Diaschismic notation

Dave Keenan wrote:

> Graham Breed, I'm glad to be able to leave the defense and development
> of
> the "Fokker-plus" interval names in your capable hands. When you get to
> write it up including the wide/narrow extension, please email it to me,
> or
> its URL. Could you post your schismic? letter # b / \ notation for
> 22-tET
> alongside your Fokker-plus interval names.

Ah, 22-equal is what I'm calling a diaschismic scale. As such, it
requires a different notation to either meantone or schismic. On my page

http://x31eq.com/diaschis.htm

I suggest two different 12-note scales relative to which notes can be
defined. The first is keyboard-oriented:

C C# D Eb E F F# G GA A Bb B C
s s s s r s s s s s s r

In 22=, s=2 and r=1. For more generality, see the website. With this
scale, fifths are perfect iff they are between two notes of the same
colour. The alternative scale has two spirals of 6 fifths, with wolves at
D-A and G#-Eb

C C# D Eb E F F# G G# A Bb B C
s s r s s s s s r s s s

The former scale is easier to remember if you've been indoctrinated into
the hegemony of the Halberstadt keyboard. The latter is more "logical"
and has a higher degree of symmetry, as well as having wolves in the same
places as my core meantone and schismic scales. It also means the C, F
and G chords can be played on the unmodified white keys. I think I prefer
the former, but the proof will come if and when somebody writes a body of
music using either.

So, having explained that I can give the scale.

But no, instead I'll define the intervals in terms of s and r.

11 tritone 7:5,10:7 5s+r
10 wide fourth 11:8 5s
9 perfect fourth 4:3 4s+r
8 wide major third 9:7 4s
7 narrow major third 5:4 3s+r
6 wide minor third 6:5 3s
5 narrow minor third 7:6 2s+r
4 wide major second 9:8 2s
3 narrow major second 10:9 s+r
2 wide minor second 16:15 s
1 narrow minor second 25:24 r
0 unison 1:1 -

Which makes the following table easier to construct

#from22 keyboard diatonic

11 tritone 5s+r F# Eb F# Eb
10 wide fourth 5s F/ D/ F/ D
9 narr fourth 4s+r F D F D\
8 wide maj third 4s E C#/ E/ C#
7 narr maj third 3s+r E\ C# E C#\
6 wide min third 3s Eb C/ Eb/ C
5 narr min third 2s+r Eb\ C Eb C\
4 wide maj second 2s D B D B
3 narr maj second s+r D\ B\ D\ B\
2 wide min second s C# Bb C# Bb
1 narr min second r C#\ Bb\ C#\ Bb\
0 unison - C A C A

The "keyboard" columns are for the keyboard-oriented base scale and the
"diatonic" columns are for the other one. This table is generally valid
for all diaschismic scales.

Now, here's where it gets complicated. The "normal" range of diaschismic
scales covers two different 7-limit mappings. The mapping of ratios to
names given in the first table is consistent with neither of these, and so
unique to 22=.

(Note that it is not, therefore, generally true that a diaschismic scale
will have an exact half-octave equal to 7:5 and 10:7. Somebody stated
this on the list way back. There will always be a half-octave, but only
in 22= does it approximate anything useful.)

That means we need another table:

22 46 ratio 46-34 34-22 22

11 22 7:5 5s+2r 6s-r 5s+r
9 19 4:3 4s+r 4s+r 4s+r
8 17 9:7 5s-r 3s+2r 4s
7 15 5:4 3s+r 3s+r 3s+r
6 12 6:5 3s 3s 3s
5 10 7:6 s+2r 3s-r 2s+r
4 9 8:7 3s-r s+2r 2s
4 8 9:8 2s 2s 2s
3 7 10:9 s+r s+r s+r
2 4 16:15 s s s
1 3 25:24 r r r
0 0 1:1 - - -

From that, I guess you can define the super/sub interval as r for 22=,
whereas the wide/narrow interval is s-r. So 3s-r is the subminor third
and 2s+r is the narrow minor third. As they're both 5 steps, you could
call them both 7/6. I think that makes the interval C-D/ a subminor
third, but it's getting late so I wouldn't guarantee it. Similarly, the
supermajor second will be C-Eb\ or C-Eb\\ depending on the base scale.

Oh boy. And for scales like 46=, the super/sub interval will be 2(s-r)
which is written as // or \\. That's that same as the schismic super/sub.
How convenient.

Only two more things before I go to bed.

Although I use the word "diaschismic" for this scale class, I don't like
it. Too many syllables, doesn't seem to mean anything. If such scales
were ever too catch on, I'd want to find a new name.

