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#1
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from Cents to Hertz and back again
I think I could figure this out if I banged my head on the wall for
long enough. Probably easier to ask though... Can anyone tell me how to covert a difference in hertz (say between 440 and 440.5) to cents? And in the other direction? |
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#2
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from Cents to Hertz and back again
apa wrote:
I think I could figure this out if I banged my head on the wall for long enough. Probably easier to ask though... Can anyone tell me how to covert a difference in hertz (say between 440 and 440.5) to cents? And in the other direction? It's not trivial. The octave intervals are easy... if middle A is 440, then the next A up on the keyboard is 880 Hz and the one below it is 220 Hz. The problem is that the note intervals are not even factors, and how close they are depends on the particular tuning of the instrument. You can find a nifty formula for even temperament at: http://www.hauptwerk.co.uk/CreatingO...50-Tuning.html however, this may not relate to the actual tuning of any given instrument. --scott -- "C'est un Nagra. C'est suisse, et tres, tres precis." |
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#3
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from Cents to Hertz and back again
There is only one way to answer this question: Thoroughly
 Given a frequency of any note, the frequency of the next (half)note equals the current one times 2^(1/12) Hz. There are 12 half-notes in an octave, at every octave the note frequency doubles. The frequency of a normal A is defined as 440 Hz. So A sharp=440* (2^(1/12)) =440* 1.05946309 =466.163762 Hz. The same goes for cents, except in an octave there arent 12 cents but 1200, so the multiplier= 2^(1/1200)=1.00057779 to get from a note to a note 1 cent higher (you'll need to calculate this repeatedly to get N cents higher). This is considering our current tonal system (if I'm not mistaken it's called the "chromatic scale". Each half note has the same mathematical distance to the next). In Pythagoras' time, note frequencies were calculated differently, based on fractions (of string lengths, to be more precise) Frequency of next octave(A4-A5) =base frequency*2/1 e.g.440*2/1=880 Hz Frequency a fifth higher (A4-E5) =base frequency*3/2 e.g. 440*3/2=660 Hz Note that in chromatic tuning this is slightly different: (a fifth is 7 half-notes): 440*((2^(1/12))^7)=659.255114. Fractional tuning seems more 'correct' here. Problem with the fractional system is that several answers are possible for the frequency of any given note. If we keep going a fifth higher over and over again until we're back at the original note ("Circle of fifths"), we get slightly different results depending on the note that we depart from. This is why classical music pieces always mentioned the tonality; they would simply have sounded out of tune in any other tonality. Needless to say, chromatic tuning differs from fractional tuning, which is why for example traditional music from the Middle East to our ears sounds a bit "out of tune" to what we're used to. It also explains why a guitar can never be tuned perfectly; the frets are spaced based on chromatic tuning, whereas the nature of strings dictates the other. As a guitarist this of course deeply frustrates me Hope this clears up things! |
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#4
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from Cents to Hertz and back again
apa wrote:
I think I could figure this out if I banged my head on the wall for long enough. Probably easier to ask though... Can anyone tell me how to covert a difference in hertz (say between 440 and 440.5) to cents? And in the other direction? Here is everything you need: http://www.sengpielaudio.com/calculator-centsratio.htm Cheers Jens |
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#5
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from Cents to Hertz and back again
I forgot to mention. The "Circle of fifths" by the way also indicates
why in western music we use 12 half-notes: after 12 steps, you're back at the note where you began (albeit several octaves higher). |
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#6
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from Cents to Hertz and back again
Exactly I needed, thanks!
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#7
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from Cents to Hertz and back again
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#8
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from Cents to Hertz and back again
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#9
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from Cents to Hertz and back again
On Mon, 24 Oct 2005 20:05:38 +0200, "Jens Rodrigo"
wrote: Here is everything you need: http://www.sengpielaudio.com/calculator-centsratio.htm Cheers Jens Really cool and really useful! I do have a gripe with them that they incorrectly identify a perfect 5th as 1.498307 : 1 and perfect fourth as 1.334840 : 1. Perfect intervals are fractions. A perfect 5th is 3/2 = 1.5000 : 1 and a perfect fourth is 4/3 = 1.3333... : 1. Julian |
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#10
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from Cents to Hertz and back again
Julian wrote: I do have a gripe with them that they incorrectly identify a perfect 5th as 1.498307 : 1 and perfect fourth as 1.334840 : Well, nobody's perfect. I think the term "perfect fifth" is like "48V phantom power" in another discussion. Everyone assumes that if it's a fifth, it's perfect. The ratios in that calculator are for an equal tempered scale, which, as everone knows, does not have "perfect" fifths or thirds. This is why your guitar's B string is always out of tune unless you're playing in the key of B (in which case every other string it out of tune). Heck, in the REAL MUSICIAN'S world, today a fifth isn't a fifth any more, it's 750 mL. |
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#11
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from Cents to Hertz and back again
kleinebre wrote ...
