46-tet

I'm writing a series of pieces in 46-tet on a dare, but I've never
used it before. If someone could provide a quick crash course, it
would be really helpful.

Thanks, Tristan.

lmtbl wrote:
> I'm writing a series of pieces in 46-tet on a dare, but I've never > used it before. If someone could provide a quick crash course, it > would be really helpful.
> > Thanks, Tristan.

A good place to start might be to take a look at a "lattice" diagram with fifths on the horizontal axis and major thirds going upward to the right.

24 05 32 13 40 21 02 29 10 37 18
09 36 17 44 25 06 33 14 41 22 03
21 02 29 10 37 18 45 26 07 34 15
06 33 14 41 22 03 30 11 38 19 00
18 45 26 07 34 15 42 23 04 31 12
03 30 11 38 19 00 27 08 35 16 43
15 42 23 04 31 12 39 20 01 28 09
00 27 08 35 16 43 24 05 32 13 40
12 39 20 01 28 09 36 17 44 25 06
43 24 05 32 13 40 21 02 29 10 37
09 36 17 44 25 06 33 14 41 22 03

From this you can see some of the commas; the most obvious is 2048/2025 (left four and diagonally down to the left two). If you start at any note and follow that path, you end up at the same note. Others include 78732/78125 (right 9, down 7), 1990656/1953125 (right 5, down 9), and 1600000/1594323 (left 13, up 5). Tempering out these commas results in the temperaments that we've named srutal, semisixths (sensi for short), valentine, and amity. (You might run across "diaschismic" as an older name of srutal. Valentine also had an older name which I've forgotten.) You can see that there are a number of ways of modulating around the grid and ending up in your original key; each of these will work in other equal temperaments, but only 46-ET supports all of them. (I haven't done anything specifically with 46-ET, but this kind of analysis can be useful with ET's in general.) You can divide the lattice into an arrangement of identical pieces that are tiled in all directions, for example:

+--------------------------+
| 36 17 |
| 40 21 02 29 |
|17 44 25 06 33 14 |
| 29 10 37 18 45 26 |
|14 41 22 03 30 11 |
| 26 07 34 15 42 23 |
|11 38 19 00 27 08 35|
| 23 04 31 12 39 20 |
| 35 16 43 24 05 32|
| 20 01 28 09 36 17 |
| 32 13 40 21 |
| 17 44 25 |
+--------------------------+

Since the fourths (and fifths) are slightly better than the thirds in this tuning, the traditional diatonic scale notation based on a chain of fourths is probably the best way to notate 46-ET. I looked at a few possibilities for notating the accidentals, and it looks like a good choice would be to use a sharp or a flat to represent an interval of 4 steps. This is convenient for the notation of major and minor thirds (such as D-F#, or F#-A), and your favorite quarter-tone notation can be used for the 2-step interval (which is close to a quarter tone). I'll use ^ and v for the quarter tones in this example; the single steps can be notated + for up and - for down:

0.........1.........2.........3.........4......
01234567890123456789012345678901234567890123456
D.^.#.v.E..F.^.#.v.G.^.#.v.A.^.b.v.B..C.^.b.v.D
*.......*......*.....*.....*....*....*....*...*

Here the asterisks mark the closest 46-ET approximations to the harmonic series in the key of D: D, E, F#, G^, A, Bv-, C-, C# (Db), D. Slide it across to get the harmonic series in other keys:

*.......*......*.....*.....*....*....*....*...*
E..F.^.#.v.G.^.#.v.A.^.b.v.B..C.^.b.v.D.^.#.v.E
F.^.#.v.G.^.#.v.A.^.b.v.B..C.^.b.v.D.^.#.v.E..F
G.^.#.v.A.^.b.v.B..C.^.b.v.D.^.#.v.E..F.^.#.v.G
A.^.b.v.B..C.^.b.v.D.^.#.v.E..F.^.#.v.G.^.#.v.A
B..C.^.b.v.D.^.#.v.E..F.^.#.v.G.^.#.v.A.^.b.v.B
C.^.b.v.D.^.#.v.E..F.^.#.v.G.^.#.v.A.^.b.v.B..C

46-ET also has excellent approximations of harmonics 17, 21, and 23.

Thank you so much. These are really helpful.

>(You might run across "diaschismic" as an older
>name of srutal.

Diaschismic is certainly a nicer name than srutal.

-Carl

--- In tuning-math@yahoogroups.com, Rozencrantz the Sane
<rozencrantz@g...> wrote:
>
> Thank you so much. These are really helpful.

Could you elaborate on what you find helpful? I like to know stuff
like chords, comma pumps, and linear temperaments myself.

On 1/2/06, Gene Ward Smith <gwsmith@svpal.org> wrote:
> --- In tuning-math@yahoogroups.com, Rozencrantz the Sane
> <rozencrantz@g...> wrote:
> >
> > Thank you so much. These are really helpful.
>
> Could you elaborate on what you find helpful? I like to know stuff
> like chords, comma pumps, and linear temperaments myself.

