Yahoo Tuning Groups Ultimate Backup tuning-math Suggestion for a linear temperament notation system

This has a somewhat long introduction, so you might want to skip to the row of asterisks if you're short on time.

The standard meantone notation has a useful property, which is also a feature of my notation for porcupine temperament (as described at http://www.io.com/~hmiller/music/temp-porcupine.html). The chromatic semitone, which is about 76 cents in TOP meantone, is smaller than any of the steps of the 7-note meantone MOS which is the basis for the notation. Furthermore, this interval is generated by 7 steps of the meantone generator, so any note in meantone temperament can be represented with one of the basic 7 notes with the addition of one or more accidentals which represent this chromatic semitone.

My porcupine notation also has this property; the "chromatic semitone" equivalent (which is 60.7 cents in TOP porcupine) is reached with 7 porcupine generators, and the basic scale has 7 notes with steps of 162.3 cents between them. With 8 nominals, one of the steps between notes would be the same size as the 60.7-cent "semitone" (as with German note names, where Hb represents the same note as B). You need to go as far as 15 steps of porcupine before you get a smaller interval of approximately 41 cents. So a 15-nominal system would also have this property of accidentals which are smaller than the steps between notes.

For most of the useful temperaments, you can find a contiguous scale with less than 24 notes that has this property. To take a random example, "muggles" <<5, 1, -7, -10, -25, -19]] (TOP tuning P=1203.15, G=379.39), which is a tuning system that hasn't received much attention but seems to have some interesting musical possibilities from my experimentation with it, has a 16-note scale with a smallest step of around 65 cents, which can be used with accidentals that represent a series of 16 muggles generators, producing an interval of around 54.5 cents. The next step in notating muggles is to assign names to each of the 16 notes and to identify which accidental is the most suitable for the 54.5 cent interval.

For naming the nominals, I've been using a system based on dividing the tempered octave into 24 equal parts, with the standard A-G assigned to the diatonic scale, H-N for notes a quarter tone below A-G, P-V for notes a quarter tone above A-G, and O W X Y Z for the black notes of the keyboard (O between G and A, W between A and B, X between C and D, Y between D and E, and Z betweeen F and G). The note between E and F could be either M or T, and the note between B and C could be either J or Q. So with this system, the 16 notes of muggles would be something like this:

(V) J L G B S N W D Z P K F A R T (H)

Depending on convention, either V or H could be the 16th note. I've tried this method on a number of the more well-known temperaments and it seems to be fairly productive. Most temperaments have at least one point where a small accidental in combination with a set of less than 24 notes can be used in this way to extend the notation beyond the basic set in both directions, and some of them have more than one.

But this naming system falls apart when you get to the myna temperament (formerly nonkleismic), <<10, 9, 7, -9, -17, -9]] (TOP tuning P=1198.83, G=309.89). After 4 steps of myna, a tiny interval of 40.7 cents shows up. This tiny interval dominates the scale until you reach 27 steps, when a smaller 24.7 cent interval finally appears. Naturally, there's no way to represent 27 notes with only 26 nominals, even if M/T and J/Q are distinguished. And the problem shows up so early in the scale that 4 steps is all that can be unambiguously named.

If this had been some obscure temperament that no one's likely to use, this wouldn't be that serious of a flaw, but myna appears to be one of the more useful temperaments when you start going to higher limits like 11 and 13. But I noticed something about the nominals that had problems with myna; they're the ones that represent the black notes of the keyboard. This brings up an interesting possibility for notation of myna, and for linear temperaments in general.

*********************************************************************

What if, instead of using single symbols for the basic notes of a temperament, we could use the standard sharp and flat symbols, along with the Tartini/Fokker accidentals for semisharps and semiflats? These could then be used in combination with the Sagittal accidentals for the more minute distinctions. In this case, the half-octave above D, which I've been representing as O, could be split into G# and Ab for scales like myna which need both, or in other cases represented as either G# or Ab interchangeably, as in 12-ET notation. So the 27 notes of myna could be represented like this (using "z" and "v" for the semisharp and semiflat symbols):

Az C# E G Bb Db Ez Gz Bv Dv Fv G# B D F Ab Bz Dz Fz Av Cv D# F# A C Eb Gv

Note that a 12-note chain of just fourths would look like this:

(Ab) Db Gb B E A D G C F A# D# (G#)

This could be a little disorienting to musicians who are used to F#-B and F-Bb being perfect fourths. So it might be better to have the higher note labeled as "G#" and the lower one as "Ab". But that causes problems with major-third based temperaments like magic: you expect a M3 above D to be written as "F#", not "Gb". If a major third above D is notated as "Z", that problem doesn't arise, but then you still have the problem of how you notate a "Z" on a musical staff.

My preference is for sharps lower than flats, and semisharps lower than semiflats, so the basic set of 24 note categories could be notated like this:

D | Dz (S) | D# Eb (Y) | Ev (L) | E | Ez Fv (T/M) |
F | Fz (U) | F# Gb (Z) | Gv (N) | G | Gz (V) |
G# Ab (O) | Av (H) | A | Az (P) | A# Bb (W) | Bv (I) |
B | Bz Cv (Q/J) | C | Cz (R) | C# Db (X) | Dv (K)

With this notation, you can get a pretty good idea of the rough pitches without having to remember that S is slightly sharper than D, or W is about halfway between A and B. The thing you have to remember is that the symbols # b z v aren't really being used as accidentals, but more like "accent marks" for notes, distinguishing one basic pitch of the scale from a slightly different pitch in another scale. What gave me the idea to use these symbols in this way is looking at the way accidentals are used in Arabic music (as described at http://www.maqamworld.com/). A note described as "E semiflat" in one tetrachord (e.g., Bayati) may be a different pitch than the "E semiflat" of a different one (e.g., Rast or Sikah). And an advantage of this system is that it uses standard musical notation with familiar symbols, in ways that suggest approximately the "right" pitches for the basic scale of each temperament.

So then, once the size of the basic scale has been determined by looking for small steps that can be used as accidentals, and names have been assigned to each of the notes in the basic scale, all that remains is to find appropriate Sagittal accidentals for the small intervals. Ideally you could find an accidental of the appropriate size that is consistent with the JI mapping of the temperament. But this is an area of the notation system that I haven't spent much time with, and needs further investigation.

🔗Ozan Yarman <ozanyarman@superonline.com>

Dear Herman, do you think my incomplete Spectral Notation proposal could help in the notation part?

http://www.ozanyarman.com/files/SPECTRAL%20NOTATION.pdf

Cordially,
Ozan Yarman
----- Original Message -----
From: Herman Miller
To: tuning-math@yahoogroups.com
Sent: 11 Şubat 2005 Cuma 6:59
Subject: [tuning-math] Suggestion for a linear temperament notation system

This has a somewhat long introduction, so you might want to skip to the
row of asterisks if you're short on time.

The standard meantone notation has a useful property, which is also a
feature of my notation for porcupine temperament (as described at
http://www.io.com/~hmiller/music/temp-porcupine.html). The chromatic
semitone, which is about 76 cents in TOP meantone, is smaller than any
of the steps of the 7-note meantone MOS which is the basis for the
notation. Furthermore, this interval is generated by 7 steps of the
meantone generator, so any note in meantone temperament can be
represented with one of the basic 7 notes with the addition of one or
more accidentals which represent this chromatic semitone.

My porcupine notation also has this property; the "chromatic semitone"
equivalent (which is 60.7 cents in TOP porcupine) is reached with 7
porcupine generators, and the basic scale has 7 notes with steps of
162.3 cents between them. With 8 nominals, one of the steps between
notes would be the same size as the 60.7-cent "semitone" (as with German
note names, where Hb represents the same note as B). You need to go as
far as 15 steps of porcupine before you get a smaller interval of
approximately 41 cents. So a 15-nominal system would also have this
property of accidentals which are smaller than the steps between notes.

For most of the useful temperaments, you can find a contiguous scale
with less than 24 notes that has this property. To take a random
example, "muggles" <<5, 1, -7, -10, -25, -19]] (TOP tuning P=1203.15,
G=379.39), which is a tuning system that hasn't received much attention
but seems to have some interesting musical possibilities from my
experimentation with it, has a 16-note scale with a smallest step of
around 65 cents, which can be used with accidentals that represent a
series of 16 muggles generators, producing an interval of around 54.5
cents. The next step in notating muggles is to assign names to each of
the 16 notes and to identify which accidental is the most suitable for
the 54.5 cent interval.

For naming the nominals, I've been using a system based on dividing the
tempered octave into 24 equal parts, with the standard A-G assigned to
the diatonic scale, H-N for notes a quarter tone below A-G, P-V for
notes a quarter tone above A-G, and O W X Y Z for the black notes of the
keyboard (O between G and A, W between A and B, X between C and D, Y
between D and E, and Z betweeen F and G). The note between E and F could
be either M or T, and the note between B and C could be either J or Q.
So with this system, the 16 notes of muggles would be something like this:

(V) J L G B S N W D Z P K F A R T (H)

Depending on convention, either V or H could be the 16th note. I've
tried this method on a number of the more well-known temperaments and it
seems to be fairly productive. Most temperaments have at least one point
where a small accidental in combination with a set of less than 24 notes
can be used in this way to extend the notation beyond the basic set in
both directions, and some of them have more than one.

But this naming system falls apart when you get to the myna temperament
(formerly nonkleismic), <<10, 9, 7, -9, -17, -9]] (TOP tuning P=1198.83,
G=309.89). After 4 steps of myna, a tiny interval of 40.7 cents shows
up. This tiny interval dominates the scale until you reach 27 steps,
when a smaller 24.7 cent interval finally appears. Naturally, there's no
way to represent 27 notes with only 26 nominals, even if M/T and J/Q are
distinguished. And the problem shows up so early in the scale that 4
steps is all that can be unambiguously named.

If this had been some obscure temperament that no one's likely to use,
this wouldn't be that serious of a flaw, but myna appears to be one of
the more useful temperaments when you start going to higher limits like
11 and 13. But I noticed something about the nominals that had problems
with myna; they're the ones that represent the black notes of the
keyboard. This brings up an interesting possibility for notation of
myna, and for linear temperaments in general.

*********************************************************************

What if, instead of using single symbols for the basic notes of a
temperament, we could use the standard sharp and flat symbols, along
with the Tartini/Fokker accidentals for semisharps and semiflats? These
could then be used in combination with the Sagittal accidentals for the
more minute distinctions. In this case, the half-octave above D, which
I've been representing as O, could be split into G# and Ab for scales
like myna which need both, or in other cases represented as either G# or
Ab interchangeably, as in 12-ET notation. So the 27 notes of myna could
be represented like this (using "z" and "v" for the semisharp and
semiflat symbols):

Az C# E G Bb Db Ez Gz Bv Dv Fv G# B D F Ab Bz Dz Fz Av Cv D# F# A C Eb Gv

Note that a 12-note chain of just fourths would look like this:

(Ab) Db Gb B E A D G C F A# D# (G#)

This could be a little disorienting to musicians who are used to F#-B
and F-Bb being perfect fourths. So it might be better to have the higher
note labeled as "G#" and the lower one as "Ab". But that causes problems
with major-third based temperaments like magic: you expect a M3 above D
to be written as "F#", not "Gb". If a major third above D is notated as
"Z", that problem doesn't arise, but then you still have the problem of
how you notate a "Z" on a musical staff.

My preference is for sharps lower than flats, and semisharps lower than
semiflats, so the basic set of 24 note categories could be notated like
this:

With this notation, you can get a pretty good idea of the rough pitches
without having to remember that S is slightly sharper than D, or W is
about halfway between A and B. The thing you have to remember is that
the symbols # b z v aren't really being used as accidentals, but more
like "accent marks" for notes, distinguishing one basic pitch of the
scale from a slightly different pitch in another scale. What gave me the
idea to use these symbols in this way is looking at the way accidentals
are used in Arabic music (as described at http://www.maqamworld.com/). A
note described as "E semiflat" in one tetrachord (e.g., Bayati) may be a
different pitch than the "E semiflat" of a different one (e.g., Rast or
Sikah). And an advantage of this system is that it uses standard musical
notation with familiar symbols, in ways that suggest approximately the
"right" pitches for the basic scale of each temperament.

So then, once the size of the basic scale has been determined by looking
for small steps that can be used as accidentals, and names have been
assigned to each of the notes in the basic scale, all that remains is to
find appropriate Sagittal accidentals for the small intervals. Ideally
you could find an accidental of the appropriate size that is consistent
with the JI mapping of the temperament. But this is an area of the
notation system that I haven't spent much time with, and needs further
investigation.

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🔗Igliashon Jones <igliashon@sbcglobal.net>

Greetings, Mr. Miller!

I've been having quite a discourse with Mr. Erlich over a naming
system for Porcupine-8 as represented in 22-tet, based on the
properties of chord formation as described by relative degrees of the
scale. I noticed that if you take a template of 1-4-6 (in degrees of
porcupine 8) you can form otonal chords in psychoacoustic root
position but utonals in first inversion; also, this template produces
two "suspended 4"-sounding chords that don't involve
actually "suspending" any intervals. Likewise, a 1-3-6 pattern
produces non-inverted utonals but inverted otonals, and "suspended
2nd" chords instead of "sus 4"'s. 1-4-7 produces first inversion
otonals, 2nd inversion utonals, and "stacked fourth" chords.

Since you have vastly more experience than myself in the field of
porcupine music, how would you propose to name the 8-note porcupine
scale such that it could be consistent the way traditional diatonic
notation is with meantone-7? What I mean by consistent is that an
interval of some type of "porcu-nth" would be represented by the same
letter-relation (say C to E or C to Eb in meantone-7 is a type of
meantone 3rd, but C to D or C to D# is a meantone 2nd). The system a
colleague of mine and I have worked out seems to suggest that to get
psychoacoustic root-position triads one cannot simply use a single
triadic template (one that would be analogous to the 1-3-5 of
diatonic), but must rather pick one for major and then consider root-
position minors as "suspensions" of the major template.

