Horagrams for TOP temperaments

Inspired by Erv Wilson and David Finnamore, I wrote a program that
draws horagrams for any generator, period, and 'multiplicity' integer
(periods per equivalence interval) -- these are probably the first
horagrams where this may be greater than 1. Also, instead of
generating always in one direction from the "tonic", I alternately
apply the generator to each of the two ends of the chain. Each ring
shows a DE scale (same thing as an MOS scale if the multiplicity is
1) of the temperament, which is therefore either symmetrical around
the tonic or about a point midway between the tonic and the first note
(s) generated beyond it.

Hopefully the several scales implied in each of the diagrams will be
self-explanatory even if you've never seen or heard of any of the
stuff mentioned above.

And Dave, I know what you're going to say, if you happen to read this
post and look at the latter few examples.

/tuning/files/Erlich/schismic.gif
/tuning/files/Erlich/kleismic.gif
/tuning/files/Erlich/diaschismic.gi
f
/tuning/files/Erlich/magic.gif
/tuning/files/Erlich/meantone.gif
/tuning/files/Erlich/augmented.gif
/tuning/files/Erlich/porcupine.gif
/tuning/files/Erlich/diminished.gif
/tuning/files/Erlich/blackwood.gif
/tuning/files/Erlich/pelogic.gif
/tuning/files/Erlich/dicot.gif
/tuning/files/Erlich/father.gif
/tuning/files/Erlich/beep.gif

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:

/tuning/topicId_51352.html#51352

> Inspired by Erv Wilson and David Finnamore, I wrote a program that
> draws horagrams for any generator, period, and 'multiplicity'
integer
> (periods per equivalence interval) -- these are probably the first
> horagrams where this may be greater than 1. Also, instead of
> generating always in one direction from the "tonic", I alternately
> apply the generator to each of the two ends of the chain. Each ring
> shows a DE scale (same thing as an MOS scale if the multiplicity is
> 1) of the temperament, which is therefore either symmetrical around
> the tonic or about a point midway between the tonic and the first
note
> (s) generated beyond it.
>
> Hopefully the several scales implied in each of the diagrams will
be
> self-explanatory even if you've never seen or heard of any of the
> stuff mentioned above.
>
> And Dave, I know what you're going to say, if you happen to read
this
> post and look at the latter few examples.
>
> /tuning/files/Erlich/schismic.gif
> /tuning/files/Erlich/kleismic.gif
>
/tuning/files/Erlich/diaschismic.gi
> f
> /tuning/files/Erlich/magic.gif
> /tuning/files/Erlich/meantone.gif
>
/tuning/files/Erlich/augmented.gif
>
/tuning/files/Erlich/porcupine.gif
>
/tuning/files/Erlich/diminished.gif
>
/tuning/files/Erlich/blackwood.gif
> /tuning/files/Erlich/pelogic.gif
> /tuning/files/Erlich/dicot.gif
> /tuning/files/Erlich/father.gif
> /tuning/files/Erlich/beep.gif

***Wow, these are impressive... just like the ones in
Xenharmonikon... I believe I understood how they worked at one time,
but now I've completely forgotten... :(

JP

>> /tuning/files/Erlich/schismic.gif
>> /tuning/files/Erlich/kleismic.gif
>> /tuning/files/Erlich/diaschismic.gif
>> /tuning/files/Erlich/magic.gif
>> /tuning/files/Erlich/meantone.gif
>>
>> /tuning/files/Erlich/augmented.gif
>> /tuning/files/Erlich/porcupine.gif
>> /tuning/files/Erlich/diminished.gif
>>
>> /tuning/files/Erlich/blackwood.gif
>> /tuning/files/Erlich/pelogic.gif
>> /tuning/files/Erlich/dicot.gif
>> /tuning/files/Erlich/father.gif
>> /tuning/files/Erlich/beep.gif
>
>***Wow, these are impressive... just like the ones in
>Xenharmonikon... I believe I understood how they worked at one time,
>but now I've completely forgotten... :(

These are a *fantastic* resource. They're super-simple. Each horogram
represents a temperament, such as "beep", "father", etc.

Each ring in a horogram represents an DE (Distributionally Even; what
we used to call MOS) scale in the temperament. The red numbers, one
for each ring, tell the number of tones in that scale.

The blue lines represent notes. Where a line occurs in more than one
ring, it means those two scales share notes. The cents values for the
notes are written in black text just outside the ring. In this case
the values given are the TOP-tempered values.

So, the significance of this is HUGE. For the first time EVER, *anyone*
can pick a temperament, pick a number of tones, and get a theoretically-
approved, grade-A1 scale in the mathematically-optimal tuning. All
without TOUCHING a calculator.

