Cents vs Hertz

[s1m0n]

What exactly is a cent, in pitch? Is this an absolute value (ie, X Hz), or a relative one (ie, a percentage of the distance to the next note?

[Doug_Tipple]

I know that there are a number of music majors here on the forum, and I never made it past beginning piano. However, my answer to your question is that cents is an absolute measurement in the equal temperment system, which is the standard in Western music. The distance between each half tone is exactly the same, and each half tone interval is divided into 100 cents. Accordingly, there are 200 cents between each whole tone interval.

[MTGuru]

It's relative, Simon.

Since an octave represents a doubling of pitch, and equal temperment divides the octave into a scale of 12 semitones, the relationship between a given note and the next semitone is a multiplicative factor of the twelfth root of 2, 2^(1/12) = 1.059 (approximately). Where ^ represents exponentiation.

So given a note with frequency f in Hz, you can calculate the frequency f' of any relative semitone n by the forumula f' = f * 2^(n/12). Or conversely, n = 12*(log(f'/f)/log(2)).

Cents divide each semitone into 100 increments, and thus the octave into 1200 increments. So one cent = 2^(1/1200) = 1.000578 (approximately).

Given a note with frequency f in Hz, you can calculate the frequency f' of any relative pitch n in cents by the formula f' = f * 2^(n/1200). Or conversely, n = 1200*(log(f'/f)/log(2)).

Intuitively, since our perception of pitch is logarithmic, a cent is 1/100 the distance to the next semitone. A relative difference of 50 cents is always a quarter tone (half a semitone), 25 cents a microtonal eighth, etc. But the absolute frequency differences will depend on the pitch.

Hope that helps!

[s1m0n]

Hope that helps!

Yes, it did.

The math is more than I want tot tackle, however. Does anyone know a site with the values in hertz for, say, every key on a standard keyboard?

~~

On a related note*, is the 1st octave A on a D whistle 220 hertz?

*they're all related notes.

[MTGuru]

The math is more than I want tot tackle

Sure, sorry about that.

Does anyone know a site with the values in hertz for, say, every key on a standard keyboard?

This one is not bad, and you can save the keyboard graphic for reference:
http://www.vibrationdata.com/piano.htm

On a related note*, is the 1st octave A on a D whistle 220 hertz?

No, on a standard high D whistle it's A=880 Hz (A5 on the piano). On a low D whistle it's A=440 Hz (A4).

[djm]

2^

What is the caret supposed to represent in your formulas?

djm

[talasiga]

Accepting the convention that sets current concert pitch of A at 440Hz, why is it that, in the artifice that is equal tempered scale,
c''''' = c'''' X 2
c'''' = c''' X 2
c''' = c'' X 2
BUT no such pattern obtains for octaves of C below c'' ?
(namely c'' , c', c, C, C' )

[fel bautista]

exponent- can't use superscripts

[Wanderer]

2^

What is the caret supposed to represent in your formulas?

djm

Caret represents exponentials in most programming languages.

for instance: 2^3 = 2 to the third power=8

[Doug_Tipple]

I usually preface my comments with "please correct me if I am wrong", but I forgot to do so in my previous post. MTGuru has clearly shown that cents is relative and not absolute, as I suggested. Since there is a doubling of the frequency (hertz) with every octave, the progression is logarithmic and not linear. Also, because every octave is divided into 1200 cents, as the octave frequency ranges become larger with ascending pitch, the cent values must also increase accordingly. Again, please correct me if I am wrong.

[coupedefleur]

cents is relative and not absolute

Well, it depends on how you're looking at it. If you're talking about the size of intervals, or the placement of notes in relation to each other, a figure in cents stays the same regardless of octave.

A pure 5th is always 702 cents. It doesn't matter if you're talking about C to G, or F# to G#. A pure major 3rd is always 386 cents. And it doesn't matter if you're using a=440 hz, a=450 hz, or a=392 hz as a reference pitch.

[djm]

Is that Equal Tuning or Just tuning?

djm

[rhulsey]

Accepting the convention that sets current concert pitch of A at 440Hz, why is it that, in the artifice that is equal tempered scale,
c''''' = c'''' X 2
c'''' = c''' X 2
c''' = c'' X 2
BUT no such pattern obtains for octaves of C below c'' ?
(namely c'' , c', c, C, C' )

Organbuilders use notation for pipework starting with CCCC for 32' C, CCC for 16' C (the lowest C on the piano with a few exceptions), etc. There's a pretty good chart here: http://www.dolmetsch.com/defso.htm
Reg

[Doug_Tipple]

cents is relative and not absolute

Well, it depends on how you're looking at it. If you're talking about the size of intervals, or the placement of notes in relation to each other, a figure in cents stays the same regardless of octave.

A pure 5th is always 702 cents. It doesn't matter if you're talking about C to G, or F# to G#. A pure major 3rd is always 386 cents. And it doesn't matter if you're using a=440 hz, a=450 hz, or a=392 hz as a reference pitch.

The original query was about cents vs. hertz. If I understand it correctly, while a pure fifth may always be 702 cents, the frequency or hertz interval from cent to cent will change as the pitch increases. For example, in a lower register there may be a 440 hertz range in one octave, whereas in the second register up from that there is a 1760 hertz range in the octave. If both registers are divided into 1200 cents, the hertz interval per cent must be larger in the higher register.

[Denny]

so...if ya had enough cents this wouldn't hertz?