What are your thoughts?

Highest regards,
-Igliashon

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
> This has a somewhat long introduction, so you might want to skip to
the
> row of asterisks if you're short on time.
>
> The standard meantone notation has a useful property, which is also
a
> feature of my notation for porcupine temperament (as described at
> http://www.io.com/~hmiller/music/temp-porcupine.html). The
chromatic
> semitone, which is about 76 cents in TOP meantone, is smaller than
any
> of the steps of the 7-note meantone MOS which is the basis for the
> notation. Furthermore, this interval is generated by 7 steps of the
> meantone generator, so any note in meantone temperament can be
> represented with one of the basic 7 notes with the addition of one
or
> more accidentals which represent this chromatic semitone.
>
> My porcupine notation also has this property; the "chromatic
semitone"
> equivalent (which is 60.7 cents in TOP porcupine) is reached with 7
> porcupine generators, and the basic scale has 7 notes with steps of
> 162.3 cents between them. With 8 nominals, one of the steps between
> notes would be the same size as the 60.7-cent "semitone" (as with
German
> note names, where Hb represents the same note as B). You need to go
as
> far as 15 steps of porcupine before you get a smaller interval of
> approximately 41 cents. So a 15-nominal system would also have this
> property of accidentals which are smaller than the steps between
notes.
>
> For most of the useful temperaments, you can find a contiguous
scale
> with less than 24 notes that has this property. To take a random
> example, "muggles" <<5, 1, -7, -10, -25, -19]] (TOP tuning
P=1203.15,
> G=379.39), which is a tuning system that hasn't received much
attention
> but seems to have some interesting musical possibilities from my
> experimentation with it, has a 16-note scale with a smallest step
of
> around 65 cents, which can be used with accidentals that represent
a
> series of 16 muggles generators, producing an interval of around
54.5
> cents. The next step in notating muggles is to assign names to each
of
> the 16 notes and to identify which accidental is the most suitable
for
> the 54.5 cent interval.
>
> For naming the nominals, I've been using a system based on dividing
the
> tempered octave into 24 equal parts, with the standard A-G assigned
to
> the diatonic scale, H-N for notes a quarter tone below A-G, P-V for
> notes a quarter tone above A-G, and O W X Y Z for the black notes
of the
> keyboard (O between G and A, W between A and B, X between C and D,
Y
> between D and E, and Z betweeen F and G). The note between E and F
could
> be either M or T, and the note between B and C could be either J or
Q.
> So with this system, the 16 notes of muggles would be something
like this:
>
> (V) J L G B S N W D Z P K F A R T (H)
>
> Depending on convention, either V or H could be the 16th note. I've
> tried this method on a number of the more well-known temperaments
and it
> seems to be fairly productive. Most temperaments have at least one
point
> where a small accidental in combination with a set of less than 24
notes
> can be used in this way to extend the notation beyond the basic set
in
> both directions, and some of them have more than one.
>
> But this naming system falls apart when you get to the myna
temperament
> (formerly nonkleismic), <<10, 9, 7, -9, -17, -9]] (TOP tuning
P=1198.83,
> G=309.89). After 4 steps of myna, a tiny interval of 40.7 cents
shows
> up. This tiny interval dominates the scale until you reach 27
steps,
> when a smaller 24.7 cent interval finally appears. Naturally,
there's no
> way to represent 27 notes with only 26 nominals, even if M/T and
J/Q are
> distinguished. And the problem shows up so early in the scale that
4
> steps is all that can be unambiguously named.
>
> If this had been some obscure temperament that no one's likely to
use,
> this wouldn't be that serious of a flaw, but myna appears to be one
of
> the more useful temperaments when you start going to higher limits
like
> 11 and 13. But I noticed something about the nominals that had
problems
> with myna; they're the ones that represent the black notes of the
> keyboard. This brings up an interesting possibility for notation of
> myna, and for linear temperaments in general.
>
>
*********************************************************************
>
> What if, instead of using single symbols for the basic notes of a
> temperament, we could use the standard sharp and flat symbols,
along
> with the Tartini/Fokker accidentals for semisharps and semiflats?
These
> could then be used in combination with the Sagittal accidentals for
the
> more minute distinctions. In this case, the half-octave above D,
which
> I've been representing as O, could be split into G# and Ab for
scales
> like myna which need both, or in other cases represented as either
G# or
> Ab interchangeably, as in 12-ET notation. So the 27 notes of myna
could
> be represented like this (using "z" and "v" for the semisharp and
> semiflat symbols):
>
> Az C# E G Bb Db Ez Gz Bv Dv Fv G# B D F Ab Bz Dz Fz Av Cv D# F# A C
Eb Gv
>
> Note that a 12-note chain of just fourths would look like this:
>
> (Ab) Db Gb B E A D G C F A# D# (G#)
>
> This could be a little disorienting to musicians who are used to F#-
B
> and F-Bb being perfect fourths. So it might be better to have the
higher
> note labeled as "G#" and the lower one as "Ab". But that causes
problems
> with major-third based temperaments like magic: you expect a M3
above D
> to be written as "F#", not "Gb". If a major third above D is
notated as
> "Z", that problem doesn't arise, but then you still have the
problem of
> how you notate a "Z" on a musical staff.
>
> My preference is for sharps lower than flats, and semisharps lower
than
> semiflats, so the basic set of 24 note categories could be notated
like
> this:
>
> D | Dz (S) | D# Eb (Y) | Ev (L) | E | Ez Fv (T/M) |
> F | Fz (U) | F# Gb (Z) | Gv (N) | G | Gz (V) |
> G# Ab (O) | Av (H) | A | Az (P) | A# Bb (W) | Bv (I) |
> B | Bz Cv (Q/J) | C | Cz (R) | C# Db (X) | Dv (K)
>
> With this notation, you can get a pretty good idea of the rough
pitches
> without having to remember that S is slightly sharper than D, or W
is
> about halfway between A and B. The thing you have to remember is
that
> the symbols # b z v aren't really being used as accidentals, but
more
> like "accent marks" for notes, distinguishing one basic pitch of
the
> scale from a slightly different pitch in another scale. What gave
me the
> idea to use these symbols in this way is looking at the way
accidentals
> are used in Arabic music (as described at
http://www.maqamworld.com/). A
> note described as "E semiflat" in one tetrachord (e.g., Bayati) may
be a
> different pitch than the "E semiflat" of a different one (e.g.,
Rast or
> Sikah). And an advantage of this system is that it uses standard
musical
> notation with familiar symbols, in ways that suggest approximately
the
> "right" pitches for the basic scale of each temperament.
>
> So then, once the size of the basic scale has been determined by
looking
> for small steps that can be used as accidentals, and names have
been
> assigned to each of the notes in the basic scale, all that remains
is to
> find appropriate Sagittal accidentals for the small intervals.
Ideally
> you could find an accidental of the appropriate size that is
consistent
> with the JI mapping of the temperament. But this is an area of the
> notation system that I haven't spent much time with, and needs
further
> investigation.

🔗Herman Miller <hmiller@IO.COM>

Igliashon Jones wrote:

> Greetings, Mr. Miller!
> > I've been having quite a discourse with Mr. Erlich over a naming > system for Porcupine-8 as represented in 22-tet, based on the > properties of chord formation as described by relative degrees of the > scale. I noticed that if you take a template of 1-4-6 (in degrees of > porcupine 8) you can form otonal chords in psychoacoustic root > position but utonals in first inversion; also, this template produces > two "suspended 4"-sounding chords that don't involve > actually "suspending" any intervals. Likewise, a 1-3-6 pattern > produces non-inverted utonals but inverted otonals, and "suspended > 2nd" chords instead of "sus 4"'s. 1-4-7 produces first inversion > otonals, 2nd inversion utonals, and "stacked fourth" chords.

I've mainly used porcupine in conjunction with 15-ET and the 15-note subset of 37-ET, so it's good to see some interest in the other end of the porcupine continuum for variety. Paul has done some nice things with 22-ET using porcupine among other temperaments (in particular, pajara), and I've been following these new developments in porcupine theory with interest. But I've been behind on my email reading, and by the time I caught up to the threads in MMM and the main tuning list, there wasn't much I could think of to add.

> Since you have vastly more experience than myself in the field of > porcupine music, how would you propose to name the 8-note porcupine > scale such that it could be consistent the way traditional diatonic > notation is with meantone-7? What I mean by consistent is that an > interval of some type of "porcu-nth" would be represented by the same > letter-relation (say C to E or C to Eb in meantone-7 is a type of > meantone 3rd, but C to D or C to D# is a meantone 2nd). The system a > colleague of mine and I have worked out seems to suggest that to get > psychoacoustic root-position triads one cannot simply use a single > triadic template (one that would be analogous to the 1-3-5 of > diatonic), but must rather pick one for major and then consider root-
> position minors as "suspensions" of the major template.
> > What are your thoughts?

I'm actually not all that familiar with porcupine[8]; previously I've mainly been concerned with the 7-note and 15-note versions, and with chord progressions that exploit the 250/243 comma. But the 8-note scale sounds like it has some interesting properties, and I'd like to become more familiar with it whenever I get the time (which has been in short supply lately).

The system of notation that I've been using for porcupine temperament is based on the 7-note scale and described on my web page:

http://www.io.com/~hmiller/music/temp-porcupine.html

The 7 notes A B C D E F G are equally spaced, with 3 step intervals between them in the 22-ET version. I use "half-sharp" and "half-flat" symbols which I've been representing in ASCII as square brackets ] [ to represent an interval of 1 step of 22-ET. So the 22-ET scale would be:

D D] E[ E E] F[ F F] G[ G G] G]]/A[[ A[ A A] B[ B B] C[ C C] D[ D

This could be extended to 8 notes by adding H as equivalent to A[, which would eliminate the need for double accidentals for the pitch a tritone above D (which could then be represented as H[). This would have the same issues as the German system of note naming with B equivalent to H-flat. But I haven't done much specifically with the 8-note scale, so I don't know if that's a problem or not.

In the alphabetic notation that Dave Keenan and I were discussing earlier on this list and in private email, the basic porcupine-7 scale would be notated A I R D L U G. The 8th note of porcupine-8 could either be H (a small step away from A) or V (a small step away from G). The fact that the H in this notation also happens to be next to A is a nice coincidence. One advantage of this system is that the microtonal nature of the scale is more readily apparent. But it's a little misleading, since most of the minor thirds are notated as if they were neutral thirds. On the other hand, my original notation has C-E as a minor third, which will be confusing to anyone who knows traditional notation.

> Highest regards,
> -Igliashon

🔗Ozan Yarman <ozanyarman@superonline.com>

🔗Herman Miller <hmiller@IO.COM>

Here are some step sizes for specific temperaments which may be used as accidentals. In each case, the number of generators is given followed by the size of the step in cents. The assumptions I'm following are: 1. no accidental larger than 200 cents, 2. no more than 31 basic notes in the scale without the use of accidentals, and 3. accidental size smaller than any interval in the basic scale. The most problematic of these temperaments for note spelling are wï¿½rschmidt, myna, and cynder.

5-limit linear temperaments:

Mavila <<1, -3, -7]] TOP P=1206.55 G=521.52
-2 (163.51), +7 (31.00)
Basic scale: I L A D G R U (or Bv Ev A D G Cz Fz; since the semiflats and semisharps are not needed to distinguish one note of the scale from another in this case, they may be omitted)

Meantone <<1, 4, 4]] TOP P=1201.70 G=504.13
-2 (193.43), +5 (117.27), -7 (76.16), +12 (41.12), -19 (35.03), +31 (6.08)
Basic scale: B E A D G C F

Schismic (Helmholtz) <<1, -8, -15]] TOP P=1200.07 G=498.28
+5 (91.26), -12 (20.98)
Basic scale: (O X Z B) E A D G C (F W Y O)

Superpyth[agorean] <<1, 9, 12]] TOP P=1197.60 G=489.43
+5 (51.94), -22 (10.97)
Basic scale: (O X Z B L H K N Q) E A D G C (M P S V R F W Y O)
Superpyth is complex enough that using only 5 nominals might be inadequate, but whether it's worth going to the full set of 22 is a good question.

Semisixths (sensipent) <<7, 9, -2]] TOP P=1199.59 G=442.98
+3 (129.36), -8 (54.89), +19 (19.59)
Basic scale: (W S G Q E) (H) R F P D N B L (V) (C M A K Z)

Wï¿½rschmidt <<8, 1, -17]] TOP P=1199.69 G=387.64
-3 (36.76), +31 (20.07)
Basic scale: E *V J L *V *B Y G *B *S N I *S *Z *W D *Z *W *K U P *K *F A X *F *H R T *H C. Lots of duplicated notes; this temperament has always been a problem to notate and the split of O, W, X, Y, Z won't help in this case.

