Notating equal temperaments

I've been thinking about the notation of equal temperaments lately, since bjc put up the page on 15-ET chord progressions. Easley Blackwood's notation uses a chain of fifths, but since this closes to a circle of 5 fifths in 15-ET and other multiples of 5 (up to 30-ET), this results in having two different names for the same note, if all seven notes are used.

D..E..G..A..C..D
..(F)......(B)..

Now this is a reasonable notation, and works out well if you're using Blackwood's decatonic scale. But fifths aren't necessarily the best intervals to use as a basis for 15-ET notation; 15-ET has better minor thirds than fifths. If you notate 15-ET with a chain of minor thirds (E# G# B D F Ab Cb), and set the size of a sharp or flat to one step of 15-ET, you end up with a porcupine notation.

D.E.F.G..A.B.C.D

So I began to think: what would happen if you use a notation based on major or minor thirds (whichever is better) for those ET's that don't work out well using a chain of fourths or fifths? I tried this out on 16-ET and ended up with:

D..EF..G.A..BC..D

Which, oddly enough, is the same as Blackwood's notation for 16-ET. I used a chain of major thirds this time, but I suspect (from the liner notes of his CD) that Blackwood may have used the circle of minor thirds that 16-ET shares with 12-ET. As I tried more of these notations, it became apparent that if you set the sizes of the sharps correctly, a chain of minor thirds often works out to be the same notation as a chain of major thirds. So these are two classes of notations: third-based notations and fourth-based notations. 17-ET works out well with a fourth-based notation.

D..EF..G..A..BC..D

While a third-based notation looks like a good possibility for 23-ET (as an alternative to the mavila-based notation).

D...E.F...G..A...B.C...D

Some tunings such as 32-ET have both of these.

D.....EF.....G.....A.....BC.....D (fourth-based)
D.....E.F.....G...A.....B.C.....D (third-based)

There also seem to be some ET's that work best with a notation based on major seconds, such as 18-ET.

D..E.F..G.A..B.C..D

Now, one thing I've noticed about all these notations is that they have two symmetrical tetrachords. DE, FG, AB, and CD are always the same size of step, and EF is the same size as BC. GA can be the same size as EF and BC (as in 18-ET), the same size as the other large steps (as in the diatonic scale), an intermediate size (as in 23-ET), or larger than the other two step sizes. Most of these also have EF and BC as being less than or equal to the other intervals in size, although a third-based notation for 33-ET has larger steps for EF and BC:

D...E....F...G......A...B....C...D

Since 33-ET also has a diatonic notation, this exception can probably be ignored (at least in this case).

D....E...F....G....A....B...C....D

So I'm beginning to wonder ... is it always possible to create a sensible notation for an ET with this kind of symmetrical tetrachord structure? This would seem like a really convenient way to notate ET's if it works in general. Some smaller ET's will need to use fewer than 7 note names, but that shouldn't be much of a problem, e.g.:

D.E.GA.C.D (9-ET)
D.E.G.A.C.D (10-ET)
D..F..GA..B..D (13-ET)

