Yahoo Tuning Groups Ultimate Backup tuning More on 22-ET-based notations

A while back I proposed a system of notation based on an extended version of "porcupine" notation using 22-ET to notate scales based on repeated intervals other than fourths or fifths. Centered on "D", the notes of the 5-limit JI lattice would be notated like this:

+---------------+
| /Q\__ |
| /V S P\_ |
| /E B U R(O) |
| /J G D A T/ |
| (O)L I F C/ |
| \N_K H/ |
| \M/ |
+---------------+

One problem with this notation is that it's incompatible with the traditional cycle of fifths (FCGDAEB). It agrees with traditional notation in the representation of most thirds (EG, GB, BD, DF, FA, AC), but CE is a minor third instead of the expected major third.

So I've been trying to come up with alternative 22-ET-based notations. If we keep "O" as the notation for the half-octave, the rest of the alphabet can be divided into three groups of 7: A-G, H-N, and P-V. If we use A-G to notate the basic chain of fifths, we can the other groups to notate chains of fifths transposed up or down by an interval of 10/9:

+---------------------+
|______________ |
|\U_R V S P T Q(O) |
| \F C G D A E B\__ |
| (O)M_J_N_K_H_L_I\|
| |
+---------------------+

Alternatively, the groups could be arranged this way:

+---------------------+
|______________ |
|\M_J N K H L I(O) |
| \F C G D A E B\__ |
| (O)U_R_V_S_P_T_Q\|
| |
+---------------------+

I have a slight preference for the first one, since in 22-ET the intervals G-H is the same size as the intervals B-C and E-F (one step each), but I don't have a really strong preference for either one.

But this isn't much improvement over the traditional notation for third-based temperaments, so it's useful to move the far corners to be closer to the middle, where they can serve to notate extra major and minor thirds.

+---------------+ +---------------+
| En____R | | En____J |
| /H L I\_ | | /P T Q\_ |
|___/S P T Q(O) | |___/K H L I(O) |
|\F C G D A E_B\| |\F C G D A E_B\|
| (O)M J N K/ | | (O)U R V S/ |
| \U_R_V/ | | \M_J_N/ |
| L Cu | | T Cu |
+---------------+ +---------------+

Note that if you go beyond five notes in a chain of major or minor thirds, or seven notes in a chain of fourths or fifths, you end up outside the bounds of this 22-ET block. For the chains of major thirds, the notation is only off by a magic comma (3125;3072, ~29.6 cents), and since magic is the temperament you're probably wanting to notate anyway if you're using a chain of major thirds, this can be ignored. If you're going beyond seven notes in a chain of fourths or fifths, you can just use the traditional sharps and flats. But chains of minor thirds run into the porcupine comma (250;243, ~49.2 cents), so you'll need to start using accidentals after a five note chain if you need to be precise. I suggest using the Sagittal symbols for 36;35 /|) \!) (or the shorthand equivalents n and u), which differ from the porcupine comma by the very small interval 4375;4374 (~0.4 cents) (in the size range for a schismina if it isn't one already). This allows for an 11-note chain of minor thirds, which is sufficient to notate kleismic:

In Qn En H P D N V Cu Mu Uu

or alternatively,

Qn In En P H D V N Cu Uu Mu

🔗Dave Keenan <d.keenan@bigpond.net.au>

Hi Herman,

I'm glad someone's still thinking about this stuff. Good work.

I think that it will be very difficult to remember the relative
pitch relationships, given that we are interleaving the rest of the
alphabet in between the existing A to G, so I wouldn't want to do
anything to make this more difficult.

So I'd prefer to keep the very logical order you gave originally,
where every step of the alphabet is 3 steps of the sequence.

O H A P I B Q J C R K D S L E T M F U N G V(O)

To preserve FCGDAEB as a chain of fifths, which I agree is
essential, I'd like to see what happens if we simply do not think of
this sequence of 22 nominals as being equally spaced (only
approximately so, e.g. they should at least be proper). But they
need not have any particular fixed spacing. The only thing that
needs to be fixed is the pitch order.

Can we agree on sequences of nominals (taken from the above) for
various generator size ranges.

To obtain FCDAEB as a sequence of fifths, from the above, we only
need to squeeze the three gaps between E and F down to the width of
two (same for B to C), and squeeze the four gaps between G and A
down to 3, so it ends up more like 19-equal. Minor thirds should
fall better out of such a pseudo-19 arrangement too. The middle 3
should be BDF for minor thirds (VBDFH would be good).

🔗Dave Keenan <d.keenan@bigpond.net.au>

Pitch order:
O H A P I B Q J C R K D S L E T M F U N G V(O)

7 fifth nominals:
F C G D A E B

7 minor-third nominals:
E# G# B D F Ab Cb
becomes
T V B D F H J

7 major third nominals:
Ebb Gb Bb D F# A# Cx
becomes
S N I D U P K
but that uses none of our familiar major thirds.

Or 10 major third nominals (centered on D):
Bbbb Dbb Fb Ab C E G# B# Dx Fx#
becomes
A R M H C E V Q L G

Or 13 major third nominals:
Fbbb Abbb Cbb Ebb Gb Bb D F# A# Cx Ex Gx# Bx#
becomes
L G B S N I D U P K F A R

Pitch order (repeated for convenience):
O H A P I B Q J C R K D S L E T M F U N G V(O)

7 minor tone or semiminor third nominals:
A B C D E F G
or
L M N O P Q R
?