Magic <<5, 1, -10]] TOP P=1201.28 G=380.80
-3 (58.89), +19 (27.46)
Basic scale: E V Q Y G B S N W D Z P K F A X M H C

Dicot <<2, 1, -3]] TOP P=1207.66 G=353.22
-3 (148.01), +7 (57.21), -17 (33.60), +24 (23.61)
Basic scale: L G I D U A R (or Ev G Bv D Fz A Cz)

Amity <<5, 13, 9]] TOP P=1199.85 G=339.47
-3 (181.43), +4 (158.04), -7 (23.40)
Basic scale: E G I D U A C

Kleismic (Oolong) <<6, 5, -6]] TOP P=1200.29 G=317.07
+4 (67.99), -15 (45.12), +19 (22.86)
Basic scale: (P R) E G W K T V B D F H J S Z A C (L N)

Orson (5-limit orwell) <<7, -3, -21]] TOP P=1200.24 G=271.65
-4 (113.62), +9 (44.40), -22 (24.82), +31 (19.59)
Basic scale: Y Z H Q D M V W X

Tetracot <<4, 9, 5]] TOP P=1199.03 G=176.11
+1 (176.11), -6 (142.34), +7 (33.77)
Basic scale: H I C D E U V

Porcupine <<3, 5, 1]] TOP P=1196.91 G=162.32
+1 (162.32), -7 (60.68), +15 (40.95), -22 (19.73)
Basic scale: A I R D L U G

Negri(pent) <<4, -3, -14]] TOP P=1201.82 G=126.15
-9 (66.52), +10 (59.63), -19 (6.89)
Basic scale: (V) A W Q R D L M Z G (H)

Superchrome <<5, 8, 1]] TOP P=1203.32 G=101.99
+1 (101.99), -11 (81.41), +12 (20.59)
Basic scale: (O) A W B C X D Y E F Z G (O)

Subchrome <<5, -4, -18]] TOP P=1198.31 G=98.40
+1 (98.40), -12 (17.51)
Basic scale: (O) A W B C X D Y E F Z G (O)

7-limit linear temperaments:

Gawel <<3, 12, 11, 12, 9, -8]] TOP P=1202.62 G=569.05
-2 (64.58) +17 (52.84) -19 (11.69)
Basic scale: (B) M I E P L A S H D V K G R N C U Q (F)
The 19-note version is attractive because of the embedded series of fourths B E A D G C F. The version with the semisharps and semiflats is interesting: (B) Fv Bv E Az Ev A Dz Av D Gz Dv G Cz Gv C Fz Bz (F). Substituting Ez for Fv would result in a series of fourths Ez Az Dz Gz Cz Fz, and the same from substituting Cv for Bz: Bv Ev Av Dv Gv Cv. This could be an argument for swapping Ez/Fv and Bz/Cv.

Flattone <<1, 4, -9, 4, -17, -32]] TOP P=1202.54 G=507.14
-2 (188.26), +5 (130.62), -7 (57.64), +19 (15.33)
Basic scale: B E A D G C F

Meantone <<1, 4, 10, 4, 13, 12]] same as 5-limit meantone

Dominant <<1, 4, -2, 4, -6, -16]] TOP P=1195.23 G=495.88
+5 (88.95), -12 (25.57)
Basic scale: (O X Z B) E A D G C (F W Y O)
Since dominant is not a very complex temperament, the 5-note version is probably adequate for notation.

Schismic (Garibaldi) <<1, -8, -14, -15, -25, -10]] TOP P=1200.76 G=498.12
+5 (89.08), -12 (26.37)
Basic scale: (O X Z B) E A D G C (F W Y O)
Essentially the same structure as 5-limit schismic.

Superpyth[agorean] <<1, 9, -2, 12, -6, -30]] same as 5-limit superpyth

Semisixths (sensisept) <<7, 9, -2]] TOP P=1199.59 G=442.98
+3 (131.09), -8 (49.89), +19 (31.32), -27 (18.57)
Basic scale: (W S G Q E) (H) R F P D N B L (V) (C M A K Z)

Magic <<5, 1, 12, -10, 5, 25]] same as 5-limit magic

Beatles <<2, -9, -4, -19, -12, 16]] TOP P=1197.1 G=354.72
-3 (132.94), +7 (88.83), -10 (44.11), +27 (0.62)
Basic scale: (O C) L G I D U A R (E O)

Kleismic (Keemun) <<6, 5, 3, -6, -12, -7]] TOP P=1203.19 G=317.83
+4 (68.15), -15 (45.23), +19 (22.92)
Basic scale: (P R) E G W K T V B D F H J S Z A C (L N)

Catakleismic <<6, 5, 22, -6, 18, 37]] TOP P=1200.54 G=316.91
+4 (67.09), -15 (48.55), +19 (18.54)
Basic scale: (P R) E G W K T V B D F H J S Z A C (L N)

Myna <<10, 9, 7, -9, -17, -9]] TOP P=1198.83 G=309.89
+4 (40.74), -27 (24.70), +31 (16.04)
Basic scale: P *X E G W *X T V I K M *O B D F *O Q S U H J *Y Z A C *Y N
Three duplicate notes which can be resolved by using the #/b notation.

Orwell <<7, -3, 8, -21, -7, 27]] TOP P=1199.53 G=271.49
-4 (113.56), +9 (44.38), -22 (24.80), +31 (19.57)
Basic scale: Y Z H Q D M V W X
Essentially the same as 5-limit orwell (orson)

Semaphore <<2, 8, 1, 8, -4, -20]] TOP P=1203.67 G=252.48
-4 (193.75), +5 (58.73), -19 (17.55)
Basic scale: (L Z H B K E N) A Q D M G (P C S F V W R)

Cynder (Mothra) <<3, 12, -1, 12, -10, -36]] TOP
-5 (39.09), +26 (37.06), -31 (2.03)
Basic scale: (O W R Y F V *P C S M G) *P J D T *N (A Q K E *N H B X L Z O). The duplicate notes in the 26-note scale are not resolved by the #/b notation.

Porcupine <<3, 5, -6, 1, -18, -28]] same as 5-limit porcupine

Negri(sept) <<4, -3, 2, -14, -8, 13]] TOP P=1203.19 G=124.84
+1 (124.84), -9 (79.61), +10 (45.23), -19 (34.38), +29 (10.85)
Basic scale: (O) A I J X D Y T U G (O)
Note how the slight difference in the TOP tuning causes this scale to have a different spelling from 5-limit negripent (A W Q R D L M Z G). This may not be a desirable feature of the system.

Miracle <<6, -7, -2, -25, -20, 15]] TOP P=1200.63 G=116.72
+1 (116.72), -10 (33.42), +31 (16.45)
Basic scale: (O) P I J X D Y T U N (O)

Nautilus <<6, 10, 3, 2, -12, -21]] TOP P=1202.66 G=82.97
+1 (82.97), -14 (41.01), +29 (0.95)
Basic scale: (O) A W I J R X D Y L T U Z G (O)

🔗Carl Lumma <ekin@lumma.org>

🔗Igliashon Jones <igliashon@sbcglobal.net>

> I'm actually not all that familiar with porcupine[8]; previously
I've
> mainly been concerned with the 7-note and 15-note versions, and
with
> chord progressions that exploit the 250/243 comma.

Actually, if I recall correctly doesn't the "famous" chord
progression from your "Mizarian Porcupine Overture" (the one that
everyone cites as an example of a progression that only works in
temps. that vanish the 250/243) use what amounts to an 8-note
porcupine scale? At any rate I remember Paul pointing out that that
progression only needs a Porcu-8 scale to work correctly...maybe you
were treating the 8th note as a "chromatic" step added to the 7-note
scale?

> D D] E[ E E] F[ F F] G[ G G] G]]/A[[ A[ A A] B[ B B] C[ C C] D[ D
>
> This could be extended to 8 notes by adding H as equivalent to A[,
which
> would eliminate the need for double accidentals for the pitch a
tritone
> above D (which could then be represented as H[).

Interesting. However, if you're basing the nominals on a "circle" of
Porcupine generators, shouldn't the flats come out lower than the
sharps, in order to have consistency in interval naming? What my
colleague and I came up with for Porcu-8 (out of 22) originally was
something like:

F Gb F# G Hb G# H Ab H# A Bb A# B Cb B# C Db C# D Eb D# E F
(#=raise by 2 steps of 22 equal, b=lower by 2 steps of 22 equal)

Such that a 3 3 3 3 3 3 3 1 scale (in steps of 22-equal) starting on
F would have all naturals, on G would have one #, H would have 2 #'s,
etc. Though after discussing with Mr. Erlich at great length we
decided it would be best to use H-P ommitting "I", and to use "^" for
sharp and "u" for flat to avoid confusion with diatonic naming.

> In the alphabetic notation that Dave Keenan and I were discussing
> earlier on this list and in private email, the basic porcupine-7
scale
> would be notated A I R D L U G. The 8th note of porcupine-8 could
either
> be H (a small step away from A) or V (a small step away from G).

So it would be A I R D L U G H or V A I R D L U G ? Interesting.
How would you fill that out with all the 22-equal accidentals in
between? I must say I find it rather confusing to have so many
nominals, especially when there isn't a linear correlation between
letter and pitch, but then I suppose if your compositions frequently
involve multiple temperaments something like this would be very
useful. This seems to be quite a dilemma in the microtonal
community: how do we refer to all these many notes outside of a
diatonic context? This is why I'm starting small and taking it one
temp. at a time! :-) Anyway, I want to eventually publish a small
essay/article on Porcupine-8 tonality as it occurs in both 15- and 22-
equal, with a target audience of non-microtonalists or those new to
the field. I worry about suggesting my own accidental/8-nominal
system if a system like yours and Mr. Keenan's will come to be
adopted as a "standard" for linear temperaments. On the one hand, a
simple 8-nominal system will be easier for the traditionalits and
neophytes to understand, but if anyone who reads my paper gets more
involved in the field then they might end up having to re-learn
everything according to the "standard". What do you think?

-Igs

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Herman Miller <hmiller@IO.COM>

Igliashon Jones wrote:

> Actually, if I recall correctly doesn't the "famous" chord > progression from your "Mizarian Porcupine Overture" (the one that > everyone cites as an example of a progression that only works in > temps. that vanish the 250/243) use what amounts to an 8-note > porcupine scale? At any rate I remember Paul pointing out that that > progression only needs a Porcu-8 scale to work correctly...maybe you > were treating the 8th note as a "chromatic" step added to the 7-note > scale?

It uses most of the notes of the 15-note scale; this was written before there was such a thing as "porcupine temperament", so it's just good luck that it happens to fit in the 15-note range. I believe it was Paul Erlich who first remarked that this chord progression also works in 22-ET, which was the beginning of the path that eventually led to the discovery of porcupine temperament. As I recall, the scale used was something like this:

c g]
a[ e b
c[ g d a e]
e[ b[ f c g]
d[ a[ e b

I didn't hear any instances of f] when listening to the chord progression, but the other 14 notes are used. The progression starts with an e[ major chord and goes something like this (up a major third and down a fifth, repeated):

e[... gm | c[... e | a[... c | fm f b[ | e[...

> Interesting. However, if you're basing the nominals on a "circle" of > Porcupine generators, shouldn't the flats come out lower than the > sharps, in order to have consistency in interval naming? What my > colleague and I came up with for Porcu-8 (out of 22) originally was > something like:
> > F Gb F# G Hb G# H Ab H# A Bb A# B Cb B# C Db C# D Eb D# E F
> (#=raise by 2 steps of 22 equal, b=lower by 2 steps of 22 equal)
> > Such that a 3 3 3 3 3 3 3 1 scale (in steps of 22-equal) starting on > F would have all naturals, on G would have one #, H would have 2 #'s, > etc. Though after discussing with Mr. Erlich at great length we > decided it would be best to use H-P ommitting "I", and to use "^" for > sharp and "u" for flat to avoid confusion with diatonic naming.

I guess that makes sense for the 8-note scale. In the 7-note notation you'd have something like this:

a b c d e f g a[
b c d e f g a[ b[
c d e f g a[ b[ c[

but if you replace "a[" with "h", that doesn't help for the transposed scales. You'd need an accidental that represents 8 porcupine generators, which in the case of 22-ET is 2 steps. So you'd end up with something like this:

a b c d e f g h
b c d e f g h a#
c d e f g h a# b#

>>In the alphabetic notation that Dave Keenan and I were discussing >>earlier on this list and in private email, the basic porcupine-7 > > scale > >>would be notated A I R D L U G. The 8th note of porcupine-8 could > > either > >>be H (a small step away from A) or V (a small step away from G). > > > So it would be A I R D L U G H or V A I R D L U G ? Interesting. > How would you fill that out with all the 22-equal accidentals in > between?

Extending the series in both directions, you'd get:

Ab Ib Rb Db Lb Ub Gb Hb A I R D L U G H A# I# R# D# L# U# G# H#

Or arranged in 22-ET order:

A Ib A# I Rb I# R Db R# D Lb D# L Ub L# U Gb U# G Hb G# H A
(H#) (Ab)

> I must say I find it rather confusing to have so many > nominals, especially when there isn't a linear correlation between > letter and pitch, but then I suppose if your compositions frequently > involve multiple temperaments something like this would be very > useful.

Eventually I guess you'd learn the sequence OHAPWIBQJCRXKDSYLETMFUNZGVO the same way you learn the alphabetic orders of languages that don't use the Roman alphabet. But I still haven't internalized this order, and I still have trouble remembering exactly which pitch a note name like "I" or "M" is supposed to represent. What interval does something like X-S or Q-T represent? That's one of the reasons I've been considering other approaches, but they all have drawbacks of one kind or another.

> This seems to be quite a dilemma in the microtonal > community: how do we refer to all these many notes outside of a > diatonic context? This is why I'm starting small and taking it one > temp. at a time! :-) Anyway, I want to eventually publish a small > essay/article on Porcupine-8 tonality as it occurs in both 15- and 22-
> equal, with a target audience of non-microtonalists or those new to > the field. I worry about suggesting my own accidental/8-nominal > system if a system like yours and Mr. Keenan's will come to be > adopted as a "standard" for linear temperaments. On the one hand, a > simple 8-nominal system will be easier for the traditionalits and > neophytes to understand, but if anyone who reads my paper gets more > involved in the field then they might end up having to re-learn > everything according to the "standard". What do you think?

There are already "standards" for particular linear temperaments: Graham Breed's decimal notation for miracle temperament is a good one that comes to mind. At this point it's too early to say what will come from attempts to create a "universal" notation system. I've had some success using alphabetic notation for lemba temperament, but then I only needed to learn 10 note names (D L T U N O P I J R), and the alphabetical sequences T-U, N-O-P, I-J make it easier to remember than some arbitrary set of letters. I still need to look at a chart to see that a major tetrad on P is written P-R#-T-N#. In some ways my original notation for lemba based on 26-ET meantone notation is easier to use.