Hi Herman,

On Tue, 20 Dec 2005, Herman Miller wrote:
>
> I've been thinking about the notation of equal temperaments lately,
> since bjc put up the page on 15-ET chord progressions. Easley
> Blackwood's notation uses a chain of fifths, but since this closes to a
> circle of 5 fifths in 15-ET and other multiples of 5 (up to 30-ET), this
> results in having two different names for the same note, if all seven
> notes are used.
>
> D..E..G..A..C..D
> ..(F)......(B)..
>
> Now this is a reasonable notation, and works out well if you're using
> Blackwood's decatonic scale. But fifths aren't necessarily the best
> intervals to use as a basis for 15-ET notation; 15-ET has better minor
> thirds than fifths. If you notate 15-ET with a chain of minor thirds (E#
> G# B D F Ab Cb), and set the size of a sharp or flat to one step of
> 15-ET, you end up with a porcupine notation.
>
> D.E.F.G..A.B.C.D
>
> So I began to think: what would happen if you use a notation based on
> major or minor thirds (whichever is better) for those ET's that don't
> work out well using a chain of fourths or fifths? I tried this out on
> 16-ET and ended up with:
>
> D..EF..G.A..BC..D
>
> Which, oddly enough, is the same as Blackwood's notation for 16-ET. I
> used a chain of major thirds this time, but I suspect (from the liner
> notes of his CD) that Blackwood may have used the circle of minor thirds
> that 16-ET shares with 12-ET. As I tried more of these notations, it
> became apparent that if you set the sizes of the sharps correctly, a
> chain of minor thirds often works out to be the same notation as a chain
> of major thirds. So these are two classes of notations: third-based
> notations and fourth-based notations. 17-ET works out well with a
> fourth-based notation.
>
> D..EF..G..A..BC..D
>
> While a third-based notation looks like a good possibility for 23-ET (as
> an alternative to the mavila-based notation).
>
> D...E.F...G..A...B.C...D
>
> Some tunings such as 32-ET have both of these.
>
> D.....EF.....G.....A.....BC.....D (fourth-based)
> D.....E.F.....G...A.....B.C.....D (third-based)
>
> There also seem to be some ET's that work best with a notation based on
> major seconds, such as 18-ET.
>
> D..E.F..G.A..B.C..D
>
> Now, one thing I've noticed about all these notations is that they have
> two symmetrical tetrachords. DE, FG, AB, and CD are always the same size
> of step, and EF is the same size as BC. GA can be the same size as EF
> and BC (as in 18-ET), the same size as the other large steps (as in the
> diatonic scale), an intermediate size (as in 23-ET), or larger than the
> other two step sizes. Most of these also have EF and BC as being less
> than or equal to the other intervals in size, although a third-based
> notation for 33-ET has larger steps for EF and BC:
>
> D...E....F...G......A...B....C...D
>
> Since 33-ET also has a diatonic notation, this exception can probably be
> ignored (at least in this case).
>
> D....E...F....G....A....B...C....D
>
> So I'm beginning to wonder ... is it always possible to create a
> sensible notation for an ET with this kind of symmetrical tetrachord
> structure? This would seem like a really convenient way to notate ET's
> if it works in general. Some smaller ET's will need to use fewer than 7
> note names, but that shouldn't be much of a problem, e.g.:
>
> D.E.GA.C.D (9-ET)
> D.E.G.A.C.D (10-ET)
> D..F..GA..B..D (13-ET)

Herman,
I'm impressed!

One thing is a little unclear, though: By
"a sensible notation", do you mean one with
no more than the seven nominals A to G?

And if so, is this really a "sensible" limitation?
Looking at, eg, your third-based notation for
32-EDO:
> D.....E.F.....G...A.....B.C.....D
it seems we have five degrees between some
nominals, so will need to use triple sharps or
triple flats. This seems to me to be both
cumbersome and "unnatural". Ideally, I would
think, a more useful notation for any n-EDO
tuning would have either:

a) One nominal for each of the n tuning steps,

- or -

b) One nominal for each scale degree of an
m-note scale drawn from the n-EDO, where
the value of m is not less than half of n,

- or -

c) One nominal roughly corresponding to each
"standard" nominal, and enough distinct signs
for intermediate accidentals to keep the
notation compact.

I think c) is probably the best approach.

The most familiar example of b) is of course
a 7-note diatonic scale drawn from 12-EDO.
The reason I suggested b) is that it's more
practical than one that potentially requires
multiple accidentals in a proper key signature;
in particular,
1) multiple accidentals are tedious to write,
2) multiple accidentals are harder to read, and
3) key signatures would be nightmares!

Imagine, for example, you've been cheerfully
modulating through a chain of keys rooted on
successive fifths. In 32-EDO, we have these
notes:
D.....E.F.....G...A.....B.C.....D
which you proposed we notate as:
D, D#, D##, D###=Ebbb, Ebb, Eb, E,
E#=Fb, F and so on.

