Talk:Mathematics of musical scales

message: 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)
thread_title:  Equal tempered 12 note scale 

message: 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)
replies: text: 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)
replies: text: 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)
thread_title:  are the sound samples correct? 

message: 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)
thread_title: Equal temperament edits of  68.15.221.177

message: 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)
replies: text: Chris, are you talking about beat frequencies and related phenomena, or what?  —Tamfang 07:52, 12 September 2006 (UTC)
thread_title:  math 

message: 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)
thread_title:  Practice-based 

message: 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)
replies: text: I take the liberty of correcting a minor error in the above (to what you presumably had in mind).  —Tamfang 06:15, 28 March 2007 (UTC)
thread_title:  The mathematics of music should discuss rationals, irrationals, and continuing fractions 

message: 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)
thread_title:  Edits by 68.253.20.252