10 minor second nominals?
Hmm. That's a tough one.
L T F N V H P B J R
?

Twin chains of 5 fifths each:
C G D A E
N R O L P

There are probably many inconsistencies with the above, depending
what temperament mapping you apply to each. But do they matter?

🔗Herman Miller <hmiller@IO.COM>

Dave Keenan wrote:

> To preserve FCGDAEB as a chain of fifths, which I agree is > essential, I'd like to see what happens if we simply do not think of > this sequence of 22 nominals as being equally spaced (only > approximately so, e.g. they should at least be proper). But they > need not have any particular fixed spacing. The only thing that > needs to be fixed is the pitch order.

Hmm... there's not much room between E=9/8 and F=32/27 for two steps, especially if the scale needs to be proper. Maybe F could be 6/5 and E could be considered as basically a 10/9, but could double as a 9/8. Using E for 9/8 would free T to represent 8/7, if the distinction needs to be made (the two are identical in 22-ET). I was wanting to make T sharper for kleismic anyway. Probably both S and L should be higher than their 22-ET equivalents, to fill in some of the gap between D and E (near the boundaries between 22-ET intervals? around 21/20 and 12/11?)

Some basic assignments seem clear:

D = 1/1 D = 2/1
S = ? K = ?
L = ? R = ?
E = 10/9 C = 9/5
T = 8/7 J = 7/4
M = 7/6 Q = 12/7
F = 6/5 B = 5/3
U = 5/4 I = 8/5
N = 9/7 P = 14/9
G = 4/3 A = 3/2
V = 11/8 H = 16/11

Actually, it could be nice to set L=16/15. In that case, one of each pair of notes in this system between D and O (SL, ET, MF, UN, GV) is the same as one of the 22 srutis of Indian music, and the other note of the pair is pretty close to 2 "narrow sruti intervals" (I don't know the technical term) away from it. (Specifically: L=16/15, E=10/9, F=6/5, U=5/4, and G=4/3). Then 25/24 suggests itself as one possibility for S. Unfortunately, this isn't a proper scale, although it is a constant structure.

> Can we agree on sequences of nominals (taken from the above) for > various generator size ranges.
> > To obtain FCDAEB as a sequence of fifths, from the above, we only > need to squeeze the three gaps between E and F down to the width of > two (same for B to C), and squeeze the four gaps between G and A > down to 3, so it ends up more like 19-equal. Minor thirds should > fall better out of such a pseudo-19 arrangement too. The middle 3 > should be BDF for minor thirds (VBDFH would be good).

This is actually pretty close to the "magic" interpretation of this notation, since 19-ET is one version of magic.

🔗Herman Miller <hmiller@IO.COM>

Dave Keenan wrote:

> 7 minor-third nominals:
> E# G# B D F Ab Cb
> becomes
> T V B D F H J

This agrees with what I have for kleismic.

> 7 major third nominals:
> Ebb Gb Bb D F# A# Cx
> becomes
> S N I D U P K
> but that uses none of our familiar major thirds.

Right.

> Or 10 major third nominals (centered on D):
> Bbbb Dbb Fb Ab C E G# B# Dx Fx#
> becomes
> A R M H C E V Q L G
> > Or 13 major third nominals:
> Fbbb Abbb Cbb Ebb Gb Bb D F# A# Cx Ex Gx# Bx#
> becomes
> L G B S N I D U P K F A R

I'd say either the 7 or the 13, to be consistent with the 22-ET interpretation (since 22 is also a magic ET). Unless you're notating Wuerschmidt (but in that case, you might as well use 31-ET notation).

> Pitch order (repeated for convenience): > O H A P I B Q J C R K D S L E T M F U N G V(O)
> > 7 minor tone or semiminor third nominals:
> A B C D E F G
> or
> L M N O P Q R > ?

The first one seems fine to me.

> 10 minor second nominals?
> Hmm. That's a tough one.
> L T F N V H P B J R
> ?

That seems to be about as good as you can do.

> Twin chains of 5 fifths each:
> C G D A E
> N R O L P
> > There are probably many inconsistencies with the above, depending > what temperament mapping you apply to each. But do they matter?

Probably not much. 12-ET music has those sorts of problems all the time (major thirds being notated as diminished fourths, etc.) You'll have the enharmonic equivalents to deal with.

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> Hmm... there's not much room between E=9/8 and F=32/27 for two
steps,
> especially if the scale needs to be proper.

OK. Scratch propriety. But maintain pitch order.

Or another way of looking at it might be, when E and F get that
close together, you have to drop the nominals in between them out of
consideration, to preserve propriety. But that sounds like cheating.

When fifth generators are as wide as Pythagorean or wider, shouldn't
we be looking at 5 nominals (or 12) instead of 7?

