Talk:Cent (music)

Question

• QUESTION:
• Hello, I once saw a formula for converting from a ratio or a frequency to cents that did not involve the use of the logarithmic function [log] on the web, but the site it was on has since vanished, after doing an extensive search on the web and in an encyclopedia and several "physics of music" books, I've come up empty handed. If some one can help out I would be very appreciative and it would be a welcome addition to this short page for such a contrived topic as this. Thanks in advance. 206.157.148.147 (talk contributions‎ ) 11:48, 1 November 2005‎

I pulled this from the article, where it is not appropriate. It seems to be part of a copy and paste - but is it a copyvio? Ian Cairns 17:37, 1 November 2005 (UTC)

I'm sorry, but you're talking about mapping from a geometric scale (frequency) to an arithmetic scale (cents) and there's no other way to do that except with logarithms. Contrived topic?

Ian, why would you think this is a copyvio? It's just a question. —Wahoofive (talk) 22:02, 1 November 2005 (UTC)

Strictly speaking, Wahoo, it makes sense a lot of the time to take pitch as the independent, and frequency as the dependent variable, making the mapping exponential. No need for logarithms there (but of course, they are implied as the inverse function.) __Just plain Bill (talk) 14:12, 12 August 2008 (UTC)

The measure was developed by A. J. Ellis around the 1870s

Hi, I am reading in this little book by M. le Baron de Prony, Instruction Elémentaire sur les moyens de calculer les intervalles musicaux ("en prenant, pour unités ou termes de comparaison, soit l'octave, soit le douzième d'octave,..." Firmin Didot, Paris. 1834) where he introduces a notation for intervals using logarithmic tables and writes about "centièmes de demiton" comparing just intonation to equal temperament. Bosanquet also proposes that semitones are convenient expressing intervals but cedes De Morgan's precedent for the method of calculation used in On the Theory of the Division of the Octave. Mireut 20:55, 7 January 2006 (UTC)

I found a relevant passage from Ellis: "The first person to propose the measuring of musical intervals by equal Semitones was, I believe, de Prony, but I have not been able to see his pamphlet ; the next was the late Professor de Morgan ("Cam. Phil. Trans.," x, 129), from whom I learned it, and I employed it in the Appendix of my translation of Helmholtz, by the advice of Mr. Bosanquet. Having found that two places of decimals sufficed for most purposes, I was led to take the second place, or hundredth of an equal Semitone as the unit, and I have extensively employed this practice, here for the first time published, with the greatest advantage. In fact, I do not know how I could have expressed the results of the present investigation in any other brief and precise, and at the same time suggestive, method." (Ellis, A. (1884) Tonometrical Observations on some existing Non harmonic Musical Scales.) Mireut 15:53, 14 February 2006 (UTC)

Decimal in a fraction

A recent edit of mine was reverted with the edit summary "why would you use a decimal in a fraction?". That's such a bizarre justification, I'm not even sure how to respond. Really, it's not even an objection; it's a question, and I have an answer: I did it because because 1/17.3 is much more accurate than 1/17. Please don't revert my edit again unless you can provide a reason that the article is better without it. --Doradus (talk) 14:25, 17 January 2008 (UTC)

Hi, I recently made an edit removing the fraction, before I saw this discussion. As a reader, I found the fraction more confusing than anything, because 1.0058 is clearly not "equal" to some fraction of one percent. After I removed the fraction I realized the intention was to represent this as 1 cent = $2^{\frac {1}{1200}}$ = 1 + $({\frac {1}{17.3}})\%$ . But I don't have a good suggestion on a clear way of phrasing that so that it would clarify rather than distracting. Maybe just putting it back in as-is is good enough, with the phrasing of "one percent" (or even "one per cent", though that risks confusion with the logarithmic unit "cent") instead of using a '%' sign, since that seems to convey the fraction 1/100. VineetKumar (talk) 20:14, 12 January 2009 (UTC)

Awkward definition in Sound Files section

The last paragraph might be better explained if it included a link to Beat (acoustics), though I'm not entirely sure if that's what the author was driving at. --scruss (talk) 11:17, 10 May 2008 (UTC)