This is considering our current tonal system (if I'm not mistaken it's called the "chromatic scale". Each half note has the same mathematical distance to the next). And that is a whole 'nuther can of worms. There are far more tuning systems, schemes, modes, etc. than we know of. I think there are well over a dozen just for pianos. |
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#12
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from Cents to Hertz and back again
Julian wrote:
On Mon, 24 Oct 2005 20:05:38 +0200, "Jens Rodrigo" wrote: Here is everything you need: http://www.sengpielaudio.com/calculator-centsratio.htm Really cool and really useful! I do have a gripe with them that they incorrectly identify a perfect 5th as 1.498307 : 1 and perfect fourth as 1.334840 : 1. Perfect intervals are fractions. A perfect 5th is 3/2 = 1.5000 : 1 and a perfect fourth is 4/3 = 1.3333... : 1. A perfect 5th is not only 1.5 : 1 as you think, 1.498307 : 1 is also correct. See: http://www.fact-index.com/p/pe/perfect_fifth.html http://www.torvund.net/guitar/Theory..._intervals.asp http://www.torvund.net/guitar/intervals/int-eb.asp Diatonic intervals: unison | minor second | major second | minor third | major third | perfect fourth | tritone | perfect fifth | minor sixth | major sixth | minor seventh | major seventh | octave http://en.wikipedia.org/wiki/Interval_(music) Cheers Jens |
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#13
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from Cents to Hertz and back again
On Tue, 25 Oct 2005 00:39:26 +0200, "Jens Rodrigo"
wrote: A perfect 5th is not only 1.5 : 1 as you think, 1.498307 : 1 is also correct. See: http://www.fact-index.com/p/pe/perfect_fifth.html Yeah I read what but I don't agree. I can also show you links where 1.5 is defined as a perfect fifth. Perfect by definition means not tempered. Calling a tempered fifth "perfect" is really a misnomer even it is common in popular usage. The reason why even temperament works at all is that 1.498 is so damn close to 1.500 that few can hear the difference and that the fourth is also extremely close to a "real" perfect 4th. Thirds aren't as close but still useable. 1.259921 is a tempered third. 1.25000 (5/4) is a perfect third. Let's see you come up with a link somewhere that claims a perfect third is not 5/4! Julian |
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#14
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from Cents to Hertz and back again
On 24 Oct 2005 15:27:11 -0700, "Mike Rivers"
wrote: The ratios in that calculator are for an equal tempered scale, which, as everone knows, does not have "perfect" fifths or thirds. This is why your guitar's B string is always out of tune unless you're playing in the key of B (in which case every other string it out of tune). I have been known to tune the B string differently for different keys because of that. Heck, in the REAL MUSICIAN'S world, today a fifth isn't a fifth any more, it's 750 mL. Can't argue with that! Julian |
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#15
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from Cents to Hertz and back again
"Julian" wrote ...
I do have a gripe with them that they incorrectly identify a perfect 5th as 1.498307 : 1 and perfect fourth as 1.334840 : 1. Perfect intervals are fractions. A perfect 5th is 3/2 = 1.5000 : 1 and a perfect fourth is 4/3 = 1.3333... : 1. Depends on which tuning you are using. There is no single answer. |
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#16
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from Cents to Hertz and back again
On Mon, 24 Oct 2005 17:27:02 -0700, "Richard Crowley"
wrote: Depends on which tuning you are using. There is no single answer. No, perfect intervals are by definition intervals that are based on pure harmonics. Perfect intervals are possible in many tuning systems. They are not possible in tempered tunings. Julian |
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#17
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from Cents to Hertz and back again
"Julian" wrote:
On Tue, 25 Oct 2005 00:39:26 +0200, "Jens Rodrigo" wrote: A perfect 5th is not only 1.5 : 1 as you think, 1.498307 : 1 is also correct. See: http://www.fact-index.com/p/pe/perfect_fifth.html Yeah I read what but I don't agree. I can also show you links where 1.5 is defined as a perfect fifth. Perfect by definition means not tempered. Calling a tempered fifth "perfect" is really a misnomer even it is common in popular usage. The reason why even temperament works at all is that 1.498 is so damn close to 1.500 that few can hear the difference and that the fourth is also extremely close to a "real" perfect 4th. Thirds aren't as close but still useable. 1.259921 is a tempered third. 1.25000 (5/4) is a perfect third. Let's see you come up with a link somewhere that claims a perfect third is not 5/4! Unison, fourth, fifth, octave. These intervals may be perfect, augmented, or diminished. A perfect fourth is five semitones, a perfect fifth is seven semitones, a perfect octave is twelve semitones. A perfect unison occurs between notes of the same pitch, so it is zero semitones. In each case, an augmented interval contains one more semitone, a diminished interval one fewer. Cheers Jens |
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#18
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from Cents to Hertz and back again