> Maybe F could be 6/5 and E
> could be considered as basically a 10/9, but could double as a
9/8.
> Using E for 9/8 would free T to represent 8/7, if the distinction
needs
> to be made (the two are identical in 22-ET). I was wanting to make
T
> sharper for kleismic anyway. Probably both S and L should be
higher than
> their 22-ET equivalents, to fill in some of the gap between D and
E
> (near the boundaries between 22-ET intervals? around 21/20 and
12/11?)
>
> Some basic assignments seem clear:
>
> D = 1/1 D = 2/1
> S = ? K = ?
> L = ? R = ?
> E = 10/9 C = 9/5
> T = 8/7 J = 7/4
> M = 7/6 Q = 12/7
> F = 6/5 B = 5/3
> U = 5/4 I = 8/5
> N = 9/7 P = 14/9
> G = 4/3 A = 3/2
> V = 11/8 H = 16/11
>
> Actually, it could be nice to set L=16/15. In that case, one of
each
> pair of notes in this system between D and O (SL, ET, MF, UN, GV)
is the
> same as one of the 22 srutis of Indian music, and the other note
of the
> pair is pretty close to 2 "narrow sruti intervals" (I don't know
the
> technical term) away from it. (Specifically: L=16/15, E=10/9,
F=6/5,
> U=5/4, and G=4/3). Then 25/24 suggests itself as one possibility
for S.
> Unfortunately, this isn't a proper scale, although it is a
constant
> structure.

OK. Lets fall back to trying to keep it CS.

But I don't see that it is necessary to define each nominal as a
specific ratio, or even sets of ratios. We are intending to notate
temperaments after all. I think what we want to know is what is the
allowable range (best expressed as rational fractions of an octave,
since ETs will be pivots) for each nominal. We'd like to minimise
the overlap of those ranges.

We can start with D and O having no allowable variation. They are
always zero and 1/2-oct respectively, and if you arrange the 22
nominals around a circle, we can insist on reflective symmetry about
the line DO. So we only need to look at the half-octave from D to O.

I assume we want to let G vary from 2/5 to 3/7-oct and E vary from
1/7 to 1/5-oct, but F isn't needed for a 5-nominal system of fifths
so it only needs to vary from 3/12 to 2/7-oct. So far it looks
something like this:

D---------E---E---F-F--------G-G----O

>
> > Can we agree on sequences of nominals (taken from the above) for
> > various generator size ranges.
> >
> > To obtain FCDAEB as a sequence of fifths, from the above, we
only
> > need to squeeze the three gaps between E and F down to the width
of
> > two (same for B to C), and squeeze the four gaps between G and A
> > down to 3, so it ends up more like 19-equal. Minor thirds should
> > fall better out of such a pseudo-19 arrangement too. The middle
3
> > should be BDF for minor thirds (VBDFH would be good).
>
> This is actually pretty close to the "magic" interpretation of
this
> notation, since 19-ET is one version of magic.

Is this good or bad? Kleismic (minor thirds) is also supported by 19-
ET.

Seems good to me. So it seems that when notating LTs supported by 19-
ET we can think in terms of the following set of 19-nominals (which
is nevertheless compatible with the larger 22).

Here's the 22 for comparison
O H A P I B Q J C R K D S L E T M F U N G V
now the 19
H A P I B J C R K D S L E T F U N G V

Can someone come up with a mnemonic like "Every Good Boy Deserves
Fruit" (or its many variations) for that sequence of 22 letters?
Feel free to start with either D or O since it's circular.

🔗Dave Keenan <d.keenan@bigpond.net.au>

> > Pitch order (repeated for convenience):
> > O H A P I B Q J C R K D S L E T M F U N G V(O)

Here's my mnemonic verse suggestion for this sequence of letters. I
thought it would be good to chunk it so you get the letter either
side of each familiar one (A to G).

O Oh
HAP happy
IBQ I be queued.
JCR JC? Ah!
KDS Kids,
LET let
MFU 'im foo.
NGV Now Go Vah!

Picture it:
I'm happily waiting in a queue for something, with a group of kids.
Is that Jesus Christ approaching the end of the queue? Ah, yes it is.
I tell the kids to let him through. I guess if anyone deserves to
queue jump, he does. Now I tell the kids to go "wah!", with a German
accent. I have no idea why. It's just the sort of thing that would
happen in a dream.

I've marked the strong beats here, so you get my intended meter.

. . . .
Oh happy I be queued.
.. . .
JC? Ah!
. . . .
Kids, let 'im foo.
. . . .
Now go vah!

O HAP I B Q
JC R
KDS LET M FU
N G V

Don't you hate it? It's so stupid you'll probably remember it for
the rest of your life. ;-)

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:>
> Note that if you go beyond five notes in a chain of major or minor
> thirds, or seven notes in a chain of fourths or fifths, you end up
> outside the bounds of this 22-ET block. For the chains of major
thirds,
> the notation is only off by a magic comma (3125;3072, ~29.6
cents), and
> since magic is the temperament you're probably wanting to notate
anyway
> if you're using a chain of major thirds, this can be ignored. If
you're
> going beyond seven notes in a chain of fourths or fifths, you can
just
> use the traditional sharps and flats. But chains of minor thirds
run
> into the porcupine comma (250;243, ~49.2 cents), so you'll need to
start
> using accidentals after a five note chain if you need to be
precise. I
> suggest using the Sagittal symbols for 36;35 /|) \!) (or the
shorthand
> equivalents n and u), which differ from the porcupine comma by the
very
> small interval 4375;4374 (~0.4 cents) (in the size range for a
schismina
> if it isn't one already).

