# Wavelength to Frequency Relationship in Musical Notes

• KiwiJosh
In summary, the Fifth Interval is halfway between two notes and sounds especially good with the root note. It is related to the Fourth Interval, which is also halfway between two notes. The Fourth and Fifth intervals are directly related because 7+5=12.
KiwiJosh
Hi people,

So I've been digging into music theory and want to understand the basic reasons for how it is constructed.
I've come across a particular relationship but I can't understand the reason it exists.

A quick musical lesson:

Imagine you've got a pure musical note: let's say C. From a low C up to a middle C (the next C up) is called an octave. From one C (or any note, for that matter) to the next octave up there are 12 notes: C, C#, D, D#, E, F, F#, G, G#, A, A#, B. Notice that there are two sharps (#s) missing from here - that's just the way music is structured. For the purpose of my question, that doesn't matter, the only point is that there are twelve notes in an octave. So don't get confused by the Letter names of the notes, which could lead you to believe there are only seven distinct notes (A through G). The 12 notes are in fact pretty close to (more on that soon) equidistant in pitch, meaning that mathematically it would be more useful to label them A through L, dispensing with the need for sharps, to avoid confusion. There are other musical reasons this is not done, but that's another story.

Halfway between any two Cs lies a particular note that sounds especially good with C: it's G. This relationship is known as a Fifth Interval (because, well, another story). Notice that G is not five but seven notes up from the root (C, C#, D, D#, E, F, F#, G)... as I said, not going to explain the reason here for naming it a Fifth. Anyway, it's halfway in between the two Cs when you're measuring pitch(aka frequency). But it's the seventh note of twelve, you say! That's not half! Except that, the twelve notes are not actually perfectly equidistant, for more reasons I won't go into here. Some gaps between notes are larger than other gaps. Suffice to say that what we now call a Fifth Interval is indeed halfway between the root and its octave, when measuring the frequency. In fact, no matter which note you start from, if you go "up a Fifth" (seven notes up) then you get the same relationship: a note that sounds especially good alongside the root note.

Another particularly pleasing Interval is a Fourth Interval, which is actually five notes up from the root. So a Fourth up from C is F.

These two intervals are directly related, because seven notes + five notes = 12 notes. Meaning if you go up a Fifth from C you get G; but if you go down a Fourth from C you also get G. Just basic counting here, in two directions. So far so good.

So, a musical note can be expressed as a pure waveform. It can be measured by its wavelength in cm.
The sound of that note, or rather the frequency - how high or low it sounds - can be measured in Hz - oscillations per second.
So there are the two ways to measure a note.

I decided to look into what makes the Fourth and Fifth intervals so special.

Here is a chart containing the Wavelengths (in cm) and the frequencies (in Hz) of many notes: Frequency/Wavelength Chart

The story goes that our ancestors took a musical string and plucked it, obtaining a musical note. Then they took the string (which we can think of as a wavelength) and halved the length, noticing that the pitch (frequency) of the note was now twice as high - the same note but a higher version (what we now call an octave higher). This relationship can be expressed as a ratio of inverse-fractions, where WL = Wavelength and FR = Frequency:
½WL → ²⁄₁FR
(I'm using an arrow instead of "=" to express this relationship, as the values are not equal. I'm not sure of the proper symbol to use here but the arrow will do. Take it to mean "This leads to That". Maybe a colon would be more appropriate?)

So then our ancestors said, how else can we split this string up? How about instead of halves, we split it into thirds?
So, they took the original string, cut a third of the length off it, and got a pitch which was 50% higher than the original. Let's say they took a string that made the note C, then cut it down to two-thirds of the length. They got the note G, which is actually halfway (in frequency) between the original C and the next C up (or, the original C + 50%).

This relationship can be expressed through another inverse fraction ratio:
⅔WL → ³⁄₂FR

Then they went on to base our whole system of Western Musical Scales on this "perfect fifth", as they called it (not to mention the ordering of the days of the week... but that is such another story, forget I even mentioned it).

Now, remember we discovered that the Fourth Interval is really just the other side of the coin to a Fifth, since going up a Fifth will give you the same note as going down a Fourth (albeit an octave higher). You can also say that a Fifth + a Fourth = an Octave (7 notes + 5 notes = 12 notes).

