46

Errors of primes in cents, to the 17-limit: 2.4, 5.0, -3.6, -3.5,
-5.75, -0.6

43+3 15/46

The mapping [11,1,-19,-17,-12,13] shows we have a major third
generator, but not magic.

41+5 9/46

The mapping [3,17,-1,13] suggests this is a good system for exploring
3 and 7 harmonies, without getting into much else. The generator is
an approximate 8/7, and related systems, with the same mapping, are
8/41 and 17/87. The 11-note scale 818181811 would allow for 7
subminor and 4 supermajor triads; there are also 7 1-3/2-7/4
incomplete tetrads.

34+12 4/23

The mapping [9,-4,-1,-3,4] suggests using this to explore 5,7, and 11
harmonies. We have a 10-note scale (44447)x2 which could work for
this, with two 7:10:11 chords; for more chords and intervals, as well
as a strange but interesting 12-tone temperament, we have (444443)x2.

31+15 3/46

The mapping [9,5,-3,7] suggests using this one for the 11-limit, and
the 15-note scale/temperament 333333333333334 works well for this.

27+19 19/46

The mapping [7,9,13,-15,10,-16] shows this one has good 7-limit
possibilities, with the addition of 13s. We have scales of size 8 and
11, namely 55755757 and 55525552552, both of which are supplied with
5:6:7 and 10:12:13:14 chords, and if we need tetrads we can go all
the way to the 19--3232322323232232322.

24+22 2/23

The mapping [-5,-8,-2,-6,8,13] shows this has interesting 11-limit
possibilities, and we can temper a 22 or 24 note scale by it, via
(2)x10 3 or (2)x11 1. One way to cure what ails quarter-tone music?

--- In tuning@y..., genewardsmith@j... wrote:

> 24+22 2/23
>
> The mapping [-5,-8,-2,-6,8,13] shows this has interesting 11-limit
> possibilities, and we can temper a 22 or 24 note scale by it, via
> (2)x10 3 or (2)x11 1. One way to cure what ails quarter-tone music?

Correct me if I'm wrong, but didn't Graham say that there would be no
11-limit hexads in the 22-tone MOS? From this, it looks like the 11-
limit complexity is only 20, so there should be a couple . . .
Graham? Am I confusing two different temperaments again?

The Shrutar scale is an omnitetrachordal variant of (2)x10 3. It
allows the entire Modern Indian Gamut to be played very well in tune,
and puts many 7- and 11-limit relationships in the positions occupied
by the hypothetical shrutis of Indian theory -- that is, all the
intervals of the Modern Indian Gamut will be subtended by the same
number of notes in this scale as they are by shrutis in the Indian
theory.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., genewardsmith@j... wrote:

> > 24+22 2/23
> >
> > The mapping [-5,-8,-2,-6,8,13] shows this has interesting 11-
limit
> > possibilities, and we can temper a 22 or 24 note scale by it, via
> > (2)x10 3 or (2)x11 1. One way to cure what ails quarter-tone
music?

> Correct me if I'm wrong, but didn't Graham say that there would be
no
> 11-limit hexads in the 22-tone MOS?

I don't know. Graham?

From this, it looks like the 11-
> limit complexity is only 20, so there should be a couple . . .
> Graham? Am I confusing two different temperaments again?

There seem to be two 11-limit hexads in the 22 MOS, and four in the
24 MOS.

> The Shrutar scale is an omnitetrachordal variant of (2)x10 3. It
> allows the entire Modern Indian Gamut to be played very well in
tune,
> and puts many 7- and 11-limit relationships in the positions
occupied
> by the hypothetical shrutis of Indian theory -- that is, all the
> intervals of the Modern Indian Gamut will be subtended by the same
> number of notes in this scale as they are by shrutis in the Indian
> theory.

Hmmm...I wonder if scales in 12+10;24+22 muddles would be good for
quasi-Indian music?

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., genewardsmith@j... wrote:

> > 24+22 2/23
> >
> > The mapping [-5,-8,-2,-6,8,13] shows this has interesting 11-
limit
> > possibilities, and we can temper a 22 or 24 note scale by it, via
> > (2)x10 3 or (2)x11 1. One way to cure what ails quarter-tone
music?

> Correct me if I'm wrong, but didn't Graham say that there would be
no
> 11-limit hexads in the 22-tone MOS?

I don't know. Graham?

From this, it looks like the 11-
> limit complexity is only 20, so there should be a couple . . .
> Graham? Am I confusing two different temperaments again?

There seem to be two 11-limit hexads in the 22 MOS, and four in the
24 MOS.

> The Shrutar scale is an omnitetrachordal variant of (2)x10 3. It
> allows the entire Modern Indian Gamut to be played very well in
tune,
> and puts many 7- and 11-limit relationships in the positions
occupied
> by the hypothetical shrutis of Indian theory -- that is, all the
> intervals of the Modern Indian Gamut will be subtended by the same
> number of notes in this scale as they are by shrutis in the Indian
> theory.

Hmmm...I wonder if scales in 12+10;24+22 muddles would be good for
quasi-Indian music?

--- In tuning@y..., genewardsmith@j... wrote:

> There seem to be two 11-limit hexads in the 22 MOS, and four in the
> 24 MOS.

Double that if you include Utonal ones.
>
> > The Shrutar scale is an omnitetrachordal variant of (2)x10 3. It
> > allows the entire Modern Indian Gamut to be played very well in
> tune,
> > and puts many 7- and 11-limit relationships in the positions
> occupied
> > by the hypothetical shrutis of Indian theory -- that is, all the
> > intervals of the Modern Indian Gamut will be subtended by the
same
> > number of notes in this scale as they are by shrutis in the
Indian
> > theory.
>
> Hmmm...I wonder if scales in 12+10;24+22 muddles would be good for
> quasi-Indian music?

Well, the Modern Indian Gamut comes out as 4 4 4 3 4 4 4 4 4 4 3 4 in
46-tET, which in terms of the hypothetical shrutis, it is stated as 2
2 2 1 2 2 2 2 2 2 1 2 . . . Indian music uses a very large number of
subsets of this, most with 7 notes apiece . . . my idea is that
dividing the 4s into 2+2 in 46 will allow quasi-Indian effects that
both conform to the hypothetical shruti divisions and give nice 7-
and 11-limit harmonies . . . now go ahead and show me how this all
related to muddles.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

> Well, the Modern Indian Gamut comes out as 4 4 4 3 4 4 4 4 4 4 3 4
in
> 46-tET, which in terms of the hypothetical shrutis, it is stated as
2
> 2 2 1 2 2 2 2 2 2 1 2 . . .

