Yahoo Tuning Groups Ultimate Backup tuning A JI realisation of Messiaen's octatonic (was: Schoenberg, serliasim, 12-edo)

Hi all,

I've given Paul's posting on Messiaen's octatonic
some thought, and have been thinking about some
musical consequences of his suggestions. In
particular, I wanted to find out how I might tune
my Roland E-28 keyboard to Paul's JI realisation
of that scale, so I could try making music using
this scale and the four major and four minor
chords that Paul's message described. I may have
succeeded.

On Thu, 08 Dec 2005, "wallyesterpaulrus" wrote:
>
> --- In tuning@yahoogroups.com, Christopher John Smith
> <christopherjohn_smith@y...> wrote:
>
> > This reminds me of a recurring idea of mine, one which has not
> >yet made it past the giving-up-after-thinking-about-it-for-thirty-
> >seconds stage - has anyone tried to make just versions of Messiaen
> >modes, specifically the second and third?
>
> Yup -- this is right up my alley, so thanks for asking! I'm pretty
> sure Gene for example has posted some on the tuning-math list, but we
> can procede without them . . .
>
> >(Which would require sets
> >of multiple sets of pitches a major diesis
> >(for the 2nd mode) or
> >minor diesis (for the third mode) apart,
> >multiple tones for scale >degrees to represent the various 11s, 13s,
> >17s, 19s, etc.)
>
> It would? I guess that depends on your view and usage of just
> intonation. Here's something explaining one possible view which I'll
> basically use here:
>
> http://lumma.org/tuning/erlich/erlich-tFoT.pdf
>
> This is a preview discusses the diatonic scale and the decatonic (you
> can think of the latter as a 7-limit version of Messaien's 10-note
> MLT). The complete work will also cover the scales you're asking
> about as well as some impossible in 12-equal, such as Porcupine.
>
> You might want to read this too:
>
> http://sonic-arts.org/td/erlich/intropblock1.htm
>
> Are you amenable to these ways of thinking?
>
> Not knowing quite what you mean by "the various 11s, 13s, 17s, 19s,
> etc.", I'll give you some 5- and 7-limit realizations:
>
> The diminished/octatonic scale (Messaien 2nd) in the 5-limit can be
> considered a periodicity block with commatic unison vector 648:625
> (like all scales in the 5-limit diminished family) and chromatic
> unison vector 16:15, such as the Fokker block:

[YA] I have introduced approximate letter names
below, in order to understand this better.

> C 1/1
> C# 25/24
> Eb 6/5
> E 5/4
> F# 25/18
> G 3/2
> A 5/3
> Bb 9/5
> C (2/1)
>
> Since you are free to transpose any of the notes by the commatic
> unison vector, we can allow for more consonant chords (4 major and 4
> minor triads instead of 3 of each) ...

[YA] What an intriguing idea! It sounds like a nice
way to expand our tonal resources - and in
just intonation, too!

> ... by providing an alternate pitch
> for each of two scale degrees by transposing by 648:625:

[YA] As above, I have introduced approximate letter
names. The alternate for C# is Db, 25/24 higher.
Similarly, the alternate for F# is Gb, 25/24 higher.
Including both "alternates" for C# and F# means
that we actually need 10 pitches per octave to
realise this tuning.

> C 1/1
> Db 25/24 * 648/625 = 27/25
> Eb 6/5
> E 5/4
> Gb 25/18 * 648/625 = 36/25
> G 3/2
> A 5/3
> Bb 9/5
> C (2/1)
>
> This allows the just major triads
>
> 1/1-5/4-3/2
> 6/5-3/2-9/5
> 36/25-9/5-27/25
> 5/3-25/24-5/4

[YA] That is, C, Eb, Gb and A major:
C-E-G
Eb-G-Bb
Gb-Bb-Db
A-C#-E

It's interesting to note that the roots of these chords
(as might be expected from the symmetry of the scale
construction) lie an exact 6/5 minor third (but not an
exact quarter octave) apart.

Starting from A 5/3, the four minor thirds upwards are:
- first C 1/1,
- second Eb 6/5,
- third Gb 36/25, and
- fourth A' 216/125.

A' is an octave and 128/125 above the A we started on.
128/125 = 640/625, which is quite a bit smaller than the
comma 648/625 used to create the two alternate scale
degrees.

So the roots of these four major chords are distributed
almost symmetrically over the span of the octave,
dividing it almost exactly into four quarters. Very
different from the way the three major chords of
common practice are spread, with roots falling on C, F and
G. Altho the F and G are symmetric about C, the three
roots do not divide the octave at all equally. This very
asymmetry lends more finality to common practice chord
progressions favouring major chords in which the last
movement of the root is upward [or downward] by a
fourth. The same kinds or quality of chord progression
will not be available using only major chords in the
octatonic scale, since the last movement of the root must
necessarily be upward [or downward] by a minor third or
tritone. So we can confidently predict a more restless,
less resolute, kind of harmonisation occurring in this
context.

How do the implications of this tuning, with its four major
chords, differ from those of 12-EDO with four major
chords built on successive minor thirds C, Eb, Gb and A?

In the 12-EDO case, these four chords are:
C-E-G
Eb-G-Bb
Gb-Bb-Db
A-C#-E

which together require eight distinct notes:
C-C#(=Db)-Eb-E-Gb-G-A-Bb

to play.

In the octatonic case, these four chords are:
C-E-G
Eb-G-Bb
Gb-Bb-Db
A-C#-E

which together require nine distinct notes:
C-C#-Db-Eb-E-Gb-G-A-Bb

to play.

So it requires only one more pitch to create the scale
supporting these chords in JI than in 12-EDO! That's
surprisingly little extra effort to provide, if we want to
hear pure JI harmonies, and has most impact on fixed
tuning instruments such as pianos, pipe organs and
metallophones. But were I moved to create tunes in
octatonic (*) and harmonise them with major chords, it's
great to know that a little "adaptive JI" could handle this
effortlessly on most instruments, including human voices.

> and the just minor triads
>
> 1/1-6/5-3/2
> 6/5-36/25-9/5
> 25/18-5/3-25/24
> 5/3-1/1-5/4

[YA] That is, C, Eb, F# and A minor:
C-Eb-G
Eb-Gb-Bb
F#-A-C#
A-C-E

To play the four major triads in C octatonic, we need
both C# and Db; similarly, to play the four minor triads
in C octatonic, we need both F# and Gb. So far we need
ten tones per octave to realise an octatonic scale in JI.

> The diminished/octatonic scale in the 7-limit can be considered a
> periodicity block with commatic unison vectors 36:35 and 50:49 (like
> all scales in the 7-limit diminished family) and chromatic unison
> vector 16:15, ...

[YA] (Paul goes on to describe the tetrads in the 7-limit
octatonic in a similar fashion, and how we can have more
of them by allowing commatic alternates for several
notes.)

It should be possible to notate the 7-limit octatonic
equally simply using letter-names ("nominals") and
accidentals or diacritics. I expect that we should find
that the same kind of adaptive JI, oriented toward the
octatonic scale, would similarly be capable of expanding
our range of expression.

How small is the difference, really, between C# and Db
above in the 5-limit case? The ratio is 648/625, so in
cents, that's 1200 * (log 648 - log 625)/log 2, or
62.565148 c.

Certainly, 63 cents is a very noticeable adjustment if
made to a held tone! Almost a third of a tone, in fact.

Db is 27/25, while D is usually taken as 9/8 which is
just 25/24 higher.

So we have:
C ......... 1/1 ............, * 25/24 .......... ==> C#
C# ..... 25/24 ...., * 648/625 .... ==> Db
Db ...... 27/25 ...., * 25/24 .......... ==> D
D ......... 9/8 .........., * 16/15 ........... ==> Eb
Eb ...... 6/5

The steps between these scale degrees & alternates
are:
25/24 ............... C# .............. 70.672427 c
648/625 ......... Db ............... 62.565148 c
25/24 ............... D .................. 70.672427 c
16/15 ................ Eb ................ 111.731285 c

(BTW, I defined a function in PowerToy Calc:
cents(r) = 1200 * ln(r) / ln(2)
to do these conversions for me. Windows users can
download the PowerToys free from Microsoft. To
find the cents in any ratio r, first define the cents(r)
function and save the definitions. Then just type
cents(r) in the Input box of the calculator, where r
is your chosen ratio, eg 25/24, and the answer will
appear, to more decimals than you'll ever need.)