How come it's "Paul E.", "Jerry E." and "Graham Breed"? Perhaps because,
in my experience, the majority of "Graham"s are also "Graham B."s. I
wonder.

>How come it's "Paul E.", "Jerry E." and "Graham Breed"? Perhaps because,
>in my experience, the majority of "Graham"s are also "Graham B."s. I
>wonder.

That's funny, the only Graham I know personally is Graham Brownstein.

Graham, do you see any problem with your table given that 34-tET is not
consistent in the 7-limit?

[Graham Breed, TD 535.25]
>Ah, 22-equal is what I'm calling a diaschismic scale. As such, it
>requires a different notation to either meantone or schismic. On my page
>
>http://x31eq.com/diaschis.htm
>
>I suggest two different 12-note scales relative to which notes can be
>defined.
...

Thanks Graham. It may take me some time to digest that. It might help if I
could see what the twin chains of fifths look like in those notations.

>Although I use the word "diaschismic" for this scale class, I don't like
>it. Too many syllables, doesn't seem to mean anything. If such scales
>were ever to catch on, I'd want to find a new name.

How about "super_Pythagorean". It's really "super-Pythagorean with a half
octave", but it can be shortened to simply "super-Pythagorean" because
super-Pythagorean without a half octave is of almost no interest (since it
takes a chain of 9 fifths (min 10 notes) to make a single major third.

I put the boundary between meantone and Pythagorean (schismic?) linear
temperaments at the size of a 12-tET fifth (700 c).

Without a half-octave, I'd put the boundary between Pythagorean and
super-Pythagorean at the size of a 17-tET (and 34-tET) fifth (705.8 c).

With a half-octave, the boundary between Pythagorean and super-Pythagorean
"bilinear" temperaments is around 702.8 c. Does this agree with
schismic/diaschismic?

>How come it's "Paul E.", "Jerry E." and "Graham Breed"? Perhaps because,
>in my experience, the majority of "Graham"s are also "Graham B."s. I
>wonder.

The fact is, I nearly wrote "Graham B." but it just didn't _sound_ right.
:-) I like the _sound_ of "Breed". I think maybe it sets up resonances in
my mind with words like "breathe", "free", and even maybe "Craig Breedlove"
(see http://www.spiritofamerica.com/).

Similarly, I would only write "Carl Lumma" (which I understand to be
pronounced loo-MAH) for similar reasons. Thanks Carl. Have fun in Berkely.

Regards,

-- Dave Keenan
http://dkeenan.com

In-Reply-To: <3.0.6.32.20000215171843.0097c6e0@uq.net.au>
Dave Keenan wrote:

> Thanks Graham. It may take me some time to digest that. It might help
> if I
> could see what the twin chains of fifths look like in those notations.

How do you mean? Does the web page cover it?

> >Although I use the word "diaschismic" for this scale class, I don't
> like >it. Too many syllables, doesn't seem to mean anything. If such
> scales >were ever to catch on, I'd want to find a new name.
>
> How about "super_Pythagorean". It's really "super-Pythagorean with a
> half
> octave", but it can be shortened to simply "super-Pythagorean" because
> super-Pythagorean without a half octave is of almost no interest (since
> it
> takes a chain of 9 fifths (min 10 notes) to make a single major third.

Sounds better than anything I could think of! I especially like capital
letters in the middle of words.

> I put the boundary between meantone and Pythagorean (schismic?) linear
> temperaments at the size of a 12-tET fifth (700 c).

Yes, that works.

> Without a half-octave, I'd put the boundary between Pythagorean and
> super-Pythagorean at the size of a 17-tET (and 34-tET) fifth (705.8 c).

46= has a 704.3 cent fifth, and so wouldn't be super-Pythagorean by this
definition.

> With a half-octave, the boundary between Pythagorean and
> super-Pythagorean
> "bilinear" temperaments is around 702.8 c. Does this agree with
> schismic/diaschismic?

You mean with 118= being schismic? There's a super-Pythagorean scale with
just 7/5 very close to your cutoff (I think 702.9 cents). 94= is clearly
schismic with a fifth of 702.13 cents. 82= is consistent with twice 41=
and a fifth of 702.44 cents. 70= is better as diaschismic than schismic,
and has a fifth of 702.85 cents. As it's suspiciously close to 702.8
cents, where did your figure come from?

Actually the simplest rule for ETs seems to be:

More than 70 steps to the octave, and it can't be super-Pythagorean.

70 or fewer steps and double-positive (number of steps (mod 12) = 10) and
it must be super-Pythagorean.

More than 70 steps and number of steps (mod 24) = 22, and it's schismic if
anything. More than 70 steps and number of steps (mod 24) = 10 and it's
two schismic scales, if anything.