That means the Gamut is actually (7+5);34+12--here I'm putting in
parenthesis because if I'm going to talk about both muddles and non-
symmetrical scales I don't want *them* muddled. This seems to explain
your question to Graham, since the mapping for 34+12 is [9,-4,-1,-3,4]
and so we have an 11-limit complexity of 22. On the other hand as I
pointed out you have a great deal going on with 5,7 and 11, leaving
apart the 3, and 17 is in there also.

This suggests looking at the 7;7+5;(7+5);34+12 muddle for starters,
which is to say the 7-note scales one gets by treating the Gamut as
if it were 12-et and we were looking at diatonic scales. We get
scales such as 0-8-15-19-27-35-42 and 0-7-15-19-27-34-38, with three
interval sizes suitable for the Stearns Tribonnaci treatment.

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > Well, the Modern Indian Gamut comes out as 4 4 4 3 4 4 4 4 4 4 3
4
> in
> > 46-tET, which in terms of the hypothetical shrutis, it is stated
as
> 2
> > 2 2 1 2 2 2 2 2 2 1 2 . . .
>
> That means the Gamut is actually (7+5);34+12--here I'm putting in
> parenthesis because if I'm going to talk about both muddles and non-
> symmetrical scales I don't want *them* muddled. This seems to
explain
> your question to Graham, since the mapping for 34+12 is [9,-4,-1,-
3,4]
> and so we have an 11-limit complexity of 22.

OK -- so you must understand what Graham meant -- but can you explain
to _me_ what this means? For example, what are the generators and
intervals of repetition for 22+24 and for 34+12?
>
> This suggests looking at the 7;7+5;(7+5);34+12 muddle for starters,
> which is to say the 7-note scales one gets by treating the Gamut as
> if it were 12-et and we were looking at diatonic scales. We get
> scales such as 0-8-15-19-27-35-42

That's the Indian Diatonic scale: in JI it's 9/8 10/9 16/15 9/8 9/8
10/9 16/15.

> and 0-7-15-19-27-34-38, with three
> interval sizes suitable for the Stearns Tribonnaci treatment.

The Stearns Tribonacci treatment (which I believe was inspired by my
question here a year ago and my subsequent reporting of the
Tribonacci constant) would be terrific (perhaps "optimal") if one
were interested in emphasizing the three-step-size nature of the
scale (as well as any "muddle" -- correct?). However, I believe that
it sounds much more "Indian" in JI as well as in tunings where the
two whole steps are equal. 46 is acceptable, and 34 is
borderline . . .

I think I'm catching on . . . so the usual specification of the
tuning of the 22 shrutis is (12+10);12+34, but the Shrutar scale is
(12+10);22+24. Have I got this right?

I wrote,

> I think I'm catching on . . . so the usual specification of the
> tuning of the 22 shrutis is (12+10);12+34, but the Shrutar scale is
> (12+10);22+24. Have I got this right?

But the fact that 12 appears on both the left and right sides of
(12+10);12+34 means that it is not a muddle at all, but an MOS --
right?

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

> But the fact that 12 appears on both the left and right sides of
> (12+10);12+34 means that it is not a muddle at all, but an MOS --
> right?

It's a MOS because the parenthesis around 12+10 mean that we take 12
generators in one set, and 10 in another, so that (12+10);12+34 would
temper 22 notes diaschismically, using the 4/23 generator. You could
then go to paultone muddles like (6+4);(12+10);12+34 if you wanted to,
meaning you treat (12+10);12+34 as if it were 22-equal and look at
what your (6+4);12+10 scales turn into--something horrid for
Halloween, perhaps, though I suspect it might actually be quite nice.

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > But the fact that 12 appears on both the left and right sides of
> > (12+10);12+34 means that it is not a muddle at all, but an MOS --
> > right?
>
> It's a MOS because the parenthesis around 12+10 mean that we take
12
> generators in one set, and 10 in another, so that (12+10);12+34
would
> temper 22 notes diaschismically, using the 4/23 generator.

Which is a perfect fifth generator modulo the half-octave interval of
repetition, right? . . . but was I right that if a number appears on
both the left and right sides of a muddle, it's an MOS?

> You could
> then go to paultone muddles like (6+4);(12+10);12+34 if you wanted
to,
> meaning you treat (12+10);12+34 as if it were 22-equal and look at
> what your (6+4);12+10 scales turn into--something horrid for
> Halloween, perhaps, though I suspect it might actually be quite
nice.

Well, the 7-limit approximations of paultone get broken (I think all
of them do), but it does seem as if the 10 tones specified in early
Indian theory (7 diatonic notes plus Antara Ga and Kakali Ni, in two
transpositions a 3/2 apart, with a syntonic comma difference in one
note between the transposed forms) are just such a muddle -- see my
paper if you care to verify this -- or tell me if the ratios

sa-grama ma-grama
Ni 1/1 Ga 4/3
kakali Ni 3/sqrt(2) [~16/15] antara Ga sqrt(2) [~45/32]
Sa 9/8 Ma 3/2
Ri 5/4 Pa 5/3
Ga 4/3 (16/9)
antara Ga 45/32 Dha 15/8
Ma 3/2 Ni 1/1
(8/5) kakali Ni 16/15
Pa 27/16 Sa 9/8
Dha 15/8 Ga 5/4

appear to agree with the muddle you proposed.

Note that the modern Indian note names have rotated by one place
relative to the modern ones. I base my conclusion that the diaschisma
was ignored upon the statement in the early texts that the two forms
sa-grama and ma-grama were identical except that Pa was a bit higher
in sa-grama (looks like a syntonic comma). I reflected this above by
using sqrt(2) to represent 45/32 tempered by part of a diaschisma (in
this case, half).

Hopefully, from the last posts, you can see why I'm using a straight-
across fretting but tuning some strings to 1/1 and some strings to
3/2 (which I mentioned in the post before that) -- each set of
strings will give one of the two gramas!

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

> > It's a MOS because the parenthesis around 12+10 mean that we take
> 12
> > generators in one set, and 10 in another, so that (12+10);12+34
> would
> > temper 22 notes diaschismically, using the 4/23 generator.

> Which is a perfect fifth generator modulo the half-octave interval
of
> repetition, right? . . .

Exactly. This is a pretty good paultone variant, I think.

but was I right that if a number appears on
> both the left and right sides of a muddle, it's an MOS?

It's not a muddle if it is a double-decker, a muddle is where it is a
sandwich (or worse.) (12+10);12+34 means we take the perfect fifth
generator within sqrt(2) you mentioned, and do a set of 12 vs a set
of 10, rather than two sets of 11, to produce a 22-tone scale.