The b sign marks the same distance down as the sharp
marks up, namely 25/24 or around 71 cents, with the
gap C#-Db being around 63 cents, giving an almost
equal division of the 204 cent wholetone C-D into
three parts.

(*) I think I'll try this, now that I know I can create
a piece in C octatonic using only this standard notation
for 10 pitches per octave:
C-C#-Db-Eb-E-F#-Gb-G-A-Bb
(0, 71, 133, 316, 386, 569, 631, 702, 884, 1018)

which I can tune on my MIDI keyboard directly, in
cents relative to 12-EDO, as:
C ............. C + 0 c
C# ......... C# -29 c
Db .......... D -71 c
Eb ........... Eb +12 c
E .............. E -14 c
F# .......... F +69 c
Gb ........... F# + 31c
G .............. G +2 c
A ............. A -16 c
Bb ........... Bb +18c

This tuning of the 12-note keyboard octave does not
use the keys commonly assigned to Ab and B. So I
need to avoid fingering them. The nearest JI
octatonic scale degrees, for both these keys, lie
outside the -64 to + 64c variation available in tuning
any key on this particular keyboard, so it is not
possible to assign those keys to any scale degree of
the JI octatonic rooted on C.

Note that the notation I've chosen for this scale does
not give a meaning to D, D#, E#, Fb, F, G#, Ab, A#,
B# or Cb.

The advantage of Paul's realisation of this scale is that
we can notate it "normally", and use any MIDI notation
software to record or generate a MIDI performance
in this scale, together with a rich set of ready-made
harmonies - all in Just Intonation. I find this opens up
some quite exciting possibilities!

Perhaps I should also look at what the tetrads offer,
and whether they can be notated and played on the
same keyboard.

Regards,
Yahya

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🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

Hello Yahya!

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:
>
>
> Hi all,
>
> I've given Paul's posting on Messiaen's octatonic
> some thought, and have been thinking about some
> musical consequences of his suggestions. In
> particular, I wanted to find out how I might tune
> my Roland E-28 keyboard to Paul's JI realisation
> of that scale, so I could try making music using
> this scale and the four major and four minor
> chords that Paul's message described. I may have
> succeeded.
>
>
> On Thu, 08 Dec 2005, "wallyesterpaulrus" wrote:
> >
> > --- In tuning@yahoogroups.com, Christopher John Smith
> > <christopherjohn_smith@y...> wrote:
> >
> > > This reminds me of a recurring idea of mine, one which has not
> > >yet made it past the giving-up-after-thinking-about-it-for-
thirty-
> > >seconds stage - has anyone tried to make just versions of
Messiaen
> > >modes, specifically the second and third?
> >
> > Yup -- this is right up my alley, so thanks for asking! I'm
pretty
> > sure Gene for example has posted some on the tuning-math list,
but we
> > can procede without them . . .
> >
> > >(Which would require sets
> > >of multiple sets of pitches a major diesis
> > >(for the 2nd mode) or
> > >minor diesis (for the third mode) apart,
> > >multiple tones for scale >degrees to represent the various 11s,
13s,
> > >17s, 19s, etc.)
> >
> > It would? I guess that depends on your view and usage of just
> > intonation. Here's something explaining one possible view which
I'll
> > basically use here:
> >
> > http://lumma.org/tuning/erlich/erlich-tFoT.pdf
> >
> > This is a preview discusses the diatonic scale and the decatonic
(you
> > can think of the latter as a 7-limit version of Messaien's 10-
note
> > MLT). The complete work will also cover the scales you're asking
> > about as well as some impossible in 12-equal, such as Porcupine.
> >
> > You might want to read this too:
> >
> > http://sonic-arts.org/td/erlich/intropblock1.htm
> >
> > Are you amenable to these ways of thinking?
> >
> > Not knowing quite what you mean by "the various 11s, 13s, 17s,
19s,
> > etc.", I'll give you some 5- and 7-limit realizations:
> >
> > The diminished/octatonic scale (Messaien 2nd) in the 5-limit can
be
> > considered a periodicity block with commatic unison vector
648:625
> > (like all scales in the 5-limit diminished family) and chromatic
> > unison vector 16:15, such as the Fokker block:
>
>
> [YA] I have introduced approximate letter names
> below, in order to understand this better.
>
> > C 1/1
> > C# 25/24
> > Eb 6/5
> > E 5/4
> > F# 25/18
> > G 3/2
> > A 5/3
> > Bb 9/5
> > C (2/1)
> >
> > Since you are free to transpose any of the notes by the commatic
> > unison vector, we can allow for more consonant chords (4 major
and 4
> > minor triads instead of 3 of each) ...
>
> [YA] What an intriguing idea! It sounds like a nice
> way to expand our tonal resources - and in
> just intonation, too!
>
>
> > ... by providing an alternate pitch
> > for each of two scale degrees by transposing by 648:625:
>
> [YA] As above, I have introduced approximate letter
> names. The alternate for C# is Db, 25/24 higher.

You mean 648/625 higher, right?

> Similarly, the alternate for F# is Gb, 25/24 higher.

Here also you mean 648/625 higher, yes?

> Including both "alternates" for C# and F# means
> that we actually need 10 pitches per octave to
> realise this tuning.
>
> > C 1/1
> > Db 25/24 * 648/625 = 27/25
> > Eb 6/5
> > E 5/4
> > Gb 25/18 * 648/625 = 36/25
> > G 3/2
> > A 5/3
> > Bb 9/5
> > C (2/1)
> >
> > This allows the just major triads
> >
> > 1/1-5/4-3/2
> > 6/5-3/2-9/5
> > 36/25-9/5-27/25
> > 5/3-25/24-5/4
>
> [YA] That is, C, Eb, Gb and A major:
> C-E-G
> Eb-G-Bb
> Gb-Bb-Db
> A-C#-E
>
> It's interesting to note that the roots of these chords
> (as might be expected from the symmetry of the scale
> construction) lie an exact 6/5 minor third (but not an
> exact quarter octave) apart.
>
> Starting from A 5/3, the four minor thirds upwards are:
> - first C 1/1,
> - second Eb 6/5,
> - third Gb 36/25, and
> - fourth A' 216/125.
>
> A' is an octave and 128/125 above the A we started on.

Yahya, I'm afraid that's incorrect:

(216/125)/(5/3) = 648/625.

> 128/125 = 640/625, which is quite a bit smaller than the
> comma 648/625 used to create the two alternate scale
> degrees.

If it were smaller, this wouldn't make any sense. Thankfully, it's
not smaller, it's "just right": 648:625.

> So the roots of these four major chords are distributed
> almost symmetrically over the span of the octave,
> dividing it almost exactly into four quarters. Very
> different from the way the three major chords of
> common practice are spread, with roots falling on C, F and
> G.

Welcome to the diminished scale! You should also try tuning it to the
8-note ring in the 'TOP Dimipent' horagram in the paper I mailed you,
which doesn't require any alternate notes, but aims at improving the
harmonies relative to 12-equal.

> Altho the F and G are symmetric about C, the three
> roots do not divide the octave at all equally. This very
> asymmetry lends more finality to common practice chord
> progressions favouring major chords in which the last
> movement of the root is upward [or downward] by a
> fourth. The same kinds or quality of chord progression
> will not be available using only major chords in the
> octatonic scale, since the last movement of the root must
> necessarily be upward [or downward] by a minor third or
> tritone. So we can confidently predict a more restless,
> less resolute, kind of harmonisation occurring in this
> context.

Sometimes.