We finally get to the key of Ebbb. Please try
writing out the key signature ... :-)

As I see it, there are two simple solutions for
the kind of mess this leads to. They are:

(A) More distinct accidental symbols, as in
Saggital.

(B) More nominals. This corresponds to the
approach b) above. 32-EDO, for example,
might use these:
(1) D.V.U.E.F.T.S.G.R.A.Z.Y.B.C.X.W.D
or these:
(2) D.V.W.E.F.X.Y.G.Z.A.R.S.B.C.T.U.D
or these:
(3) D.L.M.E.F.N.O.G.P.A.H.I.B.C.J.K.D
or even these:
(4) A.B.C.D.E.F.G.H.I.J.K.L.M.N.O.P.A
in place of these:
D.....E.F.....G...A.....B.C.....D

The first three alternates preserve the current
meanings, more or less, of the nominals A to G.
I personally think the fourth way, although more
radical, would actually work best in practice.

Still, for relatively low-numbered EDOs, solution
(A) may be best; it requires only that we have,
and learn the use of, one accidental symbol for
each of the inter-nominal degrees, such as the
dots in D.....E represent. We already have #, x
and b, so really need just two more accidentals
to fill the gaps for 32-EDO. For mine, these
could easily be such readily available and easily
written symbols as @ and *, which also happen
to be large enough to avoid confusing with fly-
specks!

So I think your 32-EDO notation:
D.....E.F.....G...A.....B.C.....D
could be filled in as follows, using the sequence
of accidentals #x@*b ascending:
D, D#, Dx, D@, E*, Eb, E, E#=Fb, F, etc.

If you need even more inter-nominal notes for
a particular n-EDO, then we need only add a
further accidental symbol or two. These
should come, in the first instance, from the
symbols available on every keyboard, and also
should not have any other musical meaning at
present, for example we might extend the list
of accidentals to read #x@*=&$%b, dropping
those from the middle of this set of nine that
we don't need (*), none of which should cause
any confusion.

(*) So we would use,
for 3 accidentals: #xb
for 4 accidentals: #x%b
for 5 accidentals: #x@%b
for 6 accidentals: #x@$%b
for 7 accidentals: #x@*$%b
for 8 accidentals: #x@*&$%b
for 9 accidentals: #x@*=&$%b

The great advantages of this approach are:

1. it builds on current practice, so needs minimal
relearning for everyone, both writer and reader;

2. it is easily extensible;

3. it is quick to write and to read;

4. it is _immediately_ usable in all notation
programs that accept text entry.

What do you reckon?

Regards,
Yahya

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Yahya Abdal-Aziz wrote:

> One thing is a little unclear, though: By > "a sensible notation", do you mean one with
> no more than the seven nominals A to G?

I left that a little vague since I want to leave the possibilities open, but basically I'm looking for notations that have something in common with the familiar system of notation in the way that intervals are notated (e.g., a sensible notation for an ET that has good fifths is one in which the fifths are notated in a chain of fifths BEADGCF, but if the minor thirds are better, a chain of minor thirds should be notated E# G# B D F Ab Cb). It may turn out that larger ET's will require more than seven nominals, although that will require some changes in the staff notation, which I'd like to avoid as far as possible. I've experimented with non-standard staff notation for rank 2 (formerly "linear") temperaments other than meantone, but I find it difficult to read (more so than just using the letter names as note heads without a staff).

> > And if so, is this really a "sensible" limitation?
> Looking at, eg, your third-based notation for
> 32-EDO:
> >>D.....E.F.....G...A.....B.C.....D
> > it seems we have five degrees between some
> nominals, so will need to use triple sharps or
> triple flats. This seems to me to be both
> cumbersome and "unnatural".

Once we start getting into the larger EDO's, the size of the sharp will need to be 2 or more steps in order for the third-based notations to make sense; the difference between a minor third and a major third gives us the size of the sharp (2 steps in the case of 32-ET). So this turns out to be the same notation as 16-ET. To cover the whole tuning system, the intermediate steps will need semisharps and sesquisharps.

