Talk:Mathematics of musical scales

Writing Tone
This article is on the whole, excellent. But the tone of the writing is not really in keeping with how an encyclopedia article should read (it's more like a friendly "How do I" column in a newspaper). Also it is authored - do we have permission to rewrite the article wholesale? When the author gave his/her permission, were they aware of that possibility? MMGB


 * On the original Talk page, it says we do have permission. --Zundark 2001 Nov 11

Could we have articles on non-western scales for comparison?

Rewrite
I'm going to try and rewrite this article - it could be a good starting point for all the musical tuning articles, but I'm not convinced it is as it stands. It could be months before I get round to it, but it will be done... --Camembert

Equal tempered 12 note scale
Hmm... is there any chance of including a brief mention that it's arguable that the equal tempered 12 note scale is a compromise? I realise 'general' music theory states that it /is/ a compromise (and that ratios of 5/4, 6/5 etc. are 'supposed' to sound the sweetest), but there's always a possibility that it isn't. --dwhite

These days it often is not treated as a compromise; historically however it arose as a compromise. That could certainly be mentioned. Gene Ward Smith 18:05, 28 May 2006 (UTC)

Table of intervals
I hope no-one minds but I edited the table of intervals a bit. In its previous form it was either wrong, in which case it needed correcting, or used some system which I (and perhaps others) did not know, and needed this explaining. I think the whole article while interesting is far from perfect, but I did feel it needed this quite prominent table sorting out as a start. Nevilley 09:11 Mar 14, 2003 (UTC)

Edits and pictures
Alright, I am editing and adding to this page a lot. I hope people like the changes. I am having some trouble getting the pictures to line up nicely. I guess because they are too big. :-\ Anyone wants to clean them up or teach me how, you're welcome to. Maybe some of this stuff should be moved to other pages. Like the harmony section to the harmony page? It is mathematical, though. Omegatron 02:08, Jan 18, 2004 (UTC)

I have no idea what this line means:

"To obtain a scale of 12 notes the major tone 9:8 is equated with the minor tone 10:9 and to two semitones 256:225."

Can someone explain it? Omegatron 16:45, Jan 18, 2004 (UTC)


 * I have since learned that they aren't saying "9:8 is equal to 10:9", as I misunderstood, but saying that the interval that approximates 9/8 is set equal to the interval that approximates 10/9. in quarter comma meantone, for instance, the two are geometrically averaged to get sqrt(5)/2. Omegatron

Errors
Of course I could. Should I?

There are egregious errors on this page, which I would like to fix, but I am afraid someone may have some of them confused with useful content. Much of the best stuff here could be moved to something on the physics of music, for which there are empty links, and as yet no article. Some actual mathematics of scale theory could be added, if that was deemed a good idea.


 * Yes. Of course you should.  I was going to continue editing down the rest of the page (equal temperament and beyond), but you sound like more of an authority than me... One thing that should definitely be changed is the description of the reasons for equal temperament, which just shows that the first two seconds in one scale are not the same, rather than explaining that the minor second will be different if you start a new scale on a different note.  I also wanted to add something on even and odd harmonics and the symmetry of waves, but you're right, a lot of this stuff should be in a physics of music article instead. Omegatron 23:18, Jan 24, 2004 (UTC)


 * I added a little bit more that I never uploaded last week, changing the "two seconds don't equal a third" example into a "two scales don't have the same intervals" example. But I fear I have turned one poor example into three poor examples.  I think it's a bit clearer overall, but it still needs lots of work. Omegatron 05:59, Jan 25, 2004 (UTC)

I made some massive changes, moving the first portion of this article into a physics of music page, which it seems was needed. A great deal of text was removed, but this was either erroneous (this page has been riddled with errors) confusing or redundent in my judgment. The article no longer claims falsely that there is such a thing as the standard justly tuned scale of Western music, nor attempts to determine the "error" in equal temperament by comparison to this mythical entity (a scale not even in the Scala archives!) I also moved it so that we are not confined to discussing Western music. Gene Ward Smith

are the sound samples correct?
Or maybe it's my ears. I don't hear the second tone in either sound sample. There's a slight click in the second sample (C-A chords). Is winamp capable of distinguishing the two notes in a .ogg file based on a tuning deviation? Blair P. Houghton 03:01, 5 Apr 2005 (UTC)