Those would be really helpful too, but this was enough of an outline
for me to start figuring those out. I'm still feeling around in the
dark, it's just a little less dark.

The lattice chart and the notation system were what proved most useful
to me, but your comments on the Sensi temperament were really usefull,
too.

--- In tuning-math@yahoogroups.com, Rozencrantz the Sane
<rozencrantz@g...> wrote:

> The lattice chart and the notation system were what proved most useful
> to me, but your comments on the Sensi temperament were really usefull,
> too.

Sensi is a good way to organize 46-et when the focus is on the
7-limit, but other systems are particularly interesting in higher
limits. Valentine, the 31&46 temperament, is good in both the 7 and 11
limits. Valentine however has a generator of 21/20, which is on the
small side. Five of these generators make up a major third, nine make
up a fifth, three make up a septimal whole tone of 8/7, and seven give
an 11/8 interval. Particularly if you are a fan of no-threes chords,
this is pretty good.

Even better as a high-limit system is leapday temperament:

/tuning-math/message/10605

As noted, this is pretty well a specialty of the house for 46-et, and
is particularly nice for people (and there seem to be a great many of
them) who enjoy no-fives harmony, since booting 5 lowers the Graham
complexity from 21 down to 16. It also has a fifth as a generator,
which for some people is a big plus. Leapday is such a good 46-et
temperament that it suggests to me that some variant of Pythagorean
notation would be a good idea for 46-et, but simply notating
everything using sharps and flats will work, if you recall the fifth
is *sharp*, not flat!

Here's another old 46 article:

/tuning-math/message/10604

Note semisixths is the old name for sensi.

>Here's another old 46 article:
>
>/tuning-math/message/10604
>
>Note semisixths is the old name for sensi.

I'll PayPal $10 to the first person who can find a post
where this change was announced / agreed upon.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> >Here's another old 46 article:
> >
> >/tuning-math/message/10604
> >
> >Note semisixths is the old name for sensi.
>
> I'll PayPal $10 to the first person who can find a post
> where this change was announced / agreed upon.

I'm pretty sure Paul announced something else, and I wanted sensi
since that was cuter. Paul is still holding out for sensisept. As for
agreeing on things, I don't think that's possible, as Dave doesn't
agree with the whole idea. If me, Paul and Herman can agree, would you
go along?

>> >Here's another old 46 article:
>> >
>> >/tuning-math/message/10604
>> >
>> >Note semisixths is the old name for sensi.
>>
>> I'll PayPal $10 to the first person who can find a post
>> where this change was announced / agreed upon.
>
>I'm pretty sure Paul announced something else, and I wanted sensi
>since that was cuter. Paul is still holding out for sensisept. As for
>agreeing on things, I don't think that's possible, as Dave doesn't
>agree with the whole idea. If me, Paul and Herman can agree, would you
>go along?

Sure. But let me rephrase that: How do people who keep track
of these things keep track of them, and if it's in a file would
they consider making it public?

-C.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Sure. But let me rephrase that: How do people who keep track
> of these things keep track of them, and if it's in a file would
> they consider making it public?

These are good questions. There's my web page, however:

http://66.98.148.43/~xenharmo/sevnames.htm

At 05:42 AM 1/3/2006, you wrote:
>--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
>> Sure. But let me rephrase that: How do people who keep track
>> of these things keep track of them, and if it's in a file would
>> they consider making it public?
>
>These are good questions. There's my web page, however:
>
>http://66.98.148.43/~xenharmo/sevnames.htm

That's a start. I wasn't aware of it.

-Carl

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
>
> lmtbl wrote:
> > I'm writing a series of pieces in 46-tet on a dare, but I've
never
> > used it before. If someone could provide a quick crash course, it
> > would be really helpful.
> >
> > Thanks, Tristan.
>
> A good place to start might be to take a look at a "lattice"
diagram
> with fifths on the horizontal axis and major thirds going upward to
the
> right.
>
> 24 05 32 13 40 21 02 29 10 37 18
> 09 36 17 44 25 06 33 14 41 22 03
> 21 02 29 10 37 18 45 26 07 34 15
> 06 33 14 41 22 03 30 11 38 19 00
> 18 45 26 07 34 15 42 23 04 31 12
> 03 30 11 38 19 00 27 08 35 16 43
> 15 42 23 04 31 12 39 20 01 28 09
> 00 27 08 35 16 43 24 05 32 13 40
> 12 39 20 01 28 09 36 17 44 25 06
> 43 24 05 32 13 40 21 02 29 10 37
> 09 36 17 44 25 06 33 14 41 22 03
>
> From this you can see some of the commas; the most obvious is
2048/2025
> (left four and diagonally down to the left two). If you start at
any
> note and follow that path, you end up at the same note. Others
include
> 78732/78125 (right 9, down 7), 1990656/1953125 (right 5, down 9),
and
> 1600000/1594323 (left 13, up 5). Tempering out these commas results
in
> the temperaments that we've named srutal, semisixths (sensi for
short),
> valentine, and amity.