🔗Herman Miller <hmiller@IO.COM>

🔗Igliashon Jones <igliashon@sbcglobal.net>

🔗Herman Miller <hmiller@IO.COM>

🔗Igliashon Jones <igliashon@sbcglobal.net>

🔗Herman Miller <hmiller@IO.COM>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Herman Miller <hmiller@IO.COM>

Herman Miller wrote:
> Here are some step sizes for specific temperaments which may be used as > accidentals. In each case, the number of generators is given followed by > the size of the step in cents. The assumptions I'm following are: 1. no > accidental larger than 200 cents, 2. no more than 31 basic notes in the > scale without the use of accidentals, and 3. accidental size smaller > than any interval in the basic scale. The most problematic of these > temperaments for note spelling are wï¿½rschmidt, myna, and cynder.

Here's another approach: start by finding an accidental or set of accidentals that can be written with conveniently available Sagittal symbols. For the case of 5-limit LT's, the basic set of accidentals would include:

/| //| /|) (|\ )||( ||\ /||\ /||| //||| )X( X\ /X\

possibly with the addition of 5-schismas if needed. In many cases there are other possibilities besides the ones mentioned.

> 5-limit linear temperaments:
> > Mavila <<1, -3, -7]] TOP P=1206.55 G=521.52

)||( [-3, -1, 2> +7 (31.00)

> Meantone <<1, 4, 4]] TOP P=1201.70 G=504.13

)||( [-3, -1, 2> -7 (76.16)

> Schismic (Helmholtz) <<1, -8, -15]] TOP P=1200.07 G=498.28

||\ [-7, 3, 1> +5 (91.26)
/||\ [-11, 7> -7 (112.25)
/| [-4, 4, -1> -12 (20.98)
)||( [-3, -1, 2> +17 (70.28)
/||| [-15, 11, -1> -19 (133.23)
/|) [1, -5, 3> +29 (49.29)

> Superpyth[agorean] <<1, 9, 12]] TOP P=1197.60 G=489.43

)||( [-3, -1, 2> -17 (62.91)
||\ [-7, 3, 1> -12 (114.85)
/||\ [-11, 7> -7 (166.80)

Note that -22 is notated as /|) but represents an interval of only 10.97 cents (in TOP tuning). This would be misleading since /|) ordinarily represents an interval of around 49.17 cents. So probably either of these three (7, 12, 17) would be better than using a set of 22.

> Semisixths (sensipent) <<7, 9, -2]] TOP P=1199.59 G=442.98

/|) [1, -5, 3> -8 (54.89)
)||( [-3, -1, 2> +11 (74.48)
/| [-4, 4, -1> +19 (19.59)

> Wï¿½rschmidt <<8, 1, -17]] TOP P=1199.69 G=387.64

)||( [-3, -1, 2> -6 (73.51)
||\ [-7, 3, 1> +25 (93.59)
/| [-4, 4, -1> +31 (20.07)

> Magic <<5, 1, -10]] TOP P=1201.28 G=380.80

||\ [-7, 3, 1> +16 (86.35)
/| [-4, 4, -1> +19 (27.46)

> Dicot <<2, 1, -3]] TOP P=1207.66 G=353.22

||\ [-7, 3, 1> +7 (57.21)

> Amity <<5, 13, 9]] TOP P=1199.85 G=339.47

/| [-4, 4, -1> -7 (23.40)

> Kleismic (Hanson) <<6, 5, -6]] TOP P=1200.29 G=317.07

/|) [1, -5, 3> -15 (45.12)
/| [-4, 4, -1> +19 (22.86)

> Orson (5-limit orwell) <<7, -3, -21]] TOP P=1200.24 G=271.65

)||( [-3, -1, 2> -13 (69.22)
/| [-4, 4, -1> +31 (19.59)
`//| [7, 0, -3> +9 (44.40)

I can't find a way to notate +9 without a schisma accent, but +18 can be notated as ||\.

> Tetracot <<4, 9, 5]] TOP P=1199.03 G=176.11

/| [-4, 4, -1> +7 (33.77)

> Porcupine <<3, 5, 1]] TOP P=1196.91 G=162.32

)||( [-3, -1, 2> -7 (60.68)
`/||\ [4, -1, -1> +8 (101.64)
`//| [7, 0, -3> +15 (40.95)

> Negri(pent) <<4, -3, -14]] TOP P=1201.82 G=126.15

)||( [-3, -1, 2> +10 (59.63)
||\ [-7, 3, 1> -9 (66.52)

> Superchrome <<5, 8, 1]] TOP P=1203.32 G=101.99

)||( [-3, -1, 2> -11 (81.41)

> Subchrome <<5, -4, -18]] TOP P=1198.31 G=98.40

)||( [-3, -1, 2> +13 (80.89)
||\ [-7, 3, 1> -11 (115.91)

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

This is great Herman. I was hoping you'd get back to it eventually.

What you're proposing is what I'd call "using pseudo-nominals",
although I had not previously considered including quartertones for
these.

I think your suggestion to use a non-sagittal quartertone notation
along with the letters A thru G, to give a set of 24 pseudo-nominals,
is a brilliant idea!

I think the alphabetic system should still be kept around somewhere
just so people can have fun hiding words and maybe whole sentences of
text in their music - like the famous B A C H. :-)

I have a number of further suggestions.

1. I suggest using the Tartini/Couper (TC) rather than Tartini/Fokker
(TF) quartertone symbols. The former appear in the fonts of the major
notation packages, the latter do not, or if they do, are not the
default. The Couper half-flat is a backwards flat, which George Secor
and I suggest should be made noticeably narrower than the forward flat
to reduce lateral confusability and make it _look_ more like a half of
something.

2. I suggest using ASCII characters other than "v" and "z" to
represent the TC semiflat and semisharp, as "v" has already been used
for many different accidentals and is used for the 11-diesis-down
arrow in sagittal. "z" is also shorthand for a sagittal, as you can
see here
http://dkeenan.com/sagittal/map/index.htm
Character pairs not used for ASCII sagittal are:
{} [] <> -+ ,` el

I'll use "[" for semiflat and "]" for semisharp below since they
already have a history of representing quartertones on this list,
albeit for the Sims symbols, not TC. But feel free to make other
suggestions. There's also the pair & % which have been assigned to
sagittals, but ones that are so unlikely to be used that we might
consider revoking their shorthand license.

3. To make it clear (in textual discussions) when we're using TC and
conventional symbols to construct pseudo-nominals, as opposed to using
them as accidentals, I suggest we put them to the _left_ of the
letters, and pronounce them that way. e.g. "Eb" has an accidental and
is pronounced "E flat" while "bE" is a pseudo-nominal and is
pronounced "flat-E" (and spelled with a hyphen to indicate the tight
binding).

But I expect they should both appear the same on the staff, or else
rely on your shaped notehead ideas.

For the benefit of readers trying to follow this thread (and earlier
ones on the same topic), here's a table showing the correspondence
between the pseudo-nominals and the full-alphabetic nominals in a more
easily remembered form. If you're using Yahoo's web interface you'll
need to hit the reply button to see this table formatted properly.

flat semiflat natural semisharp sharp
------------------------------------------
O bA H [A A A P ]A W #A
W bB I [B B B Q ]B
J [C C C R ]C X #C
X bD K [D D D S ]D Y #D
Y bE L [E E E T ]E
M [F F F U ]F Z #F
Z bG N [G G G V ]G O #G

These are listed so that normal reading order corresponds to pitch
order, but note that the rightmost pitch of each row is the same as
the leftmost pitch of the row below it, and the leftmost of the last
row is the same as the rightmost of the first row. So there are only
24 "nominals".

By looking down the columns you can see the logic behind this way of
assigning all the letters of the alphabet.

Here are some reasons for choosing 24:
(a) It's divisible by 2,3 and 4 - common numbers of chains in
(multi-)linear temperaments. For (multi-)linear temperaments having 5
to 24 chains of generators, one proposal is that there should be only
one nominal per chain, equally spaced within the octave.

(b) It is a multiple of the popular 12ET.

My suggestion to use lowercase Greek letters for the semiflats and
Hebrew letters for the semisharps, in non-email contexts, still
stands. These would be the nearest Greek and Hebrew transliterals of
the Roman/Latin A thru G, that actually look different from them.

4. You describe certain anomalies that occur when the boundaries
between (pseudo-)nominals are set halfway between the notes of 24-ET.

My approach to this is to allow unequally spaced boundaries, set in
such a way that the normal chain-of-fourth/fifth relationships are
preserved between the 12 notes of (a) meantones out to 19-ET and (b)
Pythagoreans out to 17-ET, (the limits of propriety for 12 note chains
of fourths/fifths). For temperaments with more than one chain,
different pairs of ETs delimit the region in which the familiar
notation of chains of fourths/fifths are preserved and so slightly
different boundaries result.

By determining upper and lower boundaries for these 12 pseudo-nominals
(bE bB F C G D A E B #F #C #G) the boundaries of the other 12 are
automatically decided.

Yes there will be cases where other intervals do not have their
familiar meantone letter relationships, but this is bound to happen
somewhere, outside of meantone, and I feel that it is more
disorienting to violate the familiar notational relationships of
fourths and fifths than thirds etc., since fourths/fifths are the most
consonant intervals other than the unison/octave (speaking
octave-equivalently), and fourths/fifths are the most popular
(multi-)linear temperament generators by far.

Here are the resulting boundaries. I've only shown just over half an
octave because the rest is symmetrical about D.

Number of chains
Nom- 1 2 3 4
inal Lower bound of nominal (as a fraction of an octave)
------------------------------
D D 0 0 0 0
S ]D 0 0 0 0
Y bE 1/17 1/14 1/15 1/14
L [E 2/19 1/10 2/21 1/10
E E 1/7 1/7 1/7 3/20
M [F 1/5 1/5 1/5 5/28
F F 4/17 5/22 5/21 1/4
U ]F 2/7 2/7 4/15 1/4
Z #F 6/19 3/10 1/3 7/20
N [G 6/17 5/14 1/3 9/28
G G 2/5 2/5 2/5 2/5
V ]G 3/7 3/7 3/7 3/7
O #G 9/19 1/2 10/21 1/2
H [A 9/17 1/2 8/15 1/2

Note that the natural, flat and sharp nominals are inclusive of their
boundaries, while the semiflat and semisharp ones in between are
exclusive of their boundaries.

With these boundaries, Porcupine-7 (a single chain of 160 to 163.6
cent generators) has the pseudo-nominals
A B ]C D [E F G
or A B R D L F G.

5. When nominals are unique by letter A to G alone (as in the
Porcupine-7 example above), rather than omitting the Tartini/Couper
"nomino-accidentals" completely from the staff I suggest they appear
in a "key" signature.

6. When the MOS/DE used for nominals has an even number of notes, I
propose we always have the extra note on the "generators down" side of
"D", when the generator is expressed in lowest terms. This has the
advantages of
(a) agreeing with the convention of favouring G# over Ab for a 12 note
chain of fourths, and
(b) often resulting in the nominal "O" (#G or bA) being the "zero" of
the chain. This was one of the reasons for using the letter "O" for
the note a half-octave from D in the first place.

So Porcupine-8-based pseudo-nominals would be
]G A B ]C D [E F G
or V A B R D L F G.

7. Any formula for automatically deciding the number of nominals to
use for a (multi-)linear temperament with a given generator and
period, ought to consider improper MOS/DE when their slight
impropriety (L/s > 2) is outweighed by the proximity of their
cardinality to the magic number 7.

I propose the following MOS/DE-nominal-badness formula:

L/s + k * abs(ln(N/7))

where
L is the size of the largest step
s is the size of the smallest step
N is the number of notes
k = (5/3)/abs(ln(5/7)) = 4.953356

This choice of k results in 7 nominals being preferred for
Pythagoreans out to 17-ET.

If we apply it to Porcupine-7 and Porcupine-8, with generators of 2/15
oct and 3/22 oct, we have the following L/s values:

P-7 P-8
-----------------
15 3/2 2/1
22 4/3 3/1

So of course this formula prefers Porcupine-7 in all cases, since it
has both the lowest L/s _and_ the "magic" number of notes.

That isn't to say that there might not be other reasons to prefer
Porcupine-8, but we could make a distinction between the set of
nominals that gives the best notation, and the set of notes that gives
the most useful scale.

-- Dave Keenan

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

I wrote:

> If you're using Yahoo's web interface you'll
> need to hit the reply button to see this table formatted properly.
>
> flat semiflat natural semisharp sharp
> ------------------------------------------
> O bA H [A A A P ]A W #A
> W bB I [B B B Q ]B
> J [C C C R ]C X #C
> X bD K [D D D S ]D Y #D
> Y bE L [E E E T ]E
> M [F F F U ]F Z #F
> Z bG N [G G G V ]G O #G
>
> These are listed so that normal reading order corresponds to pitch
> order, but note that the rightmost pitch of each row is the same as
> the leftmost pitch of the row below it, and the leftmost of the last
> row is the same as the rightmost of the first row.

Oops! That should have been:

"... and the rightmost of the last row is the same as the leftmost of
the first row."