Yes. It's quite valid to use /|) and \!) for 243;250 as a secondary
role. Over on tuning math we've just agreed that you can notate it
precisely if you wish, by adding a _right_ accent to indicate that
schismina:
/|)` and \!),

But I think it makes more sense to have 7 (or 11) nominals for a
minor third generator, in which case this would no longer be the
appropriate chromatic comma.

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> Dave Keenan wrote:
> > Pitch order (repeated for convenience):
> > O H A P I B Q J C R K D S L E T M F U N G V(O)
> >
> > 7 minor tone or semiminor third nominals:
> > A B C D E F G
> > or
> > L M N O P Q R
> > ?
>
> The first one seems fine to me.

Here's the problem I see with using A to G for porcupine. I assume
recognisable fifths vary from 4/7 to 3/5-oct. So, as mentioned
previously in this thread, it seems that we should have something
like the following limits (to the nearest cent).

D---------E---E---F-F--------G-G----O
0 171 240 300 343 480 514 600 cents

Taking the Porcupine generator as 163 cents, and the nominals as A
to G we get:
D = 0 c
E = 163 c
F = 326 c
G = 489 c

And we see that F and G are in the ranges allowed by recognisable
fifths, but E is not. And by symmetry, C will have the same problem
as E, but in the opposite direction, which causes the following
anomalies relative to what we're used to with the diatonic A to G.

The porcupine mapping has -3 gens for 1:3, -5 gens for 1:5 and 6
gens for 1:7. Ignore ratios of 7 for now. This means that, using A
to G for the nominals, we have minor tones (anomalous ones in
parenthesis) A:B, (B:C), C:D, D:E, (E:F), F:G. We have minor thirds
A:C, B:D, (C:E), D:F, E:G. Perfect fourths A:D, (B:E), (C:F), D:G.
Major thirds F:A and G:B.

We can eliminate all these anomalies by using R (#C) and L (bE)
instead of C and E, giving
A B R D L F G

It's a shame that we lose the correct minor tones C:D and D:E and
the correct minor thirds A:C and E:G, by doing this, but we can't
have everything.

That almost comes out even on the set of 19
. . . . . . .
H A P I B J C R K D S L E T F U N G V

Or we could use L M N O P Q R which is even on the set of 22 and has
no existing implications about what are minor thirds or fourths etc.
. . . . . . .
D S L E T M F U N G V O H A P I B Q J C R K(D)

> > There are probably many inconsistencies with the above,
depending
> > what temperament mapping you apply to each. But do they matter?
>
> Probably not much. 12-ET music has those sorts of problems all the
time
> (major thirds being notated as diminished fourths, etc.) You'll
have the
> enharmonic equivalents to deal with.

Yes. The only question is, are we doing the best we can - preserving
as many intuitions as possible from our diatonic heritage without
compromising the simplicity and uniformity of the LT-specific
notations, and while minimising the range of variation of each
nominal over multiple LTs. A tough balancing act.

🔗Herman Miller <hmiller@IO.COM>

Dave Keenan wrote:

> We can eliminate all these anomalies by using R (#C) and L (bE) > instead of C and E, giving
> A B R D L F G

One big problem with this is that "R-L" doesn't look like any kind of third; at least "C-E" is a third (if not the right kind). It's not too hard to remember that all the thirds from A-G (AC, BD, CE, DF, EG) are minor and all the seconds are neutral. Then the other two thirds (FA, GB) are major.

> Or we could use L M N O P Q R which is even on the set of 22 and has > no existing implications about what are minor thirds or fourths etc.
> . . . . . . .
> D S L E T M F U N G V O H A P I B Q J C R K(D)

I think O should be mainly used for half-octave based temperaments; scales with odd numbers of notes should generally be centered on D, but it's okay for even numbered scales (like miracle) to be centered on O.

> Yes. The only question is, are we doing the best we can - preserving > as many intuitions as possible from our diatonic heritage without > compromising the simplicity and uniformity of the LT-specific > notations, and while minimising the range of variation of each > nominal over multiple LTs. A tough balancing act.

How about temperaments based on 9 steps, like bug or mavila? Well, those are the only 9-based temperaments that come to mind, but there are probably others. 9-ET intervals aren't very close to anything, except 7/6 (which is a possible bug generator; bug is [<1 2 3 3|, <0 -2 -3 -1|] which means that 7/6 maps to 1 step).

The best I've been able to come up with for these is based on dividing the 9-note scale into 3 parts of 3 notes each:

Bug: S U A B D F G I K
( S )
( B U )
( G D A )
( I F )
( K )

Mavila: B U R G D A L I F
( B U R )
( G D A )
( L I F )

🔗Herman Miller <hmiller@IO.COM>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> Dave Keenan wrote:
>
> > We can eliminate all these anomalies by using R (#C) and L (bE)
> > instead of C and E, giving
> > A B R D L F G
>
> One big problem with this is that "R-L" doesn't look like any kind
> of third;

Well yes, but it doesn't look like much like any kind of anything
else either. It's completely without precedent as a musical
interval, as far as I know. So where's the harm in declaring it a
minor third. I guess the problem is that it's only 4 steps of the
set of 22 nominals, while all the other minor thirds are 6 steps.