Further, by following the blue lines, one can see at a glance how the
DE scales are related, just like the pentatonic scale is a subset of
the diatonic scale in 12-equal (find the 5- and 7-tone rings in the
meantone horogram and you should recognize the pattern).

Now if Paul could improve the readability of these a bit... by making
cents font a bit smaller so they don't collide, and maybe flipping the
cents print between 7 and 11 o'clock, and maybe moving the temperament
title up and out of the way a bit, and making it a bit bigger and in
bold... and maybe shading every-other ring or every 3rd ring to aid the
eye in going around...

Paul, I've noticed you're not capitalizing TOP. "Top" looks a little
weird to me. Do you have a position on this?

-Carl

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

/tuning/topicId_51352.html#51393

> >>
/tuning/files/Erlich/schismic.gif
> >>
/tuning/files/Erlich/kleismic.gif
> >>
/tuning/files/Erlich/diaschismic.gi
f
> >> /tuning/files/Erlich/magic.gif
> >>
/tuning/files/Erlich/meantone.gif
> >>
> >>
/tuning/files/Erlich/augmented.gif
> >>
/tuning/files/Erlich/porcupine.gif
> >>
/tuning/files/Erlich/diminished.gif
> >>
> >>
/tuning/files/Erlich/blackwood.gif
> >>
/tuning/files/Erlich/pelogic.gif
> >> /tuning/files/Erlich/dicot.gif
> >>
/tuning/files/Erlich/father.gif
> >> /tuning/files/Erlich/beep.gif
> >
> >***Wow, these are impressive... just like the ones in
> >Xenharmonikon... I believe I understood how they worked at one
time,
> >but now I've completely forgotten... :(
>
> These are a *fantastic* resource. They're super-simple. Each
horogram
> represents a temperament, such as "beep", "father", etc.
>
> Each ring in a horogram represents an DE (Distributionally Even;
what
> we used to call MOS) scale in the temperament. The red numbers, one
> for each ring, tell the number of tones in that scale.
>
> The blue lines represent notes. Where a line occurs in more than
one
> ring, it means those two scales share notes. The cents values for
the
> notes are written in black text just outside the ring. In this case
> the values given are the TOP-tempered values.
>
> So, the significance of this is HUGE. For the first time EVER,
*anyone*
> can pick a temperament, pick a number of tones, and get a
theoretically-
> approved, grade-A1 scale in the mathematically-optimal tuning. All
> without TOUCHING a calculator.
>
> Further, by following the blue lines, one can see at a glance how
the
> DE scales are related, just like the pentatonic scale is a subset of
> the diatonic scale in 12-equal (find the 5- and 7-tone rings in the
> meantone horogram and you should recognize the pattern).
>
> Now if Paul could improve the readability of these a bit... by
making
> cents font a bit smaller so they don't collide, and maybe flipping
the
> cents print between 7 and 11 o'clock, and maybe moving the
temperament
> title up and out of the way a bit, and making it a bit bigger and in
> bold... and maybe shading every-other ring or every 3rd ring to aid
the
> eye in going around...
>
> Paul, I've noticed you're not capitalizing TOP. "Top" looks a
little
> weird to me. Do you have a position on this?
>
> -Carl

***Thanks so much, Carl! Yes, I understand how to read these now...
they're not difficult at all.

I see what you mean about these being a terrific resource! Bravo
Paul!

JP

These are excellent Paul!

We can see from this that Magic is just as much of a bastard as
Kleismic as far as finding a reasonably even MOS with 5 to 10 notes
for the natural nominals. I wonder if there is a 6-note or 9-note
proper non-MOS in Magic like the 8-note one Carl found in Kleismic?

>We can see from this that Magic is just as much of a bastard as
>Kleismic as far as finding a reasonably even MOS with 5 to 10 notes
>for the natural nominals. I wonder if there is a 6-note or 9-note
>proper non-MOS in Magic like the 8-note one Carl found in Kleismic?

magic[6] is proper and non-MOS but it doesn't seem to have any
generalized-diatonic harmonies.