No wonder I'm always on about lateral confusability. :-)

And just for the record. Here are the names of the proposed Greek and
Hebrew letters

flat semiflat natural semisharp sharp
-----------------------------------------------------------------
Greek Greek Latin Hebrew Hebrew
capital small capital final
-----------------------------------------------------------------
O bA alpha [A A A alef ]A
THETA bB beta [B B B bet ]B
chi [C C C chaf ]C chaf sofit #C
delta [D D D dalet ]D
UPSILON bE epsilon [E E E hey ]E
phi [F F F fay ]F fay sofit #F
gamma [G G G gimel ]G O #G

I've used the Windows Character Map accessory to insert the actual
characters below, but I can't guarantee they will still be there when
you read them.

flat semiflat natural semisharp sharp
------------------------------------------
O bA Î± [A A A × ]A
Î bB Î² [B B B × ]B ?
Ï [C C C × ]C × #C
Î´ [D D D × ]D
Î¥ bE Îµ [E E E × ]E
Ï [F ? F F ×¤ ]F ×£ #F
Î³ [G G G × ]G O #G

Note that bet and chi are the same pitch and hey and phi are the same
pitch. I'd prefer to have only one letter for each nominal pitch and
it would seem best to drop those that are derived from the extreme
letters in the sequence of fourths BEADGCF, namely bet and phi, hence
the question marks. That would also mean dropping Q and T in favour of
J and M. But I don't have strong feelings about any of these.

Note that "O" (#G/bA) is still the Latin capital letter "oh", which
can also be thought of as zero.

Capital theta and capital upsilon are used instead of the capitals of
beta and epsilon, because the characters for capital beta and epsilon
are indistinguishable from the Latin capitals B and E. This also has
the advantage that we don't need to say "capital" (or the greek
equivalent "kefaleos").

The names of the letters "chi" and "chaf" are pronouced with the
back-of-the-throat "gargling" kind of "kh" sound. But if you can't
manage that, just pronounce them as "ky" and "kaf". In fact chaf is
sometimes written "kaf". Just don't use the ch sounds of "chip" or
"champagne".

Fay is also called "pe", and hey is also written "he". I have chosen
to use names that indicate the relatedness to the corresponding latin
letter while sounding sufficiently different from them and the other
letters.

-- Dave Keenan

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

🔗Herman Miller <hmiller@IO.COM>

Dave Keenan wrote:

> I think the alphabetic system should still be kept around somewhere
> just so people can have fun hiding words and maybe whole sentences of
> text in their music - like the famous B A C H. :-)

Sounds good; we could keep the names but notate them with the "accent marks" (the way that German B is notated as H-flat, for instance).

> I have a number of further suggestions.
> > 1. I suggest using the Tartini/Couper (TC) rather than Tartini/Fokker
> (TF) quartertone symbols. The former appear in the fonts of the major
> notation packages, the latter do not, or if they do, are not the
> default. The Couper half-flat is a backwards flat, which George Secor
> and I suggest should be made noticeably narrower than the forward flat
> to reduce lateral confusability and make it _look_ more like a half of
> something.

Sounds like a good idea.

> 2. I suggest using ASCII characters other than "v" and "z" to
> represent the TC semiflat and semisharp, as "v" has already been used
> for many different accidentals and is used for the 11-diesis-down
> arrow in sagittal. "z" is also shorthand for a sagittal, as you can
> see here
> http://dkeenan.com/sagittal/map/index.htm
> Character pairs not used for ASCII sagittal are:
> {} [] <> -+ ,` el

It's too bad "d" is already used for (!( , since that would be the best symbol for the semiflat. I'd rather not use <> -+ to avoid potential confusion with HEWM notation. I thought ` was a schisma accent, but I see that . is used for that. Would it be too late to suggest ` for the schisma down symbol, since that's what it looks like?

> I'll use "[" for semiflat and "]" for semisharp below since they
> already have a history of representing quartertones on this list,
> albeit for the Sims symbols, not TC. But feel free to make other
> suggestions. There's also the pair & % which have been assigned to
> sagittals, but ones that are so unlikely to be used that we might
> consider revoking their shorthand license.

& could also be a possibility for semiflat. But I think [ and ] would be satisfactory.

> 3. To make it clear (in textual discussions) when we're using TC and
> conventional symbols to construct pseudo-nominals, as opposed to using
> them as accidentals, I suggest we put them to the _left_ of the
> letters, and pronounce them that way. e.g. "Eb" has an accidental and
> is pronounced "E flat" while "bE" is a pseudo-nominal and is
> pronounced "flat-E" (and spelled with a hyphen to indicate the tight
> binding).
> > But I expect they should both appear the same on the staff, or else
> rely on your shaped notehead ideas.

I was looking through the quarter-tone section in Gardner Read's book and noticed that there's a precedent for using triangular noteheads for quarter tones. In each case, upward-pointing triangles represent a quarter tone sharp, not flat. But you bring up a good point with notation packages; probably none of them would be able to display a diamond-shaped note halfway between F and G, for instance.

> Here are some reasons for choosing 24:
> (a) It's divisible by 2,3 and 4 - common numbers of chains in
> (multi-)linear temperaments. For (multi-)linear temperaments having 5
> to 24 chains of generators, one proposal is that there should be only
> one nominal per chain, equally spaced within the octave.

2 per chain would work for the Blackwood decatonic: TOP period = 239.18, generator = 83.83, wedgie = <<0, 5, 0, 8, 0, -14]]. Centered on D, it could be notated D Y T F G V A W J R D or D L T Z G H A B J X D. (There isn't any particular reason to use D as the base pitch other than that's what traditional notation is centered on.)

> (b) It is a multiple of the popular 12ET.
> > (c) It is not greater than the number of letters in most alphabets.
> > My suggestion to use lowercase Greek letters for the semiflats and
> Hebrew letters for the semisharps, in non-email contexts, still
> stands. These would be the nearest Greek and Hebrew transliterals of
> the Roman/Latin A thru G, that actually look different from them.

A problem with using Hebrew letters is that they're written right to left; in a document, the Unicode bidirectional algorithm will take effect depending on the level of support for RTL text in the particular application you're running. So you might write a melody as "aleph gimel beth", but the reader would see "beth gimel aleph". Cyrillic letters are probably a better choice (with appropriate selection of letters to avoid confusion with the Latin and Greek). Other possibilities exist but would require fonts that some readers might not have installed. So my suggestion would be to use:

YA U+042F = A]
BE U+0411 = B]
CHE U+0427 = C]
DE U+0414 = D]
E U+042D = E] (also called "e oborotnoe")
EF U+0424 = F] (capital, to avoid confusion with Greek lowercase)
GHE U+0413 = G] (same as above)

The problem with that is that the *spoken* name of Cyrillic EF is the same as English "F"; BE is the name of the Roman "B" in some languages, etc. (The Russian letter that looks like B is actually VE.)

> 4. You describe certain anomalies that occur when the boundaries
> between (pseudo-)nominals are set halfway between the notes of 24-ET.

This issue really seems to affect very few temperaments; of the ones I've listed so far, I've only noted the discrepancy between 5- and 7-limit versions of negri temperament (negripent and negrisept in Paul's terminology), and the problems with wï¿½rschmidt, myna, and cynder (which require more than 24 nominals under the constraints I was setting for myself). I'm not convinced that arbitrarily setting the boundaries would be any better than using the equally spaced boundaries for the majority of temperaments, and exceptions can be made for the few that have problems with these.

> 5. When nominals are unique by letter A to G alone (as in the
> Porcupine-7 example above), rather than omitting the Tartini/Couper
> "nomino-accidentals" completely from the staff I suggest they appear
> in a "key" signature.

I was just thinking about that last night after I logged off. I think this is a good idea, and would help to simplify the notation.

> 6. When the MOS/DE used for nominals has an even number of notes, I
> propose we always have the extra note on the "generators down" side of
> "D", when the generator is expressed in lowest terms. This has the
> advantages of > (a) agreeing with the convention of favouring G# over Ab for a 12 note
> chain of fourths, and
> (b) often resulting in the nominal "O" (#G or bA) being the "zero" of
> the chain. This was one of the reasons for using the letter "O" for
> the note a half-octave from D in the first place.

In cases where "O" is used, it doesn't matter which way you go; they're both the same distance from D. But there are two possibilities: either the "generators down" rule, or the "lower pitch" rule (which would have the advantage of always favoring G# over Ab, not just in the case of 12-note chains of fourths, and may be easier to remember).

> 7. Any formula for automatically deciding the number of nominals to
> use for a (multi-)linear temperament with a given generator and
> period, ought to consider improper MOS/DE when their slight
> impropriety (L/s > 2) is outweighed by the proximity of their
> cardinality to the magic number 7.
> > I propose the following MOS/DE-nominal-badness formula:
> > L/s + k * abs(ln(N/7))
> > where > L is the size of the largest step
> s is the size of the smallest step
> N is the number of notes > k = (5/3)/abs(ln(5/7)) = 4.953356
> > This choice of k results in 7 nominals being preferred for
> Pythagoreans out to 17-ET.

Hmmm... I'll plug this in to the MOS/DE spreadsheet I've been working on and see what happens for certain temperaments....

Meantone = 7 (no surprise)

Orwell = 9 (again, no real surprise, although Sagittal seems to have better accidentals for orwell[13] unless I'm missing something)

Miracle = 10 (what else?)

Negri = 9 (good choice)

7-limit Kleismic = 7 (hmm.... not sure about this one; I'd prefer 11 or even 15.)

Myna = 4 (too few nominals for so complex a temperament!)

Pajara = 8 (not 10? Still, 10 is pretty close; a slight adjustment to the formula could fix this.)

Lemba = 6 (not 10! This would be sufficient for my current needs with lemba[16], but I've been thinking of lemba[10] as more the basic scale rather than lemba[6], Still, |||( for the lemba[6] accidental might be a better fit than either |) or )||( for lemba[10]... And 6 nominals is more convenient for staff notation ....)

Misty = 9

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Herman Miller <hmiller@IO.COM>

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

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> Dave Keenan wrote:
>
> > I think the alphabetic system should still be kept around somewhere
> > just so people can have fun hiding words and maybe whole sentences of
> > text in their music - like the famous B A C H. :-)
>
> Sounds good; we could keep the names but notate them with the "accent
> marks" (the way that German B is notated as H-flat, for instance).

Yes. However I think we should avoid the full-alphabetical names in
this discussion. They make it too hard for people to follow. I'd
prefer to just let them be a sort of not-so-secret code, just for fun.

I'm happy to use names like "semiflat-A" for [A etc, and occasionally
in future any Greek/Hebrew/Cyrillic names we might agree on, provided
their relationship to the Latin A thru G is easily remembered.

> It's too bad "d" is already used for (!( , since that would be the best
> symbol for the semiflat.

True. But "d" and "q" as an up/down pair were too good to pass up for
Aphrodite's curves, the 5:11 small diesis (44;45) symbol.

> I'd rather not use <> -+ to avoid potential
> confusion with HEWM notation.

Yes. That's one reason we didn't use them in sagittal either.

>I thought ` was a schisma accent, but I
> see that . is used for that. Would it be too late to suggest ` for the
> schisma down symbol, since that's what it looks like?

That was a tough call, but in the end we decided that the vertical
position was a stronger cue than the slope in the case of such tiny
glyphs. The actual staff symbols for the schisma accents rely on this
vertical position too (relative to the notehead and other
accidentals), so the schisma-down doesn't actually look like an ASCII
back-quote (grave) but rather a comma that slopes the wrong way. So we
used ' and . since at least they don't have the _wrong_ slope (since
they don't have any).

> & could also be a possibility for semiflat.

It could, but it doesn't have a partner that looks like the Tartini
semisharp, whereas "]" has at least a vague resemblance (one long
vertical stroke and two short horizontal ones).

> But I think [ and ] would be
> satisfactory.

OK. That's settled - "[" for narrow Couper semiflat and "]" for
Tartini semisharp, at least when used as "nominal accent marks" or
"nomino-accidentals".

I could make a special version of the sagittal font with these two TC
symbols in place of the Wilson sloping - and + symbols that Joseph
uses, or in place of some really obscure non-Athenian non-Trojan
sagittals (the font mapping space is otherwise full).

> I was looking through the quarter-tone section in Gardner Read's book
> and noticed that there's a precedent for using triangular noteheads for
> quarter tones. In each case, upward-pointing triangles represent a
> quarter tone sharp, not flat.

Makes sense to me.

> But you bring up a good point with
> notation packages; probably none of them would be able to display a
> diamond-shaped note halfway between F and G, for instance.

Hmm. You could define your own with half-position offsets. Easy enough
in Sibelius. As far as Sibelius is concerned they would be positioned
as either an F or a G but you'd choose one of the two diamond
noteheads you had defined, to which you had given the appropriate
half-position offsets up and down.

> 2 per chain would work for the Blackwood decatonic: TOP period =
239.18,
> generator = 83.83, wedgie = <<0, 5, 0, 8, 0, -14]]. Centered on D, it
> could be notated D Y T F G V A W J R D or D L T Z G H A B J X D.

Yes, it would work. But is it the best way to do it? When you cast the
above as accented nominals it is quite messy. When you use only the 5
nominals ACDEG and use 5-comma accidentals for moving along the
chains, it makes perfect sense relative to the temperament's 5-limit
mapping. A\ A C\ C D\ D E\ E G\ G

By the way, this is an example where using equally spaced boundaries
for the nominals doesn't do the right thing as far as maximising the
familiar spellings of the consonant intervals.

> (There
> isn't any particular reason to use D as the base pitch other than
that's
> what traditional notation is centered on.)

Well no, but that _is_ a pretty good reason to center on D, given that
we intend to keep the traditional meaning of A thru G.

> > My suggestion to use lowercase Greek letters for the semiflats and
> > Hebrew letters for the semisharps, in non-email contexts, still
> > stands. These would be the nearest Greek and Hebrew transliterals of
> > the Roman/Latin A thru G, that actually look different from them.
>
> A problem with using Hebrew letters is that they're written right to
> left; in a document, the Unicode bidirectional algorithm will take
> effect depending on the level of support for RTL text in the particular
> application you're running. So you might write a melody as "aleph gimel
> beth", but the reader would see "beth gimel aleph".

Good point!