> at least "C-E" is a third (if not the right kind). It's not too
> hard to remember that all the thirds from A-G (AC, BD, CE, DF, EG)
> are minor and all the seconds are neutral. Then the other two
> thirds (FA, GB) are major.

Not too hard for you to remember perhaps, since you've been doing it
for years, but we don't really want a system that has to be
sprinkled with too many exceptions for this or that temperament, do
we?

I guess the problem is really the one with which you started this
thread. I didn't really address it, so it's come back to haunt me. I
guess I'm not really conceiving of these 22 nominals as being
related to 22-ET at all, but more like a 22 note meantone, such that
these are equivalent

O H A P I B Q J C R K D S L E T M F U N G V
O Ab A A# Bb B B^ Cv C C# Db D D# Eb E E^ Fv F F# Gb G G#
Cb E#
where ^ and v are semisharp and semiflat.

If we look at an actual fifth-based notation for 22-equal, using
only 5 nominals (since 7 is improper), but showing where B and F
would be, we get
G} A\ A A/ A} C{ C\ C C/ D{ D\ D D/ D} E\ E E/ E} G{ G\ G G/
A{ B C} E{ F
where I'm temporarily using { and } to stand for the Sagittal !!/
and ||\ symbols.

When you line it up with the "Oh-happy" sequence we see that B, C, E
and F, do not line up.
. . . .
O H A P I B Q J C R K D S L E T M F U N G V
G} A\ A A/ A} C{ C\ C C/ D{ D\ D D/ D} E\ E E/ E} G{ G\ G G/
A{ B C} E{ F

I guess that's pretty much what you were saying at the start of this
thread, but in a different way. The more ways we look at this the
better.

What if we throw in the remaining letters of the alphabet, W X Y Z,
like this
. . . .
O H A P W I BQJC R X K D S Y L ETMF U Z N G V
G} A\ A A/ A} C{ C\ C C/ D{ D\ D D/ D} E\ E E/ E} G{ G\ G G/
A{ B C} E{ F

This lets us have nominals for all steps of 19-ET and 22-ET with
FCGDAEB being a chain of fifths in both.

Here's 19-ET
. . . . . . . . . . . . . . . . . . .
H A PW I B QJ C R XK D SY L E TM F U ZN G VO

Now the 7 nominals for porcupine temperament are every thrd step of
22-ET and so
A I R D L U G which corresponds to
A B\ C/ D E\ F/ G

> I think O should be mainly used for half-octave based
temperaments;
> scales with odd numbers of notes should generally be centered on
D, but
> it's okay for even numbered scales (like miracle) to be centered
on O.
>

OK I'll go along with that.

> How about temperaments based on 9 steps, like bug or mavila? Well,
those
> are the only 9-based temperaments that come to mind, but there are
> probably others. 9-ET intervals aren't very close to anything,
except
> 7/6 (which is a possible bug generator; bug is [<1 2 3 3|, <0 -2 -
3 -1|]
> which means that 7/6 maps to 1 step).
>
> The best I've been able to come up with for these is based on
dividing
> the 9-note scale into 3 parts of 3 notes each:
>
> Bug: S U A B D F G I K
> ( S )
> ( B U )
> ( G D A )
> ( I F )
> ( K )
>
> Mavila: B U R G D A L I F
> ( B U R )
> ( G D A )
> ( L I F )

I have no idea what "bug" is. What was it called last month? Or just
tell me the approximate generator in cents, and the period.

I understand mavila (or is it mavilla?) is another name for pelogic,
which approximates JI only vaguely. Perhaps the thing to do for it
is to find a reasonably even 23 out of our fluctuating 26, and make
its nominals every 10th step of the 23, either side of D. Is that
the right generator? The 23 should of course be compatible with the
ranges for each letter implied by the 19-ET and 22-ET sets.
. . . . . . . . . . . . . . . . . . . . . .
O H A P W I BQJC R X K D S Y L ETMF U Z N G V 22
| | | | / / // | \ \ \ | | | / / / | \\ \ \ | | |
O H A P W I B Q J C R X K D S Y L E T M F U Z N G V 19
. . . . . . . . . . . . . . . . . . .

The obvious thing to do would be to add WXYZ to those of 19-ET,
giving these 23.
H A P W I B J C R X K D S Y L E T F U Z N G V 23

and so if pelogic/mavila used 7 nominals they would be (shown with
dots)
. . . . . . .
H A P W I B J C R X K D S Y L E T F U Z N G V 23

The same as those proposed above for porcupine.

If we used 9 for pelogic (which appears to be improper so I don't
know why we would) we would add B and F. That's if I got 10/23-oct
right as a generator.

🔗Dave Keenan <d.keenan@bigpond.net.au>

Paul Erlich just sent me this (I'm still trying to decode it):

Are you aware of Liese's proposal? Liese is the guy who came up with
the temperament called Liese, formerly called Gawel (in the old
version of my new paper). There's a precedent for using all these
letters for notation so I think you should be familiar with it.