"Scott Dorsey" Can anyone tell me how to covert a difference in hertz (say between 440 and 440.5) to cents? And in the other direction? It's not trivial. The octave intervals are easy... if middle A is 440, then the next A up on the keyboard is 880 Hz and the one below it is 220 Hz. The problem is that the note intervals are not even factors, and how close they are depends on the particular tuning of the instrument. You can find a nifty formula for even temperament at: http://www.hauptwerk.co.uk/CreatingO...50-Tuning.html however, this may not relate to the actual tuning of any given instrument. ** What insane crap. The "equal tempered" scale has been built into every fretted and keyboard instrument for centuries. Every semitone is the twelfth root of 2 times the note below it. That is: 1.0594631 times. A cent is 1/00 of a semitone. ......... Phil |
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#19
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from Cents to Hertz and back again
Julian wrote:
Yeah I read what but I don't agree. I can also show you links where 1.5 is defined as a perfect fifth. Perfect by definition means not tempered. Calling a tempered fifth "perfect" is really a misnomer even it is common in popular usage. It's more than a popular misnomer. It's absolutely standard use by musicians, who use the term "perfect" to distinguish it from a diminished fifth. In that context, the fine distinctions between tuning systems are irrelevant, we're only concerned about issues like "is that note in bar 17 supposed to be an F natural or an F sharp?" In other contexts, I'm sure your usage (I'd hestiate to call it a definition) of "perfect" is valid, but it's not the only one. Anahata |
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#20
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from Cents to Hertz and back again
Jens Rodrigo wrote: "Julian" wrote: Unison, fourth, fifth, octave. These intervals may be perfect, augmented, or diminished. Augmented and diminished are musical terms meaning sharped or flatted a half-step. Do they apply to temperment as well? I don't think so, but the language of music might have become as corrupt as the language of audio engineering by people who come into the field without education about the fundamentals. |
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#21
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from Cents to Hertz and back again
BTW, the cents-versus-Hertz discussion has occurred multiple times in
rec.music.makers.squeezebox, since accordion tunings have to consider the beat frequencies between similarly-tuned sets of reeds. You might want to search there. (Yeah, I know. Spare me.) |
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#22
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from Cents to Hertz and back again
On Tue, 25 Oct 2005 11:29:04 +0100, Anahata
wrote: Julian wrote: Yeah I read what but I don't agree. I can also show you links where 1.5 is defined as a perfect fifth. Perfect by definition means not tempered. Calling a tempered fifth "perfect" is really a misnomer even it is common in popular usage. It's more than a popular misnomer. It's absolutely standard use by musicians, who use the term "perfect" to distinguish it from a diminished fifth. In that context, the fine distinctions between tuning systems are irrelevant, we're only concerned about issues like "is that note in bar 17 supposed to be an F natural or an F sharp?" I've never heard this usage of the term. Being a jazz musician I am well familiar with flatted fifths, shaped fifths and natural fifths, but we never never call them perfect fifths. Who uses that? It is common to call the intervals that are made of harmonics "pure" intervals or "perfect" intervals. I think this is the original usage of the term and the usage you refer to is a corruption. I think Mike Rivers was correct one or two messages ago when he said "Augmented and diminished are musical terms meaning sharped or flatted a half-step. Do they apply to temperament as well? I don't think so, but the language of music might have become as corrupt as the language of audio engineering by people who come into the field without education about the fundamentals." In other contexts, I'm sure your usage (I'd hestiate to call it a definition) of "perfect" is valid, but it's not the only one. I'd hesitate to call your usage a definition too. If there are historically many ways to temper a scale, how can any one temperament create THE "perfect" fifth? I still maintain the perfect fifth is based on the interval 3/2 which is the non-tempered fifth. If the even tempered fifth is called perfect now a days it is only because 1.498307 is so very very close to 3/2 it hardly matters. How can you possibly claim 1.488397 is the perfect fifth but 1.500000 is NOT the perfect fifth? I'll bet for every link you find that says 1.498 is the perfect fifth I can find 2 more that says it isn't! :-) The thing musicians who only know modern western music and are not familiar with early western temperaments of temperaments in the rest of the world don't understand is that the vast majority of scales in all world music are originally based on the same 7 notes. This is true even of Balinese music where some of the notes are so very strange there is nothing at all like them in our culture. When you talk to Balinese musicians (and I have been to Bali and done so) about how they tune their instruments they will tell you they start with the standard scale and then de-tune a certain amount untill teh amount of detuned beats reaches the desired frequency. But they DO start with the 7 note major