> How do the implications of this tuning, with its four major
> chords, differ from those of 12-EDO with four major
> chords built on successive minor thirds C, Eb, Gb and A?
>
> In the 12-EDO case, these four chords are:
> C-E-G
> Eb-G-Bb
> Gb-Bb-Db
> A-C#-E
>
> which together require eight distinct notes:
> C-C#(=Db)-Eb-E-Gb-G-A-Bb
>
> to play.
>
> In the octatonic case, these four chords are:
> C-E-G
> Eb-G-Bb
> Gb-Bb-Db
> A-C#-E

The first E can't be the same as the last E.

> which together require nine distinct notes:
> C-C#-Db-Eb-E-Gb-G-A-Bb
>
> to play.

I think that's ten.

> So it requires only one more pitch to create the scale
> supporting these chords in JI than in 12-EDO!

I think that's two more pitches. Meanwhile, if you're only looking at
the major triads, the diatonic scale requires zero more pitches.

> That's
> surprisingly little extra effort to provide, if we want to
> hear pure JI harmonies, and has most impact on fixed
> tuning instruments such as pianos, pipe organs and
> metallophones.

But is it really still the same scale to your ears? Or does 648:625
produce a note that sounds like a distinct *musical* pitch? To me,
even 81:80, which is several times smaller than 648:625, doesn't
sound right as a *melodic* unison.

> But were I moved to create tunes in
> octatonic (*) and harmonise them with major chords, it's
> great to know that a little "adaptive JI" could handle this
> effortlessly on most instruments, including human voices.

With a more "hard-core" implementation of adaptive JI, you could cut
the 648:625 distinction between different occurences of the same note
down to a considerably smaller interval.

> > and the just minor triads
> >
> > 1/1-6/5-3/2
> > 6/5-36/25-9/5
> > 25/18-5/3-25/24
> > 5/3-1/1-5/4
>
> [YA] That is, C, Eb, F# and A minor:
> C-Eb-G
> Eb-Gb-Bb
> F#-A-C#
> A-C-E
>
> To play the four major triads in C octatonic, we need
> both C# and Db; similarly, to play the four minor triads
> in C octatonic, we need both F# and Gb. So far we need
> ten tones per octave to realise an octatonic scale in JI.

It was ten even for just the major triads.

> > The diminished/octatonic scale in the 7-limit can be considered a
> > periodicity block with commatic unison vectors 36:35 and 50:49
(like
> > all scales in the 7-limit diminished family) and chromatic unison
> > vector 16:15, ...
>
> [YA] (Paul goes on to describe the tetrads in the 7-limit
> octatonic in a similar fashion, and how we can have more
> of them by allowing commatic alternates for several
> notes.)
>
> It should be possible to notate the 7-limit octatonic
> equally simply using letter-names ("nominals") and
> accidentals or diacritics. I expect that we should find
> that the same kind of adaptive JI, oriented toward the
> octatonic scale, would similarly be capable of expanding
> our range of expression.
>
> How small is the difference, really, between C# and Db
> above in the 5-limit case? The ratio is 648/625, so in
> cents, that's 1200 * (log 648 - log 625)/log 2, or
> 62.565148 c.

You got it right this time!

> Certainly, 63 cents is a very noticeable adjustment if
> made to a held tone! Almost a third of a tone, in fact.

Indeed. That was my point above.

> Db is 27/25, while D is usually taken as 9/8 which is
> just 25/24 higher.
>
> So we have:
> C ......... 1/1 ............, * 25/24 .......... ==> C#
> C# ..... 25/24 ...., * 648/625 .... ==> Db
> Db ...... 27/25 ...., * 25/24 .......... ==> D
> D ......... 9/8 .........., * 16/15 ........... ==> Eb
> Eb ...... 6/5

But D isn't part of this scale.

> The steps between these scale degrees & alternates
> are:
> 25/24 ............... C# .............. 70.672427 c
> 648/625 ......... Db ............... 62.565148 c
> 25/24 ............... D .................. 70.672427 c
> 16/15 ................ Eb ................ 111.731285 c
>
> (BTW, I defined a function in PowerToy Calc:
> cents(r) = 1200 * ln(r) / ln(2)
> to do these conversions for me. Windows users can
> download the PowerToys free from Microsoft. To
> find the cents in any ratio r, first define the cents(r)
> function and save the definitions. Then just type
> cents(r) in the Input box of the calculator, where r
> is your chosen ratio, eg 25/24, and the answer will
> appear, to more decimals than you'll ever need.)
>
> The b sign marks the same distance down as the sharp
> marks up, namely 25/24 or around 71 cents, with the
> gap C#-Db being around 63 cents, giving an almost
> equal division of the 204 cent wholetone C-D into
> three parts.
>
> (*) I think I'll try this, now that I know I can create
> a piece in C octatonic using only this standard notation
> for 10 pitches per octave:
> C-C#-Db-Eb-E-F#-Gb-G-A-Bb
> (0, 71, 133, 316, 386, 569, 631, 702, 884, 1018)
>
> which I can tune on my MIDI keyboard directly, in
> cents relative to 12-EDO, as:
> C ............. C + 0 c
> C# ......... C# -29 c
> Db .......... D -71 c
> Eb ........... Eb +12 c
> E .............. E -14 c
> F# .......... F +69 c
> Gb ........... F# + 31c
> G .............. G +2 c
> A ............. A -16 c
> Bb ........... Bb +18c
>
> This tuning of the 12-note keyboard octave does not
> use the keys commonly assigned to Ab and B. So I
> need to avoid fingering them. The nearest JI
> octatonic scale degrees, for both these keys, lie
> outside the -64 to + 64c variation available in tuning
> any key on this particular keyboard, so it is not
> possible to assign those keys to any scale degree of
> the JI octatonic rooted on C.
>
> Note that the notation I've chosen for this scale does
> not give a meaning to D, D#, E#, Fb, F, G#, Ab, A#,
> B# or Cb.
>
> The advantage of Paul's realisation of this scale is that
> we can notate it "normally", and use any MIDI notation
> software to record or generate a MIDI performance
> in this scale, together with a rich set of ready-made
> harmonies - all in Just Intonation. I find this opens up
> some quite exciting possibilities!
>
> Perhaps I should also look at what the tetrads offer,

The 'TOP Dimisept' 8-note scale might satisfy a taste for better
tetrads than the 12-equal version of this scale (the octatonic or
diminished scale) can offer, without requiring any alternate pitches.

> and whether they can be notated and played on the
> same keyboard.
>
>
> Regards,
> Yahya

Have you pursued this approach for the plain old diatonic case?
There, it works much better, in that the alternates are much closer
together, but the melodic unisons between alternates still don't work
for my ears.

Also the decatonic case (both this and the diatonic case are
discussed in my paper below) may be of interest to you:

http://lumma.org/tuning/erlich/erlich-tFoT.pdf

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

On Wed, 14 Dec 2005, "wallyesterpaulrus" wrote:
>
> Hello Yahya!

Hi Paul, long time no debate ..!

> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
> <yahya@m...> wrote:
> >
> > Hi all,
> >
> > I've given Paul's posting on Messiaen's octatonic
> > some thought, and have been thinking about some
> > musical consequences of his suggestions. In
> > particular, I wanted to find out how I might tune
> > my Roland E-28 keyboard to Paul's JI realisation
> > of that scale, so I could try making music using
> > this scale and the four major and four minor
> > chords that Paul's message described. I may have
> > succeeded.
> >
> >
> > On Thu, 08 Dec 2005, "wallyesterpaulrus" wrote:
> > >
> > > --- In tuning@yahoogroups.com, Christopher John Smith
> > > <christopherjohn_smith@y...> wrote:
> > >
> > > > This reminds me of a recurring idea of mine, one which has not
> > > >yet made it past the giving-up-after-thinking-about-it-for-
> > > >thirty-seconds stage - has anyone tried to make just versions of
> > > >Messiaen modes, specifically the second and third?
> > >
> > > Yup -- this is right up my alley, so thanks for asking! I'm
> > > pretty sure Gene for example has posted some on the tuning-
> > > math list, but we can procede without them . . .
> > >
> > > >(Which would require sets
> > > >of multiple sets of pitches a major diesis
> > > >(for the 2nd mode) or
> > > >minor diesis (for the third mode) apart,
> > > >multiple tones for scale >degrees to represent the various 11s,
> > > >13s, 17s, 19s, etc.)
> > >
> > > It would? I guess that depends on your view and usage of just
> > > intonation. Here's something explaining one possible view which
> > > I'll basically use here:
> > >
> > > http://lumma.org/tuning/erlich/erlich-tFoT.pdf
> > >
> > > This is a preview discusses the diatonic scale and the decatonic
> > > (you can think of the latter as a 7-limit version of Messaien's
> > > 10-note MLT). The complete work will also cover the scales
> > > you're asking about as well as some impossible in 12-equal, such
> > > as Porcupine.
> > >
> > > You might want to read this too:
> > >
> > > http://sonic-arts.org/td/erlich/intropblock1.htm
> > >
> > > Are you amenable to these ways of thinking?
> > >
> > > Not knowing quite what you mean by "the various 11s, 13s, 17s,
> > > 19s, etc.", I'll give you some 5- and 7-limit realizations:
> > >
> > > The diminished/octatonic scale (Messaien 2nd) in the 5-limit can
> > > be considered a periodicity block with commatic unison vector
> > > 648:625 (like all scales in the 5-limit diminished family) and
> > > chromatic unison vector 16:15, such as the Fokker block:
> >
> >
> > [YA] I have introduced approximate letter names
> > below, in order to understand this better.
> >
> > > C 1/1
> > > C# 25/24
> > > Eb 6/5
> > > E 5/4
> > > F# 25/18
> > > G 3/2
> > > A 5/3
> > > Bb 9/5
> > > C (2/1)
> > >
> > > Since you are free to transpose any of the notes by the commatic
> > > unison vector, we can allow for more consonant chords (4 major
> > > and 4 minor triads instead of 3 of each) ...
> >
> > [YA] What an intriguing idea! It sounds like a nice
> > way to expand our tonal resources - and in
> > just intonation, too!
> >
> >
> > > ... by providing an alternate pitch
> > > for each of two scale degrees by transposing by 648:625:
> >
> > [YA] As above, I have introduced approximate letter
> > names. The alternate for C# is Db, 25/24 higher.
>
> You mean 648/625 higher, right?

Right!

> > Similarly, the alternate for F# is Gb, 25/24 higher.
>
> Here also you mean 648/625 higher, yes?

Right, again! (You're so predictable! :-) )

> > Including both "alternates" for C# and F# means
> > that we actually need 10 pitches per octave to
> > realise this tuning.
> >
> > > C 1/1
> > > Db 25/24 * 648/625 = 27/25
> > > Eb 6/5
> > > E 5/4
> > > Gb 25/18 * 648/625 = 36/25
> > > G 3/2
> > > A 5/3
> > > Bb 9/5
> > > C (2/1)
> > >
> > > This allows the just major triads
> > >
> > > 1/1-5/4-3/2
> > > 6/5-3/2-9/5
> > > 36/25-9/5-27/25
> > > 5/3-25/24-5/4
> >
> > [YA] That is, C, Eb, Gb and A major:
> > C-E-G
> > Eb-G-Bb
> > Gb-Bb-Db
> > A-C#-E
> >
> > It's interesting to note that the roots of these chords
> > (as might be expected from the symmetry of the scale
> > construction) lie an exact 6/5 minor third (but not an
> > exact quarter octave) apart.
> >
> > Starting from A 5/3, the four minor thirds upwards are:
> > - first C 1/1,
> > - second Eb 6/5,
> > - third Gb 36/25, and
> > - fourth A' 216/125.
> >
> > A' is an octave and 128/125 above the A we started on.
>
> Yahya, I'm afraid that's incorrect:
>
> (216/125)/(5/3) = 648/625.

Oops! OK, you got it. So, we can now say:
"A' is an octave and a (648/125) comma above the A we
started on."

> > 128/125 = 640/625, which is quite a bit smaller than the
> > comma 648/625 used to create the two alternate scale
> > degrees.
>
> If it were smaller, this wouldn't make any sense. ...

Yep, that's what I thought, too ...

> ... Thankfully, it's
> not smaller, it's "just right": 648:625.

Fantastic!

> > So the roots of these four major chords are distributed
> > almost symmetrically over the span of the octave,
> > dividing it almost exactly into four quarters. Very
> > different from the way the three major chords of
> > common practice are spread, with roots falling on C, F and
> > G.
>
> Welcome to the diminished scale! ...

Well, _of course_ I've already played with the diminished
scale in 12-EDO ...! Also noted its restless harmonies.

> ... You should also try tuning it to the
> 8-note ring in the 'TOP Dimipent' horagram in the paper I mailed you,
> which doesn't require any alternate notes, but aims at improving the
> harmonies relative to 12-equal.

At what cost to the octave? Not too much, I hope,
or it wouldn't be "just right" ... Yes, I have it now - the
tempered octave of TOP Dimipent is just 3.36 cents flat.
Audible, but not maddening in fast passages.

How good, in TOP Dimipent, are the harmonies of the
same four major and four minor chords on roots C, Eb,
F# and A? How do they compare with the "harmonic
octatonic", or "octatonic with alternates", that you
proposed?

> > Altho the F and G are symmetric about C, the three
> > roots do not divide the octave at all equally. This very
> > asymmetry lends more finality to common practice chord
> > progressions favouring major chords in which the last
> > movement of the root is upward [or downward] by a
> > fourth. The same kinds or quality of chord progression
> > will not be available using only major chords in the
> > octatonic scale, since the last movement of the root must
> > necessarily be upward [or downward] by a minor third or
> > tritone. So we can confidently predict a more restless,
> > less resolute, kind of harmonisation occurring in this
> > context.
>
> Sometimes.

Turning it around ... I know I can't achieve quite the same
sense of finality or resolution in a cadence with roots on
the diminished seventh chord. I've spent half an hour this
morning trying to get a really satisfying cadence. Maybe
I'm just a "perfect fifth" junkie after all!

> > How do the implications of this tuning, with its four major
> > chords, differ from those of 12-EDO with four major
> > chords built on successive minor thirds C, Eb, Gb and A?
> >
> > In the 12-EDO case, these four chords are:
> > C-E-G
> > Eb-G-Bb
> > Gb-Bb-Db
> > A-C#-E
> >
> > which together require eight distinct notes:
> > C-C#(=Db)-Eb-E-Gb-G-A-Bb
> >
> > to play.
> >
> > In the octatonic case, these four chords are:
> > C-E-G
> > Eb-G-Bb
> > Gb-Bb-Db
> > A-C#-E
>
> The first E can't be the same as the last E.

That would be a (648/125) comma above it, then,
or E'. Looks like I'll be mapping that to the unused
F key then.

> > which together require nine distinct notes:
> > C-C#-Db-Eb-E-Gb-G-A-Bb
> >
> > to play.
>
> I think that's ten.

OK.

> > So it requires only one more pitch to create the scale
> > supporting these chords in JI than in 12-EDO!
>
> I think that's two more pitches. Meanwhile, if you're only looking at
> the major triads, the diatonic scale requires zero more pitches.

Sure, but the advantage of your tuning of the octatonic
was that it supplied alternates for those notes necessary
to make the given triads totally in (JI) tune. Which is the
only reason it attracted my attention in the first place.
People who like rough harmonies are welcome to them; I
prefer something smoother when possible.

> > That's
> > surprisingly little extra effort to provide, if we want to
> > hear pure JI harmonies, and has most impact on fixed
> > tuning instruments such as pianos, pipe organs and
> > metallophones.
>
> But is it really still the same scale to your ears? Or does 648:625
> produce a note that sounds like a distinct *musical* pitch? To me,
> even 81:80, which is several times smaller than 648:625, doesn't
> sound right as a *melodic* unison.

Of course not! It's 20.5 cents, after all, which
is way too big a mistuning to ignore, even in speed.
Seems to me that there are two ways to use the
resulting set of pitches - (1) as an adaptive JI
with notes that sometimes must move by a whole
(648/125) comma in order to maintain harmony;
(2) as a scale with more than eight notes (at last
count I was up to ten, so now it's eleven), some
separated by the (648/125) comma, and good
harmonisations for only some scale degrees.

