New 26-nominal alphabetic notation system

If we want to preserve the traditional labeling of the diatonic scale, and at the same time gain the ability to notate scales based on a half-octave period, using the letters of the Latin alphabet, an obvious choice is 26-ET. It's pretty far out as far as meantone-type scales go, but it is (surprisingly) consistent up to the 13-limit, which is better than any lower-numbered ET, even 22.

The rules are the same as the 19-nominal system: letters H-N represent one step below A-G, and P-V one step above. O is reserved for the note a half octave from D.

19-ET: D S L E T/M F U N G V H A P I B Q J/C R K D
26-ET: D S * L E T M F U * N G V O H A P * I B Q J C R * K D

Then fill in the gaps with the remaining letters W-Z:

D S Y L E T M F U Z N G V O H A P W I B Q J C R X K D

Clearly there are temperaments that don't work well with 26-ET, but some of those might work better with other meantones. Miracle for instance has a generator of about 2.5 steps of 26-ET, but it could use a similar style of notation to represent notes of 31-ET.

31-ET: DS*YLETMFU*ZNGVO*HAPW*IBQJCRX*KD
D..Y..T..U..N..O...P..I..J..X..D

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
> If we want to preserve the traditional labeling of the diatonic
scale,
> and at the same time gain the ability to notate scales based on a
> half-octave period, using the letters of the Latin alphabet, an
obvious
> choice is 26-ET. It's pretty far out as far as meantone-type
scales go,
> but it is (surprisingly) consistent up to the 13-limit, which is
better
> than any lower-numbered ET, even 22.
>
> The rules are the same as the 19-nominal system: letters H-N
represent
> one step below A-G, and P-V one step above. O is reserved for the
note a
> half octave from D.
>
> 19-ET: D S L E T/M F U N G V H A P I B Q J/C R K D
> 26-ET: D S * L E T M F U * N G V O H A P * I B Q J C R * K D
>
> Then fill in the gaps with the remaining letters W-Z:
>
> D S Y L E T M F U Z N G V O H A P W I B Q J C R X K D
>
> Clearly there are temperaments that don't work well with 26-ET,
but some
> of those might work better with other meantones. Miracle for
instance
> has a generator of about 2.5 steps of 26-ET, but it could use a
similar
> style of notation to represent notes of 31-ET.
>
> 31-ET: DS*YLETMFU*ZNGVO*HAPW*IBQJCRX*KD
> D..Y..T..U..N..O...P..I..J..X..D

Yes!

We're definitely converging on something here. I've got a piece of
paper with 7-ET near the top labelled DEFGABCD, and 5-ET at the
bottom labelled DEGACD, with straight lines joining these 7
nominals. F converges on E but stops just before it meets it.
Similarly B converges on C. For me, this is what sets the basic
rules of the system, i.e. FCGDAEB is a chain of approximate fifths
and lines never cross, nor do they ever quite meet.

In between 7-ET and 5-ET, going down the page I have ETs 26, 19, 31,
12, 29, 17, 22, 27.

At the same position as 7-ET we also have 14 and 21. At 5-ET we have
10, 15, 20. At 12-ET we have 24. I also go a little way above 7-ET
to include 16-ET. 23-ET is between 7 and 16, but like 27, 29 and 31-
ETs it doesn't get a letter on every degree.

In between the lines bounding the tones D to E, F to G, G to A, A to
B and C to D I have 3 other lines, and in between the semitones E to
F and B to C I have 2 other lines. These lines are not straight,
they zig-zag around a little so as to connect up degrees of the
various ETs they pass thru, but they never cross over and never
meet. Overall, they stay approximately evenly spaced so that those
in the tones gradually diverge and those in the semitones gradually
converge until they stop completely at 29-ET (two of them don't
quite make it to 29-ET).

I too, made the lines wiggle as they went thru 31-ET so as to ensure
a letter for the 10 miracle nominals centered (as much as possible)
on D. But it wasn't clear to me which way to wiggle the "O" line.
Should it be the first or last in the chain of secors? What made you
settle on making it the last?

One reason for making "O" the first is so that it corresponds to the
zero of the decimal notation for miracle so it doesn't matter if one
confuses the letter for the digit. This also makes D correspond to 5
instead of 4, and somehow 5 seems more like the center of a decimal
system than 4 does.

