the high third is 9/7 ...sometimes

>
> [Jerry Eskelin, TD 459.13]
> John [Link] did some experimenting on his guitar regarding
> the "high third" and ... In general, he experienced the
> "high third" as being considerably higher than Pythagorean.
> He played it for me over the phone (remember those?) and it
> is indeed the third that I commonly hear as the "singer's
> third."

> [John Link, TD 459.21]
> The high third is 9/7 relative to the root. Here's the
> experiment I did with my guitar to confirm that conclusion:
> <etc. ... snip>

Jerry, since you expressed such an interest in my
Robert Johnson webpage,
http://www.ixpres.com/interval/monzo/rjohnson/drunken.htm

and John, since you're a microtonal (xenharmonic?) vocalist,
and that webpage is concerning Johnson's microtonal vocals,
I'd like to point out to both of you one of the most surprising
things I found after doing my analysis:

At the end of the very first line of the song (on the word
'man' in 'I'm a drunken hearted man...'), Johnson sings
a note which is the '3rd' of the tonic D-major chord.
It begins quite a bit below the 12-tET (12-EDO?) '3rd',
then rises quite far above it.

What surprised me was that the ending pitch is *clearly*
far higher than the Pythagorean '3rd' [= 81/64 = ~408 cents].

I hear it as starting out approximately 14 cents below
2^(4/12) [= 400 cents] and rising to ~27 cents above.

The starting pitch is quite clearly implying a 5/4
[= ~386 cents], and I interpreted the ending pitch rationally
as a 32/25 [= ~427 cents].

I considered 9/7 [= ~435 cents] as the ending pitch, but
when I tuned the MIDI track up to that, it sounded too high
and did not (to my ears) accurately reflect Johnson's vocal.

I settled on 32/25 because my criteria, aside from closeness
in pitch, was lowness of prime/integer limit, and those were
the smallest ratio-numbers which fit.

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

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Joe Monzo wrote,

>I settled on 32/25 because my criteria, aside from closeness
>in pitch, was lowness of prime/integer limit, and those were
>the smallest ratio-numbers which fit.

In the context of the discussion with Gerald Eskelin, 32/25 would probably
not be a viable answer, since its numbers (32 and 25) are far to high to
produce any noticeable reduction of beating. Try tuning a 32/25 accurately
by ear, without tuning any intermediary notes. It can't be done. However,
9/7 can be tuned pretty comfortably by ear in the higher register.

As for it being the "natural" vocal tuning for a major triad, certain list
members have expressed the view (with which I agree) that 14:18:21 (or
closer to the voicing Gerald discusses, 14:21:36) is a very nasty-sounding
major triad.

I would like to encourage those of you on this list you have worked with
Indian music to please respond to Gerald's assertion that it is natural to
sing a high third over a root-fifth drone, regardless of cultural
environment.

I wrote,

>I would like to encourage those of you on this list you have worked with
Indian music to please respond to >Gerald's assertion that it is natural to
sing a high third over a root-fifth drone, regardless of cultural
>environment.

Or Renaissance choral music ("high third" meaning higher than 400 cents).

> [me, monz, to John Link]
>> John, since you're a microtonal (xenharmonic?) vocalist,

> [John Link, TD 460.8]
> I've never said or thought either of those things. I just do
> my best to sing in tune.

Well, OK, sorry to label you without your approval... but
as far as anyone on this List (except maybe Johnny Reinhard)
is concerned, your efforts to 'sing in tune' automatically
make you a microtonalist/xenharmonicist/whatever-you-call-it,
because you *sure* aren't singing in 12-tET/12-EDO!

> [Paul Erlich, TD 460.15]
> In the context of the discussion with Gerald Eskelin,
> 32/25 would probably not be a viable answer, since its
> numbers (32 and 25) are far to high to produce any noticeable
> reduction of beating. Try tuning a 32/25 accurately by ear,
> without tuning any intermediary notes. It can't be done.
> However, 9/7 can be tuned pretty comfortably by ear in the
> higher register.

OK, I'm willing to grant that you may be right that what I
said about 32/25 may not be applicable to the discussion with
Gerald. And probably also right about 9/7 being easier to
tune correctly by ear. But in Johnson's song, there's no
other low-prime/odd ratio that comes closer to the pitch he's
singing (as I hear it) than 32/25.

Paul, after reading what I have to say below (and, most
importantly, *listening* to the passage in question), tell
me what you think about why I found 64:75:96 to be so, uh...
'consonant', given the largeness of the integers involved.

Is 75/64 close enough to some other ratio with larger (but
not 'too' large) prime/odd-factors, that it is emulating it?
Or what exactly do you think is going on? I found 64:75:96,
which can be expressed utonally with slightly smaller proportions
as 50/75 : 50/64 : 50/50, to be better here than any of the
smaller-number possibilities that I tried.

You may also want to look again at my posting on direct/indirect
lattice connections in Onelist TD 132; I think what I noted
there has a direct bearing on this.

> [John Link]
> Thanks for bringing this to my attention. I will tune my guitar
> so that I can hear 32/25. Note the following:
>
> (9/7) / (32/25) = 225/224 = 1.00446
>
> 1200*(log(225/224)/log(2)) = 7.71
>
> So the two intervals vary by less than one half of one percent,
> just under 8 cents.

Ah, very interesting...

Of course, we've discussed this particular interval quite
a bit in this forum already earlier this year, partly because
it was important in Fokker's earliest musical theories (c. 1949).

I have another interesting observation on this interval. It pertains to
the new JI retuning I've just done of my piece
_3 Plus 4_:
http://www.ixpres.com/interval/monzo/3plus4/3plus4ji.mid

I was going to save this for the analytical webpage of that
tune, but since you brought it up...

At the climax of the main part of the tune (just before the
'hook' - the first occurrence is at 0:20), I wanted a darker
sound for the cadence onto D#-minor than could be gotten with
the 'usual minor' chord with proportions 10:12:15
[== 1/6 : 1/5 : 1/4].

After experimenting with lots of different 'minor 3rds', I finally
settled on 64:75 [= ~275 cents], which is 224:225 higher than 7/6
[= ~267 cents], and which I would normally go for when trying to
get that 'darker' 'minor' sound.

I had tried 7/6 first, and it didn't sound right at all - the
whole chord sounded badly out of tune. Then I tried 19/16
[= ~298 cents], and that was better in-tune, but not 'dark'
enough. So then I just altered the amount of pitch-bend by
eye/ear until I got the sound I wanted, and it was darn close
to 75/64, so that's how I finally retuned the '3rd'.

The big surprise to me was that there was such an audible
difference between 75/64 and 7/6 [= ~8 cents difference],
more so than between 19/16 and 75/64, which are ~23 cents apart!

My guess is that some of that perception depends on the
musical context, and some of it on prime-limit or something
resembling or related to it.

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------