It might help to discuss some of the scales that naturally arise in
these temperaments (though tuned in 46-equal).

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:

> Since the fourths (and fifths) are slightly better than the thirds in
> this tuning, the traditional diatonic scale notation based on a chain
of
> fourths is probably the best way to notate 46-ET. I looked at a few
> possibilities for notating the accidentals, and it looks like a good
> choice would be to use a sharp or a flat to represent an interval of
4
> steps.

It would be more standard to use 5 steps, so that the chain of pure
fifths is notated as usual.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> >(You might run across "diaschismic" as an older
> >name of srutal.
>
> Diaschismic is certainly a nicer name than srutal.
>
> -Carl

(runs to toilet with hand over mouth)

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> >> >Here's another old 46 article:
> >> >
> >> >/tuning-math/message/10604
> >> >
> >> >Note semisixths is the old name for sensi.
> >>
> >> I'll PayPal $10 to the first person who can find a post
> >> where this change was announced / agreed upon.
> >
> >I'm pretty sure Paul announced something else, and I wanted sensi
> >since that was cuter. Paul is still holding out for sensisept. As
for
> >agreeing on things, I don't think that's possible, as Dave doesn't
> >agree with the whole idea. If me, Paul and Herman can agree, would
you
> >go along?
>
> Sure. But let me rephrase that: How do people who keep track
> of these things keep track of them, and if it's in a file would
> they consider making it public?
>
> -C.

I thought you were going to put up a scan of my paper. You still
haven't. In any case, it'll be published in XH18 soon.

>I thought you were going to put up a scan of my paper. You still
>haven't. In any case, it'll be published in XH18 soon.

Is "sensi" mentioned therein?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> >I thought you were going to put up a scan of my paper. You still
> >haven't. In any case, it'll be published in XH18 soon.
>
> Is "sensi" mentioned therein?

Sensipent and Sensisept are, these being the 5-limit and 7-limit
incarnations. I needed to distinguish them, so I used limit suffixes,
and given that it seemed right to shorten the root "semisixths".
Linguistically, an "m" tends turns into an "n" when it immediately
precedes "s" in a word.

Carl Lumma wrote:
> Sure. But let me rephrase that: How do people who keep track
> of these things keep track of them, and if it's in a file would
> they consider making it public?

I have a Word document with names and basic information (tuning maps, TOP periods and generators) for a bunch of "linear" (i.e. rank 2) temperaments. Most of the names from Gene's page are in there, although I noticed that I didn't have a name for "Countercata" even though it's in the list. I didn't keep track of where I got the names from, and some of them have more than one name. But I'll put a copy up in case it could be useful.

http://www.io.com/~hmiller/music/linear-temperaments.doc

wallyesterpaulrus wrote:
> --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> > >>Since the fourths (and fifths) are slightly better than the thirds in >>this tuning, the traditional diatonic scale notation based on a chain > > of > >>fourths is probably the best way to notate 46-ET. I looked at a few >>possibilities for notating the accidentals, and it looks like a good >>choice would be to use a sharp or a flat to represent an interval of > > 4 > >>steps.
> > > It would be more standard to use 5 steps, so that the chain of pure > fifths is notated as usual.

Yes, but less convenient. You then need either a different symbol for the 4-step interval, or a combination of "sharp minus" or "flat plus", with one accidental partially canceling out the other one. Any notation system for a non-meantone system is going to notate some intervals differently from standard notation; a notation with 5-step sharps requires non-standard notation for all major and minor thirds. With 4-step sharps, you gain a number of thirds at the expense of the Bb-F (or B-F#) notation, which seems like a fair trade, and it allows you to use semisharp symbols for the quarter tones. The Sagittal "double scroll double up" symbol )||( could be used for this instead of the sharp for extra preciseness, but the traditional sharp is essentially ambiguous; it can either represent 25/24 or 2187/2048 in meantone. Of these, I'd guess that 25/24 is more often likely to be useful.

>> Sure. But let me rephrase that: How do people who keep track
>> of these things keep track of them, and if it's in a file would
>> they consider making it public?
>
>I have a Word document with names and basic information (tuning maps,
>TOP periods and generators) for a bunch of "linear" (i.e. rank 2)
>temperaments. Most of the names from Gene's page are in there, although
>I noticed that I didn't have a name for "Countercata" even though it's
>in the list. I didn't keep track of where I got the names from, and some
>of them have more than one name. But I'll put a copy up in case it could
>be useful.
>
> http://www.io.com/~hmiller/music/linear-temperaments.doc

Thanks!

