source file: m1447.txt Date: Sun, 14 Jun 1998 17:56 +0100 (BST) Subject: RE: magic chord, reply to Paul Erlich From: gbreed@cix.compulink.co.uk (Graham Breed) >>Thirds in 12-equal are mistuned 5-limit intervals. So, the chord as a >>whole will be a mistuned 5-limit chord. >And what would be the correctly tuned version? It contains three mistuned 5-limit intervals, so it is a mistuned 5-limit chord. It can't be translated consistently into JI, like the 'magic chord' in question. Which is why you gave it as an example, remember? > I strongly agree that augmented triads sound more dissonant in 31-equal, > or with just major thirds as in JI or meantone, than in 12-equal. Does > that argue against anything I said? No, it argues _for_ what you said, which is why you gave the example. >>The 9/8 in 31-eq is tuned much better than the 5/4 in 12-eq. So, it >>should be more recognisable as a consonance. Unless you think higher >>limit intervals generally require better tuning, >In this context, yes. The 5/4 is by far the strongest interpretation of >the 4deg12 interval, while the 9/8 and 10/9 have nearly equal claims to >the 5deg31 interval. This is the point I disagree with. You seem to be saying that the closeness to 10/9 makes the tone more dissonant. I say it is the amount by which it deviates from 9/8 that is important. The interval does sound worse than a 5-limit one with similar mistuning. I suggest this is because mistuning adds dissonance to an interval, which will be more noticeable the more dissonant it is to start with, until it becomes so dissonant as to be incomprehensible. Maybe "incomprehensible" isn't the right word there, but hopefully you get the idea. >>but three out of tune >>intervals will still be worse than one. >I was only making an analogy, not a direct comparison. You provided the analogy, I provided the argument. Naturally I give the counter-argument as well. >>I'll write the chord under discussion as Eb-G-A-C#. The G-A interval >>_is_ ambiguous as to 9/8 or 10/9 -- I made a mistake in my working >>before. The chord does sound worse in 31-eq than 1/5 comma meantone or >>schismic temperament, probably because of the poor 9/8. However, that >>interval isn't so bad as to be unimportant. I don't see that its >>proximity to 10/9 makes it _more_ dissonant, but maybe nobody else does >>either. The chord sounds worse in golden meantone, implying that G-A >>really should be 9/8 and not 10/9. >Can you flesh out exactly how you see that one implies the other? Don't >forget that the 28/25 is naturally closer to a 9/8 than a 10/9, so >tuning it to a 10/9 means you've had to distort the other intervals >more. Golden and 1/5 comma meantone (or 43-equal) are roughly as good in the 7-limit. So, any difference in this chord is likely to be a result of the poorer 9-limit approximation to 9/8. Between 31-equal and golden meantone, it is less clear. I generally find the 4-6-9 chord works in 31-eq, but not in golden meantone, suggesting the 9/4 is well enough tuned in 31-eq, but not in golden. However, I find 7-limit harmony usually does work in golden. So, if this chord is poor it's probably because of the 9-limit interval. To confirm this suspicion, I did this: >>I find that G-Eb-A-C# and Eb-G-C#-A sound better than A-Eb-G-C# and >>Eb-A-C#-G in 31-eq. In golden meantone, they all have roughly the same >>consonance. I suggest this is because of the 9/4 vs 10/5. >Come again? That should have been 9/5 rather than 10/5. If you stretch the tone to be a major 9th, it will be interpreted as a 9/4 rather than a 20/9. If you make it a minor 7th, it will be a 9/5 rather than 16/9. So, inversions with a major 9th should sound better than those with a minor 7th if the 9-limit approximation holds. I think this is the case in 31-eq, although this is all subjective. To be sure it's the approximation to 9/8 and octave equivalents that's making the difference, I tried in golden meantone, where the tone approximates the less consonant 10/9 better. Then, there was no contrast in dissonance between the inversions. So, the 9-limit approximation is probably relevant in 31-eq. >>The 31-eq 9/8 and 28/25 both being sharp is the most relevant thing >>here. Ideally, the tempered interval should be between the two just >>ones for them to both be well tuned. >Er, do you really think 28/25 is any kind of "just" interval in the >sense of being a "target" you could really aim for without tuning the >other intervals in the chord? Firstly, I got sharp and flat mixed up there. 9/8 is flat in 31=, or sharp in JI relative to 31=. 28/25 is just in the sense of being an integer ratio. If the other intervals are just, the tone will be exactly 28/25. We are tempering the chord so that the tone looks more like 9/8. If the tone looks less like 9/8 this isn't a good way of tempering: we'd be better off in JI. So, taking this chord in isolation, there's no point in tuning it to 31-eq. However, if you have some other reason for using 31-eq, the chord does work. Now, it may be that 28/25 _is_ perceived as a consonance it itself. However, considering its proximity to 9/8 I see no way of testing this. >>I need to do more listening to be sure of these things. Does anyone >>have a good chord progression that exploits this comma? >Anything that goes from the dominant of the dominant to the augmented >sixth chord is exploiting the vanishing of the 225/224. Okay, how about this: Bb C E C# Bb G -> A -> C# -> A -> G Eb F A G Eb Eb Or, in matrix form: (-1 1 0) (-5 3 0) (-7 0 2) (-7 2 2 0) (-2 0 1)H => (-5 2 1)H => (-5 2 1)H => (-5 2 1 0)H ( 0 0 0) (-3 2 0) (-7 3 1) (-7 2 3 -1) (-5 2 2 -1) (-7 2 2 0) (-2 0 0 1) (-1 1 0) (-5 2 1 0)H ~ ( 0 0 -1 1)H => (-2 0 1)H (-7 2 3 -1) = (-2 0 1 0) ( 0 0 0) (-5 2 2 -1) ( 0 0 0 0) I've tried it in JI and 31, 72 and 46-eq. It works best in 72=, and also works well in JI. It isn't as good in 31=, but still works. The biggest problem here seems to be the dissonance of the magic chord. It works badly in 46=, because the comma (-5 2 2 -1)H doesn't vanish. I find the best way of dealing with it is to tune the magic chord (-7 2 2) (-5 2 1)H (-2 0 1) ( 0 0 0) so the comma is hidden in the chord that implied it in the first place. However you tune that chord in 46-eq, it's going to be a mess. In JI, I hide the comma in the D major chord, but this is too blatant in 46=. On my guitar, I can only play the magic chord in one of its poorer inversions in this context. Then, the contrast in dissonance between it and the tonic is too great. So, I throw in a dominant subminor-seventh chord and that works fine. If you expand the progression to Bb C C C# C# G -> Ab -> A -> A -> A -> Bb Eb Eb F E G G Eb Eb The usual syntonic comma becomes important, so a meantone like 31 or 43= has distinct advantages. Graham Breed gbreed@cix.co.uk www.cix.co.uk/~gbreed