RE: magic chord, reply to Paul Erlich

>>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

Benjamin Tubb wrote,

>Just Intonation Ratio Limits [with respect to the denominator]

The term "limit" is normally used to refer to the highest odd or prime
factor to be found looking at BOTH the numerator and denominator. If the
denominator alone is really intended, then it is probably not valid to
use octave equivalents as you have done.

>Does anyone have any comment as to why the following list of "just"
intervals
>isn't otherwise accepted for the most "consonant" use insofar as they
are based
>on the second most consonant interval of a Perfect Fifth besides the
Octave.

The ear doesn't care what an interval is "based" on. The various
psychoacoustical models of dissonance (covered now and then on the list)
essentially state that the simplest ratios will be most easily
perceived, but beyond a certain point (about 17/13 or 19/13 in various
members' experience) the exact ratio (if there is one) ceases to be
relevant and the degree of approximation to simpler ratios is the only
important factor. Thus, in your list of Pythagorean ratios,

M3 8182/6561 384.36

(which should really be 8192/6561) is actually very consonant, despite
its complex ratio, since it is only 2 cents off a just 5/4.

>Essentially,
>I'd like to know what is the definition of "just" intonation and how
"should"
>it be applied.

Just Intonation will typically give many different ratios for a given
interval, depending on how that interval is arrived at via simple,
consonant ratios. The typical use of the term really means 5- (prime)
limit JI, as opposed to 3- (prime) limit JI which is referred to as
Pythagorean tuning, and higher limits which have no fully accepted
terminology as of yet. I think it should only be applied within the
limits of what it is possible to hear. Listen!

Paul H. Erlich wrote:
> essentially state that the simplest ratios will be most easily
> perceived, but beyond a certain point (about 17/13 or 19/13 in various
> members' experience) the exact ratio (if there is one) ceases to be
> relevant and the degree of approximation to simpler ratios is the only
> important factor.

Speaking for my own experience, I don't perceive there to be any sort
of absolute cutoff point either (beyond thus and so ratio). Again
speaking in generalities, as the ratio becomes more complex, it becomes
more difficult to attribute an audibly intuitive meaning to.

Also, simpler ratios seem to claim more space around them than more
complex ones. By that I mean that anything within something on the
order of 75c of a simple ratio like an octave seems to be perceived as
an approximation of an octave. But the "claim zone" as I've called it
in the past, of a more complex ratio, such as 5:4 is much narrower.
400c (13c sharp) clearly seems to be an approximation of 5:4, but you
don't have to go much sharper than that before it starts sounding more
like a flat 9:7.

Or once again, that's my own personal experience anyway.