Also, it's unclear to me again, whether to drop M and Q or T and J
for 19-ET. But I assume this will become clear eventually.

The other thing I'm not clear on is whether we have mapped the rest
of the alphabet (after G) onto these pitches in the best possible
way. No other way springs to mind as yet, but I'd like to remain
open on that question.

If anyone can explain Liese's system to me, I'd be much obliged.

Dave Keenan wrote:

> I too, made the lines wiggle as they went thru 31-ET so as to ensure > a letter for the 10 miracle nominals centered (as much as possible) > on D. But it wasn't clear to me which way to wiggle the "O" line. > Should it be the first or last in the chain of secors? What made you > settle on making it the last?

No particular reason. Actually, I'm not sure if it's a good idea to use "O" for anything other than the half-octave. In my more recent chart (http://www.io.com/~hmiller/png/alphabetic.png), I've left the area around "O" empty, and went back to your idea of using "O" as the invisible center of the 10 miracle nominals (which explains the positioning of W, X, Y, and Z in the 31-ET labeling).

> Also, it's unclear to me again, whether to drop M and Q or T and J > for 19-ET. But I assume this will become clear eventually.

As a first approximation, I like to look at what the nearest degree of 26-ET is (now that I'm thinking of this as a basically 26-nominal system), which in this case is T and J (dropping M and Q). But there are cases where (for instance) a note is closer to one that we're labeling as "E" than "T" (and it would be nice to label it as "E"), but it's closer to "T" in the 26-ET interpretation. So 26-ET is only a rough guide.

Here's what I've come up with for a few low-numbered ETs:

7-ET: D E F G A B C D
8-ET: D L F N O P B R D
(by convention, F and B represent quarter octaves)
9-ET: D L M Z V H W Q R D
10-ET: D L? T U G? O A? I J R? D
(The T and J might be unexpected, but seems clear if we want to use T and J at all; 2/10 is slightly sharper than 5/26 or 6/31, which are T. It would be nice to extend the range of G and A, but 4/10 and 6/10 are actually closer to the N and P regions.)
11-ET: D Y? T F N? V H P? B J X? D
(Obviously, 11-ET will be every other note of however we end up notating 22-ET. It's still not clear what the best 22-ET notation is.)
12-ET: D Y E F Z G O A W B C X D
13-ET: D Y E M U N V H P I Q C X D
14-ET: D Y E M F Z G O A W B Q C X D
15-ET: D Y L T F Z G? V H A? W B J R X D
16-ET: D Y? L T F U N G? O A? P I B J R X D
17-ET: D S? L E? M U Z G V H A W I Q C? R K? D

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
> Here's what I've come up with for a few low-numbered ETs:
>
> 7-ET: D E F G A B C D

Agreed.

> 8-ET: D L F N O P B R D
> (by convention, F and B represent quarter octaves)

I'd use every third step of 24-ET
| . . | . . | . . | . . | . . | . . | . . | . . |
D S Y L E T F U Z N G V O H A P W I B J C R X K D
M Q
giving
D L F N O P B R D
just as you have.

> 9-ET: D L M Z V H W Q R D

Every fourth step of 36-ET
| . . . | . . . | . . . | . . . | . .
D S * Y * L E T M F U * Z * N G V * O

. . | . . . | . . . | . . . | . . . |
O * H A P * W * I B Q J C R * X * K D

which can certainly be made to agree with your list. One could also
go for a different mapping as every third step of 27-ET.
| . . | . . | . . | . . | . . | . . | . . | . . | . . |
D S Y Y L E F U Z Z N G V O O H A P W W I B C R X X K D

giving
D Y F Z V H W B X D
which agrees in many places.

> 10-ET: D L? T U G? O A? I J R? D
> (The T and J might be unexpected, but seems clear if we want to
use T
> and J at all; 2/10 is slightly sharper than 5/26 or 6/31, which
are T.
> It would be nice to extend the range of G and A, but 4/10 and 6/10
are
> actually closer to the N and P regions.)