-------------- His text starts here ----------------

The NPD 36 system, wich I presented at the Warsaw Academy of Music
and
Polish Radio Program 3 in Warsaw, is continuation of the diatonic
major-minor
system and extension of the principles of functional harmony. The
system
is
based on the rules of physic and does not depend on common octave
division.
It can be achieved by the application of approximations
wich result from dividing an octave into 29, 36,41,54,72 and more.For
notation, melodies and harmony the system includes the logarytmic
octave
divisjon into 36 section and for the construction of music noises the
division into 1152 and 768.Many new terms are of similai nature to
traditional ones , which considerably facilitates the understanding
of
the NPD 36 system.

THE NPD 36
The organisation of music sound system
By Tomasz Liese, tel/fax: (+48)(22) 676 97 60, Warsaw, POLAND.

1. Measures
octave 1200 cents 1152 stripes
band 3.125 cents 3 stripes
semiton 32 bands 96 stripes
microton 33.33 cents 32 stripes
sharp + 1 microton
flat - 1 microton

2 The Microdiatonic System
Number of sharps of major keys in order 0 to 17 :
C, T.Q, G, Z, N, D, U, K, A, R, O, E, V, L, B, S, Win.
Order of sharps :Fin, Win, Min, Cin, Tin, ... etc.
Number of flat of major keys in order 1 to 18 :
M, W, F, P, Len, Ven, En, Oen, Ren, Aen, Ken, Uen, Den, Nen, Zen,
Gen,
Qen, Ten.
Order of flat : Ben, Len, Ven, En, ... etc.
For exemple : For Qmajor-0 key are sharps : Fin, Win. For F major-0
key
are flat : Ben, len, Ven.
Replacement mark : Y , Xn , P¾n , Sn , Jn.

3.Modes
Number of flat :
0 - A minor-0 key or C major-0
1 - M major-0 key or C major-1
2 - W major-0 key or C major-2
3 - F major-0 key or C major-3
4 - P major-0 key or C major-4
5 - Len major-0 key or C minor-4
6 - X major-0 key or C minor-3
7 - Y major-0 key or C minor-2
8 - Oen major key or C minor-1
9 - Ren major Key or C minor-0

4. The C major-0 key :
C, Q, Z, D, K, R, E, L, F, M, T, G, N, U, A, O, V, B, W, c.
The tierce round :C, E, G, B, D, M, A, Q, E, N, B, K, M, O, Q, L, N,
W,
K, T, O, Z, L, U, W,
R, T, V, Z, F, U, C, R, G, V, D, F, A, C.

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> Paul Erlich just sent me this (I'm still trying to decode it):
>
> Are you aware of Liese's proposal? Liese is the guy who came up with
> the temperament called Liese, formerly called Gawel (in the old
> version of my new paper).

So how many name changes is it for this one?

In case anyone is wondering, we are talking about the temperament with
a flat 7/5 as generator which uses three of these to get to an 8/3,
and which uses four of the resulting meantone fifths to get to 5. You
can get an optimized ("copop") generator from 74-et or 93-et, every
third note of the latter being 31-et meantone. In terms of comma
sequences it goes [81/80, 686/675], showing its "illegitimate" family
descent from 5-limit meantone. Another name for it would be 19&55.

> -------------- His text starts here ----------------
>
>
> The NPD 36 system, wich I presented at the Warsaw Academy of Music
> and
> Polish Radio Program 3 in Warsaw, is continuation of the diatonic
> major-minor
> system and extension of the principles of functional harmony. The
> system
> is
> based on the rules of physic and does not depend on common octave
> division.
> It can be achieved by the application of approximations
> wich result from dividing an octave into 29, 36,41,54,72 and more.

This does not sound all that much like gawel/liese. That does have a
36-note MOS, and 17/36 makes an acceptable generator if you think
12-equal is an OK meantone. 14/29 is way sharp and does not use the
best fifth of 29, and 19/41 is way flat and does not use the best
fifth of 41. A meantone system like 74 in which the number of degrees
for 8/3 is divisible by three is what we want for that. I'd like clear
evidence that Liese actually did invent this temperament, and the
above does not supply it.

🔗Herman Miller <hmiller@IO.COM>

Dave Keenan wrote:

> Not too hard for you to remember perhaps, since you've been doing it > for years, but we don't really want a system that has to be > sprinkled with too many exceptions for this or that temperament, do > we?

We've hardly even established the rules yet; it's not clear to me that this counts as an exception. It seems that in general it would be a good rule to use evenly spaced nominals (from either the 19-set or the 22-set, as appropriate) to represent equal intervals.

> What if we throw in the remaining letters of the alphabet, W X Y Z, > like this
> . . . .
> O H A P W I BQJC R X K D S Y L ETMF U Z N G V > G} A\ A A/ A} C{ C\ C C/ D{ D\ D D/ D} E\ E E/ E} G{ G\ G G/
> A{ B C} E{ F
> > This lets us have nominals for all steps of 19-ET and 22-ET with > FCGDAEB being a chain of fifths in both.
> > Here's 19-ET
> . . . . . . . . . . . . . . . . . . .
> H A PW I B QJ C R XK D SY L E TM F U ZN G VO
> > Now the 7 nominals for porcupine temperament are every thrd step of > 22-ET and so
> A I R D L U G which corresponds to
> A B\ C/ D E\ F/ G

But if this is interpreted according to the 19-ET version, it would be A Bb C# D Eb F# G. That's more of an irregularly spaced pelog than an evenly spaced porcupine. How about swapping the positions of W, X, Y, and Z:

O H A P I W B JC X R K D S L Y ET F Z U N G V
G} A\ A A/ A} C{ C\ C C/ D{ D\ D D/ D} E\ E E/ E} G{ G\ G G/
A{ B C} E{ F

Then porcupine would be A W X D Y Z G. (I left out M and Q, since neither of the 19 or 22-based notations uses them.)

