# Why do notes repeat an octave above/below?



## sammyooba

I want to know the acoustics reasoning for this.

Is it because

A) Waves repeat after a certain period

B) A spiral completes a revolution in 360 degrees as shown in this diagram: 
http://www.schillerinstitute.org/graphics/fidelio/foundations_sci_tuning/42pct/f8_cone_intervals.jpg

or C) Both, theta and sin waves are interrelated; and thus, they are both responsible for a note repeating after an octave in an interrelated way.


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## Kopachris

Examples? Are you talking about the series of overtones?


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## Philip

Kopachris said:


> Examples? Are you talking about the series of overtones?


I think sammyooba wants to know why it's

_do ré mi fa sol la si do_ 
and not 
_do ré mi fa sol la si po dé ri ma fol sa li_ etc.


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## sammyooba

Yes what Philip said.


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## Polednice

i.e. why are 220hz, 440 hz, and 880 hz all 'A's?

P.S. No idea.


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## Jeremy Marchant

Try the Wikipedia article "Octave"


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## Philip

It's matter of psychoacoustics, which can be explained by math...

In the end, it all comes down to: don't you think these notes sound the same, but at a higher/lower pitch? Hence the same name.


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## Rasa

There's a more or less physical relation: each octave above is double the frequency. (is supposed to be)


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## sammyooba

I get that there is a harmonic reasoning for this, but what I really want to know is more of an accurate mathematical way to visually graph it.

From psychoacoustics on wiki: "The semitone scale used in Western musical notation is not a linear frequency scale but logarithmic."

And it reminds me of this link below:

http://www.schillerinstitute.org/graphics/fidelio/foundations_sci_tuning/42pct/f8_cone_intervals.jpg

It shows the equal tempered scale graphed onto some sort of spiral that I'm thinking is logarithmic. The author claims that a 360 degree revolution of the spiral is equivalent to that of an octave.

I guess my real question is if visually representing the mini tone scale by some logarithmic spiral or by some sin wave would accurately depict an octave because they both have the characteristics of repeating patterns, and this pattern perhaps being equivalent to one octave?


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## Philip

sammyooba said:


> I get that there is a harmonic reasoning for this, but what I really want to know is more of an accurate mathematical way to visually graph it.
> 
> From psychoacoustics on wiki: "The semitone scale used in Western musical notation is not a linear frequency scale but logarithmic."
> 
> And it reminds me of this link below:
> 
> http://www.schillerinstitute.org/graphics/fidelio/foundations_sci_tuning/42pct/f8_cone_intervals.jpg
> 
> It shows the equal tempered scale graphed onto some sort of spiral that I'm thinking is logarithmic. The author claims that a 360 degree revolution of the spiral is equivalent to that of an octave.
> 
> I guess my real question is if visually representing the mini tone scale by some logarithmic spiral or by some sin wave would accurately depict an octave because they both have the characteristics of repeating patterns, and this pattern perhaps being equivalent to one octave?


The spiral is a 'good' way of geometrically representing the intervals because it a generalization of the circle of fifths, which is widely used to teach various principles such as temperament and harmonic relations.

Personally i think it's rather complicated, being a 3D graph, at least for the purpose of illustrating frequency with respect to the interval. You can use whatever method is convenient. The classical method is the vibrating string, as it was used by Pythagoras, Zarlino, and so on.

Your sine wave example is not very practical, because _all_ pure tones are sinusoidal waves, which also implies a "repeating pattern". Thus it wouldn't reveal as adequately as other methods, at least not visually, the true essence of the intervals. You must consider that all sounds are a priori made up of sine waves (see Fourier), then find out what makes an octave an octave versus a fifth or a fourth.

Consequently, from the vibrating string example, it is easy to visually determine what are the characteristic ratios of each interval. The octave is 2:1, the perfect fifth 3:2 (1.5:1), the fourth 4:3 (1.33:1), etc.


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## samurai

Say what?


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## science

It's fairly simple. 

Tones have wavelengths; different tones have different wavelengths. 

For some neurological reason that I think we don't understand yet, there are fairly simple patterns to the way we hear tones in relation to each other. 

If you double the wavelength (~halve the frequency) of a tone, it sounds to us like the same note, just lower. If you halve the wavelength (~double the frequency), it sounds like the same note, just higher. 

The next question is why we break the intermediate notes into 8ths (so that we have C, D, E, F, G, A, B, and C rather than more or less notes in between the octaves). That seems to be basically arbitrary, varying from culture to culture.

Incidentally, as I am not an expert, if any part of this is not exactly right, feel free to correct me! I'd love it!


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## chee_zee

I always assumed it's because our brains best chunk things into 7. I know in indian music it's all 7 note scales, even the 6 and 5 note scales they use are taught by simply saying to remove a note or two from a known scale. the quarter notes aren't an integral part, they are approached by one of the 7 notes as vibrato/ornamentation. I think in certain mid-east music that uses 'dastgahs/modes' the performer has the option of changing up the modes by varying the placement of any of the notes by a quarter tone.


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## sammyooba

How does our brains recognize that a scale has 7 notes? There are so many scales. Is a scale (or an ideal natural scale) already out there programmed in nature that is saying to our brain, "look at me, I have 7 notes!" or are scales man-made? 

What is most interesting is that it is exactly at multiples of 2 in which we hear notes repeating. Mathematics is always nasty numbers like 3.14 and 1.618. Whole numbers rarely have their spotlight in anything except for octaves


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## itywltmt

We have to keep in mind that "octave" (as the name suggests" is a sequence of "eight".

Furthermore, it is widely accepted in electronics that an octave implies "doubling a frequency".

From wave mechanics, the frequency of a wave is how many times a second a wave goes through a full "cycle" - think of a sine wave or a pendulum swinging back and forth. An octiave means that we double the frequency of the previous note (from 220 to 440 Hz.).


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## Amfibius

If you want to understand why it is so, read The Music of the Spheres. It explains how music and sound is related to mathematics.


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## Scarpia

As has been mentioned above, an octave is a note whose frequency (cycles per second) is a factor of 2 greater (or a factor of 1/2). There are physical reasons why your ear and brain interpret an octave as closely related to the original note. When a tuned instrument produces a note, it naturally produces overtones that are factors of 2, 3, 4, 5, 6, etc, above. A note which is a factor of 2 higher in frequency matches the first overtone of the fundamental note, and is recognized by the human auditory system as closely related. Other intervals, such as the perfect fifth and perfect fourth also have simple relationships between the pitches (2:3 for the fifth, 3:4 for the fourth).

As to why the major/minor scales are the way they are, that is more complicated. There is more than one way to define a scale, but the scale of western music works well because it contains a lot of combinations of pitches which have simple relationships and sound well together.


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