If 5-ET is D E G A C D as I'd have it, then 10-ET would be
D Y E Z G O A X C Y D
although you could go for every 5th step of 50-ET or every sixth
step of 60-ET, in which case you'd probably end up with what you got.

> 11-ET: D Y? T F N? V H P? B J X? D
> (Obviously, 11-ET will be every other note of however we end up
notating
> 22-ET.

Agreed.

> It's still not clear what the best 22-ET notation is.)

For me it's still
D S Y L E F U Z N G V O H A P W I B C R X K D
so that FCGDAEB represents a chain of best fifths.

> 12-ET: D Y E F Z G O A W B C X D

Agreed.

> 13-ET: D Y E M U N V H P I Q C X D

Agreed. Every second step of 26-ET.

> 14-ET: D Y E M F Z G O A W B Q C X D

Agreed, with the usual uncertainty about whether to use M and Q, or
T and J.

> 15-ET: D Y L T F Z G? V H A? W B J R X D

Could try for every 6th step of 60-ET, but the simplest is to
subdivide 5-ET (D E G A C D) as

D S L E U N G V H A P I C R K D

> 16-ET: D Y? L T F U N G? O A? P I B J R X D

If we allow fifths smaller than those of 7-ET then it's
D Y E T M F Z C O A W B Q J C X D
otherwise you could try for every third step of 48-ET
| . . | . . | . . | . . | . . | . . | . . | . . |
D * S * Y * L * E * T * F * U * Z * N * G * V * O
| . . | . . | . . | . . | . . | . . | . . | . . |
O * H * A * P * W * I * B * J * C * R * X * K * D
which gives me a question mark for every second one, but at least my
non-question-marks agree with what you have.

> 17-ET: D S? L E? M U Z G V H A W I Q C? R K? D

That one's easy since it has good fifths.

D S L E F U N G V H A P I B C R K D
which confirms all your question marks but disagrees on a few
others. F instead of M, N instead of Z, P instead of W, B instead of
Q.

This is like 22-ET, where I let the size of the FCDAEB fifths vary,
but you are looking at nearest to 26-ET.

OK. I just realised that my preference for allowing the fifths FCGDAEB to vary widely has serious problems, which you probably already saw.

Take the Magic temperament with an approx 381 c generator and octave period. Its generator can equally be considered 6/19-oct or 7/22-oct. Let's look at these using the 19-ET set we both agree on, and the 22-ET set I was proposing. Starting from D we end up a fifth away, on A, after 5 generators in both cases (as the mapping tells us we should), but we have slightly different nominals in between.

0 41 52 3
19 DSLETFUNGVHAPIBJCRKD
M Q

0 41 52 3
22 DSYLEFUZNGVOHAPWIBCRXKD

FU in one case becomes UZ in the other.

So maybe you're on the right track there.

I had the thought, instead of basing it on a particular meantone ET, how about 26 notes of an open temperament, Golden meantone, which will guarantee that no ET will ever have a step that is exactly halfway between steps of the reference. In fact such decisions will be as clear as they can be.

The fifth of golden meantone is (2+phi)/(3+2*phi) octaves,
where phi = (sqrt(5)+1)/2.

That's not particularly near 26-ET but between 19 and 31-ETs, so it might not be best as 26 notes in a single chain of fifths, but might be 19 notes in a single chain of fifths, plus some choice of 7 out of the remaining 12 that would make it up to 31. But a straight chain of 26 would be the first thing to try.

Paul Erlich emailed me the following:

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> If anyone can explain Liese's system to me, I'd be much obliged.

Liese's system -- I remember it well. The old archived posts were
shared with me by Robert personally, and you know how he feels about
them, so I'm not sure it should have been forwarded to the list. Now
that that can of worms has been opened up, and the system questioned
perhaps you should forward this to the list as well:

Tomasz Liese wrote:

>I propose to give the names 12 basic sorts of chords:
>C or CEGO = 0+12+9+8
>c or CRenGOen = 0+9+12+7
>Cd or CYGOen = 0+11+10+7
>Cm or CRGO = 0+10+11+8
>C* or CYGenOen = 0+11+9+8
>c* or CRenGenOen = 0+9+11=8
>Ct or CETO = 0+12+7+10
>ct or CRenTOen = 0+9+10+9
>Ctd or CYTOen = 0+11+8+9
>Ctm or CRTO = 0+10+9+10
>C" or CLGO = 0+14+7+8
>c" or CLenGOen = 0+13+8+7.