Ideally, I would
> think, a more useful notation for any n-EDO > tuning would have either:
> > a) One nominal for each of the n tuning steps,

This gets to be a little awkward after 26-ET....

> - or -
> > b) One nominal for each scale degree of an > m-note scale drawn from the n-EDO, where
> the value of m is not less than half of n,

This could go as far as 52-ET before needing to use characters from non-roman alphabets....

> - or -
> > c) One nominal roughly corresponding to each
> "standard" nominal, and enough distinct signs
> for intermediate accidentals to keep the > notation compact.
> > I think c) is probably the best approach.

Yes, I think it's easier to learn new accidentals than a whole new staff notation.

> The most familiar example of b) is of course
> a 7-note diatonic scale drawn from 12-EDO.
> The reason I suggested b) is that it's more
> practical than one that potentially requires > multiple accidentals in a proper key signature;
> in particular,
> 1) multiple accidentals are tedious to write, > 2) multiple accidentals are harder to read, and > 3) key signatures would be nightmares!

Key signatures aren't much use in a case where your basic scale has e.g. an A sharp and an A flat between some variety of G and some variety of B (or in general, more than 7 notes, like the octatonic scale or the 9-note orwell MOS). And key signatures are probably going to be a bit messy in any EDO that isn't compatible with meantone. But I agree that key signatures with multiple accidentals would be a mess to deal with.

> (*) So we would use,
> for 3 accidentals: #xb
> for 4 accidentals: #x%b
> for 5 accidentals: #x@%b
> for 6 accidentals: #x@$%b
> for 7 accidentals: #x@*$%b
> for 8 accidentals: #x@*&$%b
> for 9 accidentals: #x@*=&$%b

I'd like to relate the meaning of sharps and flats to traditional notation in some way (whatever makes sense for the particular ET). For instance, a fourth above F could be notated Bb; the flat size is the difference between that note and the note named B. Or, the note a major third above D could be written F#. Then if the sharp turns out to be 2 steps of the ET, you'd use a semisharp for one step; finer divisions could use accidentals from 72-ET notation, Sagittal, or whatever works out for the temperament in question. On the other hand, I don't want the "sharps" to go negative, as they do if you notate 16-ET as a chain of fifths.

Hi Herman,

On Wed, 21 Dec 2005, Herman Miller wrote:
>
> Yahya Abdal-Aziz wrote:
>
> > One thing is a little unclear, though: By
> > "a sensible notation", do you mean one with
> > no more than the seven nominals A to G?
>
> I left that a little vague since I want to leave the possibilities open,
> but basically I'm looking for notations that have something in common
> with the familiar system of notation in the way that intervals are
> notated (e.g., a sensible notation for an ET that has good fifths is one
> in which the fifths are notated in a chain of fifths BEADGCF, ...

These are fourths! :-)

I do tend to think of a chain of ascending fifths:
(FbCbGbDbAbEbBb)FCGDAEB(F#C#G#D#A#E#)

> ... but if the
> minor thirds are better, a chain of minor thirds should be notated E# G#
> B D F Ab Cb). It may turn out that larger ET's will require more than
> seven nominals, although that will require some changes in the staff
> notation, which I'd like to avoid as far as possible.

D'accord!

> I've experimented
> with non-standard staff notation for rank 2 (formerly "linear")
> temperaments other than meantone, but I find it difficult to read (more
> so than just using the letter names as note heads without a staff).

I'm sure that using non-standard nominals,
in contexts that require standard interpretations
(eg the fifth of "D" is "A"), would cause more
pain to most users than it's worth.