 * No offense intended, it's not winamp, it's you. Or rather the limits of human perception as the pitches are audibly indistinguishable from each other as the just noticeable difference for pitch is five cents.  They should, however, create interference beating. Hyacinth 05:41, 5 Apr 2005 (UTC)


 * The description in the article says the first recording is a half-second at C#JI followed by a half-second at C#ET, hence no overlap and no beating. I expected to hear that difference, but I was listening on a laptop, so mea culpa; I'm listening now on my home computer with a Klipsch THX-certified sound system in the loop.  Still can't hear any difference in the first recording, but that's not surprising any more.  The second recording does depend on the beating of the two C# tones with the constant A, and the click in the middle appears to be a result of the sudden change in phase (or maybe just the inability of the software and/or hardware to keep phase when changing frequency, though that'd be a good way to ruin your own sales figures, to add clicks to every recording with a bent note in it).  And though I perceive a change in the resultant beat tone, I perceive it as being lower in the second half, when it should be about 4% higher...which I have no doubt is a problem only of perception.  If I had the skillz, I'd rerecord these to make the switch several times over several seconds.  Blair P. Houghton 22:24, 5 Apr 2005 (UTC)

Added a new link to show graphical comparisons of harmonic ratios to mathematical progressions.

Equal temperament edits of 68.15.221.177
On October 7, 2005 68.15.221.177 added material with false claims, which I have removed. In particular I mean the claim:

"Equal tempered scales have been built using 19 equally spaced tones, and also 24 equally spaced tones. These scales have their uses, but the 12 tone scale does the best job approximating the perfect fifth, perfect fourth, minor third, major third, minor sixth, and major sixth" can easily be shown to be false for 19, and even falser for other equal temperaments. People adding material like this should at least do the math. Gene Ward Smith 18:05, 28 May 2006 (UTC)

math
The practice of copying someone's text onto a page has not served this subject well. Perhaps it is just poorly titled, but I'm amazed that there is not even a simple explanation about adding and subtracting ratios of frequencies on a page entitled "Mathematics of musical scales". I could easily do this myself but am reluctant to edit a text that may have been meant to stand on its own. Perhaps a new page would be better for this purpose?

Comments?

Chris Saetti September 11th 2006


 * Hi Chris, if you're not sure whether the text you want to contribute should go in the article, put it on the talk page here first, and then we can all decide whether it should go in or not. Cheers, Madder 20:03, 11 September 2006 (UTC)


 * Chris, are you talking about beat frequencies and related phenomena, or what? &mdash;Tamfang 07:52, 12 September 2006 (UTC)

visual aids
I'd like to see graphs - perhaps I can make some and submit them. Pictures of different chords would tell more of the story (I mean, graphs of the wave function, not the finger positions) --Golda


 * I added a jpg comparing the accuracy of various equal tempered scales. I don't believe there is one "right" way to calculate this, so someone may disagree with my methods.  But I believe it is an area where reasonable people can disagree.  I stand behind the methodology. GaulArmstrong 11:15, 5 November 2006 (UTC)


 * What is the scale of this chart? by which I mean (since "scale" is ambiguous in context) what is the unit of the vertical axis? &mdash;Tamfang 16:32, 11 November 2006 (UTC)


 * I used $$(100\% * (Observed - Expected) / Expected).$$ The Observed equals the note's step number divided by the total number of steps in the scale.  For example, the note closest to a P5 in 12-tet is the 7th note of the scale -- 7/12, or 0.58333.  The note closest to a M3 in 19-tet is the 6th step -- 6/19, or 0.3157.
 * For the P5 Expected, I calculated as $$Log_2(3/2)$$ or 0.584963. See the Normalizing the Musical Scale wiki, for the reasons why I used $$Log_2(3/2).$$  Similarly, the M3 Expected is $$Log_2(5/4)$$ and the P7 Expected is $$Log_2(7/4).$$ GaulArmstrong 19:47, 28 November 2006 (UTC)