-C.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:

> Yes, but less convenient. You then need either a different symbol for
> the 4-step interval, or a combination of "sharp minus" or "flat plus",
> with one accidental partially canceling out the other one. Any notation
> system for a non-meantone system is going to notate some intervals
> differently from standard notation; a notation with 5-step sharps
> requires non-standard notation for all major and minor thirds.

I'm not following this. Anything using fifths as generators will use 7
generator steps for 2187/2048; what 25/24 comes to varies widely. In
the case of 46, 25/24 comes to *three* steps of 46, not four, and so
is -5 generator steps.

Using 2187/2048 sharps, the major third is C-C###. Using 25/24 sharps,
the major third is C-Gbbbb. Using a four generator step sharp symbol,
which is -22 generator steps, it's C-Fb, which is familar from
schmatic notation, I presume.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
>
> > Yes, but less convenient. You then need either a different symbol
for
> > the 4-step interval, or a combination of "sharp minus" or "flat
plus",
> > with one accidental partially canceling out the other one. Any
notation
> > system for a non-meantone system is going to notate some
intervals
> > differently from standard notation; a notation with 5-step sharps
> > requires non-standard notation for all major and minor thirds.
>
> I'm not following this. Anything using fifths as generators will
use 7
> generator steps for 2187/2048; what 25/24 comes to varies widely. In
> the case of 46, 25/24 comes to *three* steps of 46, not four,

Ha! I should have checked that instead of assuming Herman had it
right . . .

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
>
> wallyesterpaulrus wrote:
> > --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
> >
> >
> >>Since the fourths (and fifths) are slightly better than the
thirds in
> >>this tuning, the traditional diatonic scale notation based on a
chain
> >
> > of
> >
> >>fourths is probably the best way to notate 46-ET. I looked at a
few
> >>possibilities for notating the accidentals, and it looks like a
good
> >>choice would be to use a sharp or a flat to represent an interval
of
> >
> > 4
> >
> >>steps.
> >
> >
> > It would be more standard to use 5 steps, so that the chain of
pure
> > fifths is notated as usual.
>
> Yes, but less convenient.

That's a judgment call, but Wilson, Secor/Keenan, D. Wolf and others
have called it the other way.

> a notation with 5-step sharps
> requires non-standard notation for all major and minor thirds.

If by "major and minor thirds" you mean the approximate ratios of 5,
then this is true, but then C-E, F-A, and G-B are *not* major thirds,
and B-D, D-F, E-G, and A-C are *not* minor thirds, under either of
our proposals, and both proposals would require non-standard notation
for the nearest intervals that actually *are* major and minor thirds.
So I don't think we'd have any business *expecting* standard notation
for approximate-ratio-of-5-major-and-minor-thirds anyway!

> With
> 4-step sharps, you gain a number of thirds at the expense of the Bb-
F
> (or B-F#) notation, which seems like a fair trade,

To you, perhaps :)

and it allows you to
> use semisharp symbols for the quarter tones. The Sagittal "double
scroll
> double up" symbol )||( could be used for this instead of the sharp
for
> extra preciseness,

Isn't it kind of silly to use Sagittal accidentals when you're
violating its basic premises about the nominals?

> but the traditional sharp is essentially ambiguous;
> it can either represent 25/24 or 2187/2048 in meantone.
> Of these, I'd
> guess that 25/24 is more often likely to be useful.

To the extent that it's even valid to make such a distinction, I'd
think other ratios -- 135/128, for one -- would come into play as a
sharp a lot more often than 2187/2048! As Carl Lumma pointed out, if
you transpose the usual JI major scale up a fifth, F->F# is a rise of
135/128 . . .

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus" <perlich@a...>
wrote:

> > I'm not following this. Anything using fifths as generators will
> use 7
> > generator steps for 2187/2048; what 25/24 comes to varies widely. In
> > the case of 46, 25/24 comes to *three* steps of 46, not four,
>
> Ha! I should have checked that instead of assuming Herman had it
> right . . .

16/15 comes to four steps, and using it seems to be a good plan in
connection with a diaschismic/srutal point of view.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus" <perlich@a...>
> wrote:
>
> > > I'm not following this. Anything using fifths as generators will
> > use 7
> > > generator steps for 2187/2048; what 25/24 comes to varies widely.
In
> > > the case of 46, 25/24 comes to *three* steps of 46, not four,
> >
> > Ha! I should have checked that instead of assuming Herman had it
> > right . . .
>
> 16/15 comes to four steps, and using it seems to be a good plan in
> connection with a diaschismic/srutal point of view.

Following Herman, 16/15 "should" be a diatonic semitone (the distance
from E to F and B to C), not a chromatic semitone.

Anyway, all this contradicts the whole Sagittal/HEWM approach, where
one chain of fifths is supposed to be notated conventionally. The dream
of a standardized microtonal notation drifts further away . . .