Thanks for including the Indian scales, I'll take a look at them.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

> sa-grama ma-grama
> Ni 1/1 Ga 4/3
> kakali Ni 3/sqrt(2) [~16/15] antara Ga sqrt(2) [~45/32]
> Sa 9/8 Ma 3/2
> Ri 5/4 Pa 5/3
> Ga 4/3 (16/9)
> antara Ga 45/32 Dha 15/8
> Ma 3/2 Ni 1/1
> (8/5) kakali Ni 16/15
> Pa 27/16 Sa 9/8
> Dha 15/8 Ga 5/4
>
> appear to agree with the muddle you proposed.

These both turn out to be 4444474447 when rendered into 46-et, or in
other words (6+4);34+12--not a muddle at all, but scales I would have
mentioned if I was mentioning these types of scales, though I've been
avoiding doing that. It's 46-tempered Standard Pentachordal Major.

Hello, there, everyone, and I find it hard to resist adding another
view of 46-tET, just to fill out the discussion and show how different
people can view these tunings in different ways.

In an approach of the kind described here, reflecting a neo-Gothic
style based largely on 13th-14th century European music, a subset such
as 12 or 24 notes generated by a regular chain of fifths makes an
excellent tuning, and one very characteristic of this style.

---------------------------------------
1. 46-tET and the 11:14 eventone tuning
---------------------------------------

The 46-tET division is virtually identical to a regular or "eventone"
tuning with major thirds at a pure 11:14 (~417.51 cents), with fifths
at ~704.38 cents, or ~2.42 cents wide.

In 46-tET, fifths are 27/46 octave or ~704.35 cents, or ~2.39 cents
wide, producing regular major thirds at 16/46 octave or ~417.39 cents.

Fifths are tempered at or extremely close to 1/4 of the 891:896 comma
(~9.69 cents) defining the difference between 11:14 and the regular
Pythagorean major third produced by four pure fifths at 64:81 (~407.82
cents).

While 46-tET has a mathematically precise closure, an 11:14 eventone
can likewise be carried to a circulating 46-note system, with one
"odd" fifth tempered at about 1.342 cents less than the others, or
~703.035 cents, or ~1.080 cents wide.

This very small difference over a 46-note tuning circle may further
suggest the near-identical nature of these two systems.

--------------------------------------------
2. Main features in a 12-note regular tuning
--------------------------------------------

Either of these almost identical tunings, like other regular
temperaments in the "704-cent neighborhood" ranging from around 29-tET
(~703.45 cents) to somewhere around the "e-based temperament" (~704.61
cents), features mildy tempered fifths and two main families of thirds
in a 12-note tuning:

(1) Regular major and minor thirds in the general
vicinity of 11:14 and 11:13 or 28:33; and

(2) "Submajor/supraminor thirds" -- actually
diminished fourths and augmented seconds --
in the vicinity of 17:21 and 14:17.

In 46-tET, the virtually just major thirds at ~11:14 are
accompanied by minor thirds (11/46 octave) of around 286.96 cents,
about midway between 11:13 (~289.21 cents) and 28:33 (~284.45 cents).

Submajor/supraminor thirds at around 365.22 cents (14/46 octave) and
339.13 cents (13/46 octave) nicely exemplify this category, and might
be described as "reverse Pythagorean" thirds about as far from the
simplest ratios of 4:5 and 5:6 as the regular thirds of Pythagorean
intonation, but in the opposite directions. This gives them an
engaging and active quality inviting effective cadential resolutions.

Along with these two excellent types of thirds go very pleasant
melodic qualities. Diatonic semitones are around 78 cents (3/46
octave), and whole-tones around 209 cents (8/46 octave), providing
melodic contrast; the compact semitones make directed cadences more
efficient (e.g. major third expanding to fifth and major sixth to
octave).

Chromatic semitones in 46-tET at 5/46 octave (~130.43 cents) play a
very important role in resolutions of sonorities involving the
submajor and supraminor thirds, often respectively expanding to fifths
and contracting to unisons.

-----------------------------------
3. Going to 24 notes: near-7 ratios
-----------------------------------

While all these features are evident in a regular 12-note tuning,
corresponding to a Renaissance meantone, extending the tuning to 24
notes reveals some other features such as the melodic diesis and some
intervals approaching ratios of 7 (2-3-7-9 odd).

Approximate ratios of 7 involve chains of 13, 14, or 15 fifths up or
down, with the diesis of around 52.17 cents (2/46 octave) serving as a
kind of cadential semitone for resolutions of the large major third
(17/46 octave, ~443.48 cents) to a stable fifth, or of the large major
sixth (36/46 octave, ~939.13 cents) to an octave, etc.

These ratios, although not as accurate as those obtainable by slightly
increasing the tempering of the fifths to around 704.61 cents (~2.65
cents wide), have their own "mood," with the large major third having
some qualities of either a 7:9 (~435.08 cents) or a more complex ratio
such as 17:22 (~446.36 cents). The near-4:7 seventh of 37/46 octave at
~965.22 cents (~3.61 cents narrow), and the near-6:7 of 10/46 octave
at ~260.87 cents (~6.00 cents narrow), are closer to 7-based ratios.

Carrying the tuning to 24 notes also generates some other types of
intervals, for example thirds at 21 fifths up or 15/46 octave (~391.30
cents) and 20 fourths up or 12/46 octave (~313.04 cents), quite close
to 4:5 (~386.31 cents) and 5:6 (~315.64 cents).

In an "11:14 eventone" approach to 46-tET, these are rather exotic
intervals, occurring in a few positions in a 24-note tuning and often
inviting diesis shifts to other flavors of thirds then resolving in
more conventional cadences.

In other approaches to 46-tET, these intervals may be defined as the
usual thirds, resulting in quite different musical approaches.

---------------------------------------
4. A 17-note well-temperament variation
---------------------------------------

A regular 12-note tuning in 46-tET or an 11:14 eventone (e.g. Eb-G#)
may be converted to a circulating 17-note well-temperament by adding
six more "remote" fifths (G#-D#-A#-Gb/E#-Db/B#-Ab-F##)[1] tempered at
around 1/4 of the 63:64 septimal comma (~27.26 cents), with the
mathematics slightly varying depending on which system one is using.
For example, with an 11:14 eventone, we might temper five of these
fifths by precisely 1/4 septimal comma (708.77 cents, ~6.82 cents
wide), with the last "odd" fifth at ~708.00 cents (~6.04 cents wide).

Here the more remote transpositions have a mood like that of 22-tET,
with near-7:9 major thirds, rather heavily tempered fifths, and narrow
semitones in the range of 56-60 cents (compare ~54.55 cents in 22-tET),
providing an interesting contrast to the "usual" conditions over the
nearer 12-note range.

In this kind of well-temperament, the fifths, as the main concords in
a neo-Gothic kind of style, seem analogous to thirds in an
18th-century well-temperament. We have mildly tempered fifths in the
"near" part of the circle, and heavily tempered ones in the remote
part of the circle, somewhat like the meantone-like and Pythagorean
thirds of an historical 12-note unequal temperament.