But these rules are two complex to be of any real use. You can work out
the 5-limit approximation for whatever scale you're interested in.

Paul Erlich wrote:

>Graham, do you see any problem with your table given that 34-tET is not
>consistent in the 7-limit?

No. The fact that it supports two different 7-limit mappings is entirely
consistent with its inconsistency. It supports both of them equally
badly, and so is the worst possible 7-limit super-Pythagorean, except for
silly ones.

The best 7-limit super-Pythagoreans have a fifth of around 703.9 cents for
the 46-34 range or 707.6 cents for the 34-22 range. These are accurate to
about 6 and 7 cents respectively.

>How about "super_Pythagorean". It's really "super-Pythagorean with a half
>octave", but it can be shortened to simply "super-Pythagorean" because
>super-Pythagorean without a half octave is of almost no interest (since it
>takes a chain of 9 fifths (min 10 notes) to make a single major third.

What about "doubly positive"? That's the term used for these tunings by
Bosanquet, Wilson, Chalmers, et all... since 1875!

>Similarly, I would only write "Carl Lumma" (which I understand to be
>pronounced loo-MAH) for similar reasons. Thanks Carl. Have fun in Berkely.

Weird huh? It's Finnish. Thanks Dave. Best wishes to you and your
family. Thanks for leaving us with this interesting topic.

-Carl

In-Reply-To: <20000216142408854.AAA330@scotty.nni.com@nietzsche>
Carl Lumma wrote:

> What about "doubly positive"? That's the term used for these tunings by
> Bosanquet, Wilson, Chalmers, et all... since 1875!

Wilson, and AFAIK Bosanquet, defined "doubly positive" scales as having a
Pythagorean comma equal to 2 steps. That would include 94 and 118=, while
excluding 56=. So it doesn't mean the same thing.

In regard to tunings with two super-Pythagorean chains of fifths, Carl
wrote,

>> What about "doubly positive"? That's the term used for these tunings by
>> Bosanquet, Wilson, Chalmers, et all... since 1875!

Graham Breed wrote,

>Wilson, and AFAIK Bosanquet, defined "doubly positive" scales as having a
>Pythagorean comma equal to 2 steps. That would include 94 and 118=, while
>excluding 56=. So it doesn't mean the same thing.

That's correct. Wilson followed Bosanquet in this.

There's also the interesting case of tunings with two sub-meantone chains of
fifths, which Dave Keenan explored in some depth, inspired by my example of
26=.

>> What about "doubly positive"? That's the term used for these tunings by
>> Bosanquet, Wilson, Chalmers, et all... since 1875!
>
>Wilson, and AFAIK Bosanquet, defined "doubly positive" scales as having a
>Pythagorean comma equal to 2 steps. That would include 94 and 118=, while
>excluding 56=. So it doesn't mean the same thing.

Looks like I'd better read your web page again...

-Carl

[Graham Breed, TD 537.2]
>Dave Keenan wrote:
>
>> Thanks Graham. It may take me some time to digest that. It might help
>> if I
>> could see what the twin chains of fifths look like in those notations.
>
>How do you mean? Does the web page cover it?

All I get from the web page is that part of one chain is
F C G D A E B
and part of the other is
F# C# GA Eb Bb
But what happens outside these ranges? i.e. 11 in each chain in 22-tET.

>> With a half-octave, the boundary between Pythagorean and
>> super-Pythagorean
>> "bilinear" temperaments is around 702.8 c. Does this agree with
>> schismic/diaschismic?
>
>You mean with 118= being schismic? There's a super-Pythagorean scale with
>just 7/5 very close to your cutoff (I think 702.9 cents). 94= is clearly
>schismic with a fifth of 702.13 cents. 82= is consistent with twice 41=
>and a fifth of 702.44 cents. 70= is better as diaschismic than schismic,
>and has a fifth of 702.85 cents. As it's suspiciously close to 702.8
>cents, where did your figure come from?

See the righthand side of
http://dkeenan.com/Music/2ch5limb.gif

To see this in context see
http://dkeenan.com/Music/2ChainOfFifthsTunings.htm

I'll be unsubscribed when you read this. You'll have to email me if you
want me to read something.

Regards,

-- Dave Keenan
http://dkeenan.com

In-Reply-To: <3.0.6.32.20000217060531.009f5100@uq.net.au>
Dave Keenan wrote:

> All I get from the web page is that part of one chain is
> F C G D A E B
> and part of the other is
> F# C# GA Eb Bb
> But what happens outside these ranges? i.e. 11 in each chain in 22-tET.

Eb\ Bb\ F C G D A E B F#/ C#/

A\ E\ B\ F# C# GA Eb Bb F/ C/ G/