Once again, Yahoo sent spam to my account, which was bounced by SpamCop, causing Yahoo to stop sending mail. So I've probably missed some of this discussion. There's really no excuse for Yahoo to do this sort of thing. Here's the reply I originally wrote before Yahoo bounced it.

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> > >>Yes, but less convenient. You then need either a different symbol for >>the 4-step interval, or a combination of "sharp minus" or "flat plus", >>with one accidental partially canceling out the other one. Any notation >>system for a non-meantone system is going to notate some intervals >>differently from standard notation; a notation with 5-step sharps >>requires non-standard notation for all major and minor thirds.
> > > I'm not following this. Anything using fifths as generators will use 7
> generator steps for 2187/2048; what 25/24 comes to varies widely. In
> the case of 46, 25/24 comes to *three* steps of 46, not four, and so
> is -5 generator steps. 46-ET isn't necessarily using fifths as generators.

> Using 2187/2048 sharps, the major third is C-C###.

Ugh!!!!!!! The major third should always be C - E(something).* What the
(something) is will depend on the notation.

*except in cases where the music is so full of comma pumps that the
occasional major third may need to be notated something like C - Fb.

(I mistakenly said 25/24 when what I meant was 135/128. This doesn't
happen to be important for the sake of this comparison, but it makes a
difference if you want to use a Sagittal symbol for this interval, which
is ||\ for 135/128.)

The point is, the "4-step interval" is the thing that needs a symbol for
notating major and minor thirds, if the notation is based on a chain of
fourths B-E-A-D-G-C-F. D-F is an 11-step interval, and a major third is
15 steps. So to notate a major third above D, it needs to be F + a
4-step interval. If a sharp represents 5 steps, that will be something
like F#-, which is awkward. You could also use F^^, or something like
that, if ^ represents a 2-step interval (approximately a quarter tone).
But notating it as F# is more convenient.

The "5-step interval" is less frequently needed; mainly in cases like Bb
- F or B - F#. This can be represented with a "4-step interval" sign
plus a "1-step interval" sign, which is less confusing than it would be
to represent a 4-step interval with a "5-step interval" sign in one
direction and a "1-step interval" sign in the opposite direction.

Gene Ward Smith wrote:
> --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> > >>Yes, but less convenient. You then need either a different symbol for >>the 4-step interval, or a combination of "sharp minus" or "flat plus", >>with one accidental partially canceling out the other one. Any notation >>system for a non-meantone system is going to notate some intervals >>differently from standard notation; a notation with 5-step sharps >>requires non-standard notation for all major and minor thirds.
> > > I'm not following this. Anything using fifths as generators will use 7
> generator steps for 2187/2048; what 25/24 comes to varies widely. In
> the case of 46, 25/24 comes to *three* steps of 46, not four, and so
> is -5 generator steps. 46-ET isn't necessarily using fifths as generators.

> Using 2187/2048 sharps, the major third is C-C###.

Ugh!!!!!!! The major third should always be C - E(something).* What the (something) is will depend on the notation.

*except in cases where the music is so full of comma pumps that the occasional major third may need to be notated something like C - Fb.

(I mistakenly said 25/24 when what I meant was 135/128. This doesn't happen to be important for the sake of this comparison, but it makes a difference if you want to use a Sagittal symbol for this interval, which is ||\ for 135/128.)

The point is, the "4-step interval" is the thing that needs a symbol for notating major and minor thirds, if the notation is based on a chain of fourths B-E-A-D-G-C-F. D-F is an 11-step interval, and a major third is 15 steps. So to notate a major third above D, it needs to be F + a 4-step interval. If a sharp represents 5 steps, that will be something like F#-, which is awkward. You could also use F^^, or something like that, if ^ represents a 2-step interval (approximately a quarter tone). But notating it as F# is more convenient.

The "5-step interval" is less frequently needed; mainly in cases like Bb - F or B - F#. This can be represented with a "4-step interval" sign plus a "1-step interval" sign, which is less confusing than it would be to represent a 4-step interval with a "5-step interval" sign in one direction and a "1-step interval" sign in the opposite direction.

Once again, Yahoo sent spam to my account, which was bounced by SpamCop, causing Yahoo to stop sending mail. So I've probably missed some of this discussion. There's really no excuse for Yahoo to do this sort of thing. Here's the reply I originally wrote before Yahoo bounced it.

Herman Miller wrote:

> Yes, but less convenient. You then need either a different symbol for > the 4-step interval, or a combination of "sharp minus" or "flat plus", > with one accidental partially canceling out the other one. Any notation > system for a non-meantone system is going to notate some intervals > differently from standard notation; a notation with 5-step sharps > requires non-standard notation for all major and minor thirds. With > 4-step sharps, you gain a number of thirds at the expense of the Bb-F > (or B-F#) notation, which seems like a fair trade, and it allows you to > use semisharp symbols for the quarter tones. The Sagittal "double scroll > double up" symbol )||( could be used for this instead of the sharp for > extra preciseness, but the traditional sharp is essentially ambiguous; > it can either represent 25/24 or 2187/2048 in meantone. Of these, I'd > guess that 25/24 is more often likely to be useful.