Compare:

> I have no idea what "bug" is. What was it called last month? Or just > tell me the approximate generator in cents, and the period.

I did give the mapping [<1 2 3 3|, <0 -2 -3 -1|] and noted that 7/6 is a possible generator. This is the temperament that tempers out 27;25; the TOP tunings are: g=260.26, p=1200.00 for the 5-limit version, and g=254.90, p=1194.64 for the 7-limit version (such as it is; I prefer the 5-limit version).

> I understand mavila (or is it mavilla?) is another name for pelogic, > which approximates JI only vaguely. Perhaps the thing to do for it > is to find a reasonably even 23 out of our fluctuating 26, and make > its nominals every 10th step of the 23, either side of D. Is that > the right generator? The 23 should of course be compatible with the > ranges for each letter implied by the 19-ET and 22-ET sets.
> . . . . . . . . . . . . . . . . . . . . . .
> O H A P W I BQJC R X K D S Y L ETMF U Z N G V 22
> | | | | / / // | \ \ \ | | | / / / | \\ \ \ | | |
> O H A P W I B Q J C R X K D S Y L E T M F U Z N G V 19
> . . . . . . . . . . . . . . . . . . .
> > The obvious thing to do would be to add WXYZ to those of 19-ET, > giving these 23.
> H A P W I B J C R X K D S Y L E T F U Z N G V 23
> > and so if pelogic/mavila used 7 nominals they would be (shown with > dots)
> . . . . . . .
> H A P W I B J C R X K D S Y L E T F U Z N G V 23

Actually, coming up with a 23-ET scheme would allow us to notate what I call "superpelog", which is kind of a hybrid of bug and mavila:

generators 0 5 10 1 6 11 2 7 12 3 8 13 4 9 0
degree of 23-ET 0 2 4 5 7 9 10 12 14 15 17 19 20 22 23
"bug" equiv. 0 5 1 6 2 7 3 8 4 0
"mavila" equiv. 0 5 3 1 6 4 2 0

If we swap I/W, R/X, L/Y, and U/Z, like I did for the 22-ET version, we get this:

notation H A P I W B J C X R K D S L Y E T F Z U N G V
mavila gen. -1 -3 2 0 -2 3 1
bug gen. -2 3 -1 4 0 -4 1 -3 2

> The same as those proposed above for porcupine.
> > If we used 9 for pelogic (which appears to be improper so I don't > know why we would) we would add B and F. That's if I got 10/23-oct > right as a generator.

There's two 7-limit versions, one on either side of 9-ET. The better one, which is compatible with 16- and 23-ET (7/16 or 10/23), has been called "hexadecimal", and has a mapping of [<1, 2, 1, 5|, <0, -1, 3, -5|]; the other one is [<1, 2, 1, 1|, <0, -1, 3, 4|].

In any case, it does look like 7 nominals are more appropriate for mavila.

🔗Herman Miller <hmiller@IO.COM>

Herman Miller wrote:

> But if this is interpreted according to the 19-ET version, it would be A > Bb C# D Eb F# G. That's more of an irregularly spaced pelog than an > evenly spaced porcupine. How about swapping the positions of W, X, Y, and Z:
> > O H A P I W B JC X R K D S L Y ET F Z U N G V
> G} A\ A A/ A} C{ C\ C C/ D{ D\ D D/ D} E\ E E/ E} G{ G\ G G/
> A{ B C} E{ F
> > Then porcupine would be A W X D Y Z G. (I left out M and Q, since > neither of the 19 or 22-based notations uses them.)

On second thought, swapping the positions of W, X, Y, and Z would conflict with the 26-ET version. Instead, I think it might be better to accept that the 7-cent sharp 22-ET fifths don't fit well with a meantone based notation, and try a different mapping. Here's what I get by rounding to the nearest degree of 26-ET:

O H A pW I B Q J C rX K D S Yl E T M F U Zn G V
G} A\ A A/ A} C{ C\ C C/ D{ D\ D D/ D} E\ E E/ E} G{ G\ G G/
A{ B C} E{ F

I'm starting to fill in some suggested ranges for the nominal values of the 26-note system.

http://www.io.com/~hmiller/png/alphabetic.png

This shows the labeling of 19-, 26-, and 31-ET, and includes your suggested notation for 23-ET, but with M and Q in place of T and J.

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

The six extra notes of 31-ET are numbered 1-6 for reference (not meaning to imply that they should be notated "1-6").

D S Y 4 L E T M F U 5 Z N G V 6 1 H A P W 2 I B Q J C R 3 X K D

The spacing of W, X, Y, and Z is chosen so that we can notate miracle as every 3rd degree of 31: Y E F Z V H W B C X.