Paragraph rewritten. __Just plain Bill (talk) 14:03, 12 August 2008 (UTC)

Small interval audio files

The three uploaded small interval audio files for 1 cent, 6 cents, and 10 cents, sound to me constant pitch, that is, zero interval. I have listened to Ten_Cents_Interval.ogg repeatedly using both Cortado (Java) and QuickTime. I always hear just one tone. I strongly believe I would be able to detect an interval of 10 cents. I suspect that these files were not constructed correctly. Does anyone hear actual pitch intervals? Anomalocaris (talk) 03:05, 13 February 2009 (UTC)

Since some people can't hear the 10-cent difference and other people barely can, it seems we could use some files with bigger differences, say 20 cents and 50 cents.CountMacula (talk) 00:31, 15 July 2012 (UTC)

Whoops, since it is an exponential scale, I would make the sequence of files linear in the exponent. I would think that 5, 10, 15, 20, 25, 30 cents should be adequate for most people to find their approximate threshold.CountMacula (talk) 00:40, 15 July 2012 (UTC)

Playing them with XMMS I hear the ten-cents sharp tone as just a fraction of a fine hair brighter than the reference tone. I can convince myself that I hear a similar but smaller difference with the six cents example. The one-cent difference sounds the same to me. In all three files, the third tone shows beats that demonstrate two different frequencies played together. __Just plain Bill (talk) 12:25, 13 February 2009 (UTC)

I think it would be better if the samples were pure tones rather than that foghorn-like thing. Also, they should last longer, maybe 2x as long. I can hear a slight difference in the 10-cent sample but not the other two. I can hear beats in the mixed samples at 6 and 10 cents but not the 1 cent. Beats in the 1 cent mixed sample would probably be audible if it were played longer. The other thing is that the unmixed samples seem to have beats of their own, making the whole thing slightly uncertain. It might be better to use FLAC rather than ogg for these samples, to avoid introducing coding artifacts 67.122.211.205 (talk) 04:31, 6 September 2009 (UTC)

They should be louder, too. I have my speakers and my software volume control both turned up all the way, and still the files do not play nearly as loud as I want. A file with normal volume fills the room turned up as loud as I have it set.CountMacula (talk) 00:31, 15 July 2012 (UTC)

Regarding the foghorn timbre: see http://pom.sagepub.com/content/early/2010/08/23/0305735610373602.abstract "Does timbre affect pitch? Estimations by musicians and non-musicians". The answer seems to be that it does.CountMacula (talk) 04:35, 22 October 2012 (UTC)

I'm relieved to find it's not just me that finds the 6 cents difference imperceptible. I think I can detect a tiny difference in the 10 cent interval, but of course this is not a blind test. I have seen some research suggesting that about 14 cents is the smallest interval that most people can reliably distinguish in a double-blind randomised test. All this casts doubt on the claim in the present text that a 6 cent difference is usually perceptible.86.159.83.224 (talk) 18:35, 2 November 2009 (UTC)

Wouldn't a lossless audio format be better for such sample files? Who knows what psychoacoustic tricks the Vorbis encoder might have used that hide the difference in pitch? TorLillqvist (talk) 11:01, 15 January 2010 (UTC)

If (for the sake of argument) one plays a 440 Hz tone for one second (the spacing of the notes in these samples), followed by the same thing a cent higher, there are 440 cycles in the first and 440*1.00057 = 440.25 cycles in the second. Hence there is only a quarter of a cycle difference. This puts it a factor of 2 under the Nyquist sampling limit of half a cycle, and that doesn't even take into account the decay, which makes that 1/4 cycle that much harder to hear. For 264 Hz, middle C, the difference shrinks to 0.15 cycles difference, impossible to pick out of the noise. --Vaughan Pratt (talk) 04:40, 3 June 2010 (UTC)

It would help if the sound files wouldn't had VIBRATO. Btw, the 10 cent one crashed my Firefox. I sent a report to Mozilla already. — Preceding unsigned comment added by 81.203.50.227 (talk) 18:20, 25 May 2011 (UTC)