Oops, this should of course be 135/128, which is ||\ in sagittal
notation, not 25/24....

In tuning-math@yahoogroups.com, "wallyesterpaulrus" <perlich@a...> wrote:

>>>It would be more standard to use 5 steps, so that the chain of > > pure > >>>fifths is notated as usual.
>>
>>Yes, but less convenient.
> > > That's a judgment call, but Wilson, Secor/Keenan, D. Wolf and others > have called it the other way.

Specifically in the context of 46-ET? I can understand if they were
talking about schismic temperaments, but this isn't the same case. Even
so, I'm not too fond of the schismic notation of 5/4 as a diminished
fourth, and so on. If there wasn't much precedent for notating it that
way already, I'd look for something different. But 46-ET doesn't allow
this simple hack, since Fb would be a step too low (with 5-step flats)
to be a major third above C. So you're stuck with a cumbersome notation
for thirds if you don't have a symbol for 4 steps of 46-ET.

>>a notation with 5-step sharps >>requires non-standard notation for all major and minor thirds.
> > > If by "major and minor thirds" you mean the approximate ratios of 5, > then this is true, but then C-E, F-A, and G-B are *not* major thirds, > and B-D, D-F, E-G, and A-C are *not* minor thirds, under either of > our proposals, and both proposals would require non-standard notation > for the nearest intervals that actually *are* major and minor thirds. > So I don't think we'd have any business *expecting* standard notation > for approximate-ratio-of-5-major-and-minor-thirds anyway!

All that's saying is that 46-ET isn't a meantone, which we already knew.
Either the thirds or the fourths have to give. In this case, the fourths
are just barely better than the minor thirds, so a chain of fourths
makes as much sense as anything. This means that comma-sharp and
comma-flat symbols will be needed to notate some thirds.

> Isn't it kind of silly to use Sagittal accidentals when you're > violating its basic premises about the nominals?

The nominals are a chain of fourths in this case, so what's the problem?

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
>
> In tuning-math@yahoogroups.com, "wallyesterpaulrus" <perlich@a...>
wrote:
>
> >>>It would be more standard to use 5 steps, so that the chain of
> >
> > pure
> >
> >>>fifths is notated as usual.
> >>
> >>Yes, but less convenient.
> >
> >
> > That's a judgment call, but Wilson, Secor/Keenan, D. Wolf and
others
> > have called it the other way.
>
> Specifically in the context of 46-ET?

In some cases this is explicit, in other cases, implicit.

> So you're stuck with a cumbersome notation
> for thirds if you don't have a symbol for 4 steps of 46-ET.

It seems you are either way. See below.

> >>a notation with 5-step sharps
> >>requires non-standard notation for all major and minor thirds.
> >
> >
> > If by "major and minor thirds" you mean the approximate ratios of
5,
> > then this is true, but then C-E, F-A, and G-B are *not* major
thirds,
> > and B-D, D-F, E-G, and A-C are *not* minor thirds, under either
of
> > our proposals, and both proposals would require non-standard
notation
> > for the nearest intervals that actually *are* major and minor
thirds.
> > So I don't think we'd have any business *expecting* standard
notation
> > for approximate-ratio-of-5-major-and-minor-thirds anyway!
>
> All that's saying is that 46-ET isn't a meantone, which we already
knew.
> Either the thirds or the fourths have to give. In this case, the
fourths
> are just barely better than the minor thirds, so a chain of fourths
> makes as much sense as anything. This means that comma-sharp and
> comma-flat symbols will be needed to notate some thirds.

Then why not extend that to all thirds? At least there's some
consistency there.

> > Isn't it kind of silly to use Sagittal accidentals when you're
> > violating its basic premises about the nominals?
>
> The nominals are a chain of fourths in this case, so what's the
>problem?

I guess I shouldn't have said "nominals". One problem is that
Sagittal expects the sharp and flat symbols to represent 2187:2048 in
all cases; the definitions and usages of symbol "complements" in
Sagittal all come from this assumption. Perhaps Dave or George can
chime in here.