The mixture of fifths also generates a continuum of neutral thirds and
sixths, further enriching the intonational pallette.

----
Note
----

1. This spelling suggests looking at the added fifths in a sharpward
direction -- we could also spell these fifths and more remote
accidentals closing the circle Eb-Ab-Db-Gb-Cb/A#-Fb/D#-Bbb/G#, taking
a flatward point of view.

Most appreciatively,

Margo Schulter
mschulter@value.net

--- In tuning@y..., mschulter <MSCHULTER@V...> wrote:

> The 46-tET division is virtually identical to a regular
or "eventone"
> tuning with major thirds at a pure 11:14 (~417.51 cents), with
fifths
> at ~704.38 cents, or ~2.42 cents wide.

It seems a little strange to me to call 14/11 a major third, though
from the point of view of this mapping it's a third of some kind, and
not a "unidecimal diminished fourth", which is what Manuel lists it
as. Why not just call it a 14/11?

It's interesting to see this posting, as I thought of including the
29+17 system, thinking of you and knowing it would be something like
this, but when I looked at the mapping of [1,21,15,11,8] it didn't
seem so interesting. The 14/11 and 13/11 seemed a little extreme to
me, but we also have some 7/5 (or 13/9) in the span of seven fifths,
and out a little farther we find 13/8, 13/12, and 11/9.

> Either of these almost identical tunings, like other regular
> temperaments in the "704-cent neighborhood" ranging from around 29-
tET
> (~703.45 cents) to somewhere around the "e-based temperament"
(~704.61
> cents),

How does e get into it, or do you mean something other than exp(1)?

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > > It's a MOS because the parenthesis around 12+10 mean that we
take
> > 12
> > > generators in one set, and 10 in another, so that (12+10);12+34
> > would
> > > temper 22 notes diaschismically, using the 4/23 generator.
>
> > Which is a perfect fifth generator modulo the half-octave
interval
> of
> > repetition, right? . . .
>
> Exactly. This is a pretty good paultone variant, I think.

Not really . . . it's not using 46-tET's best approximations of
ratios of 7 (if I'm understanding correctly). Paultone requires that
64:63 vanish.
>
> > but was I right that if a number appears on
> > both the left and right sides of a muddle, it's an MOS?
>
> It's not a muddle if it is a double-decker, a muddle is where it is
a
> sandwich (or worse.)

You've lost me.

> (12+10);12+34 means we take the perfect fifth
> generator within sqrt(2) you mentioned, and do a set of 12 vs a set
> of 10, rather than two sets of 11, to produce a 22-tone scale.

I think I understand that . . . but what's the double-decker sandwich
business?

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > sa-grama ma-grama
> > Ni 1/1 Ga 4/3
> > kakali Ni 3/sqrt(2) [~16/15] antara Ga sqrt(2) [~45/32]
> > Sa 9/8 Ma 3/2
> > Ri 5/4 Pa 5/3
> > Ga 4/3 (16/9)
> > antara Ga 45/32 Dha 15/8
> > Ma 3/2 Ni 1/1
> > (8/5) kakali Ni 16/15
> > Pa 27/16 Sa 9/8
> > Dha 15/8 Ga 5/4
> >
> > appear to agree with the muddle you proposed.
>
> These both turn out to be 4444474447 when rendered into 46-et, or
in
> other words (6+4);34+12--not a muddle at all,

Why is that not a muddle? (But I don't think it's correct -- see
bottom of this message)

> but scales I would have
> mentioned if I was mentioning these types of scales, though I've
been
> avoiding doing that.

How is this different from scales you've mentioned so far?

>It's 46-tempered Standard Pentachordal Major.

Only if you identify the 22 shrutis with 22-tET, and even then not
quite. Here's the above table with the ancient shruti specifications,
which I left out before:

shruti sa-grama ma-grama
0 Ni 1/1 Ga 4/3
2 kakali Ni 3/sqrt(2) [~16/15] antara Ga sqrt(2) [~45/32]
4 Sa 9/8 Ma 3/2
7 Ri 5/4 Pa 5/3
9 Ga 4/3 (16/9)
11 antara Ga 45/32 Dha 15/8
13 Ma 3/2 Ni 1/1
15 (8/5) kakali Ni 16/15
17 Pa 27/16 Sa 9/8
20 Dha 15/8 Ga 5/4

Standard Pentachordal Major is 0 2 4 7 9 11 13 16 18 20 -- a (6+4)
scale. This, however, is a (5+5) rather than (6+4) scale (if I'm
using that notation correctly).

Hi Margo . . .

Certainly a chain of fifths in 46 is interesting, but what Gene and I
have been noting is that, from the point of view of ratios of 7 and
ratios of 11, you can do more interesting things with two or four
chains of fifths in 46. Try two chains of fifths a half-octave or
2/23-oct. apart, or four chains at 1/23-oct. intervals from one
another.

P.S. If you haven't yet, you simply must play with two 19-tET chains
a half-octave apart on your two keyboards.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

> > Exactly. This is a pretty good paultone variant, I think.

> Not really . . . it's not using 46-tET's best approximations of
> ratios of 7 (if I'm understanding correctly). Paultone requires
that
> 64:63 vanish.

Setting 64/63~1 puts strong limits on intonation; if you are going to
improve on 22-et in some significant way you must abandon it. The 22
and 27 systems are the way to go to keep this useful approximation
we've grown attached to from 12, I think.

> I think I understand that . . . but what's the double-decker
sandwich
> business?

31;41+31 has two parts, so it's double-decker (and in fact Canasta.)
12;19+12;41+31 has three parts, so it's a sandwich. It means to take
the 12-note scales using circle-of-fifths out of a 31-note scale
(because of the 12;12+19), and it tells us the 31-note scale is
Canasta (because of the 41+31 top slice of bread.) Because 19+12=31
we can join 12;19+12 to 31;41+31 and get the muddle.

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > > Exactly. This is a pretty good paultone variant, I think.
>
> > Not really . . . it's not using 46-tET's best approximations of
> > ratios of 7 (if I'm understanding correctly). Paultone requires
> that
> > 64:63 vanish.
>
> Setting 64/63~1 puts strong limits on intonation; if you are going
to
> improve on 22-et in some significant way you must abandon it.

It's impossible for a paultone to improve significantly on 22-tET.
Remember, the optimal paultone generator is 708.8 cents.

Hello, there, Gene, and thanks for asking some very helpful questions,
and reminding me that the expression "e-based tuning" is not
necessarily so familiar; I'll explain it below.