So, let's see how our WL → FR looks with a Fourth Interval (again, don't get distracted by the names Fourth and Fifth - those numbers are irrelevant to this working and can confuse you. We'd be better off calling them Bob and Jane...)
To achieve this, we cut a quarter of the length off the original string, and get a pitch one-third higher than the original. So:
¾WL → ⁴⁄₃FR

Another inverse-fraction.

Here's a quick example using the chart linked above.

We'll go with C4, also known as middle C, halfway up a piano keyboard.
Wavelength: 131.87cm; Frequency: 261.63Hz.
Make the wavelength a third shorter and we get G(88.01cm), which has a frequency 50% higher (392.00Hz). This is a Fifth Interval up from C.

Or we could go a Fourth Interval up from C, cutting only a quarter off the original wavelength, and we get F(98.79cm) which has a frequency one-third higher(349.23Hz).

These both confirm the inverse-fraction WL → FR relationships stated above.

So, this finally brings me to my question: why do these relationships exist? Is there a basic physical reason why taking a third off the wavelength gives a 50% increase in frequency, while taking a quarter off a wavelength gives a one-third increase in frequency?
It seems arbitrary until you express it as:
WL → ³⁄₂FR (Fifth Interval)
¾WL → ⁴⁄₃FR (Fourth Interval)
not to mention
½WL → ²⁄₁FR (Octave Interval).
Expressed like this it shows this seemingly-divine symmetry of inverse-fractions in each relationship.
Then add to it the fact that all three relationships are directly related: A Fifth + a Fourth = an Octave.
To me now it seems like there must be some underlying formula that justifies these relationships. Something basic at the level of math and/or physics.

But my father, an ex Math teacher, seems to think it is just a set of coincidentally pleasing relationships which were observed by our ancestors and seized upon for their seeming perfection. Is he right? I mean, I know that the concepts of Intervals, and even Notes, are human constructs. But the fundamental relationship between a wavelength and its frequency?

I'm struggling to see how these relationships could co-exist without some underlying reason.

*Now, you'll have to bear in mind you're delivering answers to someone with not-far-above junior-high school level physics understanding :/ sorry guys..

One possibility in my head is that the answers can be found in studying Waveforms. Now I did study calculus in Senior Highschool, and back then I could work with Sin, Cos and Tan. These days I really can't remember any of that. If they form part of your answer, I may be asking for some serious reschooling for clarification..!

Ok thanks. Peace
Josh

quiet thought
The Frequency of a sound is how many times a wave occurs in a second. A plucked string vibrates back-and-forth at some rate. The count of how many times it vibrates in one second is its Frequency in units of 'Cycles Per Second (CPS)', however several years ago CPS was renamed 'Hertz (Hz)' to honor Heinrich Hertz.

Wavelength of a sound is a measure of how far apart the peaks (or more usually zeros) are when the sound travels in a medium. Of course this depends on the medium. sound travel slower in air (343m/s) than it does in water (1500m/s).

For instance F4, piano key 45, has a frequency of 349+ Hz. In air the wavelength would be 98cm. In water the wavelength would be 4.3m.

If you stay in one medium, such as air, the wavelengths of two different frequencies are related to each other as the reciprocal of their frequencies. i.e. if two frequencies are related by a ratio of 2 to 5 their wavelengths are in the ratio of 1/2 to 1/5.

A Google search that may supply more than you care to read is:
Have Fun!

Cheers,
Tom

Hey thanks for the reply! Good to know that extra info about the medium through which a wave passes.
And yeah, my history of scales may be inaccurate, but it's a good story that illustrates the result..
Anyway I'm having a bit of trouble relating your answer to my question. Can you please show me how that explains the inverse fractions I observed?
Thanks
Josh

KiwiJosh said:
Hey thanks for the reply! Good to know that extra info about the medium through which a wave passes.
And yeah, my history of scales may be inaccurate, but it's a good story that illustrates the result..
Anyway I'm having a bit of trouble relating your answer to my question. Can you please show me how that explains the inverse fractions I observed?
Thanks
Josh

I'm having a bit of trouble finding your question.

If you are asking about frequencies and wavelengths, then:

1) The speed of sound in air is a constant - for a fixed temperature and pressure. As it is in any homogeneous medium.

2) The speed of a wave is the product of its wavelength time its frequency.

3) Wavelength is, therefore, inversely proportional to frequency.

Note that the frequency is more fundamental to music, as the frequency of a vibrating instrument is defined by the instrument and our ears will use frequency to hear.