> Cyrillic letters are
> probably a better choice (with appropriate selection of letters to
avoid
> confusion with the Latin and Greek). Other possibilities exist but
would
> require fonts that some readers might not have installed. So my
> suggestion would be to use:
>
> YA U+042F = A]

Looks like a backwards "R". Fine.

> BE U+0411 = B]

Like a "B" with a gap at the top right. How do you pronounce the name?
I hope it isn't a long "e" as in "bee". Likewise for others below.

> CHE U+0427 = C]
> DE U+0414 = D]
> E U+042D = E] (also called "e oborotnoe")
> EF U+0424 = F] (capital, to avoid confusion with Greek lowercase)
> GHE U+0413 = G] (same as above)
>
> The problem with that is that the *spoken* name of Cyrillic EF is the
> same as English "F"; BE is the name of the Roman "B" in some languages,
> etc. (The Russian letter that looks like B is actually VE.)

Looks reasonable, provided the pronunciations of the names are
sufficiently different from those of the Latin and Greek characters
we'd be using, and the symbols look different.

We need two varieties of sharpened F and C, both semisharp and sharp.
For the others we only need the semisharps. Could we use capitals for
one and smalls for the other. I'm concerned that small Cyrillic be
looks too much like small delta.

You didn't comment on my earlier suggestion to put the nominal accent
("[" or "]") to the left of the letter, to distinguish it from an
accidental. Do you think that's a bad idea? If so why?

Admittedly it hasn't mattered much so far, but I thk that when we
start using these in conjunction with ASCII sagittal accidentals it
will make things much more readable.

> > 4. You describe certain anomalies that occur when the boundaries
> > between (pseudo-)nominals are set halfway between the notes of 24-ET.
>
> This issue really seems to affect very few temperaments; of the ones
> I've listed so far, I've only noted the discrepancy between 5- and
> 7-limit versions of negri temperament (negripent and negrisept in
Paul's
> terminology), and the problems with würschmidt, myna, and cynder (which
> require more than 24 nominals under the constraints I was setting for
> myself). I'm not convinced that arbitrarily setting the boundaries

There isn't much that's arbitrary about the boundaries I gave. But I
know I haven't explained them very well yet.

> would
> be any better than using the equally spaced boundaries for the majority
> of temperaments, and exceptions can be made for the few that have
> problems with these.

OK. Well I guess the onus is on me to show those cases where the
chain-of-fifth-based boundaries give a better result. I've shown one
above (Blackwood).

> > 5. When nominals are unique by letter A to G alone (as in the
> > Porcupine-7 example above), rather than omitting the Tartini/Couper
> > "nomino-accidentals" completely from the staff I suggest they appear
> > in a "key" signature.
>
> I was just thinking about that last night after I logged off. I think
> this is a good idea, and would help to simplify the notation.

And is a reason to prefer notations with 5 to 7 or nominals.

> In cases where "O" is used, it doesn't matter which way you go; they're
> both the same distance from D. But there are two possibilities: either
> the "generators down" rule, or the "lower pitch" rule (which would have
> the advantage of always favoring G# over Ab, not just in the case of
> 12-note chains of fourths, and may be easier to remember).

The "lower pitch" rule would result in Miracle having "O" as the last
in the chain, i.e. corresponding to the "9" rather than the zero of
Graham's decimal notation. This seems likely to breed confusion (no
pun intended).

> > 7. Any formula for automatically deciding the number of nominals to
> > use for a (multi-)linear temperament with a given generator and
> > period, ought to consider improper MOS/DE when their slight
> > impropriety (L/s > 2) is outweighed by the proximity of their
> > cardinality to the magic number 7.
> >
> > I propose the following MOS/DE-nominal-badness formula:
> >
> > L/s + k * abs(ln(N/7))
> >
> > where
> > L is the size of the largest step
> > s is the size of the smallest step
> > N is the number of notes
> > k = (5/3)/abs(ln(5/7)) = 4.953356
> >
> > This choice of k results in 7 nominals being preferred for
> > Pythagoreans out to 17-ET.
>
> Hmmm... I'll plug this in to the MOS/DE spreadsheet I've been
working on
> and see what happens for certain temperaments....
>
> Meantone = 7 (no surprise)
>
> Orwell = 9 (again, no real surprise, although Sagittal seems to have
> better accidentals for orwell[13] unless I'm missing something)
>
> Miracle = 10 (what else?)
>
> Negri = 9 (good choice)
>
> 7-limit Kleismic = 7 (hmm.... not sure about this one; I'd prefer 11 or
> even 15.)
>
> Myna = 4 (too few nominals for so complex a temperament!)

Yes, but we have serious problems no matter what the formula, with any
generator so close to 1/2, 1/3 or 1/4 octave. How many do you propose
for Myna?

> Pajara = 8 (not 10? Still, 10 is pretty close; a slight adjustment to
> the formula could fix this.)

Hmm. I wonder if Paul Erlich has ever considered that. As you say,
reducing k to 4.49 will take care of it. But that will also cause
17-ET (considered as Pythagorean) to have only 5 nominals. Maybe
that's not so bad, or maybe we just need to add other factors to
refine the formula, or maybe we just need to use our common sense in
addition to the formula.

> Lemba = 6 (not 10! This would be sufficient for my current needs with
> lemba[16], but I've been thinking of lemba[10] as more the basic scale
> rather than lemba[6], Still, |||( for the lemba[6] accidental might
be a
> better fit than either |) or )||( for lemba[10]... And 6 nominals is
> more convenient for staff notation ....)
>
> Misty = 9

Maybe a factor pertaining to the complexity of the temperament could
be added to the formula.

-- Dave Keenan

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

🔗Herman Miller <hmiller@IO.COM>

Dave Keenan wrote:

> I'm happy to use names like "semiflat-A" for [A etc, and occasionally
> in future any Greek/Hebrew/Cyrillic names we might agree on, provided
> their relationship to the Latin A thru G is easily remembered.

I've been thinking instead of "semiflat" and "semisharp" we should call them something simpler like "upper-A", "lower-A", or something of the sort.

> OK. That's settled - "[" for narrow Couper semiflat and "]" for
> Tartini semisharp, at least when used as "nominal accent marks" or
> "nomino-accidentals". I've just gone through and replaced the extended-alphabetical nominals with the [] symbols to see what they look like: a few examples:

Mavila: [B [E A D G ]C ]F
Porcupine: A [B ]C D [E ]F G
Hanson: ]E ]G B D F [A [C
Amity: E G [B D ]F A C
Orson: bE #F [A ]B D [F ]G bB #C

Hmm, there's the potential of confusing [ with ], but after all this time with the extended alphabet, I still can't keep all those letters straight; this is certainly an improvement.

> I could make a special version of the sagittal font with these two TC
> symbols in place of the Wilson sloping - and + symbols that Joseph
> uses, or in place of some really obscure non-Athenian non-Trojan
> sagittals (the font mapping space is otherwise full).

Probably best to wait until we have a better idea which sagittal symbols would be most appropriate or necessary for a wide range of temperaments.

> Yes, it would work. But is it the best way to do it? When you cast the
> above as accented nominals it is quite messy. When you use only the 5
> nominals ACDEG and use 5-comma accidentals for moving along the
> chains, it makes perfect sense relative to the temperament's 5-limit
> mapping. A\ A C\ C D\ D E\ E G\ G
> > By the way, this is an example where using equally spaced boundaries
> for the nominals doesn't do the right thing as far as maximising the
> familiar spellings of the consonant intervals.

This only helps two of the fifths, and gives you a C-E interval which looks like a major third but is really a fourth. With the accented nominals, you'd have D ]E G A [C. Since fifths in blackwood are very sharp (about 14.6 cents in the TOP version), this doesn't seem like a very big deal. There are also melodic issues: D-E-G suggests a whole step followed by a minor third, but in fact these are two equally sized intervals, and the notation D-]E-G better represents that.

You could make a case for superpyth, with its better (6.2 cent sharp) fifths. A 7-note chain would be ]B E A D G C [F with the evenly spaced option. But a 7-note chain of superpyth is improper, and a set of 5 nominals appears to be a better option for notation.

The other reason I prefer to keep the octave evenly divided is to allow for tunings centered around pitches other than D. I ran into this problem when trying to translate music written in my original lemba notation based on A = 440 Hz; I needed to set the pitch of D to 264 Hz. It would have been better if I could just notate 440 Hz as "A" and build the rest of the scale around that.

>>BE U+0411 = B]
> > > Like a "B" with a gap at the top right. How do you pronounce the name?
> I hope it isn't a long "e" as in "bee". Likewise for others below.

I believe it's /b,e/ with a soft (palatalized) "b" and an "eh" sound similar to the way "e" is pronounced in many other languages (German, French, Italian, etc.) But it might be /be/ with a hard "b". (English doesn't have this distinction, so it's hard to describe in non-technical terms, but a soft "b" sounds a bit as if it has a "y" sound after it, like in the English word "beauty"; the hard b is more like the "b" in "boot".)

> Looks reasonable, provided the pronunciations of the names are
> sufficiently different from those of the Latin and Greek characters
> we'd be using, and the symbols look different.

The pronunciation of the letters is the biggest problem; many of them sound like the names of Latin letters (at least in some languages). The difference between Russian "ef" and English "F" is that the first is said with a Russian accent. :-) We'd have to use different names for these characters when talking about them, which is a disadvantage.

> We need two varieties of sharpened F and C, both semisharp and sharp.
> For the others we only need the semisharps. Could we use capitals for
> one and smalls for the other. I'm concerned that small Cyrillic be
> looks too much like small delta.

I guess it does in the handwritten form, but more likely the printed form would be mistaken for the number 6. For the two varieties of C, we could use TSE for one and CHE for the other. (The /ts/ sound written with the letter TSE is spelled "c" in Slavic languages that use the Latin alphabet.) I don't know what you'd do about two versions of F, unless you want to admit the archaic letter FITA (which looks a bit like the Greek theta, and won't be in most fonts).

> You didn't comment on my earlier suggestion to put the nominal accent
> ("[" or "]") to the left of the letter, to distinguish it from an
> accidental. Do you think that's a bad idea? If so why?

No, I agree that it's a good idea.

>>In cases where "O" is used, it doesn't matter which way you go; they're >>both the same distance from D. But there are two possibilities: either >>the "generators down" rule, or the "lower pitch" rule (which would have >>the advantage of always favoring G# over Ab, not just in the case of >>12-note chains of fourths, and may be easier to remember).
> > > The "lower pitch" rule would result in Miracle having "O" as the last
> in the chain, i.e. corresponding to the "9" rather than the zero of
> Graham's decimal notation. This seems likely to breed confusion (no
> pun intended).

That's assuming that the center of the scale is on D; Graham's page (http://69.10.138.114/~microton/decimal_notation.htm) suggests that the note 0 should be tuned to C. Thus, Graham's decimal notation could be written C bD ]D [E [F #F G bA ]A [B. In this case, bA (7) would correspond with "O".

>>Myna = 4 (too few nominals for so complex a temperament!)
> > > Yes, but we have serious problems no matter what the formula, with any
> generator so close to 1/2, 1/3 or 1/4 octave. How many do you propose
> for Myna?

Ideally 27, but you could get by with 23.

>>Pajara = 8 (not 10? Still, 10 is pretty close; a slight adjustment to >>the formula could fix this.)
> > > Hmm. I wonder if Paul Erlich has ever considered that. As you say,
> reducing k to 4.49 will take care of it. But that will also cause
> 17-ET (considered as Pythagorean) to have only 5 nominals. Maybe
> that's not so bad, or maybe we just need to add other factors to
> refine the formula, or maybe we just need to use our common sense in
> addition to the formula.

Perhaps the formula could take into account the complexity of the temperament. Bug, with a complexity of 2.55 according to Paul's paper, should have 9 nominals according to the formula, and wï¿½rschmidt (complexity 10.10) should have 3, but there's not much point in using more than 9 notes of bug, so 4 or 5 nominals would be adequate. Anything written in wï¿½rschmidt is going to require at least 16 notes to allow for a spin around the cycle of thirds; 16 notes requires at least 7 nominals plus accidentals. (On the other hand, wï¿½rschmidt is so problematic that it's probably going to end up as an exception to the general system; notating only enough pitches so that wï¿½rschmidt[65] can be notated with a single accidental pair, not every pitch in the horagram ring.)

> Maybe a factor pertaining to the complexity of the temperament could
> be added to the formula.

Ah, I see you've already thought of that. :-)

🔗monz <monz@tonalsoft.com>

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

🔗Paul Erlich <perlich@aya.yale.edu>

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

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> I've been thinking instead of "semiflat" and "semisharp" we should call
> them something simpler like "upper-A", "lower-A", or something of
the sort.
>

Agreed. How about "up-A" and "down-A"?

And by the way, I can't help thinking we should retain the name "O"
rather than "#G", at least in the case of temperaments with an even
number of chains, since it seems wrong to choose either one of "#G" or
"bA" in that case.

> Hmm, there's the potential of confusing [ with ], but after all this
> time with the extended alphabet, I still can't keep all those letters
> straight; this is certainly an improvement.

I've been having second thoughts about the confusability of [ and ]
too. How about we use "e" for semiflat? So "eF" is pronounced
"down-F". At least it's rounded (like the Couper backwards flat) and
it agrees with the Dutch suffix (although we'd be using it as a prefix).

Ces = C flat
Ceh = C semiflat
Cih = C semisharp
Cis = C sharp

These Dutch suffixes also give us another option for pronouncing our
compound nominals. Maybe "compound nominals" is a better term than
"pseudo-nominals".

I'll use "e" and "down-" for semiflat in this message to see how it
looks and feels.