Anyway, Liese's posts concerned an extension of the diatonic scale
to 19 notes, typically realized in 36-tone equal temperament (the
numbers above refer to steps in 36-equal). As you know, the
generator of the system is 19/36 (or 17/36) oct. (and period =
octave), so that the usual diatonic generator comes out as three of
the new generator. Hence two new nominals are inserted between each
of the old nominals in the chain of generators. Liese gives his 'C
major' scale as

C, Q, Z, D, K, R, E, L, F, M, T, G, N, U, A, O, V, B, W, c.

a 19-note MOS of this generator. This allows all the diatonic triads
to be completed into 7-limit tetrads (while preserving conventional
tuning for the triads).

From the above, it's possible to infer the following chain of 36
generators (where "en" is a micro-flat and "in" is a micro-sharp):

Ten = 18
Qen = 1
Gen = 20
Zen = 3
Nen = 22
Den = 5
Uen = 24
Ken = 7
Aen = 26
Ren = 9
Oen = 28
En = 11 = Y
Ven = 30 = X
Len = 13
Ben = 32 = P
F = 15
W = 34
M = 17
C = 0
T = 19
Q = 2
G = 21
Z = 4
N = 23
D = 6
U = 25
K = 8
A = 27
R = 10
O = 29
E = 12
V = 31
L = 14
B = 33
Fin = 16 = S
Win = 35
Min = 18

He apparently wanted to use J for something too but I can't figure
it out from his post.

Maybe you'd like to e-mail his friend, Andrzej Gawel:
<gawel@sejm.gov.pl>

I wish I had time to get into the notation thing with you guys. I'd
recommend trying whatever you come up with out on all the systems in
my paper.

[We certainly will. And thanks for that explanation Paul. - DK]

Dave Keenan wrote:

> That's not particularly near 26-ET but between 19 and 31-ETs, so it might > not be best as 26 notes in a single chain of fifths, but might be 19 notes > in a single chain of fifths, plus some choice of 7 out of the remaining 12 > that would make it up to 31. But a straight chain of 26 would be the first > thing to try.

Good idea, but I think the half-octave symmetry is critical for this system. So my suggestion would be two chains of 13, one centered around D and the other around O.

Dave Keenan wrote:

>>9-ET: D L M Z V H W Q R D

> which can certainly be made to agree with your list. One could also > go for a different mapping as every third step of 27-ET.
> | . . | . . | . . | . . | . . | . . | . . | . . | . . |
> D S Y Y L E F U Z Z N G V O O H A P W W I B C R X X K D
> > giving
> D Y F Z V H W B X D
> which agrees in many places.

I'd like to avoid overlap unless there seems to be a very good reason for it; 1/9 is greater than 2/19, which we've agreed should be called L, so calling it Y would cause the range of Y to overlap with L.

> If 5-ET is D E G A C D as I'd have it, then 10-ET would be
> D Y E Z G O A X C Y D
> although you could go for every 5th step of 50-ET or every sixth > step of 60-ET, in which case you'd probably end up with what you got.

I could go with Y for 1/10; I think the boundary between Y and L will probably end up somewhere in the neighborhood of 3/31, and 1/10 seems to be about as good a place as any. But I'm not convinced that extending E into the range of T is a good idea. The fifths are already off by 18 cents, and 2/10 is an interval that splits the 4/10 fourth in half. 2/10 could as easily be notated F as E; it really isn't either one, but something in between.

>>It's still not clear what the best 22-ET notation is.)
> > > For me it's still
> D S Y L E F U Z N G V O H A P W I B C R X K D
> so that FCGDAEB represents a chain of best fifths.

The problem with this notation is that F goes into the range of M, and U overlaps with F. We could redefine the range of F so that U doesn't overlap with it, or just allow the overlap in this one case, but F overlapping with M is a problem for half-octave symmetry. The main use of 22-ET notation is for scales based on the half octave, and you're going to want QZXOYWM as a chain of fifths if you have FCGDAEB. So you have F swapping places with M.