If, for example, you varied the air temperature and pressure, so that the speed of sound changed, then musical notes would still have the same frequency and sound the same. We wouldn't notice the change in speed and wavelength.

vanhees71 and KiwiJosh
PeroK said:
I'm having a bit of trouble finding your question.

If you are asking about frequencies and wavelengths, then:

1) The speed of sound in air is a constant - for a fixed temperature and pressure. As it is in any homogeneous medium.

2) The speed of a wave is the product of its wavelength time its frequency.

3) Wavelength is, therefore, inversely proportional to frequency.

Note that the frequency is more fundamental to music, as the frequency of a vibrating instrument is defined by the instrument and our ears will use frequency to hear.

If, for example, you varied the air temperature and pressure, so that the speed of sound changed, then musical notes would still have the same frequency and sound the same. We wouldn't notice the change in speed and wavelength.
Hey, thanks again, that looks basic enough for my brain to grasp, I'm just going to run some numbers through that equation until I can see it working and then hopefully I will wrap my head around it :)
I'll let you know if I'm still confused but looks good! Cheers!

KiwiJosh said:
Can you please show me how that explains the inverse fractions I observed?

If you mean this...
KiwiJosh said:
Is there a basic physical reason why taking a third off the wavelength gives a 50% increase in frequency, while taking a quarter off a wavelength gives a one-third increase in frequency?
Yeah, it's high school (secondary school) mathematics.

The best I can do is suggest you try playing with some numbers on a calculator.
1/1 = 1
1/0.75 = 1.33 . . . that's 1/4 off the denominator
1/0.67 = 1.5 . . . . that's 1/3 off the denominator
1/0.5 = 2.0 . . . . .that's 1/2 off the denominator
After doing some more of them maybe try graphing them.

Something to remember, the reciprocal of a fraction is found by exchanging the numerator and the denominator. For instance the reciprocal of 5/16 is 16/5.

On the Google page, click on 'Images" in the upper right of the screen.
Then search for reciprocal.
Also try a normal text search for reciprocal without clicking on Images.

Cheers,
Tom

KiwiJosh
Tom.G said:
If you mean this...

Yeah, it's high school (secondary school) mathematics.

The best I can do is suggest you try playing with some numbers on a calculator.
1/1 = 1
1/0.75 = 1.33 . . . that's 1/4 off the denominator
1/0.67 = 1.5 . . . . that's 1/3 off the denominator
1/0.5 = 2.0 . . . . .that's 1/2 off the denominator
After doing some more of them maybe try graphing them.

Something to remember, the reciprocal of a fraction is found by exchanging the numerator and the denominator. For instance the reciprocal of 5/16 is 16/5.

On the Google page, click on 'Images" in the upper right of the screen.
Then search for reciprocal.
Also try a normal text search for reciprocal without clicking on Images.

Cheers,
Tom
Roger that, yes until now I didn't have the other side of the equation - the Speed part. Now with Speed = WL X FR, I can actually run some numbers through that to make sense of it as you say..

@KiwiJosh I'm afraid your OP is too long and rambling to maintain my attention throughout. It is always a good idea to start off with no more than a paragraph or two with the main idea. That can maintain people's attention. (You could put this down to imminent dementia on my part!)

But there is one thing which that you say which is nothing to do with Music and that is the relationship between frequency and wavelength in a medium. That inverse relationship λ=c/f is basic Physics when the wave speed is constant with frequency (no dispersion). See this link and a dozen others.

Musical notes and the musical scale are a totally different thing. The (Western) scales that you get from overtones on (ideal) strings are simple ratios. The equal tempered scale (where each semitone has the ratio 12√2 to the next) doesn't fit the 'natural' scale of high order overtones. Musicians fudge the notes they play on many musical instruments in order to make them sound 'right'. Otoh, strings on a piano are individually tuned and the tuner (the person) uses skill to fit them to a well tempered scale (including off-tuning the three strings on most of the notes) The basic 12√2 rule is actually only a 'guide'.

PS Afaiaa, music was initially produced with the voice ( long before strings of constant thickness were available) , followed by the sounds produced by blowing on and through containers of various volumes and lengths. The modes of air vibration in a long tube are far from accurate harmonics (even Hollywood got that right with the ghastly sounding Roman Horns that you get in every film of that period).

KiwiJosh
sophiecentaur said:
@KiwiJosh I'm afraid your OP is too long and rambling to maintain my attention throughout. It is always a good idea to start off with no more than a paragraph or two with the main idea. That can maintain people's attention. (You could put this down to imminent dementia on my part!)