I propose that these audio files be removed from the article. They have numerous issues, above all of which is unverifiability. We need somebody who knows how to make such files, who will make new ones and explain on the talk page how they were made.CountMacula (talk) 04:03, 22 October 2012 (UTC)

It is very hard to tell the difference between notes with such a small cent difference. When both are played together, however, it is easy to hear the out-of-phase like effect. There are still many flaws with the audio samples, however. I also have a problem with the effects on the audio. I see no reason to use such a harmonically rich sound, with vibrato as someone else added, when just a sine wave or triangle wave makes it easier to hear the difference, and I don't understand why reverb was also added to this audio. Even worse, the audio is cut off very abruptly. I do have a copy of FL Studio and can easily create some examples with and would be fine with replacing the audio with simple, longer, high quality samples, using the 3xOSC plugin's fine-tuner to create samples playing at different cent differences. EDIT: As I am working on creating new files, I now know why harmonically rich files were used. Using just two sine waves with a slight difference in cents really only creates one sine wave going up and down in amplitude rather than hearing two at the same time. To solve this, I used 3 octaves of sine waves playing at once. This allows the listener to hear the intended out-of-phase-ness of the two waveforms. Anyway, I replaced the audio files (I see I'm about 9 years late however) but if anyone is still paying attention, I'd love to get some feedback on how well I solved these problems and what I could do to help improve it even more. --Imagination deprivation (talk) 18:42, 24 March 2021 (UTC)

Your files sound good. Thanks for going to the trouble. Binksternet (talk) 22:15, 24 March 2021 (UTC)

The audio sounds good, but the frequency differences in the desciptions under the files and in the bullet points above do not match. Which are the correct ones?−Woodstone (talk) 08:57, 26 March 2021 (UTC)

Thank you Binksternet! Also, Woodstone, I'm not exactly sure what you mean. Are you talking about the three audio file links above the audio files that say 1 cent difference, 10.06 cent difference, and 25 cent difference? Those are separate MIDI files of a piano, I didn't add those. Sorry if that's not what you're talking about! -Imagination deprivation (talk) 14:50, 5 April 2021 (UTC)

Yes, the bullet points (or rather speaker points) confused me. Why not remove them. It does not seem useful to have two very similar sets of examples. Your artificially constructed ones demonstrate the effect smore clearly.−Woodstone (talk) 15:27, 5 April 2021 (UTC)

I do agree that they do confuse the reader but I'm not sure if they should be deleted or if the formatting should just be changed due to them actually adding a bit of information on beat frequencies. I think it might be better to reach a consensus with more editors. -Imagination deprivation (talk) 00:52, 9 April 2021 (UTC)

Duplicate links

How many links to do we need to the same site? The article could look like this:
==External links==

• Cent conversion: Whole number ratio to cent
• Cent conversion: Online utility with several functions
• "Adaptive pitch test", Tonometric.com.
• "Adaptive pitch test", Tonometric.com.
• "Adaptive pitch test", Tonometric.com.

but we may all agree that it is unnecessary. Before accusing me of having given no reason for removing the duplicate link, why don't you address the reason I gave, or even give a reason for including it in the first place? Hyacinth (talk) 20:30, 29 October 2012 (UTC)

Please check the Music intervals requency ratio equal tempered pythagorean comparison file

File:Music intervals frequency ratio equal tempered pythagorean comparison.svg.

File:Music intervals frequency ratio equal tempered pythagorean comparison on C.png.

Can some folks in the know please check the graph File:Music intervals frequency ratio equal tempered pythagorean_comparison.svg embedded in the present article at Cent_(music)#Use for accuracy? It seems to me as that it does not agree with the table at Pythagorean_tuning#Method.CountMacula (talk) 08:12, 5 May 2013 (UTC)

As the comment in the file says: all intervals are tuned up from the tonic. So you will notice that the deviation grows with the circle of fifths, starting from C (C, G, D, A, E, B, F#, C#, G#, D#, A#, E#=F). In the table, the start is chosen at D and part of the scale (A, E, B, F#, C#, G#) is tuned forward (using factors 3 and 1/2), the other part (G, C, F, Bb, Eb, Ab) backward (using factors 1/3 and 2). Why these different paths are taken I do not know, but it's only a matter of calibration. The scale steps are the same. −Woodstone (talk) 16:43, 5 May 2013 (UTC)