>>15-ET: D Y L T F Z G? V H A? W B J R X D
> > > Could try for every 6th step of 60-ET, but the simplest is to > subdivide 5-ET (D E G A C D) as
> > D S L E U N G V H A P I C R K D

That's assuming you want to notate 5-ET as DEGACD, and that you're thinking of 15-ET as a subdivided 5-ET. But I think a better alternative for 15-ET is a minor third-based notation, and if a chain of minor thirds is notated TVBDFHJ, you want a 15-ET notation that's consistent with that. Then you could fill in the gaps DYLT - FZNV - HPWB - JRXD.

>>16-ET: D Y? L T F U N G? O A? P I B J R X D
> > > If we allow fifths smaller than those of 7-ET then it's
> D Y E T M F Z C O A W B Q J C X D

The overlap problem once again; 2/16 is less than 4/31, so it should be L instead of E, unless there's a compelling reason to call it E. 5/16 falls well into the range that I'd like to use for U (9/31 for miracle notation), but it only overlaps slightly if we set the lower limit for U at 8/26. I'd like to avoid overlaps entirely if possible, but I don't want to rule them out if it turns out to be inevitable.

> otherwise you could try for every third step of 48-ET
> | . . | . . | . . | . . | . . | . . | . . | . . |
> D * S * Y * L * E * T * F * U * Z * N * G * V * O > | . . | . . | . . | . . | . . | . . | . . | . . |
> O * H * A * P * W * I * B * J * C * R * X * K * D
> which gives me a question mark for every second one, but at least my > non-question-marks agree with what you have.

It looks like 1/16 is a toss-up between S and Y; it could go either way. Using G for 10/23 is already a stretch, but it's pretty much the only thing that works; 7/16 is getting closer to the boundary between G and V, so it could almost go either way.

>>17-ET: D S? L E? M U Z G V H A W I Q C? R K? D
> > > That one's easy since it has good fifths.
> > D S L E F U N G V H A P I B C R K D
> which confirms all your question marks but disagrees on a few > others. F instead of M, N instead of Z, P instead of W, B instead of > Q.

Since there isn't any problem with half-octave symmetry in this case, this might be acceptable, although it introduces an overlap between N and Z, if we use 11/31 as Z for miracle (11/31 > 6/17, but just barely).

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
> Good idea, but I think the half-octave symmetry is critical for
this
> system. So my suggestion would be two chains of 13, one centered
around
> D and the other around O.

Maybe. But then what about 3 and 4 chain temperaments?

I was wrong that the fifths size could be an input to the nominal-
assigning algorithm, but I don't see why the number of chains can't
be. i.e. we can have slighly different boundaries depending on the
number of chains?

We could first get something that works well for all the single-
chain temperaments in Pauls's draft paper, and then look at how it
should be adjusted to work for double-chain and so on.

Dave Keenan wrote:
> --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> > wrote:
> >>Good idea, but I think the half-octave symmetry is critical for > > this > >>system. So my suggestion would be two chains of 13, one centered > > around > >>D and the other around O.
> > > Maybe. But then what about 3 and 4 chain temperaments?

It looks like in all these cases with multiple chains of golden meantone fifths, you end up with 12 note classes, with small variations. But a chain of 25 notes could be useful for octave-based temperaments.

If we want to keep the 19-ET notation consistent with a chain of 19 golden meantone fifths, we'll need to rearrange some of the letters:

19: D S L E T F U N G V H A P I B J C R K D
31: D YS L E MT F U NZ G V H A WP I B JQ C R KX D

> I was wrong that the fifths size could be an input to the nominal-
> assigning algorithm, but I don't see why the number of chains can't > be. i.e. we can have slighly different boundaries depending on the > number of chains?

That sounds like something worth trying out. It's possible that some other kind of temperament would be a better basis for 1/2-octave or 1/3-octave scale notation. Golden lemba (map = [<2 2 5 6|, <0 3 -1 -1|]) would have a generator of 229.1796068 cents (almost exactly 5/26), which looks useful.

26: D Y S L E M T F U N Z G V O H A W P I B J Q C R K X D
* * * * * * * * * * *