I highlighted the areas around each note name, which I made to be symmetrical within the half octave to make sure that this scheme would be useful for half-octave repeating scales. The boundary between T and M, and similarly between Q and J, is an interesting case. It almost looks like 19-ET E#/Fb should be notated as T rather than M. But it looks as if we can set the T/M boundary somewhere between 13/62 octave (251.6 cents) and 4/19 (252.6 cents). That would put the F/U boundary between 11/38 (347.4 cents) and 9/31 (348.4 cents). Alternatively, we could extend the range of T to include 5/23, or leave the boundaries where they are and use T for 19-ET, M for 23-ET.

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> We've hardly even established the rules yet; it's not clear to me
that
> this counts as an exception.

Good point.

> It seems that in general it would be a good
> rule to use evenly spaced nominals (from either the 19-set or the
> 22-set, as appropriate) to represent equal intervals.

I understand that you now think 26-ET is most appropriate, but I
don't think you have to choose any particular one. I believe the
size of the LT's approximation to 2:3 (or rather what fraction that
is of its approximation to 1:2) should be an input to the nominal-
assigning algorithm. The only requirement I see is that those
letters filling in the diatonic tones (e.g. DSYLE) should be
approximately evenly spaced, and so should those filling in the
diatonic semitones (e.g. ETMF), but there is no requirement that LE
should be the same size as ET, or even that T or M need to exist
(depending on the fifth size).

> > Now the 7 nominals for porcupine temperament are every thrd step
of
> > 22-ET and so
> > A I R D L U G which corresponds to
> > A B\ C/ D E\ F/ G
>
> But if this is interpreted according to the 19-ET version, it
would be A
> Bb C# D Eb F# G. That's more of an irregularly spaced pelog than
an
> evenly spaced porcupine.

Yes. But why would you interpret it according to the 19-ET version?
Its fifth is nowhere near that of 19-ET.

> Compare:
>
> ..... 19-ET ..... ..... 22-ET .....
> +---------------+ +---------------+
> | V S P T | | V S P |
> | E B U R | | Y W U R(O) |
> | F C G D A E B | | F C G D A E B |
> | L I F C | | (O)L I Z X |
> | J N K H | | N K H |
> +---------------+ +---------------+

I understand you have probably changed these now, but I have to
admit I'm not really following them since I don't see how one can
make a fixed assignment of the letters to JI beyond the 3-limit
without seriously limiting your options.

But I'm happy that you are approaching this business from a
different direction, and yet we are still finding agreement.

> > I have no idea what "bug" is. What was it called last month? Or
just
> > tell me the approximate generator in cents, and the period.
>
> I did give the mapping [<1 2 3 3|, <0 -2 -3 -1|] and noted that
7/6 is a
> possible generator.
> This is the temperament that tempers out 27;25; the
> TOP tunings are: g=260.26, p=1200.00 for the 5-limit version, and
> g=254.90, p=1194.64 for the 7-limit version (such as it is; I
prefer the
> 5-limit version).

OK. Thanks. I'd notate it as if its generator were 3/14-oct, and
maybe it should only have 5 nominals, not 9 since 9 is improper in
the 7-limit case and the resulting C:E interval would give the wrong
idea. So that's A Q D M G, although as usual it's debatable whether
to use J and T instead of Q and M. If you did go to 9 nominals I get
EZAQDMGWC as the chain of generators.

By my current way of thinking, if we actually believe
mavila/pelogic's or hexadecimal's claim to approximate fifths, then
whether it is notated with a generator of 7/16-oct or 10/23-oct or
4/9-ET it will still come out as B E A D G C F. But I'm inclined to
deny claims of valid fifth approximations when they are outside the
range of 4/7 to 3/5 octave. If we deny these, we could notate it as
having a generator of 4/9-oct, but do it relative to a 12-ET chain
of fifths by treating 4/9-oct as 16/36-oct. This results in the
chain of generators being W L H D V R Z.

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> On second thought, swapping the positions of W, X, Y, and Z would
> conflict with the 26-ET version. Instead, I think it might be
better to
> accept that the 7-cent sharp 22-ET fifths don't fit well with a
meantone
> based notation, and try a different mapping. Here's what I get by
> rounding to the nearest degree of 26-ET:
>
> O H A pW I B Q J C rX K D S Yl E T M F U Zn G V
> G} A\ A A/ A} C{ C\ C C/ D{ D\ D D/ D} E\ E E/ E} G{ G\ G G/
> A{ B C} E{ F

I'm afraid I don't like this as it means that when the temperament's
approximation of 2:3 actually _is_ near 22-ET's, or even wider, say
27 or 37-ET's, then they miss out on being notated in the usual
manner with FCGDAEB.

I prefer Erv Wilson's "fluctuating nominals". However I'm quite
willing to say, where he wasn't quite, that FCGDAEB form a "linear
set". And to only allow them to fluctuate from 4/7-oct (686 c) [or
maybe 9/16-oct (675 c)] to 3/5-oct (720 cents).

And then I want to divide up the diatonic tones and diatonic
semitones uniformly with the new letters

I guess this should continue on tuning-math.