I don't see why anyone should have to care where on the circle of fifths the sequence starts. Unless I am wrong, we are talking about generating a generalized 12-tone scale. In my objection above, I have ignored the letter names and taken heed of only the interval names. The table in Pythagorean tuning shows that half the Pythagorean intervals are larger than the corresponding ET intervals, whereas the table and the graph in Cents (music) show all the Pythagorean intervals as larger than the corresponding ET intervals.CountMacula (talk) 03:54, 17 July 2014 (UTC)

The Circle of Fifths defined by factors 3/2 (and reduced by factors 2 to stay within one octave) is not actually a circle. After 12 steps it does not meet the start: 3¹² = 5314413, while 2¹⁹ = 524288, a difference of 1.36%, a substantial fraction of a half step being 5.95%. So the closing step to force only 12 notes is shorter than the other ones. The location of this break is important. For each choice, a different map to the equidistant scale is obtained. The table and the graph use different starting points. −Woodstone (talk) 17:21, 18 July 2014 (UTC)

The figure includes very odd intervals. There is no reason, for instance that the deviation between Pythagorean and ET be larger for the 4th than for the 5th. The "4th" in the figure actually is an augmented 3d (C-E#); similarly, the minor 3d actually is an augmented second (C-D#), and the minor 7th actually an augmented 6th (C-A#). That is to say that the labels in the figure are wrong, and to say that "all intervals are tuned up from the tonic" doesn't justify these. The only solution would be to show the intervals in the order of the cycle of fifths rather than as an ascending scale, as in the figure illustrating the List of pitch intervals. — Hucbald.SaintAmand (talk) 20:00, 27 November 2014 (UTC)

See: File:Music intervals frequency ratio equal tempered pythagorean comparison on C.png. Hyacinth (talk) 12:45, 16 February 2016 (UTC)

Please PM me on how I can fix the first image. The replacement by Hyacinth has issues, such as being a bitmap instead of a vector image and not showing the logarithm curve IMO. SharkD  Talk  00:34, 13 November 2016 (UTC)

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External link suppressed

Woodstone, you suppressed the link to a Microsoft Excel function that converts ratios to cents, on the argument that "WP is not a "how to" guide; apart from that the calculation is trivial". Not only is the calculation indeed trivial, but il also already was given in the article. However, you may not have read the linked page to its end: its main purpose is to propose Excel functions that can be embedded in an Excel spreadsheet and that perform the conversion automatically. To have this as an automated function (in the form =Cent() for ratios to cents, or =Acent() for cents to ratios) is so much easier. I use it daily since I embedded it in my spreadsheet. Are you sure you shouldn't reconsider your suppression? Do you think webpages offering the same trivial conversion are more useful? — Hucbald.SaintAmand (talk) 16:10, 27 December 2017 (UTC)

WP is also not a platform for advocacy. And even when, this function is readily expressed in existing functions. Having it as a separate function is at most of marginal interest. −Woodstone (talk) 08:45, 28 December 2017 (UTC)

Do you mean that this function already exists in Excel? I don't think so, at least I never found it. — Hucbald.SaintAmand (talk) 10:35, 28 December 2017 (UTC)

They do not exist under a name, but can be simply expressed (as shown in the French page)

"=1200*LOG(X,2)", where X is a frequency ratio

"=2^(C/1200), where C is an interval in cents

The named functions save only a handful of keystrokes. −Woodstone (talk) 17:16, 28 December 2017 (UTC)

Flash-only

Free, online web sites for self-testing are available.

I removed the above because the citation links to a web resource that is Flash-only.

• Adobe Flash

Although Flash was previously a dominant platform for online multimedia content, it is slowly being abandoned as Adobe favors a transition to HTML5. Flash Player has been deprecated and has an official end-of-life at the end of 2020.

...

Starting from Chrome 76 and Firefox 69, Flash is disabled by default and browsers do not even show a prompt to activate Flash content.