> It seems a little strange to me to call 14/11 a major third, though
> from the point of view of this mapping it's a third of some kind,
> and not a "unidecimal diminished fourth", which is what Manuel lists
> it as. Why not just call it a 14/11?

Of course, "14/11" has the advantage of being recognizable in any
musical context, and I'd agree that it's the safest as well as most
precise way of describing this interval in a "style-independent"
fashion.

However, in a neo-Gothic setting, 11:14 as an eventone or 46-tET is
the "major third" not only because it's formed from four fifths up,
because it fits the stylistic category of a major third in terms of
its musical qualities and function.

In neo-Gothic music, a conventional "major third" tends to have a size
ranging from around the Pythagorean 64:81 (~407.82 cents) to around
7:9 (~435.05 cents) or a bit larger. These sizes correspond with the
range of regular tunings from Pythagorean (fifths pure, ~701.955
cents) to 22-tET (fifths ~7.14 cents wide, ~709.09 cents).

Here 11:14 is very nicely in this range, maybe leaning a bit to the
Pythagorean side.

In terms of musical "function" -- not in an 18th-19th century sense of
"functional harmony," but in terms of directed cadential action -- an
11:14 third indeed serves as one ideal type of major third, very
efficiently expanding to a stable fifth, just as does a Pythagorean
third in usual medieval European intonation.

Thus I'd say it's a neo-Gothic major third, just as a 4:5 is a
Renaissance major third, or a major third in other styles where this
ratio sets the norm. It's a matter of how a given style draws the map
of interval categories.

This discussion reminds me of an experience a bit more than a year ago
when I tuned an 11:14 eventone for the first time, and then a few days
later played a bit for a close friend.

My first impressions of those 11:14 thirds were that they were sunny,
bright, and beautiful; I was very happy about the tuning, and I would
say the same today. Of course, part of my viewpoint might reflect my
expectations for "major thirds" in a medieval or neo-medieval context:
that they are active and often rather complex intervals, which sooner
or later resolve to stable intervals such as fifths.

Then, as I mention, my friend came a few days later, and one of her
first impressions, closely paraphrased, was that "these are different
from the 'thirds' I've come to expect. Thirds are supposed to be
sweet, and these are 'dark.'"

This is especially interesting because I was using a rather gentle
timbre which can "pastelize" concord/discord, as Paul and I have
discussed; but my friend nevertheless found these thirds quite
different than the norm.

For me, the continuum of major thirds around 11:14 or 7:9 is a
wonderful universe at another energy level, a different and
luminescent realm, with its own musical beauty, when compared for
example to the world of Renaissance meantone near 4:5. Either realm
has its own poise and beauty, but suddenly juxtaposing them can
sometimes be a surprising "quantum leap."

> It's interesting to see this posting, as I thought of including the
> 29+17 system, thinking of you and knowing it would be something like
> this, but when I looked at the mapping of [1,21,15,11,8] it didn't
> seem so interesting. The 14/11 and 13/11 seemed a little extreme to
> me, but we also have some 7/5 (or 13/9) in the span of seven fifths,
> and out a little farther we find 13/8, 13/12, and 11/9.

Please let me admit that I need some education on how to read the
matrix or array [1,21,15,11,8], so maybe this is a good opportunity
for me to learn. Since I'm familiar with lots of the intervals of
46-tET, or of a tuning almost identical to it, this might be an ideal
example to use in explaining your system.

A possibly rather wild guess: could these be numbers of fifths in a
chain for certain intervals? Here 21 fifths up would be what I'd
consider in a neo-Gothic setting the rather "exotic" third at 15/46
octave or about 391 cents; 15 fifths up would be the near-4:7; 11
fifths up would be the augmented third quite close to 8:11; and fifths
up would give what I'd call a supraminor sixth or "Phi-sixth" very
close to Phi or 21:34, and not too far from 8:13.

If I read this correctly, then your matrix shows the simplest prime
ratios; in neo-Gothic music, priorities are often different.

Thus my matrix to express the main neo-Gothic families of intervals in
this tuning might be [1,4,8,13]. Here 4 fifths are a usual 11:14 major
third; 8 fourths or fifths, a submajor third near 17:21 or supraminor
sixth near 21:34; and 13 fourths, a near-7:9 or 7-based major third.

We could also write [1,4,8,14], taking the near-6:7 minor third, 14
fifths up, as characteristic of the near-7 family; this reflects the
ten locations in a 24-note tuning for 7:9:12 or 6:7:9. If we're
looking for a full 12:14:18:21 or 14:18:21:24, however, then it might
be best to write [1,4,8,15], telling us that these sonorities are
available in nine positions of a 24-note tuning.

Likewise, if we're looking for complete submajor/supraminor sonorities
of the 14:17:21 type, it might be best to write [1,4,9,15], taking the
supraminor third with the longer chain of fifths -- here 9 -- as the
interval of that family to use for the matrix.

If I've understood your matrix notation correctly, I like it a very
great deal, and find it a wonderful tool to show how a regular tuning
for 46-tET neatly fits neo-Gothic priorities, but might not be so
convenient for priorities based on simple harmonic ratios.

Why don't I comment on the 29+17 concept from my own perspective, and
then on some of the specific ratios you mention.

One approach I often use in describing regular temperaments is Easley
Blackwood's R, the ratio between the whole-tone and diatonic
semitone. In an equal temperament, R will be a rational fraction; in
other kinds of systems such as Pythagorean, it will be irrational.

For 46-tET, we have a whole-tone of 8 steps, and a diatonic semitone
of 3 steps, giving us R=8/3.

Following your 29+17, we can derive R for 46-tET by combining these
ratios for 29-tET (5/2) and 17-tET (3/1):

5 3 8
- + - = -
2 1 3

Thus 46-tET has an R of 8/3, or ~2.67, fitting its intermediate
position on the continuum of regular or eventone temperaments at
~704.35 cents, in comparison to 29-tET (~703.45 cents) and 17-tET
(~705.88 cents).

Explaining Blackwood's R sets the stage for my explanation of what the
"e-based tuning" is -- but first the ratios you mention in 46-tET.

First of all, from a neo-Gothic point of view, 11:13 and 11:14 thirds
are favorite ratios, along with supraminor/submajor thirds at 14:17
and 17:21, and here 46-tET is a very nice optimization for all these
ratios.

The supraminor thirds and sixths of 46-tET at around 339 and 835
cents, might suggest the simpler and more neutral ratios of 9:11 or
8:13; but in a neo-Gothic setting, I'd describe them in terms of the
more complex ratios of 14:17 and 21:34, the latter quite close to Phi
at ~833.09 cents.

The diminished fifths and augmented fourths or tritones, as you note,
are somewhere between 5:7 and 7:10, for which we get close
approximations in 29-tET; and 13:18 and 9:13, for which get fine
approximations in 17-tET.