> > But were I moved to create tunes in
> > octatonic (*) and harmonise them with major chords, it's
> > great to know that a little "adaptive JI" could handle this
> > effortlessly on most instruments, including human voices.
>
> With a more "hard-core" implementation of adaptive JI, you could cut
> the 648:625 distinction between different occurences of the same note
> down to a considerably smaller interval.

What particular kinds of implementation did you
have in mind?

> > > and the just minor triads
> > >
> > > 1/1-6/5-3/2
> > > 6/5-36/25-9/5
> > > 25/18-5/3-25/24
> > > 5/3-1/1-5/4
> >
> > [YA] That is, C, Eb, F# and A minor:
> > C-Eb-G
> > Eb-Gb-Bb
> > F#-A-C#
> > A-C-E
> >
> > To play the four major triads in C octatonic, we need
> > both C# and Db; similarly, to play the four minor triads
> > in C octatonic, we need both F# and Gb. So far we need
> > ten tones per octave to realise an octatonic scale in JI.
>
> It was ten even for just the major triads.

So now it's eleven.

> > > The diminished/octatonic scale in the 7-limit can be considered a
> > > periodicity block with commatic unison vectors 36:35 and 50:49
> > > (like all scales in the 7-limit diminished family) and chromatic
> > > unison vector 16:15, ...
> >
> > [YA] (Paul goes on to describe the tetrads in the 7-limit
> > octatonic in a similar fashion, and how we can have more
> > of them by allowing commatic alternates for several
> > notes.)
> >
> > It should be possible to notate the 7-limit octatonic
> > equally simply using letter-names ("nominals") and
> > accidentals or diacritics. I expect that we should find
> > that the same kind of adaptive JI, oriented toward the
> > octatonic scale, would similarly be capable of expanding
> > our range of expression.
> >
> > How small is the difference, really, between C# and Db
> > above in the 5-limit case? The ratio is 648/625, so in
> > cents, that's 1200 * (log 648 - log 625)/log 2, or
> > 62.565148 c.
>
> You got it right this time!

Score one for the Yo-yo! :-)

> > Certainly, 63 cents is a very noticeable adjustment if
> > made to a held tone! Almost a third of a tone, in fact.
>
> Indeed. That was my point above.

Indeed. The question becomes: if you're stuck with it,
what can you do with it? I've given two approaches
above. I think I'd tend to use the second, if only
because it's hard to avoid using notes that are just lying
there, waiting for us to take notice of them.

> > Db is 27/25, while D is usually taken as 9/8 which is
> > just 25/24 higher.
> >
> > So we have:
> > C ......... 1/1 ............, * 25/24 .......... ==> C#
> > C# ..... 25/24 ...., * 648/625 .... ==> Db
> > Db ...... 27/25 ...., * 25/24 .......... ==> D
> > D ......... 9/8 .........., * 16/15 ........... ==> Eb
> > Eb ...... 6/5
>
> But D isn't part of this scale.

Didn't mean to imply that it was ... just working out
the numbers, and how I should notate 27/25. At
first, I chose to label it C#', ie one (648/625)
comma above C#. But when I noticed that it was as
far below D as C# was above C, it seemed more
logically to use the b as a sort of negative #.

> > The steps between these scale degrees & alternates
> > are:
> > 25/24 ............... C# .............. 70.672427 c
> > 648/625 ......... Db ............... 62.565148 c
> > 25/24 ............... D .................. 70.672427 c
> > 16/15 ................ Eb ................ 111.731285 c
> >
> > (BTW, I defined a function in PowerToy Calc:
> > cents(r) = 1200 * ln(r) / ln(2)
> > to do these conversions for me. Windows users can
> > download the PowerToys free from Microsoft. To
> > find the cents in any ratio r, first define the cents(r)
> > function and save the definitions. Then just type
> > cents(r) in the Input box of the calculator, where r
> > is your chosen ratio, eg 25/24, and the answer will
> > appear, to more decimals than you'll ever need.)
> >
> > The b sign marks the same distance down as the sharp
> > marks up, namely 25/24 or around 71 cents, with the
> > gap C#-Db being around 63 cents, giving an almost
> > equal division of the 204 cent wholetone C-D into
> > three parts.
> >
> > (*) I think I'll try this, now that I know I can create
> > a piece in C octatonic using only this standard notation
> > for 10 pitches per octave:
> > C-C#-Db-Eb-E-F#-Gb-G-A-Bb
> > (0, 71, 133, 316, 386, 569, 631, 702, 884, 1018)

[At this point, I had forgotten that I also needed A#
to make F# major.]

> > which I can tune on my MIDI keyboard directly, in
> > cents relative to 12-EDO, as:
> > C ............. C + 0 c
> > C# ......... C# -29 c
> > Db .......... D -71 c
> > Eb ........... Eb +12 c
> > E .............. E -14 c
> > F# .......... F +69 c
> > Gb ........... F# + 31c
> > G .............. G +2 c
> > A ............. A -16 c
> > Bb ........... Bb +18c
> >
> > This tuning of the 12-note keyboard octave does not
> > use the keys commonly assigned to Ab and B. So I
> > need to avoid fingering them. The nearest JI
> > octatonic scale degrees, for both these keys, lie
> > outside the -64 to + 64c variation available in tuning
> > any key on this particular keyboard, so it is not
> > possible to assign those keys to any scale degree of
> > the JI octatonic rooted on C.
> >
> > Note that the notation I've chosen for this scale does
> > not give a meaning to D, D#, E#, Fb, F, G#, Ab, A#,
> > B# or Cb.
> >
> > The advantage of Paul's realisation of this scale is that
> > we can notate it "normally", and use any MIDI notation
> > software to record or generate a MIDI performance
> > in this scale, together with a rich set of ready-made
> > harmonies - all in Just Intonation. I find this opens up
> > some quite exciting possibilities!
> >
> > Perhaps I should also look at what the tetrads offer,
>
> The 'TOP Dimisept' 8-note scale might satisfy a taste for better
> tetrads than the 12-equal version of this scale (the octatonic or
> diminished scale) can offer, without requiring any alternate pitches.

If it needs no alternates, then it automatically solves
the problem of the large amount of "adapatation"
needed by the JI octatonic-with-alternates. My
only remaining concern would be the quality of the
harmonies.

> > and whether they can be notated and played on the
> > same keyboard.
>
> Have you pursued this approach for the plain old diatonic case?
> There, it works much better, in that the alternates are much closer
> together, but the melodic unisons between alternates still don't work
> for my ears.

So, to your ears, JI diatonic requires too much
adaptation for the alternate notes to be considered
essentially the same? Are there any tunings in which
adaptive JI "works" for you?

> Also the decatonic case (both this and the diatonic case are
> discussed in my paper below) may be of interest to you:
>
> http://lumma.org/tuning/erlich/erlich-tFoT.pdf

Of course it will. I remember being quite interested
in this paper the first time I read it. Perhaps, now
having come head to head with the pain of adaptive
JI, I will get some new insights from rereading it.

Since I've now spent some hours trying to make
listenable, playable music in the adaptive JI octatonic,
and have been unhappy with the melodic progressions
in particular, it's good to have some other possibilities
that offer better hope of both good harmony and good
melody together.

Paul, thanks for all the trouble you've taken to comment
on my fumbling (micro) steps in Tuning Land.

Regards,
Yahya

--
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🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:

> > (216/125)/(5/3) = 648/625.
>
> Oops! OK, you got it. So, we can now say:
> "A' is an octave and a (648/125) comma above the A we
> started on."

648/625.

> > ... You should also try tuning it to the
> > 8-note ring in the 'TOP Dimipent' horagram in the paper I mailed
you,
> > which doesn't require any alternate notes, but aims at improving
the
> > harmonies relative to 12-equal.
>
> At what cost to the octave? Not too much, I hope,
> or it wouldn't be "just right" ... Yes, I have it now - the
> tempered octave of TOP Dimipent is just 3.36 cents flat.
> Audible, but not maddening in fast passages.

If you prefer, stretch the entire tuning so that octaves are pure.