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<perlich@a...> wrote:
>
> --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
> >
> > In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<perlich@a...> wrote:
> >
> > >>>It would be more standard to use 5 steps, so that the chain of
pure fifths is notated as usual.
> > >>
> > >>Yes, but less convenient.
> > >
> > > That's a judgment call, but Wilson, Secor/Keenan, D. Wolf and
others have called it the other way.
> >
> > Specifically in the context of 46-ET?
>
> In some cases this is explicit, in other cases, implicit.
>
> > So you're stuck with a cumbersome notation
> > for thirds if you don't have a symbol for 4 steps of 46-ET.
>
> It seems you are either way. See below.
>
> > >>a notation with 5-step sharps
> > >>requires non-standard notation for all major and minor thirds.
> > >
> > > If by "major and minor thirds" you mean the approximate ratios
of 5,
> > > then this is true, but then C-E, F-A, and G-B are *not* major
thirds,
> > > and B-D, D-F, E-G, and A-C are *not* minor thirds, under either
of
> > > our proposals, and both proposals would require non-standard
notation
> > > for the nearest intervals that actually *are* major and minor
thirds.
> > > So I don't think we'd have any business *expecting* standard
notation
> > > for approximate-ratio-of-5-major-and-minor-thirds anyway!
> >
> > All that's saying is that 46-ET isn't a meantone, which we
already knew.
> > Either the thirds or the fourths have to give. In this case, the
fourths
> > are just barely better than the minor thirds, so a chain of
fourths
> > makes as much sense as anything. This means that comma-sharp and
> > comma-flat symbols will be needed to notate some thirds.
>
> Then why not extend that to all thirds? At least there's some
> consistency there.
>
> > > Isn't it kind of silly to use Sagittal accidentals when you're
> > > violating its basic premises about the nominals?
> >
> > The nominals are a chain of fourths in this case, so what's the
> >problem?
>
> I guess I shouldn't have said "nominals". One problem is that
> Sagittal expects the sharp and flat symbols to represent 2187:2048
in
> all cases; the definitions and usages of symbol "complements" in
> Sagittal all come from this assumption. Perhaps Dave or George can
> chime in here.

I'm having a little trouble figuring out exactly what's the nature of
the difficulty you're having. Sagittal gives you symbols for at
least 2 alternate spellings for every step of 46, so it will allow
you to notate any interval that sounds like a third as a third on the
staff. Respelling may be necessary in some places, but that's often
the case where your scale is best expressed by something other than 7
nominals (but I'm not sure what you have in mind in that regard --
should have read farther back in this thread?).

You also have three different kinds of 5-limit "sharp" symbols (given
here in pure Sagittal, since Herman has previously indicated that's
what he prefers):

/||\ for 2048:2187
||\ for 128:135
)||( for 24:25

While the particular flags used in symbol complements were originally
determined on the basis of an apotome-difference, you can simply
think of these as three different "flavors" of a sharp (each having a
sharp-resemblance by virtue of two vertical shafts). I think it's a
nice thing that an even | odd number of flags corresponds to an even
| odd power of 5 in the ratio.

So what *is* the problem, anyway?

The only question Dave & I might have with this is that //| and )||(
are not what we've already selected for inclusion in the standard set
of symbols for 46 (and also 39):
39, 46: /| /|\ (|) ||\ /||\
This is also what we originally had for 53, until we concluded
recently that intervals involving 5^2 would be used more often in 53-
ET than ratios of 11 (particularly since 53 isn't 11-limit
consistent). We have therefore changed the standard set for 53 to:
53: /| //| )||( ||\ /||\
which also has the advantage of notating 53 as a subset of 159 & 212-
ET.

So might I take this opportunity to ask whether you think the
standard set for 46 should also be changed to agree with 53? (39
would not change, since it's not 1,5,25-consistent.)

--George

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:

> You also have three different kinds of 5-limit "sharp" symbols (given
> here in pure Sagittal, since Herman has previously indicated that's
> what he prefers):
>
> /||\ for 2048:2187
> ||\ for 128:135
> )||( for 24:25

Yipe! What's that in one character for each ascii?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
> wrote:
>
> > You also have three different kinds of 5-limit "sharp" symbols
(given
> > here in pure Sagittal, since Herman has previously indicated that's
> > what he prefers):
> >
> > /||\ for 2048:2187
> > ||\ for 128:135
> > )||( for 24:25
>
> Yipe! What's that in one character for each ascii?

We don't have one-character ascii abbreviations for each one in pure
Sagittal. You have to express them in mixed Sagittal, which usually
requires 2 characters, in this case #, #\, and #_, respectively.

--George

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@y...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> >
> > --- In tuning-math@yahoogroups.com, "George D. Secor"
<gdsecor@y...>
> > wrote:
> >
> > > You also have three different kinds of 5-limit "sharp" symbols
> (given
> > > here in pure Sagittal, since Herman has previously indicated
that's
> > > what he prefers):
> > >
> > > /||\ for 2048:2187
> > > ||\ for 128:135
> > > )||( for 24:25
> >
> > Yipe! What's that in one character for each ascii?
>
> We don't have one-character ascii abbreviations for each one in
pure
> Sagittal. You have to express them in mixed Sagittal, which
usually
> requires 2 characters, in this case #, #\, and #_, respectively.
>
> --George

Primarily, I was just pointing out that Herman's proposal is
inconsistent with this, because B-F# and Bb-F are not examples of 46-
equal's best 2:3 when Herman's notation is used, but they are with
this Sagittal notation.

wallyesterpaulrus wrote:

> --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
>>Either the thirds or the fourths have to give. In this case, the > > fourths
> >>are just barely better than the minor thirds, so a chain of fourths
>>makes as much sense as anything. This means that comma-sharp and
>>comma-flat symbols will be needed to notate some thirds.
> > > Then why not extend that to all thirds? At least there's some > consistency there.