While the chromatic semitone at about 130 cents isn't too far from
12:13, it's very close to 13:14 (~128.30 cents). I really like this
step size, roughly a "2/3-tone."

What I realize here is that how one associates 46-tET intervals with
integer ratios may in part be a matter of style: one approach is to
look mainly for comparatively simple ratios like 9:11 or 8:13; another
is to look for more complex categories such as 14:17 or 21:34, which
might be heard in many contexts more in effect as shades of
temperament than as distinct relations between harmonic partials.

> How does e get into it, or do you mean something other than exp(1)?

What I'm actually referring to -- better understood once explained
<grin> -- is the tuning where Blackwood's R, the ratio between
whole-tone and diatonic semitone, is equal precisely to Euler's e,
roughly 2.71828.

In 46-tET, R is 8/3 or ~2.67, so we're quite close to the e-based
tuning. In this region, R increases along with the tempering of the
fifth in the wide direction, so we need to make the fifths just a bit
larger to get to Euler's e, around 704.609 cents.

When I first came up with this tuning, it was by simple curiosity and
maybe a bit of intuition, having read Blackwood's discussion of R.
This was in June of 2000, the same month that I was getting to know
and like 29-tET.

After a few months, I came to the realization that if carried to 24
notes, this e-based tuning would produce virtually pure ratios of 4:7,
or 15 fifths up, and reasonably close approximations of the other
ratios of 2-3-7-9.

Like 46-tET, we might write the basic families of intervals in my
interpretation (however accurate) of your matrix notation,
conservatively specified, as [1,4,9,15].

Note that in choosing a number of fifths for the 7-based family, I
find it good "conservative" practice to use whichever ratio -- 7:9,
6:7, or 4:7 -- requires the largest number of fifths.

Thus for Pythagorean, we might write [1,4,9,16] -- 4 fifths for the
usual Pythagorean major third at 64:81; 9 fifths for the "schisma
minor third" near 5:6; and 16 fifths for the near-7:9 major third.
This last number tells us that we'll get 12:14:18:21 or 14:18:21:24 in
eight locations of a 24-note tuning.

Note that with my version, the types of interval families can vary
with the tuning; there isn't one fixed order set by the harmonic
series or the like. Maybe one helpful cue would be to give the size of
the interval in cents for each chain of fifths given in the matrix,
e.g. for 46-tET: [1(704),4{418),9(339),15(965)]. For the e-based
tuning, we'd have [1(705),4(418),9(341),15(969)]. For Pythagorean,
we'd have [1(702),4(408),9(318),16(431)].

In fact, this notation is a neat way to sum up neo-Gothic families of
intervals quite concisely for a given tuning! Since the size of the
fifth is given first in rounded cents, we can use this to calculate
the size of related intervals in a given family to the nearest cent or
two. For example, for 46-tET, by taking the 7-based minor seventh at
965 cents and subtracting the fifth at 704 cents, we get a near-6:7 at
around 261 cents.

An important point which our dialogue may illustrate is how
"optimization" can mean different things to different people: are we
going for simple ratios, or complex ones, or some mixture? The variety
of musical perspectives, together with the variety of tunings, gives
this forum lots of things to keep the discussion going.

Most appreciatively,

Margo Schulter
mschulter@value.net

Hi Margo,

Although we've discussed this before, perhaps some different ideas
might flow if we took this up again at the present time.

My contention is that, based on my understanding of your use of them
in your music, ratios like 34:21 in no way can represent
audible "targets" toward which one might wish to optimize tuning
systems. In the context in which you're using these intervals, I'd
say there's no qualitative difference, other than particular size,
between these intervals and intervals a few cents wider or narrower.
If one were to tune two notes at an interval a bit narrower than one
of these ratios, and gradually increase the interval under it was a
bit wider than said ratio, one would not hear anything distinct
happen _at_ the ratio in question, and would not be able to
distinguish the point at which the ratio was reached.

I have the fullest respect for your aesthetic goals and musical
talents, but wouldn't it make more sense, and be simpler for those
following along, to simply specify cents values or ranges where the
intervals take on the characteristics you're interested in? In fact,
I hope it wouldn't be too presumptuous to suggest that such an
approach would greatly liberate your theoretical and compositional
endeavors, in that you'd be able to consider any interval on its own
aesthetic merits, and not be confined to an RI "straightjacket",
either conceptually or compositionally, when constructing systems and
listening to intervals? Perhaps that's incorrect, and you are
fully "liberated" in your thought and music, but then I'd have to
wonder whether your persistent reference to complex ratios that are
not presented in a context where their "justness" could possibly be
appreciated might be misleading to those less experienced than you,
trying to find their way in this complex subject.

Certainly your own discussion of "pastelization", and your use of 13-
tET to produce effects that might be described as "xeno-Gothic",
points to a conceptual framework in which the simplest ratios preside
over fairly large swaths of interval space, and hence (according to a
harmonic entropy sort of model of consonance, with a large value of s
to reflect that) there is simply no room left over for very complex
ratios to represent anything but arbitrarily selected points upon a
gradually changing continuum.

I'll leave it at that for now and hope you will understand that this
is simply intended to provoke discussion, particularly from some of
our newer members, and not intended to denigrate your wonderful work
and incredible contributions to this list in any way.

-Paul

--- In tuning@y..., mschulter <MSCHULTER@V...> wrote:

> However, in a neo-Gothic setting, 11:14 as an eventone or 46-tET is
> the "major third" not only because it's formed from four fifths up,
> because it fits the stylistic category of a major third in terms of
> its musical qualities and function.

If you had two circles of 46-et fifths a tritone apart, you end up
with both of these. Now what do you do?

> In neo-Gothic music, a conventional "major third" tends to have a
size
> ranging from around the Pythagorean 64:81 (~407.82 cents) to around
> 7:9 (~435.05 cents) or a bit larger. These sizes correspond with the
> range of regular tunings from Pythagorean (fifths pure, ~701.955
> cents) to 22-tET (fifths ~7.14 cents wide, ~709.09 cents).

Have you ever experimented with 27-et (9.16 cents sharp?)

> For me, the continuum of major thirds around 11:14 or 7:9 is a
> wonderful universe at another energy level, a different and
> luminescent realm, with its own musical beauty, when compared for
> example to the world of Renaissance meantone near 4:5. Either realm
> has its own poise and beauty, but suddenly juxtaposing them can
> sometimes be a surprising "quantum leap."

Of course the 46-et would be good for doing exactly that, and this
46-et shrutar would be the obvious place to start. It wouldn't be
either Renaissance or neo-Gothic, of course.