-Carl

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
> /tuning/topicId_51352.html#51352
>
> > Inspired by Erv Wilson and David Finnamore, I wrote a program
that
> > draws horagrams for any generator, period, and 'multiplicity'
> integer
> > (periods per equivalence interval) -- these are probably the
first
> > horagrams where this may be greater than 1. Also, instead of
> > generating always in one direction from the "tonic", I
alternately
> > apply the generator to each of the two ends of the chain. Each
ring
> > shows a DE scale (same thing as an MOS scale if the multiplicity
is
> > 1) of the temperament, which is therefore either symmetrical
around
> > the tonic or about a point midway between the tonic and the first
> note
> > (s) generated beyond it.
> >
> > Hopefully the several scales implied in each of the diagrams will
> be
> > self-explanatory even if you've never seen or heard of any of the
> > stuff mentioned above.
> >
> > And Dave, I know what you're going to say, if you happen to read
> this
> > post and look at the latter few examples.
> >
> >
/tuning/files/Erlich/schismic.gif
> >
/tuning/files/Erlich/kleismic.gif
> >
>
/tuning/files/Erlich/diaschismic.gi
> > f
> > /tuning/files/Erlich/magic.gif
> >
/tuning/files/Erlich/meantone.gif
> >
>
/tuning/files/Erlich/augmented.gif
> >
>
/tuning/files/Erlich/porcupine.gif
> >
>
/tuning/files/Erlich/diminished.gif
> >
>
/tuning/files/Erlich/blackwood.gif
> >
/tuning/files/Erlich/pelogic.gif
> > /tuning/files/Erlich/dicot.gif
> > /tuning/files/Erlich/father.gif
> > /tuning/files/Erlich/beep.gif
>
>
> ***Wow, these are impressive... just like the ones in
> Xenharmonikon... I believe I understood how they worked at one
time,
> but now I've completely forgotten... :(
>
> JP

The center ring shows an exceedingly simple scale, just 1 note or
several equally-spaced notes per 'octave' (however that happens to be
tuned). Each ring corresponds to the next higher 'order' of scale
obtained by repeatedly applying the generator.

Why don't we start by looking at the *meantone* example, which should
be fairly clear and familiar. In particular, you should recognize the
5-, 7-, and 12-tone rings as *very* familiar scales . . . no?

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:

/tuning/topicId_51352.html#51577

> >
> > ***Wow, these are impressive... just like the ones in
> > Xenharmonikon... I believe I understood how they worked at one
> time,
> > but now I've completely forgotten... :(
> >
> > JP
>
> The center ring shows an exceedingly simple scale, just 1 note or
> several equally-spaced notes per 'octave' (however that happens to
be
> tuned). Each ring corresponds to the next higher 'order' of scale
> obtained by repeatedly applying the generator.
>
> Why don't we start by looking at the *meantone* example, which
should
> be fairly clear and familiar. In particular, you should recognize
the
> 5-, 7-, and 12-tone rings as *very* familiar scales . . . no?

***Thanks, Paul! Actually, Carl Lumma already "illuminated" the
clueless... They're easy to read... (very cool, too...)

JP

on 1/11/04 3:01 PM, wallyesterpaulrus <paul@stretch-music.com> wrote:

> --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:
>> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
> wrote:
>>
>> /tuning/topicId_51352.html#51352
>>
>>> Inspired by Erv Wilson and David Finnamore, I wrote a program
> that
>>> draws horagrams for any generator, period, and 'multiplicity'
>> integer
>>> (periods per equivalence interval) -- these are probably the
> first
>>> horagrams where this may be greater than 1. Also, instead of
>>> generating always in one direction from the "tonic", I
> alternately
>>> apply the generator to each of the two ends of the chain. Each
> ring
>>> shows a DE scale (same thing as an MOS scale if the multiplicity
> is
>>> 1) of the temperament, which is therefore either symmetrical
> around
>>> the tonic or about a point midway between the tonic and the first
>> note
>>> (s) generated beyond it.
>>>
>>> Hopefully the several scales implied in each of the diagrams will
>> be
>>> self-explanatory even if you've never seen or heard of any of the
>>> stuff mentioned above.
>>>
>>> And Dave, I know what you're going to say, if you happen to read
>> this
>>> post and look at the latter few examples.
>>>
>>>
> /tuning/files/Erlich/schismic.gif
>>>
> /tuning/files/Erlich/kleismic.gif
>>>
>>
> /tuning/files/Erlich/diaschismic.gi
>>> f
>>> /tuning/files/Erlich/magic.gif
>>>
> /tuning/files/Erlich/meantone.gif
>>>
>>
> /tuning/files/Erlich/augmented.gif
>>>
>>
> /tuning/files/Erlich/porcupine.gif
>>>
>>
> /tuning/files/Erlich/diminished.gif
>>>
>>
> /tuning/files/Erlich/blackwood.gif
>>>
> /tuning/files/Erlich/pelogic.gif
>>> /tuning/files/Erlich/dicot.gif
>>> /tuning/files/Erlich/father.gif
>>> /tuning/files/Erlich/beep.gif
>>
>>
>> ***Wow, these are impressive... just like the ones in
>> Xenharmonikon... I believe I understood how they worked at one
> time,
>> but now I've completely forgotten... :(
>>
>> JP
>
> The center ring shows an exceedingly simple scale, just 1 note or
> several equally-spaced notes per 'octave' (however that happens to be
> tuned). Each ring corresponds to the next higher 'order' of scale
> obtained by repeatedly applying the generator.
>
> Why don't we start by looking at the *meantone* example, which should
> be fairly clear and familiar. In particular, you should recognize the
> 5-, 7-, and 12-tone rings as *very* familiar scales . . . no?