> > And if so, is this really a "sensible" limitation?
> > Looking at, eg, your third-based notation for
> > 32-EDO:
> >
> >>D.....E.F.....G...A.....B.C.....D
> >
> > it seems we have five degrees between some
> > nominals, so will need to use triple sharps or
> > triple flats. This seems to me to be both
> > cumbersome and "unnatural".
>
> Once we start getting into the larger EDO's, the size of the sharp will
> need to be 2 or more steps in order for the third-based notations to
> make sense; the difference between a minor third and a major third gives
> us the size of the sharp (2 steps in the case of 32-ET). So this turns
> out to be the same notation as 16-ET. To cover the whole tuning system,
> the intermediate steps will need semisharps and sesquisharps.

Yes, I see; I had only been thinking of tuning
by fifths. And I had understood you to mean
that one step would be a # or b, so that any
multiples could be notated with just those two
accidentals.

> Ideally, I would
> > think, a more useful notation for any n-EDO
> > tuning would have either:
> >
> > a) One nominal for each of the n tuning steps,
>
> This gets to be a little awkward after 26-ET....

Start using the IPA? :-) Agreed, too messy.

> > - or -
> >
> > b) One nominal for each scale degree of an
> > m-note scale drawn from the n-EDO, where
> > the value of m is not less than half of n,
>
> This could go as far as 52-ET before needing to use characters from
> non-roman alphabets....

Which would almost suffice for most of the
EDOs that people actually use.

> > - or -
> >
> > c) One nominal roughly corresponding to each
> > "standard" nominal, and enough distinct signs
> > for intermediate accidentals to keep the
> > notation compact.
> >
> > I think c) is probably the best approach.
>
> Yes, I think it's easier to learn new accidentals than a whole new staff
> notation.

Good! I'm glad we agree on this.

> > The most familiar example of b) is of course
> > a 7-note diatonic scale drawn from 12-EDO.
> > The reason I suggested b) is that it's more
> > practical than one that potentially requires
> > multiple accidentals in a proper key signature;
> > in particular,
> > 1) multiple accidentals are tedious to write,
> > 2) multiple accidentals are harder to read, and
> > 3) key signatures would be nightmares!
>
> Key signatures aren't much use in a case where your basic scale has e.g.
> an A sharp and an A flat between some variety of G and some variety of B
> (or in general, more than 7 notes, like the octatonic scale or the
> 9-note orwell MOS). ...

True. They won't help at all,for example, when
someone has tuned up 53-EDO and regards it as
a scale of 53 notes.

> ... And key signatures are probably going to be a bit
> messy in any EDO that isn't compatible with meantone. But I agree that
> key signatures with multiple accidentals would be a mess to deal with.
>
> > (*) So we would use,
> > for 3 accidentals: #xb
> > for 4 accidentals: #x%b
> > for 5 accidentals: #x@%b
> > for 6 accidentals: #x@$%b
> > for 7 accidentals: #x@*$%b
> > for 8 accidentals: #x@*&$%b
> > for 9 accidentals: #x@*=&$%b
>
> I'd like to relate the meaning of sharps and flats to traditional
> notation in some way (whatever makes sense for the particular ET). For
> instance, a fourth above F could be notated Bb; the flat size is the
> difference between that note and the note named B. Or, the note a major
> third above D could be written F#. Then if the sharp turns out to be 2
> steps of the ET, you'd use a semisharp for one step; finer divisions
> could use accidentals from 72-ET notation, Sagittal, or whatever works
> out for the temperament in question. ...

That makes eminently good sense. However ...

Do you know whether there are any cases (for
moderately small n) where interpreting n-EDO
as chains of fifths or chains of thirds would
select different steps from the gamut for the
same note spelling? If so, you'd want a convention
as to which chain of generators took precedence.