But there is one thing which that you say which is nothing to do with Music and that is the relationship between frequency and wavelength in a medium. That inverse relationship λ=c/f is basic Physics when the wave speed is constant with frequency (no dispersion). See this link and a dozen others.

Musical notes and the musical scale are a totally different thing. The (Western) scales that you get from overtones on (ideal) strings are simple ratios. The equal tempered scale (where each semitone has the ratio 12√2 to the next) doesn't fit the 'natural' scale of high order overtones. Musicians fudge the notes they play on many musical instruments in order to make them sound 'right'. Otoh, strings on a piano are individually tuned and the tuner (the person) uses skill to fit them to a well tempered scale (including off-tuning the three strings on most of the notes) The basic 12√2 rule is actually only a 'guide'.

PS Afaiaa, music was initially produced with the voice ( long before strings of constant thickness were available) , followed by the sounds produced by blowing on and through containers of various volumes and lengths. The modes of air vibration in a long tube are far from accurate harmonics (even Hollywood got that right with the ghastly sounding Roman Horns that you get in every film of that period).
Ah thank you for this, yes you're right about the length of the post it certainly got out of hand... I wasn't quite sure how much info I needed to supply until I saw everyone's answers and realized how damn simple the problem was! :D
Thanks again

sophiecentaur and berkeman
KiwiJosh said:
Is there a basic physical reason why taking a third off the wavelength gives a 50% increase in frequency, while taking a quarter off a wavelength gives a one-third increase in frequency?
This is all about Arithmetic and not Physics. It applies wherever AB=C. Multiplication and Division follow easy rules, once you know them but expressing them in terms of Addition and Subtraction gets very 'lumpy' and it's far from intuitive.
There can be confusion when you try to work out problems like "If I pay £12 for an item and its price was reduced by 50%, what was the original price? (£20 or £18)" etc. etc.
The modern habit of talking about "ten times less" instead of "one tenth" really doesn't help; particularly now kids are taught about numbers with Number Lines. Why doesn't "ten times less" means minus nine times?

sophiecentaur said:
This is all about Arithmetic and not Physics. It applies wherever AB=C. Multiplication and Division follow easy rules, once you know them but expressing them in terms of Addition and Subtraction gets very 'lumpy' and it's far from intuitive.
There can be confusion when you try to work out problems like "If I pay £12 for an item and its price was reduced by 50%, what was the original price? (£20 or £18)" etc. etc.
The modern habit of talking about "ten times less" instead of "one tenth" really doesn't help; particularly now kids are taught about numbers with Number Lines. Why doesn't "ten times less" means minus nine times?
Yeah, I had to do some hand drawings of amounts today to prove to myself that 3/4 X 4/3 = 1 :D but now it all makes sense

Tom.G

## 1. What is the wavelength to frequency relationship in musical notes?

The wavelength to frequency relationship in musical notes is a mathematical concept that explains the relationship between the physical properties of sound waves and the perceived pitch of musical notes. In simple terms, the shorter the wavelength of a sound wave, the higher its frequency and the higher the perceived pitch of the musical note.

## 2. How is the wavelength to frequency relationship calculated?

The wavelength to frequency relationship is calculated using the formula: frequency = speed of sound / wavelength. The speed of sound is a constant value, so as the wavelength decreases, the frequency increases and vice versa.

## 3. Why is the wavelength to frequency relationship important in music?

The wavelength to frequency relationship is important in music because it helps musicians understand how different notes are related to each other and how they can create harmonious sounds. It also helps in the design and construction of musical instruments, as the length and thickness of strings or tubes are based on this relationship to produce specific musical notes.

## 4. How does the wavelength to frequency relationship affect the quality of sound?

The wavelength to frequency relationship affects the quality of sound by determining the pitch of a musical note. As the frequency increases, the pitch of the note becomes higher, and as the frequency decreases, the pitch becomes lower. This relationship is what allows us to distinguish between different musical notes and create melodies and harmonies.

## 5. Can the wavelength to frequency relationship be applied to all musical instruments?

Yes, the wavelength to frequency relationship can be applied to all musical instruments, as it is a fundamental principle of sound waves. However, the exact relationship may vary slightly depending on the specific instrument and its design. For example, a guitar string may have a different wavelength to frequency relationship than a flute or a piano key.

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