scale. Classical Indian Music is definitely based on the natural scale found in early western music. After studying sitar for 20 some odd years I am sure of it. Sa Re Ga Ma Pa Dha Ni Sa is exactly the same in tonality as Do Re Mi Fa Sol La Ti Do. From there they take off and create a very complicated system of flatting and sharping these notes using microtones. But they do originally start with the same 7 notes that early western music also started with. This most simple 7 note scale harmonically was discovered by various musical cultures around the world. It was not invented. It was discovered. It is harmonically the simplest way a single note and it's simplest harmonics combine to produce a scale. The octave and the fifth are the simplest harmonic intervals. The fourth is merely the fifth inverted. So the Tonic fourth and fifth are the most basic intervals harmonically. Starting from those 3 notes alone the other 4 notes of the 7 note major scale are generated by using the perfect harmonic fifth and perfect harmonic third of those notes as well. It is joked how folk singers only know 3 chords - C F G. That's because all the notes of a major scale are in one of those chords. If you know those 3 chords you can use them and sound OK for most songs written in a major key. This simplest of all tuning systems is called just intonation. http://www.justintonation.net/whatisji.html They teach in western music there are 7 modes - Ionian, Dorian, Phrygian, Locrian, Lydian, Aeolian, Mixolydian. These scales are formed by using the same 7 notes harmonically derived as I just described. If you start on an instrument tuned to a 7note scale starting on what we call C today, you get the natural scale - Ionian. Then if you start on D you get Dorian, E - Phrygian etc. However we already created a problem with tuning. The ear wants to hear the fifth and tonic as the same interval (perfect) regardless of key or mode which is why there is all the hub bub about perfect fifths in the first place. If you leave all the strings of your lyre (back in the day) tuned to the just intoned C scale and now start your scale on D you get the Dorian mode, a beautiful minor mode that much British Isles Folk music is written in. However we have already created the first IMPERFECT fifth. If you do the math you'll see that the note A which was originally derived as the perfect third of F is NOT also the perfect fifth of D. (9/8*3/2 = 27/16 is the A tuned to D, 4/3*5/4= 20/12 = 5/3 is the A tuned to F). The A in a D chord wants to be sharper than the A in an F chord. This difference in requirement of the A note in C Ionian and D Dorian is called a comma. In some cases commas are hardly noticeable and are ignored. In some early western music they are actually featured and strictly specified. In other cases they are god awful sounding and of no use musically. These harmonic clinkers in western music are called wolf tones. It is because of these wolf tones that all sorts of ways to tune developed in western music. Back in the day Pythagoras was one of the first and most successful to address the problem: http://www.medieval.org/emfaq/harmony/pyth.html There are have been historically in western music many many ways to deal with this. A good summary of most methods is he http://pages.globetrotter.net/roule/temper.htm#_nr_308 Also of interest is this article by the musician who taught most of this theory to me personally: http://www.michaelharrison.com/harmonic-piano.html I'll save my explanation of how Indian and Middle eastern music solved the temperament problems. The main difference between world music scales is how each culture chose to deal with these inharmonies. Julian |
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#23
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from Cents to Hertz and back again
Phil Allison wrote:
Every semitone is the twelfth root of 2 times the note below it. That is: 1.0594631 times. A cent is 1/00 of a semitone. So, the semitones are along a nonlinear function ( t[n] = 1,0594631 x t[n-1] ), but the cents are linearly interpolated between them? Lars -- lars farm // http://www.farm.se lars is also a mail-account on the server farm.se aim: |
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#24
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from Cents to Hertz and back again
Lars Farm wrote:
So, the semitones are along a nonlinear function ( t[n] = 1,0594631 x t[n-1] ), but the cents are linearly interpolated between them? I doubt it - I should think a cent is actually 1/1200th root of 2. Not that it makes a lot of difference for practical purposes, as "small difference" approximations apply. -- Anahata -+- http://www.treewind.co.uk Home: 01638 720444 Mob: 07976 263827 |
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#25
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from Cents to Hertz and back again
Hi Anahata,
You are correct. It is not a linear interpolation between semitones. At least according to the definition of a cent given in Harry Olsen's "Music, Physics, and Engineering". It matches your definition exactly. Mark "anahata" wrote in message ... Lars Farm wrote: So, the semitones are along a nonlinear function ( t[n] = 1,0594631 x t[n-1] ), but the cents are linearly interpolated between them? I doubt it - I should think a cent is actually 1/1200th root of 2. Not that it makes a lot of difference for practical purposes, as "small difference" approximations apply. -- Anahata -+- http://www.treewind.co.uk Home: 01638 720444 Mob: 07976 263827 |
Linear Mode