> I've just gone through and replaced the extended-alphabetical nominals
> with the [] symbols to see what they look like: a few examples:
>
> Mavila: [B [E A D G ]C ]F
> Porcupine: A [B ]C D [E ]F G
> Hanson: ]E ]G B D F [A [C
> Amity: E G [B D ]F A C
> Orson: bE #F [A ]B D [F ]G bB #C

Because of the different boundaries, I get

Mavila (-1 3) : eB eE eA D ]G ]C ]F
Porcupine (-3 -5) : A B ]C D eE F G
Hanson ( 6 5) : ]E ]G B D F eA eC
Amity: (-5 -13) : E ]G B D F eA C
Orson: ( 7 -3) : bE #F eA eC D ]E ]G bB #C

Abbreviated 5-limit mappings (generators to primes 3 and 5) are given
above in parenthesis.

But why do I think these are better?

Mavila: Equal-spaced-boundaries notate only the middle two generators
as familiar fifths. It seems better to me to have all or none. But
"all" would produce ridiculous melodic relationships, so I'm happy
that the boundaries I'm using result in "none". Mavila is somewhat
borderline _as_ a 5-limit temperament due to its huge errors.

Porcupine: It is useful that, according to the mapping, B:D and D:F
are minor thirds. I note that even with equally spaced boundaries, a
Porcupine generator only slightly smaller than TOP, results in the
same set as I give above.

Hanson: We agree. Hoorah!

Amity: Neither of our proposals agree very well with the mapping. We
both have some examples of +2 generators being notated as fifths. In
your case G:D and D:A. In mine E:B and F:C.

Amity is complex. If we take it out to 11 nominals, equal spacing
would give
A ]C E G eB D ]F A C eE G (note that G and A appear twice)
While fifth-based boundaries would give
A ]C E ]G B D F eA C eE G

Orson: We agree. Hoorah!

This agreement is obscured because you used up-B instead of down-C and
down-F instead of up-E. I note that you used down-C and up-E in
Hanson. Can we standardise on these?

> > I could make a special version of the sagittal font with these two TC
> > symbols in place of the Wilson sloping - and + symbols that Joseph
> > uses, or in place of some really obscure non-Athenian non-Trojan
> > sagittals (the font mapping space is otherwise full).
>
> Probably best to wait until we have a better idea which sagittal
symbols
> would be most appropriate or necessary for a wide range of temperaments.
>

Agreed.

> > Yes, it would work. But is it the best way to do it? When you cast the
> > above as accented nominals it is quite messy. When you use only the 5
> > nominals ACDEG and use 5-comma accidentals for moving along the
> > chains, it makes perfect sense relative to the temperament's 5-limit
> > mapping. A\ A C\ C D\ D E\ E G\ G
> >
> > By the way, this is an example where using equally spaced boundaries
> > for the nominals doesn't do the right thing as far as maximising the
> > familiar spellings of the consonant intervals.
>
> This only helps two of the fifths,

How so? I get C:G, G:D, D:A and A:E all being approximate 2:3's
according to the Blackwood temperament's prime mapping. And of course
C\:G\, G\:D\, D\:A\ and A\:E\. That's 8 out of 10 fifths named in a
familiar manner.

> and gives you a C-E interval which
> looks like a major third but is really a fourth.

But that's just the usual problem in notating 5-ET. One of the usual
ways of dealing with it is to treat E and F as interchangeable names
for the same pitch. Similarly B and C. But this can just cause more
confusion (in thinking they are different pitches). Better just to get
used to C:E being a fourth in this context, IMHO.

Anyway, no one using the system we're developing should expect C:E to
be a major third (approximate 4:5), unless they're using a temperament
where the 5-comma vanishes. They should expect C:E\ to be one, as it
is here. Unfortunately the other major thirds don't fare so well -
G:C\, D:G\, A:D\, E:A\. But something has to give. One can't expect to
find meantone notational relationships in non-meantone temperaments.

I still feel that this 5-nominal notation is the best solution for
Blackwood, and that in general, preserving the familiar fifth
notations is more important than the thirds. And that's what the
following boundaries do. After all, the A-G notation began with, and
applies equally well to, Pythagorean tunings, where approximations to
4:5 are irrelevant.

Number of chains
Nom- 1 2 3 4
inal Lower bound of nominal (as a fraction of an octave)
------------------------------
D 0 0 0 0
]D 0 0 0 0
bE 1/17 1/14 1/15 1/14
eE 2/19 1/10 2/21 1/10
E 1/7 1/7 1/7 3/20
eF 1/5 1/5 1/5 5/28
F 4/17 5/22 5/21 1/4
]F 2/7 2/7 4/15 1/4
#F 6/19 3/10 1/3 7/20
eG 6/17 5/14 1/3 9/28
G 2/5 2/5 2/5 2/5
]G 3/7 3/7 3/7 3/7
O 9/19 1/2 10/21 1/2
eA 9/17 1/2 8/15 1/2

Naturals, sharps and flats are inclusive of boundaries, ups and downs
are exclusive.

The only things I'm not sure about here are the asymmetric boundaries
for "O" with odd numbers of chains.

> With the accented
> nominals, you'd have D ]E G A [C. Since fifths in blackwood are very
> sharp (about 14.6 cents in the TOP version), this doesn't seem like a
> very big deal.

No, not a big deal. But the cycle of fifths would be eC G D A ]E (eC).
I think it better to have the single "notational wolf" C:E than have
fifths with varying combinations of ups and downs.

Yes the fifths are very wide, but does the mapping describe
recognisable approximations to 2:3 fifths or doesn't it? If it does,
then shouldn't the notation support this?

In "exo-temperaments" like "father", I feel no compulsion to make the
notation consistent with the mapping of fifths because they are
unrecognisable.

> There are also melodic issues: D-E-G suggests a whole
> step followed by a minor third, but in fact these are two equally sized
> intervals, and the notation D-]E-G better represents that.

That's quite true, and again it's just the usual 5-ET problem, and at
the other extreme of course we have 7-ET, where E-F-G is unexpectedly
uniform. Since the whole idea of temperament is predicated on harmony,
it seems to me that our temperament notations should make the notation
of harmonies as consistent as possible, by recognising the
temperament's mapping of fifths, provided the cost to melody is never
worse than those of 5-ET and 7-ET.

> You could make a case for superpyth, with its better (6.2 cent sharp)
> fifths. A 7-note chain would be ]B E A D G C [F with the evenly spaced
> option. But a 7-note chain of superpyth is improper, and a set of 5
> nominals appears to be a better option for notation.

Yes. It looks like we agree on E A D G C for superpyth.

> The other reason I prefer to keep the octave evenly divided is to allow
> for tunings centered around pitches other than D. I ran into this
> problem when trying to translate music written in my original lemba
> notation based on A = 440 Hz; I needed to set the pitch of D to 264 Hz.

Why is that a problem?

> It would have been better if I could just notate 440 Hz as "A" and
build
> the rest of the scale around that.

This issue of absolute pitch is an important one. It also relates to
the arguments you give below regarding "O" =/= zero for Miracle.

I believe there are two different ideas here that need to be teased
apart. In some ways this would be easier to see if we were still using
the full alphabetic nominals.

If someone had written something in meantone using a nonstandard
reference frequency that was 50 cents lower than standard, I hope you
would not then suggest that they used the nominals eA eB eC eD eE ]E
eG. The fact is, the nominals for meantone are ABCDEFG. If you don't
want to simply annotate the score regarding this transposition, then
you move up or down the chain of generator until you find a pitch that
is sufficiently close, and use that as the tonic. It will be notated
as one of the nominals A thru G folowed by an accidental. It will not
use a nominal from outside the meantone set.

You give the following nominals for Lemba, in the two chains.
]A eC D ]E eG
eE ]F O eB ]C

"A" does not appear in this set of nominals and so should not be used.
The note that is a fifth above D is ]A>5 where ">5" is temporarily
used as the accidental corresponding to going up 5 generators (which
will lower the pitch by about 50 cents). If it is a problem to set D =
264 Hz, you can use ]A>5 = 440 Hz.

The two different things here are:
(a) Changing key by moving the tonic up or down the chain of
generators, notated with a fixed set of nominals plus accidentals.
(b) Changing the set of nominals used for the temperament.

I think (a) should be allowed, but (b) rejected. It's difficult enough
to learn the workings of _one_ set of nominals for a temperament. And
if you shift the center of the nominal set away from D then the names
we've given to the nominals won't make so much sense and we'll be
wanting to rename bE to #D and so on causing even more confusion.

I suggest we try to agree on a single set of nominals for each
temperament, just as the set ABCDEFG is the only set that will be used
for meantone. And given this set for meantone, centering the others on
the same center should give the simplest results.

> > The "lower pitch" rule would result in Miracle having "O" as the last
> > in the chain, i.e. corresponding to the "9" rather than the zero of
> > Graham's decimal notation. This seems likely to breed confusion (no
> > pun intended).
>
> That's assuming that the center of the scale is on D; Graham's page
> (http://69.10.138.114/~microton/decimal_notation.htm) suggests that the
> note 0 should be tuned to C. Thus, Graham's decimal notation could be
> written C bD ]D [E [F #F G bA ]A [B. In this case, bA (7) would
> correspond with "O".

So, after the above argument, I hope you can see why I think the
question of absolute pitch is irrelevant to this. In decimal notation
the Miracle nominals are (as a chain of generators)
0 1 2 3 4 5 6 7 8 9
in our universal notation it would be nice if the first nominal in the
chain was "O", i.e #G. With your boundaries:
O ]A eB eC #C D bE ]E ]F eG. With mine:
O ]A eB C #C D bE E ]F eG.

... we
> could use TSE for one and CHE for the other. (The /ts/ sound written
> with the letter TSE is spelled "c" in Slavic languages that use the
> Latin alphabet.) I don't know what you'd do about two versions of F,
> unless you want to admit the archaic letter FITA (which looks a bit
like
> the Greek theta, and won't be in most fonts).

I think I'll leave this Greek/Latin/Hebrew/Cyrillic stuff to simmer in
the background for a while.

> Perhaps the formula could take into account the complexity of the
> temperament.

Good idea. :-)

How about

L/s + k*abs(ln(N/sqrt(7*C)))

Where
L is large step size
s is small step size
N is number of nominals
C is TOP complexity
k is a constant with a value around 4.5 to 5

So where we used to just have the magic number 7, we now have the
geometric mean of the complexity and the magic number 7.

I'm sorry I haven't even begun to look at your proposed sagittal
accidentals for the various temperaments. I'm kind of hoping to first
sort out
(a) The number of nominals used for each one, except possibly those
whose generators are close to 1/2, 1/3 or 1/4 octave which may require
more thought and I'm happy to ignore for now.
(b) Nominal boundaries.
This is so I don't have to consider so many different options.

Equally spaced boundaries do have the attraction of simplicity. I'm
actually hoping you can convince me that they are the right way to go.
But so far I'm seeing their disrespect of familiar fifths as a problem.

One point regarding the choice of sagittals: The rational comma used
must be validly approximated by the relevant number of generators,
according to the temperament's map (except, possibly for temperaments
with really big errors). And the simplest and lowest prime-limit comma
will probably be preferred.

-- Dave Keenan

🔗Herman Miller <hmiller@IO.COM>

Dave Keenan wrote:
> > --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> >>I've been thinking instead of "semiflat" and "semisharp" we should call >>them something simpler like "upper-A", "lower-A", or something of
> > the sort.
> > > Agreed. How about "up-A" and "down-A"?

Sounds good.

> And by the way, I can't help thinking we should retain the name "O"
> rather than "#G", at least in the case of temperaments with an even
> number of chains, since it seems wrong to choose either one of "#G" or
> "bA" in that case.

If we have a way to represent "O" on the staff (as in Joe Monzo's quarter-tone staff, my diamond-shape note heads, or specialized staff configurations for each half-octave based temperament), that would be fine. Using conventional notation, we could allow the possibility of writing "#G" or "bA" and referring to both of these as "O".

> I've been having second thoughts about the confusability of [ and ]
> too. How about we use "e" for semiflat? So "eF" is pronounced
> "down-F". At least it's rounded (like the Couper backwards flat) and
> it agrees with the Dutch suffix (although we'd be using it as a prefix).

Actually, since they're on the left, they won't be confused with the Sagittal shorthand symbols. So how about using ^ (up) and v (down)?

> Ces = C flat
> Ceh = C semiflat
> Cih = C semisharp
> Cis = C sharp
> > These Dutch suffixes also give us another option for pronouncing our
> compound nominals. Maybe "compound nominals" is a better term than
> "pseudo-nominals".

I've tried something similar with variations on the familiar "do re mi" scale, using all 5 vowels. For instance, "du" would be a quarter tone below "do", and "da" would be a quarter tone above. The full set (with alternatives for some pitches) turns out as:

DO(=ut) da/ru de/ro di/ra RE/mu ri/mo ma me MI/fu fo FA fe fi su SO(L) sa se/li si*/lo LA/tu le/to li/ta te TI(=si) du DO

*one problem is that "ti" can be "si" in some languages, so the "sol" series might be better as "sul sol sal sel sil" to avoid confusion, or just avoid "si" and use "lo".

>>I've just gone through and replaced the extended-alphabetical nominals >>with the [] symbols to see what they look like: a few examples:
>>
>>Mavila: [B [E A D G ]C ]F
>>Porcupine: A [B ]C D [E ]F G
>>Hanson: ]E ]G B D F [A [C
>>Amity: E G [B D ]F A C
>>Orson: bE #F [A ]B D [F ]G bB #C
> > > Because of the different boundaries, I get
> > Mavila (-1 3) : eB eE eA D ]G ]C ]F
> Porcupine (-3 -5) : A B ]C D eE F G
> Hanson ( 6 5) : ]E ]G B D F eA eC
> Amity: (-5 -13) : E ]G B D F eA C
> Orson: ( 7 -3) : bE #F eA eC D ]E ]G bB #C
> > Abbreviated 5-limit mappings (generators to primes 3 and 5) are given
> above in parenthesis.
> > But why do I think these are better?
> > Mavila: Equal-spaced-boundaries notate only the middle two generators
> as familiar fifths.