It was a bit of an odd duck to begin with. Not a fatal crime per se, as I personally tilt toward moderation in all things. Even so, an odd duck dragging a limp wing surely crosses a line in the sand. — MaxEnt 20:22, 2 August 2019 (UTC)

Decibels

There's a mention of decibels. Other than being a logarithm, there's no connection, and I'd say that its mention is more harmful than helpful (since both have something to do with sound, but are otherwise unrelated). Gvanrossum (talk) 05:46, 1 December 2019 (UTC)

Agree and remove some more possibly confusing and unnecessary text. −Woodstone (talk) 06:39, 1 December 2019 (UTC)

Centitones

Are centitones, etc., common? It’s easy to find lists and definitions of all these measures but has someone published a survey which ones are used the most and where? The reference used supporting the lede seems just to mention cents and Savarts - a more focused reference could support both a stronger statement about cents being standard at least in English language stuff, as well as provide context for those less frequently encountered systems. Mireut (talk) 15:57, 2 December 2019 (UTC)

Neither the New Grove Online nor the Oxford Dictionaries (Oxford Dictionary of Music and Oxford Companion to Music) have an entry for "centitone", and they never mention the term in any of their articles. I don't think that centitones could be said "common", but others will certainly think otherwise. The savart has some historical importance because, as it is based on log₁₀: it was easily computed in the time of logarithmic tables, but with the advent of electronic computers giving logarithms to any base, it generally fell in disuse. It certainly is strange that the article has a separate section for centitones, but none for other musical logarithms which may have been more common, e.g. Sauveur's heptamerides (virtually identical with savarts). However, I don't think anybody ever published a survey of how common they were. There are works on the history of musical logarithms, though (e.g. David Wright, Mathematics and music, 2009, chapter 5), or more specific papers on the origins and early history (e.g. Benjamin Wardhaugh, "Musical Logarithms in the Seventeenth Century: Descartes, Mercator, Newton." Historia Mathematica 35, 2008, pp. 19–36).

Your question could therefore be reformulated as follows: "Doesn't this article deserve a historical section?" – I am not sure it does, but if people feel that it does, I'll happily collaborate. — Hucbald.SaintAmand (talk) 17:43, 2 December 2019 (UTC)

I added the text that’s in the lede that summarizes Ellis’ comments from his 1884 paper, you can trace my comments up top here on the talk page, basically to add some context to what was a pretty bare statement about the origins of cents.

I’m not sure what source does make direct statements about cents’ basis in other systems, and wonder how to better integrate parallel things like centitones which are obviously conceptually related but maybe independent and kind of unused - I’m not suggesting they should be removed, just that some kind of supportable context be added so it’s clear how much they are or were used. Mireut (talk) 23:03, 4 December 2019 (UTC)

Thanks for this addition, Mireut. I fail to see, however, what "acoustic logarithms decimal semitone system" may mean. The expression can be found in some recent publications, but not in Ellis, nor in any 19th-century sources that I know. Prony, as you probably know better than many of us (you published about Prony here above in 2006), measured in tempered semitones (¹²√2), as did Ellis in his "On the History of Musical Pitch" of 1880. Ellis apparently used Cents (¹²⁰⁰√2) for the first time in his "On the Musical Scales of Various Nations" in 1885, then of course in his translation of Helmholtz in 1895. Logarithms on base 2 can be described as "musical", or possibly as "acoustic", but what is the meaning of "decimal semitone system"? Shouldn't it be "duodecimal"? I could have added a {clarification needed} label in the main text, but I thought it better to question you here. — Hucbald.SaintAmand (talk) 09:25, 5 December 2019 (UTC)

Yes, that should be rephrased. I meant in the sense he used decimals to get finer pitch graduations for measuring than twelve semitones to an octave - Prony’s “centièmes de demiton” translates to “hundredths of a semitone”, which would be clearer. I don’t have the text handy, I think “acoustic logarithms” was a term he used, here’s a couple results I get searching that in French.

https://gallica.bnf.fr/ark:/12148/bpt6k9382147.image

https://books.google.com/books?id=qvh62MEkOYIC&lpg=PA107&ots=HdU5GWVYCH&dq=%22logarithmes%20acoustiques%22%20prony&pg=PA107#v=onepage&q=%22logarithmes%20acoustiques%22%20prony&f=false