> A possibly rather wild guess: could these be numbers of fifths in a
> chain for certain intervals? Here 21 fifths up would be what I'd
> consider in a neo-Gothic setting the rather "exotic" third at 15/46
> octave or about 391 cents; 15 fifths up would be the near-4:7; 11
> fifths up would be the augmented third quite close to 8:11; and
fifths
> up would give what I'd call a supraminor sixth or "Phi-sixth" very
> close to Phi or 21:34, and not too far from 8:13.

This is exactly what it is.

> If I read this correctly, then your matrix shows the simplest prime
> ratios; in neo-Gothic music, priorities are often different.

The point of using primes is that rational numbers factor into
primes, so by looking at this array, you can tell where any interval
you are interested can be found, including 14/11. If I look at
[1,21,15,11,8] I see that a 7 maps to 15, and an 11 to 11, so 7/11
maps to 15-11, or 4.

> If I've understood your matrix notation correctly, I like it a very
> great deal, and find it a wonderful tool to show how a regular
tuning
> for 46-tET neatly fits neo-Gothic priorities, but might not be so
> convenient for priorities based on simple harmonic ratios.

Graham was doing this sort of thing before I showed up, so I can
hardly take credit for it.

> Following your 29+17, we can derive R for 46-tET by combining these
> ratios for 29-tET (5/2) and 17-tET (3/1):
>
> 5 3 8
> - + - = -
> 2 1 3

This is what mathematicians call a mediant, so that
med(5/2, 3/1)=8/3. It's a very handy operation in music theory!

> The supraminor thirds and sixths of 46-tET at around 339 and 835
> cents, might suggest the simpler and more neutral ratios of 9:11 or
> 8:13; but in a neo-Gothic setting, I'd describe them in terms of the
> more complex ratios of 14:17 and 21:34, the latter quite close to
Phi
> at ~833.09 cents.

I'm not clear why.

> What I'm actually referring to -- better understood once explained
> <grin> -- is the tuning where Blackwood's R, the ratio between
> whole-tone and diatonic semitone, is equal precisely to Euler's e,
> roughly 2.71828.

This is what I would call benign numerology, like Lucy tuning--
meaning it works, but e (or pi) has nothing really to do with the
matter.

> An important point which our dialogue may illustrate is how
> "optimization" can mean different things to different people: are we
> going for simple ratios, or complex ones, or some mixture?

I understand what it means to go for simple ratios, but how can you
go for complex ones? Anywhere you pick, there are ratios as complex
as you like as near as you like.

--- In tuning@y..., mschulter <MSCHULTER@V...> wrote:

> One approach I often use in describing regular temperaments is
Easley
> Blackwood's R, the ratio between the whole-tone and diatonic
> semitone. In an equal temperament, R will be a rational fraction; in
> other kinds of systems such as Pythagorean, it will be irrational.
>
> For 46-tET, we have a whole-tone of 8 steps, and a diatonic semitone
> of 3 steps, giving us R=8/3.

In the 46-et, 9/8 is 8 steps, 16/15 is 4 steps, and 25/24 is three
steps, so I'm wondering if R is supposed to be the ratio between the
major whole tone of 9/8 and the chromatic semitone of 25/24?

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., mschulter <MSCHULTER@V...> wrote:
>
> > One approach I often use in describing regular temperaments is
> Easley
> > Blackwood's R, the ratio between the whole-tone and diatonic
> > semitone. In an equal temperament, R will be a rational fraction; in
> > other kinds of systems such as Pythagorean, it will be irrational.
> >
> > For 46-tET, we have a whole-tone of 8 steps, and a diatonic semitone
> > of 3 steps, giving us R=8/3.
>
> In the 46-et, 9/8 is 8 steps, 16/15 is 4 steps, and 25/24 is three
> steps, so I'm wondering if R is supposed to be the ratio between the
> major whole tone of 9/8 and the chromatic semitone of 25/24?

Interesting points, but I think the "regular temperments" caveat
captures it. The diatonic scale in 46 is 8838883 which has the
R=8/3. The "JI" scale is 8748784, which has R of ???

Bob Valentine

Gene wrote:
>In the 46-et, 9/8 is 8 steps, 16/15 is 4 steps, and 25/24 is three
>steps, so I'm wondering if R is supposed to be the ratio between the
>major whole tone of 9/8 and the chromatic semitone of 25/24?

Not 25/24, it's the ratio between the Pythagorean whole tone
and the Pythagorean diatonic semitone (or limma) which is
3 octaves - 5 fifths.
Scala prints it in EQUALTEMP/DATA and SHOW DATA (if appropriate).

Manuel

--- In tuning@y..., BVAL@I... wrote:

> Interesting points, but I think the "regular temperments" caveat
> captures it. The diatonic scale in 46 is 8838883 which has the
> R=8/3. The "JI" scale is 8748784, which has R of ???

I think the real generalization is to other family of MOS--this ratio
is actually a measure which compares on version of a MOS scale to
alterative ones. For any MOS, the Easley Number could be a sort of
dimensionless constant which tells us something about what sort of
scale it is, melodically.

Hello, there, Gene Ward Smith, and I'd like to answer at least part of
a very interesting question you raised about how one might approach a
tuning with different sizes of thirds, and also to say a bit more
about the "Blackwood Convention" -- that is, Blackwood's R, and why it
might be more immediately relevant to some uses of a tuning than
others.

Why don't I first try to answer the question of how I approach a
tuning with different sizes of thirds of the kind one finds in
46-tET. Here I'll actually take as my example a somewhat different
tuning in the same general region of the spectrum which might be of
special interest to you in regard to some of the ratios were were
discussing.

While 46-tET has rather close ratios of 5, an even closer, I might say
"near-just," approximation occurs in the Erv Wilson/Keenan Pepper
tuning with fifths at around 704.096 cents, so that the ratio between
the chromatic and diatonic semitones is equal to Phi (~1.618). This
has come to be called the "Noble Fifth" tuning.

From my perspective, the "usual" major third of this tuning is the
regular one at around 416.38 cents, quite close to 11:14 (~417.51
cents). Additionally there are diminished fourths or submajor thirds
at around 367.24 cents, large major thirds at ~446.76 cents, and also,
more remotely, "schisma-like" thirds at ~386.01 cents, only about 0.31
cents narrow of a just 4:5 (~386.31 cents).

(The term "schisma-like" refers to a certain resemblance between these
thirds and Pythagorean schisma thirds -- diminished fourths and
augmented seconds -- very close to ratios of 5, differing from them by
what I call the "3-5 schisma" of 32805:32768, ~1.95 cents. Pythagorean
schisma thirds have ratios of 8196:6561 or ~384.36 cents, and
19683:16384 or ~317.60 cents.)

