octave span looks like this:

      C-D 9:8
      C-E 5:4
      C-F 4:3
      C-G 3:2
      C-A 5:3
      C-B 15:8
      C-C 2:1

      And since the 1600s we’ve found that 12^2 is the best way to approximate the above and have usuable semi-tones and scales in each key.

      The golden ratio struggles to explain any of this simple mathematics and adds little if nothing to our understanding of harmony.

      • says

        The article does not say that musical scales are based on the golden ratio. It says they are related to the Fibonacci numbers found in nature. As you noted, tempered musical scales are an APPROXIMATION to produce the same result in any key. (That mathematical relationship, by the way, is based on 2 ^ (1/12), not 12^2.) The ACTUAL tonal relationships that you would find if you plucked a string at various harmonic points are based on the integers, just as you listed. Frequency relationships created by ratios of Fibonacci numbers do in fact produce the true harmonic frequencies of the notes in the scale, as illustrated by the chart above.

  13. says

    I don’t get where some of the numbers are coming from when you start putting them in ratios. But what I do find interesting is that when you go to the 432 scale, all the note frequencies are all divisible by nine. That just seems right to me. Nine is an awesome number in its properties, so it’s gotta mean something that all those 432 numbers are nines, right?

  14. says

    The reason the frequencies or ratio calculations don’t line up exactly (but are close) is what is called the Pythagorean comma. The Pythagorean comma is a factor that is used to multiply fundamentals (root frequencies) by to get enharmomics to sound “in tune” when using the circle of 5ths. If you google or youtube “pythagorean comma” you’ll be able to easily see the reason for its existence.

  15. Lucien Windrich says

    The diatonic scale can be generated using successive phi numerals squared, as follows;-

    (start with 16^2 = 256)

    16.618^2 = 276.16
    17.618^2 = 310.39
    18.618^2 = 346.63
    19.618^2 = 385
    20.618^2 = 425.1
    21.618^2 = 467.33
    22.618^2 = 512

    You can check the above scale values against the 12th root of 2 values.

    I don’t know why it does this though…

    Lucien

  16. Monica Collier says

    I bought an allen organ old one made 1970 or so. I am playing different pieces from Bach well tempered clavichord. I also have a Wurlitzer baby grand made in China or out of USA, just got it tuned. The Bach sounded horrible.
    I thought to myself, I bet Bach has a different sense of turning. I started by looking up what the differences between “well tempered” and modern tuning.
    I decided to get the tuner to tune my piano exactly to the principle 8 on the
    bottom keyboard. BINGO The bach sounds great on my piano. Hope the organ
    is tuned to the well tempered scale.
    I built a very complex series of pyramids interfacing one another. They were
    all constructed golden section wise as great pyramid. It emitted such a force field
    that I was almost floating. I started psychic channeling and it wrote a book.
    which I called Open Door Prayer. Anyhow my life has since evolved around the golden section including enjoying playing the organ and the Bach etc

  17. Sean Round says

    Not sure if it counts because I didn’t do the math. Resonator pipes for marimbas, and other air columns, seem to fit along a section of the Golden Spiral. It sure is neat to the see the Fibonacci sequence popup all over the Universe.

    Good Day

  18. says

    Hi,
    A previous comment mentioned A440 X I.6I8 ends up between F natural and F#.
    What pitch would A432 x I.6I8 give?? Would it be F natural?
    Very interesting site. Thanks.Gary.
    .

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