> How good, in TOP Dimipent, are the harmonies of the
> same four major and four minor chords on roots C, Eb,
> F# and A? How do they compare with the "harmonic
> octatonic", or "octatonic with alternates", that you
> proposed?

Well, that's exactly what I was hoping you'd decide through
listening. How would you like me to answer this question (what sense
of "good" did you have in mind)?

> > > Altho the F and G are symmetric about C, the three
> > > roots do not divide the octave at all equally. This very
> > > asymmetry lends more finality to common practice chord
> > > progressions favouring major chords in which the last
> > > movement of the root is upward [or downward] by a
> > > fourth. The same kinds or quality of chord progression
> > > will not be available using only major chords in the
> > > octatonic scale, since the last movement of the root must
> > > necessarily be upward [or downward] by a minor third or
> > > tritone. So we can confidently predict a more restless,
> > > less resolute, kind of harmonisation occurring in this
> > > context.
> >
> > Sometimes.
>
> Turning it around ... I know I can't achieve quite the same
> sense of finality or resolution in a cadence with roots on
> the diminished seventh chord. I've spent half an hour this
> morning trying to get a really satisfying cadence. Maybe
> I'm just a "perfect fifth" junkie after all!

OK. In jazz, the octatonic scale is often played as a single chord --
one diminshed seventh chord a major seventh higher than another --
but resolves to a tonic not in that scale.

> > > So it requires only one more pitch to create the scale
> > > supporting these chords in JI than in 12-EDO!
> >
> > I think that's two more pitches. Meanwhile, if you're only
looking at
> > the major triads, the diatonic scale requires zero more pitches.
>
> Sure, but the advantage of your tuning of the octatonic
> was that it supplied alternates for those notes necessary
> to make the given triads totally in (JI) tune. Which is the
> only reason it attracted my attention in the first place.
> People who like rough harmonies are welcome to them; I
> prefer something smoother when possible.

Did you find the TOP Dimipent and stretched TOP Dimipent harmonies
rough? And if you want pure harmonies, you can use adaptive JI, you
don't need strict JI.

> > > That's
> > > surprisingly little extra effort to provide, if we want to
> > > hear pure JI harmonies, and has most impact on fixed
> > > tuning instruments such as pianos, pipe organs and
> > > metallophones.
> >
> > But is it really still the same scale to your ears? Or does
648:625
> > produce a note that sounds like a distinct *musical* pitch? To
me,
> > even 81:80, which is several times smaller than 648:625, doesn't
> > sound right as a *melodic* unison.
>
> Of course not! It's 20.5 cents,

plus 1

> after all, which
> is way too big a mistuning to ignore, even in speed.
> Seems to me that there are two ways to use the
> resulting set of pitches - (1) as an adaptive JI
> with notes that sometimes must move by a whole
> (648/125)

648/625

> comma in order to maintain harmony;

I guess my question is: don't these sound like different pitches
entirely when they are so moved?

> > > But were I moved to create tunes in
> > > octatonic (*) and harmonise them with major chords, it's
> > > great to know that a little "adaptive JI" could handle this
> > > effortlessly on most instruments, including human voices.
> >
> > With a more "hard-core" implementation of adaptive JI, you could
cut
> > the 648:625 distinction between different occurences of the same
note
> > down to a considerably smaller interval.
>
> What particular kinds of implementation did you
> have in mind?

For example, you can pin the roots to 4-equal, and harmonize each of
those purely within itself. This distributes the 648:625 'comma' into
four 15.6 cent 'shifts' which don't sound nearly as disturbing as a
full 648:625 (62.6) cent shift.

> > > > and the just minor triads
> > > >
> > > > 1/1-6/5-3/2
> > > > 6/5-36/25-9/5
> > > > 25/18-5/3-25/24
> > > > 5/3-1/1-5/4
> > >
> > > [YA] That is, C, Eb, F# and A minor:
> > > C-Eb-G
> > > Eb-Gb-Bb
> > > F#-A-C#
> > > A-C-E
> > >
> > > To play the four major triads in C octatonic, we need
> > > both C# and Db; similarly, to play the four minor triads
> > > in C octatonic, we need both F# and Gb. So far we need
> > > ten tones per octave to realise an octatonic scale in JI.
> >
> > It was ten even for just the major triads.
>
> So now it's eleven.

No, it should still be ten. Those same ten notes that gave you the
four pure major triads also give you the four pure minor triads.

> > The 'TOP Dimisept' 8-note scale might satisfy a taste for better
> > tetrads than the 12-equal version of this scale (the octatonic or
> > diminished scale) can offer, without requiring any alternate
>pitches.
>
> If it needs no alternates, then it automatically solves
> the problem of the large amount of "adapatation"
> needed by the JI octatonic-with-alternates. My
> only remaining concern would be the quality of the
> harmonies.

What do your ears say about TOP Dimisept?

> > > and whether they can be notated and played on the
> > > same keyboard.
> >
> > Have you pursued this approach for the plain old diatonic case?
> > There, it works much better, in that the alternates are much
closer
> > together, but the melodic unisons between alternates still don't
work
> > for my ears.
>
> So, to your ears, JI diatonic requires too much
> adaptation for the alternate notes to be considered
> essentially the same?

The strict JI approach does. The adaptive JI approach tyically
distributes the syntonic comma into 4 unobjectionable shifts of 5.4
cents each.

> Are there any tunings in which
> adaptive JI "works" for you?

Adaptive JI works great for me for diatonic music, while in many
cases, strict JI doesn't. I think you meant to ask about strict JI
rather than adaptive JI?

> > Also the decatonic case (both this and the diatonic case are
> > discussed in my paper below) may be of interest to you:
> >
> > http://lumma.org/tuning/erlich/erlich-tFoT.pdf
>
> Of course it will. I remember being quite interested
> in this paper the first time I read it. Perhaps, now
> having come head to head with the pain of adaptive
> JI, I will get some new insights from rereading it.

I think you may still mean "strict JI" when you say "adaptive JI".

> Since I've now spent some hours trying to make
> listenable, playable music in the adaptive JI octatonic,

You mean strict JI again, I think.

> and have been unhappy with the melodic progressions
> in particular, it's good to have some other possibilities
> that offer better hope of both good harmony and good
> melody together.
>
> Paul, thanks for all the trouble you've taken to comment
> on my fumbling (micro) steps in Tuning Land.

Looking forward to much more!

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

Hi Paul,

On Wed, 14 Dec 2005, "wallyesterpaulrus" wrote:
>
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
> <yahya@m...> wrote:
> >
> > Hi again,
> >
> > Further correction: playing with the keyboard
> > in this tuning, I found that F# major is unplayable.
> > Why? Because I forgot that A# is not Bb. It
> > should be a major third of 386 c above the F#
> > tuned to 603 c (when C is standard +34c), that is,
> > 989 c, which we can tune as (standard) Bb -11c.
> > This gives the necessary 11 notes as -
>
> Something's wrong, Yahya. You only need 10 notes, as my original
> message indicated.
>
> > C ............. C +34c
> > C# .......... C# +5c
> > Db .......... D -33c
> > Eb ........... Eb +50c
> > E ............. E +20c
> > F# .......... F# +3c
> > Gb ........... G -35c
>
> Are you sure you don't mean G + 36 cents?

No, the gaps C# to Db and F# to Gb should be
identical, at (648/625) comma, around 63 c.
That is, if the notation we choose is to have a
consistent meaning for a 'b' as being exactly
a negative '#', as I think it ought.

I don't think I've miscalculated - this time!

You haven't forgotten that I want a Gb as the
minor third of the Eb Minor triad?

> > G .............. G# -64c
>
> Because then that would be identical with this note, and you'd have
> 10 notes, not 11 . . .

Indeed it would, and I would have no Eb minor
playable.

> > A ............. A +18c
> > A# ......... Bb -11c
> > Bb ........... B -48c
> >
> > Which indeed makes four major AND four
> > minor triads playable. Tho using G# as G,
> > and B as Bb, is decidedly difficult at first!

