Yahoo Tuning Groups Ultimate Backup tuning bob v's 12-tone (3,19) tuning (was: G Bv D)

> From: Robert C Valentine <BVAL@IIL.INTEL.COM>
> To: <tuning@yahoogroups.com>
> Sent: Wednesday, February 06, 2002 11:47 PM
> Subject: [tuning] Re: G Bv D
>
>
> [Gerry Eskelin] came to this list in search of the
> "high third" (which he seemed to believe was a third
> that would 'lock' IN ADDITION to the 5/4, beat-free
> third). After listening to a bunch of major triads,
> the one he seemed to believe best matched his experience
> had a third of around 404c. (He did seem to think that
> 408 was too high).
>
> It may well be that all of Western music theory should
> be based on
>
> 1/1 9/8 24/19 4/3 3/2 32/19 36/19
>
> or casting it into a 12 tone system
>
> 1/1 19/18 9/8 19/6 24/19 4/3 27/19 3/2 19/12 32/19 57/32 36/19

that 4th number should be 19/16

> with a 3 and 19 lattice
>
> Db Ab Eb Bb
> F C G D
> A E B F#
>
> and that {3,19} is the spiritual coupling with all of creation
> rather than just little integers, but thats probably a different
> discussion.

nice, bob, i like this! i've been an enthusiastic advocate
of 19-limit tuning for a long time.

the beauty of construction in your scale shows in the
lattice, but not in the ratios. here's a complete breakdown:

2 3 19 ratio ~cents

[ 2 2 -1] 36/19 B 1106.396986
[-5 1 1] 57/32 Bb 999.468017
[ 5 0 -1] 32/19 A 902.4869839
[-2 -1 1] 19/12 Ab 795.5580153
[-1 1 0] 3/2 G 701.9550009
[ 0 3 -1] 27/19 F# 608.3519865
[ 2 -1 0] 4/3 F 498.0449991
[ 3 1 -1] 24/19 E 404.4419847
[-4 0 1] 19/16 Eb 297.5130161
[-3 2 0] 9/8 D 203.9100017
[-1 -2 1] 19/18 Db 93.6030144
[ 0 0 0] 1/1 C 0

i just thought i'd note that a straight Pythagorean 3^(-5...6)
system deviates no more than +/- ~3.38 cents from your (3,19)
lattice above.

comparison of Pythagorean with your lattice involves the 3==19
xenharmonic bridge (... at least i think it does; paul and i
are having a discussion about my meaning for this term over
on tuning-math this week).

as i quote from my Dictionary entry for "schisma",
http://www.ixpres.com/interval/dict/schisma.htm
i also call this interval the ...

>> nondecimal schisma, which is the difference between
>> 19^1 [= 19/16 = 2.98 Semitone = ~ 297.513 cents]
>> and the 'standard' Pythagorean "minor 3rd"
>> 3^-3 [= 32/27 = 2.94 Semitone = ~ 294.134 cents],
>> and which also has an interval size of approximately
>> 0.04 Semitones [3^3 * 19^1 = 513/512 = ~ 3.378 cents].

you say that Gerry felt that the Pythagorean third 81:64
= ~408 cents was too high, which means that to him there
is a perceptible difference between 81:64 and 24:19.
this difference is precisely the same nondecimal schisma
i refer to above. so here's an example of someone who
audibly perceives and places some importance on this
small interval. paul, please take note.

note that Eratosthenes made use of a different 3==19 bridge:
[2 3 5 19] [ 6 -5 -1 1] = 1216:1215 = ~1.424297941 cents
as long ago as the 200s BC (during Macedonian rule of Greece),
as i wrote in message 26618 (Thu Aug 2, 2001 6:31pm)
/tuning/topicId_26618.html#26618?expand=1

-monz

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🔗monz <joemonz@yahoo.com>

> From: monz <joemonz@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Thursday, February 07, 2002 1:15 AM
> Subject: [tuning] bob v's 12-tone (3,19) tuning (was: G Bv D)
>

>
> > From: Robert C Valentine <BVAL@IIL.INTEL.COM>
> > To: <tuning@yahoogroups.com>
> > Sent: Wednesday, February 06, 2002 11:47 PM
> > Subject: [tuning] Re: G Bv D
> >
> >
> > ...
> >
> > It may well be that all of Western music theory should
> > be based on
> >
> > ...
> >
> > 1/1 19/18 9/8 19/6 24/19 4/3 27/19 3/2 19/12 32/19 57/32 36/19
>
>
> that 4th number should be 19/16
>
>
> > with a 3 and 19 lattice
> >
> > Db Ab Eb Bb
> > F C G D
> > A E B F#
> >
> > and that {3,19} is the spiritual coupling with all of creation
> > rather than just little integers, but thats probably a different
> > discussion.
>
>
> comparison of Pythagorean with your lattice involves the 3==19
> xenharmonic bridge ... [3^3 * 19^1 = 513/512 = ~ 3.378 cents].

>
> ...
>
> note that Eratosthenes made use of a different 3==19 bridge:
> [2 3 5 19] [ 6 -5 -1 1] = 1216:1215 = ~1.424297941 cents
> as long ago as the 200s BC (during Macedonian rule of Greece),
> as i wrote in message 26618 (Thu Aug 2, 2001 6:31pm)
> /tuning/topicId_26618.html#26618?expand=1

since that post to which i provided a link refers to this one
/tuning/topicId_7159.html#7159?expand=1

i decided that i should mention also that the 3==19 bridge
in Bob's system, [3 19] [3 1], which i also call the
"nondecimal schisma", was a feature of the tuning system
of the venerable theorist Boethius, c. 505 AD.

it appeared between his "fixed note" _paramese_
= [2 3 19] [-3 2 0] = 9/8 = ~203.9 cents, and the
nearby "variable note" _chromatic paranete synemmenon_
= [2 3 19] [6 -1 -1] = 64/57 = ~200.5 cents.

(see my 'Tutorial on ancient Greek Tetrachord-theory'
if you don't understand this)
http://www.ixpres.com/interval/monzo/aristoxenus/tutorial.htm

-monz

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🔗paulerlich <paul@stretch-music.com>

🔗monz <joemonz@yahoo.com>

> From: paulerlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Thursday, February 07, 2002 3:24 AM
> Subject: [tuning] Re: bob v's 12-tone (3,19) tuning (was: G Bv D)
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > so here's an example of someone
>
> maybe -- we're speculating quite a bit. when i originally suggested
> the 19-limit Utonal explanation of Gerry's major chord, he didn't buy
> it for a second.
>
> > who
> > audibly perceives and places some importance on this
> > small interval.
>
> so that would be the opposite of a xenharmonic bridge.

i just posted something big to tuning-math about xenharmonic
bridges in the "comma pump". let's discuss and tighten up
the definition of "xenharmonic bridge" before i respond to this.

-monz

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