Yahoo Tuning Groups Ultimate Backup tuning tetrachordality

I'm not so sure about tetrachordality.

I've been doing some listening tests, and to me,
the 5-limit JI scale that sounds most like
the 'normal' diatonic 'major' scale is:

1/1 : 9/8 : 5/4 : 4/3 : 3/2 : 27/16 : 243/128 : 1/1
\ / \ / \ / \ / \ / \ / \ /
8:9 9:10 15:16 8:9 8:9 8:9 256:243

where the 5/4 makes a 4:5:6 triad with the 1/1 and 3/2,
and a 'syntonic' tetrachord, but the higher tetrachord
is completely Pythagorean, and gives not only the sharpened
'leading tone' that musicians generally use, but also
a sharper '6th' degree.

This makes sense to me, on analogy with the upper
tetrachord of the 'melodic minor' scale. I wouldn't be
surprised if I tried that with the same tuning as above,
except for 6/5 in place of 5/4, and it sounded 'right'.

Joseph L. Monzo....................monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
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🔗Kraig Grady <kraiggrady@xxxxxxxxx.xxxx>

🔗monz@xxxx.xxx

[Kraig Grady, TD 157.8]
> I believed that Ptolemy mentioned the habit of musicians to use
> higher pitches in one tetrachord than the other.

I would have to check Ptolemy's book to see if that's true,
but I do know that all of the scales he illustrates use
identical tetrachords.

> But possibly you just hear the raised seventh in the harmonic
> context of the dominant!

There was no harmonic context here - I was just playing scales
solo with no other accompaniment.

It's still possible that I was hearing it that way, but
the surprising thing to me was that the *'6th'* sounded
flat if I used 5/3 instead of 27/16!

🔗Kraig Grady <kraiggrady@xxxxxxxxx.xxxx>

monz@juno.com wrote:

> I would have to check Ptolemy's book to see if that's true,
> but I do know that all of the scales he illustrates use
> identical tetrachords.

In Greek musical writingsII Andrew Barker (Cambridge) has charts of mixed
tetrachords and quote Ptolemy as saying
"In every case of a mixture, the "softer" tetrachord (that with the greater
highest interval) falls below the disjunctive tone, the "tenser" above it.
Page 358
-- Kraig Grady
North American Embassy of Anaphoria Island
www.anaphoria.com

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

Ditto on this one.

Joe Monzo wrote,

>I'm not so sure about tetrachordality.

>I've been doing some listening tests,

It is crucial that your listening tests involve playing actual music, rather
than just going up and down the scale. You didn't mention which you were
doing. I suspect you listened to the I chord but not other chords. If you
were purely playing melody, try again with 81/64 instead of 5/4 and see if
you don't like that better.

Putting that aside, let's assume, with Rothenberg (or his supporters), that
differences of a comma are not important in describing the melodic
properties of a scale.

Then in your scale,

>1/1 : 9/8 : 5/4 : 4/3 : 3/2 : 27/16 : 243/128 : 1/1
> \ / \ / \ / \ / \ / \ / \ /
> 8:9 9:10 15:16 8:9 8:9 8:9 256:243

the fact that every octave species has two identical tetrachords, if 5/4 is
replaced with 81/64, makes this scale tetrachordal.

As I see it, the primary opponent of tetrachordality in scale theory is
maximal evenness. It is very fashionable among academic music theorists
right now to view the major scale as a maximally even set of 7 in a 12-tone
universe. But in investigating the decatonic scales, there were two distinct
scale forms that presented themselves, and it so happened that one was
maximally even, while the other was tetrachordal (pentachordal, actually).
In 22-tET, the maximally even scale is (any mode of)

0 2 4 7 9 11 13 15 18 20 (22)

while the pentachordal scale is (any mode of)

0 2 4 7 9 11 13 16 18 20 (22).

The former scale has 8 consonant tetrads, while the latter has 6. Both have
the same number of 709-cent "perfect fifths". But playing around melodically
with both scales, using many modes of each, the latter sounds more pleasing,
comprehensible, and musical. So I think I can conclude that tetrachordality
is more important than maximal evenness for melodic considerations of scale
structure.

🔗monz@xxxx.xxx

[Kraig Grady, TD 158.6]
> In Greek musical writingsII Andrew Barker (Cambridge) has charts
> of mixed tetrachords and quote Ptolemy as saying
> "In every case of a mixture, the "softer" tetrachord (that with
> the greater highest interval) falls below the disjunctive tone,
> the "tenser" above it.
> Page 358

Interesting... exactly what I found myself in my own
listening tests.

[Paul Erlich, TD 158.10]
> If you were purely playing melody, try again with 81/64 instead
> of 5/4 and see if you don't like that better.

Sure, I like the scale with 81/64 (all Pythagorean tetrachords),
but I was coming at it from the other direction: I listened
first to the 5-limit JI scale. It was the '6th' and '7th' lowered
by a comma that didn't sound 'right' to me - they were noticeably
flat in comparison with what I'm used to hearing musicians do.

So first I changed the '7th', knowing that the sharpened 'leading
tone' is a standard part of musical practice. That still didn't
sound 'right', so I raised to '6th' from 5/3 to 27/16, and voila!,
a beautiful-sounding scale, even with the 5/4.

Admittedly, there was no 'musical' context here - it was just
a scale-playing and -listening experiment.

-monz

🔗Carl Lumma <clumma@xxx.xxxx>

>>He depicts some of them on rectangular lattices, and surely the modified
>>linear series keyboard mappings are to be considered intermediate reps.
>
>Such as?

17-tone Tubulong, genus 3^8 * 5 comes to mind.

>>Is there any MOS that isn't "tetrachordal" by its generator?
>
>I don't think so, but there are plenty of "tetrachordal" scales that are not
>MOS.

That's what I thought.

>Doesn't bother me. How about the augmented scale, 0 3 4 7 8 11 in 12-tET (or
>0 4 5 9 10 14 in 15-tET or 0 7 9 16 18 25 in 27-tET)? Two closed chains of
>400 cents separated by a 3/2.

Paul, did you see my subsequent post on this? I first thought you meant
MOS only, but later realized that you meant tetrachordal scales in general.
I'll check out this scale...

>>* Balzano's 9-tone subset of 20-tET. It's not tetrachordal, but what the
>>hell!
>
>Carl, are you using a consistent definition of "tetrachordality" here?

I'm insisting that the tetrachordality occur at a consonance. Otherwise
lots and lots of scales would be "tetrachordal". I guess it still would
require some simplification of the interval matrix, but lots of things do
that; tetrachordality is supposed to work because some intervals are
"special".

-Carl

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

>>>He depicts some of them on rectangular lattices, and surely the modified
>>>linear series keyboard mappings are to be considered intermediate reps.
>
>>Such as?

>17-tone Tubulong, genus 3^8 * 5 comes to mind.

Of course. I was just remembering one paper where he seemed to imply that
the linear and high-limit representations were the two most important ones.

>I'm insisting that the tetrachordality occur at a consonance.

OK -- 7:5 seems a pretty weak melodic consonance though.