 * So a unit length on the chart means 1% difference in the logarithms, rather than in the raw frequencies? Okay.  &mdash;Tamfang 05:23, 2 December 2006 (UTC)


 * I used log units, because the ear hears all octaves as equally sized -- and the log function changes frequency into equally-sized octaves. So, the accuracy/error that is heard should be phrased in log terms.GaulArmstrong 04:00, 6 December 2006 (UTC)


 * I like this chart, but I don't agree with the way the differences are measured. Using percent of the frequency should make it consistent with this table.  I understand about pitch perception being logarithmic, but I'm not sure it matters as much for small pitch differences.  Normally, one would use cents as a log unit for small pitch differences, but that might be confusing here since that implies 12-tet. Maybe log2 units would be better: simply $$(Observed - Expected)$$.  Also, it would be nice if there was a little more description of the methodology on the Image page. &mdash; Rick Burns 00:49, 23 February 2007 (UTC)

Practice-based
Is "practice-based" a made up term, or is this used somewhere? Can we come up with a better name than this? - Rainwarrior 02:20, 24 November 2006 (UTC)

The mathematics of music should discuss rationals, irrationals, and continuing fractions
I won't go into detail, but at the very least rationals and irrationals should be discussed because the tempered scale is based on irrational ratios, and all harmonic relationships are due to harmonic ratios all of which are rational ratios.

Also, the fact that the circle of fifths can be perfect in a tempered system an can NOT be in ANY harmonic system is directly tied to the mathematic concepts of rationals and irrationals. In particular no multiple power of 3/2 (harmonic perfect fifth), can be a multiple power of 2, and from another view the solution to tempering the scale is based on the 12th root of 2 which is an irrational and as such can not be expressed exactly by any fractional (harmonic) ratio.

Continuing fractions are not well documented in the literature in regard to their use in musical concepts, but they provide a connection between tempered and harmonic system in such a way as to be able to convert from one to the other mathematically. For example, starting with the tempered scaled based on the 12th root of 2, it is possible to derive Just intonation system from first principles of mathematics using continuing fractions!! What continuing fractions do is enable one to discover the simplest and best harmonic relationship to ANY note of ANY frequency as continuing fractions is the necessary method to find the best fractional approximations to any note.

Also the difference between a tempered perfect 5th and a harmonic perfect 5th provides a way of expressing most harmonic systems in terms of their tempered equivalent. It turns out that the difference between a note in one system or the other are an exact multiple of this difference...that is because all the notes can be generated by going around the circle of fifths and at all points in this process, the differences between the two systems is a small integer multiple of the above mentioned difference.

There is another point which may be off topic or not...and should be somewhere included in wikipedia...Many so called experts claim that no music is intrinsic and is learned...but they are wrong because they don't know about continuing fractions and don't know that the way we hear notes is influenced by the "closeness" of any note to its simplest harmonic approximations because the way that anything with a natural resonance frequency responses is related to these considerations. This is certainly true of the frequency detectors in the ear. This can be analyzed in a direct manner using continuing fractions and in this way it is possible to predict things about what can be discriminated to hear and what is not. There has been an argument against intrinsic musical scales which uses micro tones as an argument, but it turns out that continuing fractions can be used to answer this argument.

Sorry about the sloppiness of these comments and the lack of references, but my time is VERY limited and the proof of these comments is straight forward, and can derived from the comments themselves...given a little knowledge of the fundamentals of mathematics readily available in wikipedia...so I guess that is my reference. JKSellers 01:26, 6 March 2007 (UTC)


 * I take the liberty of correcting a minor error in the above (to what you presumably had in mind). &mdash;Tamfang 06:15, 28 March 2007 (UTC)

Edits by 68.253.20.252
Can someone please check that the edits by unregistered user 68.253.20.252 are correct? I am not an expert in this field, but the changes that they have made to the article seem fairly significant. Madder 13:10, 30 May 2007 (UTC)