To achieve this "standard interpretation" of
#s, bs, and other accidentals, we could simply
permute the list of "user-friendly" accidentals
I suggested earlier. Instead of:
> > for 3 accidentals: #xb
> > for 4 accidentals: #x%b
> > for 5 accidentals: #x@%b
> > for 6 accidentals: #x@$%b
> > for 7 accidentals: #x@*$%b
> > for 8 accidentals: #x@*&$%b
> > for 9 accidentals: #x@*=&$%b

we would have something like:
for 3 accidentals: #xb
for 4 accidentals: %#xb
for 5 accidentals: %#xb@
for 6 accidentals: %#$xb@
for 7 accidentals: %#$x*b@
for 8 accidentals: %#$x&*b@
for 9 accidentals: %#$x&*=b@

but where the actual sequence of
accidentals would depend on "whatever
works out for the temperament in
question", as you wrote.

With all due respect to George and Dave
and anyone else that has contributed to
the development of Sagittal, I do think
that providing a set of accidentals that
can be found on everyone's (computer)
keyboard would have a much better chance
of success (= widespread adoption) than
any system that requires new symbols.

> ... On the other hand, I don't want the
> "sharps" to go negative, as they do if you notate 16-ET as a chain of
> fifths.

Yes, that's a nonsense, isn't it?

Regards,
Yahya

Fr�hliche Weihnachtzeit!

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On Fri, 23 Dec 2005 14:01:23 +1100, "Yahya Abdal-Aziz"
<yahya@melbpc.org.au> wrote:

>Do you know whether there are any cases (for
>moderately small n) where interpreting n-EDO
>as chains of fifths or chains of thirds would
>select different steps from the gamut for the
>same note spelling? If so, you'd want a convention
>as to which chain of generators took precedence.

Yes, that happens frequently. A chain of fourths for 16-ET, for instance,
gives a notation that matches the basic scale of what some of us are
calling "mavila temperament" (after Wilson's "meta-mavila"), which consists
of 7 small steps and 2 large steps (the opposite of the diatonic scale).
But the third-based notation matches Blackwood's notation from his
microtonal etudes. Since the major thirds are better than the fourths in
16-ET, a third-based notation might be preferable. With a fourth-based
notaiton, the interval notated as a "sharp" would end up going one step
down, which conflicts with the traditional usage of sharps.

There are also cases where the notation based on major thirds conflicts
with the notation based on minor thirds (although I'm away from home for a
while and didn't bring my tuning notes with me). As a general rule, it
seems that the better notations are the ones based on the intervals that
deviate the least from the just intervals, but I expect that some of the
ET's could have conflicting notations that each have some desirable
properties without any one of them standing out as obviously better.

--- In tuning@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
>
> On Fri, 23 Dec 2005 14:01:23 +1100, "Yahya Abdal-Aziz"
> <yahya@m...> wrote:
>
> >Do you know whether there are any cases (for
> >moderately small n) where interpreting n-EDO
> >as chains of fifths or chains of thirds would
> >select different steps from the gamut for the
> >same note spelling? If so, you'd want a convention
> >as to which chain of generators took precedence.
>
> Yes, that happens frequently. A chain of fourths for 16-ET, for
instance,
> gives a notation that matches the basic scale of what some of us are
> calling "mavila temperament" (after Wilson's "meta-mavila"), which
consists
> of 7 small steps and 2 large steps (the opposite of the diatonic
scale).

I think you meant 5 small steps and 2 large steps.

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
> With all due respect to George and Dave
> and anyone else that has contributed to
> the development of Sagittal, I do think
> that providing a set of accidentals that
> can be found on everyone's (computer)
> keyboard would have a much better chance
> of success (= widespread adoption) than
> any system that requires new symbols.

Hi Yahya,

That's a good point. Which is one reason we long ago defined single-
character ASCII equivalents for all the most useful single-shaft
Sagittal symbols, to be used in conjunction with the conventional b
bb # and x.

See the column marked "Mixed short ASCII" at
http://dkeenan.com/sagittal/map/index.htm#mid
for the most common ones.

You will notice that a few of the less useful symbols in that chart
do not show a single ASCII characters. These are out of date. _All_
unaccented single-shaft sagittal symbols now have single-character
ASCII or 8-bit ASCII (ISO Latin-1) equivalents. See Scala's
Help:Legend:Accidentals, or email me if you need others.

-- Dave Keenan