You can use a notation based on a chain of minor thirds, E# G# B D F Ab Cb, but since a major third is 15 steps, you run into problems with C-E if the notation is symmetrical around D. This means the fifths A-E and C-G are one step smaller than perfect fifths. Another problem with this approach is that there's less of a difference between the large and small steps in the notation.

D..#b..E....F..#b..G..#.b..A..#b..B....C..#b..D..#b..E
|--major third-|...|minor third|......................
.......|--major third-|....|minor third|..............
............|--major third-|......|minor third|.......
...................|--major third-|....|minor third|..
|minor third|..............|--major third-|...........
.......|minor third|..............|--major third-|....
............|minor third|.............|--major third-|

A minor third-based notation may be useful for some ET's (such as 23-ET), but I think that overall the fourth-based notation works best for 46-ET (and in general, any ET with better fourths than thirds).

If you have a symbol for the 4-step interval (I'm using a sharp only for convenience, since I'm limited to symbols that I can type), then there are two kinds of thirds, just as with standard notation: thirds that require "sharps" (major DF#, EG#, AC#, BD# and minor F#A, G#B, C#E) and thirds that don't (the rest of the thirds; written without accidentals in meantone notation, but requiring a comma sharp or comma flat symbol in this system: major FA-, GB-, CE- and minor DF+, EG+, AC+, BD+). The other nice thing is that 4 steps is easily divisible in half, so you can use quartertone symbols for 2 steps.

D.^.b.v.E..F.^.#.v.G.^.b.v.A.^.b.v.B..C.^.#.v.D.^.b.v.E
|--major third-|...|minor third|.......................
........|--major third-|...|minor third|...............
...........|--major third-|........|minor third|.......
...................|--major third-|...|minor third|....
|minor third|..............|--major third-|............
........|minor third|..............|--major third-|....
...........|minor third|..............|--major third-|.

> I guess I shouldn't have said "nominals". One problem is that > Sagittal expects the sharp and flat symbols to represent 2187:2048 in > all cases; the definitions and usages of symbol "complements" in > Sagittal all come from this assumption. Perhaps Dave or George can > chime in here.

Sagittal uses the double arrows for those. I guess you're talking about the shorthand ASCII-Sagittal notation?

wallyesterpaulrus wrote:

> Then why not extend that to all thirds? At least there's some > consistency there.

I think I misinterpreted this the first time around. The reason it's awkward to use comma symbols in conjunction with apotome symbols is that the one partially cancels out the other. You have to look at something like F#- and calculate it as being 5 steps above F, plus 1 step down (resulting in 4 steps up from F). I'd prefer to have a single symbol for the 135/128 semitone (call it something else if you don't like using the sharp for this).

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
>
> wallyesterpaulrus wrote:
>
> > --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
> wrote:
> >>Either the thirds or the fourths have to give. In this case, the
> >
> > fourths
> >
> >>are just barely better than the minor thirds, so a chain of
fourths
> >>makes as much sense as anything. This means that comma-sharp and
> >>comma-flat symbols will be needed to notate some thirds.
> >
> >
> > Then why not extend that to all thirds? At least there's some
> > consistency there.
>
> You can use a notation based on a chain of minor thirds, E# G# B D
F Ab
> Cb,

Not at all what I was suggesting, so the problems you point out with
this notation don't weaken my point at all.

> A minor third-based notation may be useful for some ET's (such as
> 23-ET), but I think that overall the fourth-based notation works
best
> for 46-ET

I was proposing a fourth-based notation, but one where the ratios of
5 are always accompanied by the same clear indication of alteration,
whether one or both of the notes are within the central seven notes
in the chain of fourths, or otherwise.

> (and in general, any ET with better fourths than thirds).

I don't see that it's the "goodness" of these intervals that matters
for notation -- perhaps it's their closeness to some implied diatonic
standard?

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
>
> wallyesterpaulrus wrote:
>
> > Then why not extend that to all thirds? At least there's some
> > consistency there.
>
> I think I misinterpreted this the first time around.

OK!

> The reason it's
> awkward to use comma symbols in conjunction with apotome symbols is
that
> the one partially cancels out the other.

Doesn't bother me that much.

> You have to look at something
> like F#- and calculate it as being 5 steps above F, plus 1 step
down
> (resulting in 4 steps up from F).

You're assuming that reading notation is going to involve this kind
of calculation in the first place . . .