- Mireut (talk) 14:42, 6 December 2019 (UTC)

Thanks, Mireut. According to Fétis, the logarithms used by Prony were either on base 2 (which I think corresponds to those used by Caramuel de Lobkowitz in the 17th century), or on base ¹²√2, the semitone. Delezenne suggests a division of the octave in syntonic commas (81/80), resulting in 55,79763048 (!) units in the octave. I think that all these have been defined, at one point or another, as "acoustic logarithms", an expression that I think therefore should be avoided. If Prony's logarithms (named pronys) were on base ¹²√2 (which is what I thought up to now, but I never read his book; I will, as soon as possible), then Ellis' cents are mere hundredths of pronys. I would therefore write something like this: "Alexander J. Ellis based his measure on the hundredth part of the logarithms developed by Gaspard de Prony in the 1830s", or something like that. Is it certain that Ellis knew the prony, however? Ellis did use cents with decimals, I think – certainly not with 8 decimals as Delezenne did! – I'll check that. But is it so important that it deserves being mentioned in the lede of the article? Would it not be better to say that the precision in cents is such that decimals usually are unneeded (even although Ellis at times used them)? — Hucbald.SaintAmand (talk) 21:44, 6 December 2019 (UTC)

I tried a different description of Ellis' cents, to replace what had been added by Mireut. This probably is perfectible: don't hesitate. — Hucbald.SaintAmand (talk) 11:51, 7 December 2019 (UTC)

I should have added that, in view of the history of musical logarithms, I think that the section on centitones should disappear. — Hucbald.SaintAmand (talk) 11:53, 7 December 2019 (UTC)

Seems okay, though is the math necessary? I think I had wanted to be explicit and stick close to Ellis’ explanation, to prevent undue emphasis either as a new invention or part of a chain kind of synthesized out of a lot of references. Mireut (talk) 14:09, 9 December 2019 (UTC)

Is it ¹²⁰⁰√2 that you call "the math"? It could be replaced by "the 1200^th part of the octave", but that may seem redudant with "the hundredth part of a semitone" that comes just before. It may not be needed at all... — Hucbald.SaintAmand (talk) 17:11, 9 December 2019 (UTC)

Centitones, savarts, pronys, etc.

I note that the Centitone section of the article compares centitones with savarts, which otherwise are only mentioned in footnote a. Footnote a, on the other hand, mentions Caramuel, Newton, Sauveur, Prony and Savart. The question then arises why centitones are mentioned in the article and not in the note, while the musical logarithms mentioned in the note may be better known. We didn't decide about whether the Centitone section must be maintained and I wonder whether the question should not be rephrased as follows:

• Why is there a section on Centitones while there is none on heptamerides, on pronys, on savarts, etc.?

Or else, should we not replace the Centitone section by one on the history of musical logarithms? It may be interesting to note that the French wikipedia, which has an article on "Cent et savart", because savarts are sometimes still in use in French, has a section on History that seems reasonablty complete (I am among its authors, that's why ;–)). If others agree (or if they say nothing against the idea), I could translate it from the French and replace with it the Centitone section (while mentioning centitones in the History section, of course). — Hucbald.SaintAmand (talk) 10:28, 14 August 2021 (UTC).

I made in the article itself a proposal concerning other logarithms. Feel free to improve. — Hucbald.SaintAmand (talk) 10:39, 29 August 2021 (UTC)

Too much math

This article isn't about mathematical methods to calculate the value of a cent, but a big chunk of the article is devoted to that. It seems like it's mainly a way for WP editors to show off. I propose chopping at least half of it. —Wahoofive (talk) 03:41, 18 April 2022 (UTC)

A big chunk? That must be your subjective anti-math bias. There are only two formulas. One to determine the cents between two frequencies, and one to get the new frequency after adding some cents to a frequency. That seems to be bare minimum.

I would agree however, to scrap the superfluous and outdated section on linear approximation.

−Woodstone (talk) 06:21, 18 April 2022 (UTC)