I'm getting used to it ... I do have one favourite
progression that sounds very nice and smooth,
with minimal stepwise motion in three parts.
It is this (top three lines in close position):

Bb .| C ..| C# .| C#
G ....| G .| A .....| A#
Eb .| E ..| E .....| F#
Eb .| C ..| A .....| F#

However, it is _played_ as if it were this:

C .....| C ....| C# .| C#
Ab .| Ab .| A .....| A#
Eb ..| E ....| E .....| F#
Eb ..| C ....| A .....| F#

The nicest 4 minor-chord progression I've
found so far is this:

F# .| E .| Eb .| Eb .| E
C# .| C .| C ....| Bb .| C
A .... |A .| G ....| Gb .| G
F# .| A .| C ....| Eb .| C

_played_ as this:

F# .| E .| Eb .| Eb .| E
C# .| C .| C ....| B ....| C
A .... |A .| Ab..| G ....| Ab
F# .| A .| C ....| Eb .| C

They _sound_ fairly beat-free to me, using
a typical Roland concert grand piano patch.

It's not quite as nice as is could be, since,
as you pointed out, the E' of the A minor
chord should be a (648/625) comma higher.

So to fix that, I've had to shift the entire
tuning up by another 19 cents, thus:

No. ... Note ..... Play .. Adjust by
1 ......... C ............. C ....... +53c
2-k ... C# .......... C# . +24c
2 ......... Db .......... D ..... -14c
3 ......... Eb ........... E ..... -31c
4 ......... E .............. F ..... -61c
5 ......... Gb ........... G ..... -16c
6 ......... G .............. G# . -45c
7 ......... A ............. A ...... +37c
7+k ... Bbb ......... Bb ... +8c
8 ......... Bb ........... B ...... -31c

which works quite well. In doing this, I've
taken the opportunity to revise the notation
from scratch. (You'll notice that I'm now
using Gb in place of F#, and so on. And you
will be pleased to notice that I now only find
ten notes necessary, as you wrote.) I've used
a spreadsheet to simplify calculations as well,
available if you should want to see it. I will
detail my thinking on this in a separate, fresh
message, free of any baggage from this
conversation! 8-0

----------

But one last comment on the issue of (what I
thought were) the "irresolute" harmonies of
octatonic - what I was missing was the familiar
movement of the root by a fourth, and especially
of course, the leading tone resolution provided
by dominant seventh chords leading to the tonic
and (in abundance) by using secondary sevenths
to feign closure.

As you would know, and as I have disovered,
the (diminished seventh) tetrad of 12-EDO here
becomes a pure tertian chord that functions
perfectly well in the role of a dominant seventh;
for example Gb-A-C-Eb resolves stepwise to
E-A-C#-E, or to E-G-C-E and so on.

Regards,
Yahya

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🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

Hi Paul,

On Thu, 15 Dec 2005, you wrote:
>
> --- In tuning@yahoogroups.com, "Yahya Abdal-Aziz"
> <yahya@m...> wrote:
>
> > > (216/125)/(5/3) = 648/625.
> >
> > Oops! OK, you got it. So, we can now say:
> > "A' is an octave and a (648/125) comma above the A we
> > started on."
>
> 648/625.

OK! :-)

> > > ... You should also try tuning it to the
> > > 8-note ring in the 'TOP Dimipent' horagram in the paper I mailed
> > > you, which doesn't require any alternate notes, but aims at
> > > improving the harmonies relative to 12-equal.
> >
> > At what cost to the octave? Not too much, I hope,
> > or it wouldn't be "just right" ... Yes, I have it now - the
> > tempered octave of TOP Dimipent is just 3.36 cents flat.
> > Audible, but not maddening in fast passages.
>
> If you prefer, stretch the entire tuning so that octaves are pure.

That's yet _another_ temperament to find
the time to listen to!

> > How good, in TOP Dimipent, are the harmonies of the
> > same four major and four minor chords on roots C, Eb,
> > F# and A? How do they compare with the "harmonic
> > octatonic", or "octatonic with alternates", that you
> > proposed?
>
> Well, that's exactly what I was hoping you'd decide through
> listening. How would you like me to answer this question (what sense
> of "good" did you have in mind)?

Purity of concord.

> > > > Altho the F and G are symmetric about C, the three
> > > > roots do not divide the octave at all equally. This very
> > > > asymmetry lends more finality to common practice chord
> > > > progressions favouring major chords in which the last
> > > > movement of the root is upward [or downward] by a
> > > > fourth. The same kinds or quality of chord progression
> > > > will not be available using only major chords in the
> > > > octatonic scale, since the last movement of the root must
> > > > necessarily be upward [or downward] by a minor third or
> > > > tritone. So we can confidently predict a more restless,
> > > > less resolute, kind of harmonisation occurring in this
> > > > context.
> > >
> > > Sometimes.
> >
> > Turning it around ... I know I can't achieve quite the same
> > sense of finality or resolution in a cadence with roots on
> > the diminished seventh chord. I've spent half an hour this
> > morning trying to get a really satisfying cadence. Maybe
> > I'm just a "perfect fifth" junkie after all!
>
> OK. In jazz, the octatonic scale is often played as a single chord --
> one diminshed seventh chord a major seventh higher than another --
> but resolves to a tonic not in that scale.
>
> > > > So it requires only one more pitch to create the scale
> > > > supporting these chords in JI than in 12-EDO!
> > >
> > > I think that's two more pitches. Meanwhile, if you're only
> > > looking at the major triads, the diatonic scale requires zero
> > > more pitches.
> >
> > Sure, but the advantage of your tuning of the octatonic
> > was that it supplied alternates for those notes necessary
> > to make the given triads totally in (JI) tune. Which is the
> > only reason it attracted my attention in the first place.
> > People who like rough harmonies are welcome to them; I
> > prefer something smoother when possible.
>
> Did you find the TOP Dimipent and stretched TOP Dimipent harmonies
> rough? And if you want pure harmonies, you can use adaptive JI, you
> don't need strict JI.

Although I've spent much of my "spare" time this week
on tuning the octatonic acceptably, I still haven't had
time to try these. Give me a little time.

> > > > That's
> > > > surprisingly little extra effort to provide, if we want to
> > > > hear pure JI harmonies, and has most impact on fixed
> > > > tuning instruments such as pianos, pipe organs and
> > > > metallophones.
> > >
> > > But is it really still the same scale to your ears? Or does
> > > 648:625 produce a note that sounds like a distinct *musical*
> > > pitch? To me, even 81:80, which is several times smaller than
> > > 648:625, doesn't sound right as a *melodic* unison.
> >
> > Of course not! It's 20.5 cents,
>
> plus 1

21.506 cents, OK.

> > after all, which
> > is way too big a mistuning to ignore, even in speed.
> > Seems to me that there are two ways to use the
> > resulting set of pitches - (1) as an adaptive JI
> > with notes that sometimes must move by a whole
> > (648/125)
>
> 648/625

How tired was I? 8-0

> > comma in order to maintain harmony;
>
> I guess my question is: don't these sound like different pitches
> entirely when they are so moved?

Yes.

> > > > But were I moved to create tunes in
> > > > octatonic (*) and harmonise them with major chords, it's
> > > > great to know that a little "adaptive JI" could handle this
> > > > effortlessly on most instruments, including human voices.
> > >
> > > With a more "hard-core" implementation of adaptive JI, you could
> > > cut the 648:625 distinction between different occurences of the
> > > same note down to a considerably smaller interval.
> >
> > What particular kinds of implementation did you
> > have in mind?
>
> For example, you can pin the roots to 4-equal, and harmonize each of
> those purely within itself. This distributes the 648:625 'comma' into
> four 15.6 cent 'shifts' which don't sound nearly as disturbing as a
> full 648:625 (62.6) cent shift.

So, you would erect perfect fifths and major thirds
over each of the 4-EDO roots? Worth a try ...