I consider [B-[E and ]C-]F as perfectly good and recognizable notations for fifths. So both schemes represent 4 out of 6 fifths. I prefer the even-spaced version because it's a little more even in the split between chains of fifths (2-3-2 vs. 3-1-3).

> Porcupine: It is useful that, according to the mapping, B:D and D:F
> are minor thirds. I note that even with equally spaced boundaries, a
> Porcupine generator only slightly smaller than TOP, results in the
> same set as I give above.

You could also count ]C-eE as a reasonable (if unconventional) spelling of a minor third. But the set based on even-spaced nominals gives correct spellings for four fifths (D-A, [E-[B, ]F-]C, and G-D), vs. only two if the uneven spacing is used. So you're trading two fifths for two minor thirds. Neither of these is any better than the original porcupine notation ABCDEFG, with four correct minor thirds, two correct major thirds, *and* four correct fifths. So if we're going to be adjusting the boundaries to suit particular temperaments, why take the simpler porcupine notation into consideration as well?

It's largely to avoid these sorts of disputes that I decided to settle on the evenly spaced set of 24 nominals per octave. This approach doesn't favor any one temperament at the expense of others, and can be used without change for temperaments involving two, three, or four chains per octave.

> Amity: Neither of our proposals agree very well with the mapping. We
> both have some examples of +2 generators being notated as fifths. In
> your case G:D and D:A. In mine E:B and F:C.

It may be a better approach with complex temperaments like this (which are pretty close to JI) to use pure Sagittal notation and notate the fifths directly. I don't have a good feel for this one since I haven't written in amity or even played with it much. (Wï¿½rschmidt either, for that matter.)

Pure Sagittal notation is always an option with most of these temperaments, but you end up with a confusing array of enharmonically equivalent notations for each pitch. It's also not very suitable for MOS/DE scales that aren't temperaments; in some cases there can be two different notes that could equally well be considered as "G" above "D".

> Orson: We agree. Hoorah! > > This agreement is obscured because you used up-B instead of down-C and
> down-F instead of up-E. I note that you used down-C and up-E in
> Hanson. Can we standardise on these?

These were from earlier versions of my notes and I've since standardized on using down-C and up-E.

>>>mapping. A\ A C\ C D\ D E\ E G\ G
>>This only helps two of the fifths, > > > How so? I get C:G, G:D, D:A and A:E all being approximate 2:3's
> according to the Blackwood temperament's prime mapping. And of course
> C\:G\, G\:D\, D\:A\ and A\:E\. That's 8 out of 10 fifths named in a
> familiar manner.

I meant two of the fifths in the chain of 5 nominals. My notation has D-A and G-D spelled "correctly", while yours adds C-G and A-E.

>>and gives you a C-E interval which >>looks like a major third but is really a fourth.
> > > But that's just the usual problem in notating 5-ET. One of the usual
> ways of dealing with it is to treat E and F as interchangeable names
> for the same pitch. Similarly B and C. But this can just cause more
> confusion (in thinking they are different pitches). Better just to get
> used to C:E being a fourth in this context, IMHO. Isn't it easier to see that when it's written [C-]E ?

> Anyway, no one using the system we're developing should expect C:E to
> be a major third (approximate 4:5), unless they're using a temperament
> where the 5-comma vanishes.

I think they should expect C:E to be an interval of more or less the *size* of a major third, not necessarily a just major third or its tempered approximation. In any case, something smaller than a fourth.

> I still feel that this 5-nominal notation is the best solution for
> Blackwood, and that in general, preserving the familiar fifth
> notations is more important than the thirds. And that's what the
> following boundaries do. After all, the A-G notation began with, and
> applies equally well to, Pythagorean tunings, where approximations to
> 4:5 are irrelevant.

One problem, as you've already noted, is that these boundaries don't work well on the other end of the fifth range, for temperaments like mavila. Mavila fifths are only a little worse than blackwood fifths (TOP tuning is -16.9 cents vs. +14.6 for 5-limit blackwood and +15.6 for 7-limit). And 4:5 approximations are just as irrelevant to mavila with its very flat "major thirds" as they are to Pythagorean.

Equal spacing allows a chain of up to 9 fifths of 17-ET or 5 of 22-ET. Chains of 7 fifths up to 708.33 cents are supported. This might not be to everyone's liking, but it doesn't seem possible to please everyone in a system that's intended for so many possible uses. And one of the reasons to want a generalized notation system is precisely for those temperaments based on chains of major or minor thirds, or fractions of those, which are awkward to represent in traditional fifth-based notation.

> Yes the fifths are very wide, but does the mapping describe
> recognisable approximations to 2:3 fifths or doesn't it? If it does,
> then shouldn't the notation support this?

This is one temperament out of many. I can't see how it's possible to satisfy this condition for all temperaments. See my comments regarding mavila above.

>>There are also melodic issues: D-E-G suggests a whole >>step followed by a minor third, but in fact these are two equally sized >>intervals, and the notation D-]E-G better represents that.
> > > That's quite true, and again it's just the usual 5-ET problem, and at
> the other extreme of course we have 7-ET, where E-F-G is unexpectedly
> uniform. Since the whole idea of temperament is predicated on harmony,
> it seems to me that our temperament notations should make the notation
> of harmonies as consistent as possible, by recognising the
> temperament's mapping of fifths, provided the cost to melody is never
> worse than those of 5-ET and 7-ET.

The uniform division system would notate 7-ET as A [B ]C D [E ]F G. Considered as 24-ET, this alternates 3/4-step intervals with whole steps, which at least is better than the 12-ET notation.

One thing I might not have pointed out is that one of my goals is to create notations for MOS/DE scales in general, not just temperaments. For that purpose, the melodic approach is preferable. But I realize that this system will mainly be used for temperaments, and I believe that the symmetrical division will tend to give as good or better results when a wide range of temperaments (not just those based on chains of fifths) is taken into consideration. I could be wrong on that, but it seems to give pretty good results so far.

> This issue of absolute pitch is an important one. It also relates to
> the arguments you give below regarding "O" =/= zero for Miracle.
> > I believe there are two different ideas here that need to be teased
> apart. In some ways this would be easier to see if we were still using
> the full alphabetic nominals.
> > If someone had written something in meantone using a nonstandard
> reference frequency that was 50 cents lower than standard, I hope you
> would not then suggest that they used the nominals eA eB eC eD eE ]E
> eG. The fact is, the nominals for meantone are ABCDEFG.

Historically, reference pitches for meantone have been lower than the modern standard, and it makes sense to continue that tradition. But when writing new music, the option to set any note to any desired pitch would be a useful feature to have. Easley Blackwood specified that middle C should be tuned to 264 Hz; other composers might want to use A=440 Hz, C=256 Hz, or some other standard. The point is, as long as we're using the traditional notation for A-G, we ought to be able to use any of these notes as the basis for a notation according to the preference of the composer. Insisting on having everything centered around D will make things unnecessarily more difficult for transcribing existing music, and limit the options for new music.

On the other hand, I do agree that it would be useful to have a "standard" set of nominals centered around D for each temperament. Composers who want to use some "non-standard" version could then transpose this to base it on any other pitch.

>>Perhaps the formula could take into account the complexity of the >>temperament.
> > > Good idea. :-)
> > How about
> > L/s + k*abs(ln(N/sqrt(7*C)))
> > Where > L is large step size
> s is small step size
> N is number of nominals
> C is TOP complexity
> k is a constant with a value around 4.5 to 5

I'll have to give that a try.

> Equally spaced boundaries do have the attraction of simplicity. I'm
> actually hoping you can convince me that they are the right way to go.
> But so far I'm seeing their disrespect of familiar fifths as a problem.

Well, I like to think of mavila fifths as "familiar", but I can understand if they're too flat for some; they're near the edge of what I'd consider as recognizable fifths. And the even-spaced version is better for the fifths of porcupine temperament. Not all temperaments with good fifths are based on chains of fifths. Try it out on nautilus (TOP P=1202.66, G=82.97) and see what you get. I get 7 of 8 fifths of nautilus[14] correctly notated, or all 8 if [F is allowed as an alternative to ]E.

map: [<1, 2, 3, 3], <0, -6, -10, -3]]
#G A bB [B [C ]C #C D bE [E ]E ]F #F G
Fifths: C#-G# D-A bE-bB [E-[B ]E-[C ]F-]C #F-#C G-D

> One point regarding the choice of sagittals: The rational comma used
> must be validly approximated by the relevant number of generators,
> according to the temperament's map (except, possibly for temperaments
> with really big errors). And the simplest and lowest prime-limit comma
> will probably be preferred.

Another consideration is the size of the tempered accidental in comparison to the untempered Sagittal version. That's why, for instance, I suggest the 25/24 )||( instead of the apotome as the basic meantone accidental (the two are equivalent in meantone).

I haven't taken schisminas into account, except for the use of the 125 M-diesis /|) in some of the 5-limit temperaments. But for the 7-limit temperaments, I'm using this symbol only for the 35 M-diesis. Actually, a good list of known schisminas would be nice to have. Is there one online anywhere?

🔗yahya_melb <yahya@melbpc.org.au>

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

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...> wrote:
> I've tried something similar with variations on the familiar "do re mi"
> scale, using all 5 vowels. For instance, "du" would be a quarter tone
> below "do", and "da" would be a quarter tone above. The full set (with
> alternatives for some pitches) turns out as:
>
> DO(=ut) da/ru de/ro di/ra RE/mu ri/mo ma me MI/fu fo FA fe fi su SO(L)
> sa se/li si*/lo LA/tu le/to li/ta te TI(=si) du DO
>
> *one problem is that "ti" can be "si" in some languages, so the "sol"
> series might be better as "sul sol sal sel sil" to avoid confusion, or
> just avoid "si" and use "lo".

A 24 tone solfa is a cute idea. I can follow your proposal better like
this:

vC du
C DO(ut)
^C da ru
#C de ro
vD di ra
D mu RE
^D mo ri
bE ma
vE me
E fu MI
^E fo m ?
F FA
^F fe
#F fi
vG su
G SO
^G sa
O se lu
vA si lo
A tu(su) LA
^A to(so) le
bB ta(sa) li
vB te(se)
B TI(SI)
vC du
C DO

I agree with Yahya that for those used to singing si instead of ti, it
would be a bad idea to expect them to distinguish SOL and sil from so
and SI, when they are sung. But I don't understand the point of
suggesting they use HI instead of SI. If you can get them to change
from SI to HI then why can't you get them to change from SI to TI? Is
it because they don't recognise much of a distinction between the
sounds of S and T?

I suggest we should not allow all the possible alternatives above. I
suggest the initial consonant should change whenever the letter
changes in our standard list of compond nominals. The only exception
might be for "O" (#G bA) which might sometimes be better as lu instead
of se. [Herman, you had a typo where you gave this as another "li"]

I expect Herman had this in mind, because (assuming we are limited to
5 vowels) his vowel ordering is an optimum ordering for this purpose
(one of 2 out of 120).

It ends up looking like this:

vC du
C DO(ut)
^C da
#C de
vD ra
D RE
^D ri
bE ma
vE me
E MI
^E my ?
F FA
^F fe
#F fi
vG su
G SO
^G sa
O se lu
vA lo
A LA
^A le
bB ta(sa)
vB te(se)
B TI(SI)
vC du
C DO

Herman's ordering of the 5 vowels is already optimal for this, but
still misses out on a name for one note. The order of e and i could be
swapped, but that would just mean that we wouldn't have a RE-derived
name for ^D instead of not having a MI-derived name for ^E. Any other
re-ordering gives us more unnamed notes. So we need a 6th singable
vowel to come after e and i.

This led me off on a Google odyssey to learn about vowel sounds, and
in particular singable ones.

The most useful single thing I found was the section "The Vowel
Systems of Four English Dialects" on this page.
http://www.ling.mq.edu.au/units/ling210-901/phonetics/ausenglish/auseng_vowels.html

The singable vowel that is most distinct from the 5 already used is
probably the diphthong "oy" as in "boy". Incidentally, DO SO RE use
diphthongs, while MI TI FA LA use monophthongs. But in the end I
thought it was probably important that it be representable by a single
letter, like the others, which only leaves "y". So ^E could be "my".

Unfortunately we still have two possible meanings for sa and se. It is
possible to order vowels in such a way that there are no such clashes,
but only if we have 7 vowels! And they must be arranged with the four
existing vowels in one of two ways.

- -
e -
i i
a or a
o o
- e
- -

Unused singable monophthongs are those in
"who" which you already represented with "u"
"haw"
"her"
"hair"

Trouble is, the diphthong in DO and SO ("ho") passes fairly close to
"hair" depending on your accent, and the diphthong in RE passes close
to either "her" or "haw" depending on your accent. So I'll fall back
on the diphthong "oy", which would be better spelled "oi" in this
context, given how it's starting and ending monphthongs are spelled.
If you really had to use a single letter for it I suppose you could
use "j".

So, for the least number of changes from Herman's 5 vowel scheme I'll use

y (high)
u (who)
i (hee)
a (ha)
o (hoe)
e (hay)
j(oi) (hoy) (I didn't put "y" "high" here because it's too similar to
"e" "hay").

So we end up with:

vC da
C DO(ut)
^C de
#C doi
vD ro
D RE
^D roi
bE my
vE mu
E MI
^E ma
F FA
^F fo
#F fe
vG sa
G SO
^G se
O soi lu
vA li
A LA
^A lo
bB ty(sy)
vB tu(su)
B TI(SI)
vC da
C DO

So we've eliminated the SO SI problem that way. But is it worth the
cost of 7 vowels. I doubt it.

What a pity there was no rhyme or reason to the original allocation of
vowels to the diatonic. OK, there _is_ a rhyme (or three), but I can't
see any reason for those rhymes. It would have made sense to me if the
two tetrachords rhymed.

-- Dave Keenan