RE: [tuning] Replies to Dan and David on locally concordant tetra ds

Harald Kügler wrote,

>If I compare the harmonic on the 4th fret on the G-String with the
>harmonic on the 5th fret of the B-String,what kind of Komma do I hear?

That's not a comma if you tuned your guitar in 12-tET, since commas are just
intervals. The interval you hear is about 14¢, the difference between a just
major third and a 12-tET major third.

Dan wrote,

>One of the more interesting results of this are the instances of
>subtle tempering that you point out, what I'm not clear on is how much
>this is a part of the process... in other words if you were to set the
>larger limit of the adjacent notes to something on the order of say
>1350¢, would the most concordant stack of 1:2s have slightly tempered
>octaves;

No -- why would you think that???

>If the results did not give an unusually, or
>undesirably tempered stack of 1:2s, then might this also be a good
>model for an adaptive retuning model for other common
>stack-of-an-interval chords, such as the stack of minor and major
>thirds, and the stack of fifths?

Sure!

>something on the order of a synergistic melodic entropy I
>suppose is what I'm imagining here...

Melodic entropy . . . hmm . . . well I understand that what you're trying to
get at is melodic suitability, and I'd be tempted to use a criterion such as
what percentage of the octave species contain two identical tetrachords . .
. but I can't think of a melodic suitability measure that might be related
to an entropy measure . . . can you?

Paul H. Erlich wrote,

> No -- why would you think that???

Just a hunch.

> Sure!

Sure sounds like something John deLaubenfels would be interested in
then. (I'd also be interested in some things like, relative to some of
the 20-tET work that I've done, what say the most concordant stack of
near 240� fifths of an octave are for instance.)

> I'd be tempted to use a criterion such as what percentage of the
octave species contain two identical tetrachords

How would you generalize tetrachordality? Couldn't you take a scale
like the 9-tone 14-tET scale that I recently posted, and using 11 as
the MOS generator, call L s the "tetrachord"? If so, wouldn't this
LsLLsLsLs scale then be "tetrachordal" in every rotation?

Dan

Dan wrote,

>Sure sounds like something John deLaubenfels would be interested in
>then. (I'd also be interested in some things like, relative to some of
>the 20-tET work that I've done, what say the most concordant stack of
>near 240¢ fifths of an octave are for instance.)

This is actually well defined -- start at the stack of 5-tET intervals, and
roll down (Matlab uses the Nelder-Mead simplex [direct search] method) the
total entropy hypersurface to the bottom of the basin. Sometimes, the
starting point is equally close to two basins, one of which is the
mirror-inversion of the other, in which case I listed both.

Using the Farey-100, s=1% diadic curve:

two of them -- 0 226 498 or 0 272 498
three of them -- 0 201 500 701
four of them -- 0 310 505.5 701 1011
five of them -- 0 190 499 699 889 1201 or 0 312 502 702 1011 1201
six of them -- 0 193 502 697 892 1201 1394
seven of them -- 0 195 503 697 1003 1202 1395 1704 or 0 309 502 701 1007
1201 1509 1704
eight of them -- 198 504 699 1007 1201 1398 1705 1900 or 0 195 502 699 893
1201 1396 1702 1900
nine of them -- 0 311 505 701 1011 1201 1511 1707 1901 2212

As you can see, for more than two of them, they all turn into mild meantone
major seconds or minor thirds.

>How would you generalize tetrachordality? Couldn't you take a scale
>like the 9-tone 14-tET scale that I recently posted, and using 11 as
>the MOS generator, call L s the "tetrachord"? If so, wouldn't this
>LsLLsLsLs scale then be "tetrachordal" in every rotation?

Personally, I'd want two "tetrachords" (of however many notes), separated by
a near-4:3, but of course you're free to disagree. However, whatever your
formulation, unless the two "tetrachords" between them subtend most of the
notes of the scale, it doesn't seem like a very useful rule, as most of the
scale is left "free". I think LsL would be better than Ls in your case, for
this reason.

Paul Erlich wrote,

> Personally, I'd want two "tetrachords" (of however many notes),
separated by a near-4:3, but of course you're free to disagree.

In some ways I do agree, however (as it stands; as a handy rule of
thumb), that's just not specific enough. I'm looking for a type of a
'generalized tetrachordality' that is both somewhat "clean" and seems
to match up well with most commonsense expectations.

> However, whatever your formulation, unless the two "tetrachords"
between them subtend most of the notes of the scale, it doesn't seem
like a very useful rule, as most of the scale is left "free".

Right, and again I pretty much agree.

> I think LsL would be better than Ls in your case, for this reason.

Incidentally, the first tetrachordally symmetrical version of this
9-tone scales that I gave (where each scale degree is connected to a
centralized tonic, and where t:a:b was an 18:21:23):

a-b
|
|
|
|
|
bi | a
\ | /
\ | /
\ | /
\ | /
\|/
(a+b)i----------t---------a+b
/|\
/ | \
/ | \
/ | \
/ | \
ai | b
|
|
|
|
|
(a-b)i

gave identical LsL, 23/21 49/46 23/21 "tetrachords," and a 9-tET out
of 14-tET tetrachordal structure that is somewhat synonymous with the
7-tET out of 12-tET dorian tetrachords:

(s L L) L (s L L)
16/15 9/8 10/9 9/8 16/15 9/8 10/9

(L s L) S L S (L
23/21 49/46 23/21 3888/3703 25921/23323 3888/3703 23/21
s L)
49/46 23/21

The tetrachordal arrangement I had given earlier today of this scale:
LsLsLLsLs, could also be said to have two LsLs "tetrachords" separated
by the fourth (or rather the nearest 3:4):

529/324
/|\
/ | \
/ | \
/ | \
/ | \
23/18---+-3703/1944
/|\ | /|\
/ | \ | / | \
/ | \ | / | \
/ | \ | / | \
/ | \|/ | \
1/1----+--161/108--+25921/23328
\ | /|\ | /
\ | / | \ | /
\ | / | \ | /
\ | / | \ | /
\|/ | \|/
7/6----+-1127/648
\ | /
\ | /
\ | /
\ | /
\|/
49/36

Dan