> > > > > and the just minor triads
> > > > >
> > > > > 1/1-6/5-3/2
> > > > > 6/5-36/25-9/5
> > > > > 25/18-5/3-25/24
> > > > > 5/3-1/1-5/4
> > > >
> > > > [YA] That is, C, Eb, F# and A minor:
> > > > C-Eb-G
> > > > Eb-Gb-Bb
> > > > F#-A-C#
> > > > A-C-E
> > > >
> > > > To play the four major triads in C octatonic, we need
> > > > both C# and Db; similarly, to play the four minor triads
> > > > in C octatonic, we need both F# and Gb. So far we need
> > > > ten tones per octave to realise an octatonic scale in JI.
> > >
> > > It was ten even for just the major triads.
> >
> > So now it's eleven.
>
> No, it should still be ten. Those same ten notes that gave you the
> four pure major triads also give you the four pure minor triads.

I'm with you.

> > > The 'TOP Dimisept' 8-note scale might satisfy a taste for better
> > > tetrads than the 12-equal version of this scale (the octatonic or
> > > diminished scale) can offer, without requiring any alternate
> > >pitches.
> >
> > If it needs no alternates, then it automatically solves
> > the problem of the large amount of "adapatation"
> > needed by the JI octatonic-with-alternates. My
> > only remaining concern would be the quality of the
> > harmonies.
>
> What do your ears say about TOP Dimisept?

Still to find out. Doesn't look too bad on paper ..

> > > > and whether they can be notated and played on the
> > > > same keyboard.
> > >
> > > Have you pursued this approach for the plain old diatonic case?
> > > There, it works much better, in that the alternates are much
> > > closer together, but the melodic unisons between alternates
> > > still don't work for my ears.
> >
> > So, to your ears, JI diatonic requires too much
> > adaptation for the alternate notes to be considered
> > essentially the same?
>
> The strict JI approach does. The adaptive JI approach tyically
> distributes the syntonic comma into 4 unobjectionable shifts of 5.4
> cents each.
>
> > Are there any tunings in which
> > adaptive JI "works" for you?
>
> Adaptive JI works great for me for diatonic music, while in many
> cases, strict JI doesn't. I think you meant to ask about strict JI
> rather than adaptive JI?

I was asking about the "strict JI with alternates"
approach you outlined.

> > > Also the decatonic case (both this and the diatonic case are
> > > discussed in my paper below) may be of interest to you:
> > >
> > > http://lumma.org/tuning/erlich/erlich-tFoT.pdf
> >
> > Of course it will. I remember being quite interested
> > in this paper the first time I read it. Perhaps, now
> > having come head to head with the pain of adaptive
> > JI, I will get some new insights from rereading it.
>
> I think you may still mean "strict JI" when you say "adaptive JI".
>
> > Since I've now spent some hours trying to make
> > listenable, playable music in the adaptive JI octatonic,
>
> You mean strict JI again, I think.
>
> > and have been unhappy with the melodic progressions
> > in particular, it's good to have some other possibilities
> > that offer better hope of both good harmony and good
> > melody together.
> >
> > Paul, thanks for all the trouble you've taken to comment
> > on my fumbling (micro) steps in Tuning Land.
>
> Looking forward to much more!

It's fun going off on adventures, but OH!
so tiring ....

Regards,
Yahya

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🔗Petr Parízek <p.parizek@chello.cz>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

Hi Yahya!

--- In tuning@yahoogroups.com, "Yahya Abdal-Aziz" <yahya@m...> wrote:

> > > > ... You should also try tuning it to the
> > > > 8-note ring in the 'TOP Dimipent' horagram in the paper I
mailed
> > > > you, which doesn't require any alternate notes, but aims at
> > > > improving the harmonies relative to 12-equal.
> > >
> > > At what cost to the octave? Not too much, I hope,
> > > or it wouldn't be "just right" ... Yes, I have it now - the
> > > tempered octave of TOP Dimipent is just 3.36 cents flat.
> > > Audible, but not maddening in fast passages.
> >
> > If you prefer, stretch the entire tuning so that octaves are pure.
>
> That's yet _another_ temperament to find
> the time to listen to!
>
>
> > > How good, in TOP Dimipent, are the harmonies of the
> > > same four major and four minor chords on roots C, Eb,
> > > F# and A? How do they compare with the "harmonic
> > > octatonic", or "octatonic with alternates", that you
> > > proposed?
> >
> > Well, that's exactly what I was hoping you'd decide through
> > listening. How would you like me to answer this question (what
sense
> > of "good" did you have in mind)?
>
> Purity of concord.

Well, in that sense, I would assess the temperament as being perhaps
a little better than in 12-equal, but still most of the way away from
JI (as in the 10-pitch "octatonic with alternates" I proposed). If we
had been discussing the diatonic scale or Messiaen's 10-note scale,
though, the situation would be quite different, and a temperament
much purer than 12-equal could be applied.

> > > comma in order to maintain harmony;
> >
> > I guess my question is: don't these sound like different pitches
> > entirely when they are so moved?
>
> Yes.

So music that is written in this scale, when retuned in this way,
won't really sound like it's in any such scale any more, will it?

> > > > > But were I moved to create tunes in
> > > > > octatonic (*) and harmonise them with major chords, it's
> > > > > great to know that a little "adaptive JI" could handle this
> > > > > effortlessly on most instruments, including human voices.
> > > >
> > > > With a more "hard-core" implementation of adaptive JI, you
could
> > > > cut the 648:625 distinction between different occurences of
the
> > > > same note down to a considerably smaller interval.
> > >
> > > What particular kinds of implementation did you
> > > have in mind?
> >
> > For example, you can pin the roots to 4-equal, and harmonize each
of
> > those purely within itself. This distributes the 648:625 'comma'
into
> > four 15.6 cent 'shifts' which don't sound nearly as disturbing as
a
> > full 648:625 (62.6) cent shift.
>
> So, you would erect perfect fifths and major thirds
> over each of the 4-EDO roots? Worth a try ...

That's a start (let me know if you want to get into more detail
though) . . . Ideally, a computer would adjust the tuning of the note
you play according to which chord you're holding down, so that you
wouldn't need separate keys for all the alternate pitches . . .

> > > > > and whether they can be notated and played on the
> > > > > same keyboard.
> > > >
> > > > Have you pursued this approach for the plain old diatonic
case?
> > > > There, it works much better, in that the alternates are much
> > > > closer together, but the melodic unisons between alternates
> > > > still don't work for my ears.
> > >
> > > So, to your ears, JI diatonic requires too much
> > > adaptation for the alternate notes to be considered
> > > essentially the same?
> >
> > The strict JI approach does. The adaptive JI approach tyically
> > distributes the syntonic comma into 4 unobjectionable shifts of
5.4
> > cents each.
> >
> > > Are there any tunings in which
> > > adaptive JI "works" for you?
> >
> > Adaptive JI works great for me for diatonic music, while in many
> > cases, strict JI doesn't. I think you meant to ask about strict
JI
> > rather than adaptive JI?
>
> I was asking about the "strict JI with alternates"
> approach you outlined.

As I suspected. So . . .

"Strict JI with alternates" works for me for music in Miracle tunings
and scales. I've proposed several "Blackjust" scales inspired by this
fact. Here's a centered one:

/tuning/files/perlich/scales/blackjust4.g
if

Each little tetrahedron in this lattice represents a 7-limit
consonant tetrad. In Miracle temperament (specifically, the 21-note
Blackjack scale), all of them are tuned very well, within 3 cents of
JI. If you use the 21-note JI scale, only the colored intervals on
the lattice are pure; the gray ones are "out-of-tune" by either
2401:2400 (0.7 cents), 225:224 (7.0 cents), or 16875:16807 (7.7
cents), as shown. But by adding new JI pitches to the original set of
21, we may provide alternates to serve as "corrections" for all
these "errors", allowing us to keep all the tetrads pure. And the key
point is that it'll still sound like the same basic melodic
structure, as shifts of less than 8 cents don't tend to disturb that
(say my ears).

> It's fun going off on adventures, but OH!
> so tiring ....

Hope I'm not contributing to tiring you! :) Keep it fun!!

A JI realisation of Messiaen's octatonic (was: Schoenberg, serliasim, 12-edo)

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

🔗Petr Parízek <p.parizek@chello.cz>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Petr Parízek <p.parizek@chello.cz>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Petr Parízek <p.parizek@chello.cz>

🔗Petr Parízek <p.parizek@chello.cz>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>