Here the order of presenting these thirds reflects my own usual
musical priorities, but from a 5-limit or higher n-limit kind of
perspective of the kind which your analysis often seems to imply, the
near-4:5 category might well get the highest priority.

From a 5-limit perspective, in fact, we get an approximation of 4:5:6
at around 0-386.01-704.10, with a minor third of ~318.09 cents, about
2.45 cents wide; the fifth is about 2.14 cents wide.

Now we come to your interesting question: "What do you do with this
kind of mixture of thirds in a tuning?"

Here what one _might_ do could have many answers, in fact as many
answers as there are musical viewpoints among the people responding to
the question. What I'll try to share is what I have done in this kind
of situation, or at least in the type where this tuning is presented
in a regular 24-note arrangement.

First, of course, it's possible to "throw in" a 0-386-704 or 0-318-704
sonority here and there as a kind of coloristic element, without any
immediate sense direction: the music continues in one way or another,
having been varied by a bit of a diversion. Sometimes this is more or
less my attitude toward the supraminor sixth or "Phi-sixth" found in
the general region around 833 cents, although both with that interval
and the intervals we're discussing now I often to have some kind of
cadential "bridge" back to more usual intervals.

One cadential approach I like, and which can also apply with the two
major thirds of 22-tET at around 382 and 436 cents, is to use a
"schisma-like" or near-4:5 third in this kind of three-sonority
progression, with an asterisk (*) showing a note raised by a diesis or
"12-comma" of about 49.15 cents in the "Noble Fifth" tuning:

Bb3 C4 D4 C4 D4 E4 C4 D4 Eb4
F#*3 G3 A3 G#*3 A3 B3 G#*3 A3 Bb3
Eb3 D3 or F3 E3 or F3 Eb3

Here we start with a 0-386-704 sonority, and have the middle voice
shift by a "subdiesis" or "17-comma" of about 30.38 cents to the
regular 416-cent major third while the upper voice moves from the
fifth to the regular 912.29-cent major sixth. From there, we have a
usual resolution to 0-704-1200 cents, with directed progressions of
major third to fifth and major sixth to octave.

(Note: the "17-comma" is the difference between 17 fifths and 10 octaves,
the comma dispersed in 17-tET.)

This can also be done in 22-tET, where both the interval for the shift
in the middle voice (a kind of diesis resembling 35:36 or ~48.77
cents, the difference between 4:5 and 7:9) and the usual diatonic
semitone (maybe in effect a tempered 27:28, ~62.96 cents) are equal to
1/22 octave or ~54.55 cents:

C4 D4 E4 C4 D4 Eb4
G#3 A3 B3 G#3 A3 Bb3
F3 E3 or F3 Eb3

Here we have, in rounded cents, a "schisma-like" sonority at
0-382-709, a regular (or, as Paul Erlich often says, "Pythagorean")
cadential sonority of 0-436-927 cents, and a usual resolution to
0-709-1200.

Returning to the "Noble Fifth" tuning, there's another treatment of
0-386-704 cents that I very much like, here involving the shift of a
diesis or "12-comma" in the middle voice (~49.15 cents):

Bb3 B4
F#*3 F#3 E3
Eb3 E3

Here the 0-386-704 sonority moves by a downward diesis shift of the
middle voice, F#*3-F#3, to a usual supraminor/submajor third sonority
at 0-337-704 cents, with a standard progression with the outer voices
each ascending by a chromatic semitone so that the three voices arrive
at a stable fifth.

One might interpret the diesis, in this context, as approximating the
34:35 difference (~50.18 cents) between a 14:17 supraminor third
(~336.13 cents) and a 4:5 third.

An important point here is that how one handles mixed thirds in this
or other tunings is largely a question of style. Similarly, how one
approaches such concepts as the sizes of "diatonic" and "chromatic"
semitones, or of "whole-tones," is also a matter of musical style, and
maybe also of "intonational" style insofar as one may have preferences
regarding tuning structures as they relate to a given type of music.

For example, as you observe, if one is using 46-tET in a 5-limit or
higher n-limit kind of setting (meaning that ratios of 5 are taken as
having a primary role), then it is natural to view the "diatonic
semitone" as 4 steps, a kind of "15:16," and the "whole-tone" as an
interval of either 7 or 8 steps, with these unequal steps together
forming a near-5:4 third of 15 steps.

In this type of setting, Blackwood's R may not strictly be applicable,
at least in the most obvious way as a guide to the usual
characteristics of melody and verticality or harmony. The R concept
compares the "regular" whole-tone (two fifths up) and diatonic
semitone (five fourths up), not necessarily the most relevant
relationships for a musical application more closely suggesting the
syntonic diatonic than a meantone or eventone.

(Meantone tunings are a subset of eventone tunings. In meantone
tunings, four fifths up approximate 4:5 more or less closely; in
eventone tunings, more generally, four fifths up approximate the
"usual" major third in a given style, for example ~11:14 in 46-tET or
the Noble Fifth tuning. Thus whether or not a regular tuning is an
"eventone" may be partly in the eyes or ears of the user in a given
stylistic context.)

From my perspective, however, R is a very convenient yardstick, since
the regular steps are the most common ones in a neo-Gothic kind of
style. Thus "R=8/3" describes the regular diatonic scale quite fitting
for my usual purposes, with diesis or subdiesis shifts providing a
variety of melody and vertical "flavor" alike, but seen as pleasant
excursions from a basic eventone structure.

Indeed, such tunings in this kind of neo-medieval style are analogous
to me to a Renaissance meantone: both systems avoid comma
complications for the "usual diatonic scale," but offer different kind
of diesis or sometimes also smaller intervals sometimes used as direct
melodic steps, and often used to distinguish different "flavors" of
the same general categories of vertical intervals and sonorities.

Thus if I'm looking for a "usual" major third around 4:5, I tend to
tune meantone; for a "usual" 11:14 major third, I tune an eventone
yielding this ratio for four fifths up (less two octaves), arriving at
something almost identical to 12-of-46-tET or 24-of-46-tET, etc.

From my perspective, either R=~1.65 for 1/4-comma meantone, or R=8/3
for 46-tET, is musically as well as theoretically the measure of a
"usual diatonic scale."

However, from other perspectives, other measures might be more
relevant -- for example, the 5-limit steps you have suggested for
46-tET.

What one does, and how one defines the intervals used to do it, may be
a matter of both musical style (what sizes of thirds best fit a given
style?) and intonational style (the contrast between meantone/eventone
and "syntonic diatonic" approaches, etc.).

Communication about musical and intonational assumptions alike can
sometimes be a patient process, especially when the assumptions may
take a while to clarify on all sides. Thank you for taking an interest
in this kind of process, and for your most generous contributions to
this and other forums of our tuning community.

Most appreciatively,

Margo Schulter
mschulter@value.net