The intervals are exactly 1/4-comma meantone. But how the intervals are
sorted in the octave doesn't look like what I initially would have expected.
But my expectation was based on "canonical" organ use which from what I can
tell goes in the flat direction only so far as Eb, so that the first tone in
the 12-tone octave would be C# at 76 cents. Monz's meantone page also
indicates that the "typical" 12-tone meantone goes from Eb to G#.

Since these tops structures are being described as a cononical form, this
just brings up the question whether to be useful the canonical form should
take some key center specification as an argument in order to be most useful
as a reference for actual use. Alternatively if there were a "canonical"
way of relating a key center to a note range (e.g. Eb to G#) then the whole
picture could be rotated as necessary. Mind you this whole assumption of
relation of key range relationship to a key center is entirely tentative,
and I don't even know for sure what it normally means to tune meantone for
"C". Put another way I don't know what key center is associated with the Eb
to G# range that I am familiar with, although I had always assumed this was
for a key center of "C".

-Kurt

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> > Why don't we start by looking at the *meantone* example, which
should
> > be fairly clear and familiar. In particular, you should recognize
the
> > 5-, 7-, and 12-tone rings as *very* familiar scales . . . no?
>
> The intervals are exactly 1/4-comma meantone.

Nope -- everything's wider than in 1/4-comma meantone by a fixed
percentage, as discussed on tuning-math . . . note that the octave,
major third, and minor sixth are all just in 1/4-comma meantone but
wider than just here . . .

> But how the intervals are
> sorted in the octave doesn't look like what I initially would have
>expected.

As I mentioned, I extended the chain of generators alterately in
the "up" and "down" directions. More typical is to extend always in
one direction, but then there would be even less correspondence with
the rotations you're used to.

> But my expectation was based on "canonical" organ use which from
what I can
> tell goes in the flat direction only so far as Eb, so that the
first tone in
> the 12-tone octave would be C# at 76 cents. Monz's meantone page
also
> indicates that the "typical" 12-tone meantone goes from Eb to G#.

My horagram should agree with that if you identify 0 cents with the
note A.

> Since these tops structures are being described as a cononical
form, this
> just brings up the question whether to be useful the canonical form
should
> take some key center specification as an argument in order to be
most useful
> as a reference for actual use.

These are only meant to be canonical *without* a choice of key
center, and without even the assumption that the key center is
unchanged from ring to ring.

> Alternatively if there were a "canonical"
> way of relating a key center to a note range (e.g. Eb to G#) then
the whole
> picture could be rotated as necessary. Mind you this whole
assumption of
> relation of key range relationship to a key center is entirely
tentative,
> and I don't even know for sure what it normally means to tune
meantone for
> "C". Put another way I don't know what key center is associated
with the Eb
> to G# range that I am familiar with, although I had always assumed
this was
> for a key center of "C".

Basically, this 12-note meantone range that you're familiar with
allows for the following key signatures:

2b = Bb major, G minor
1b = F major, D minor
0b/# = C major, A minor
1# = G major, E minor
2# = D major, B minor
3# = A major, F# minor

Since modulations to the sharp keys were slightly more common than
modulations to the flat keys, the configuration you're familiar with
corresponds to a 'best center' of C major / A minor, or 'C' for
short. That's all there is to it, and it carries no more meaning than
that. Trying to determine the most 'tonal' (major?) mode for one of
the rings in each of the horagrams, in order to extend this kind of
thinking to them, seems to me like 'speculational overkill' right
now . . .

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:

/tuning/topicId_51352.html#51824

> Basically, this 12-note meantone range that you're familiar with
> allows for the following key signatures:
>
> 2b = Bb major, G minor
> 1b = F major, D minor
> 0b/# = C major, A minor
> 1# = G major, E minor
> 2# = D major, B minor
> 3# = A major, F# minor
>
> Since modulations to the sharp keys were slightly more common than
> modulations to the flat keys, the configuration you're familiar
with
> corresponds to a 'best center' of C major / A minor, or 'C' for
> short. That's all there is to it, and it carries no more meaning
than
> that.

***This became clear to me when, armed with Kyle Gann's "good key"
1/4 comma meantone chart, I orchestrated several pieces from the
Elizabethan period (most from the Fitzwilliam Virginal Book) for a
production of The Tempest. The pieces were all, of course, in these
keys...

J. Pehrson

on 1/15/04 1:45 PM, wallyesterpaulrus <paul@stretch-music.com> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>
>>> Why don't we start by looking at the *meantone* example, which
> should
>>> be fairly clear and familiar. In particular, you should recognize
> the
>>> 5-, 7-, and 12-tone rings as *very* familiar scales . . . no?
>>
>> The intervals are exactly 1/4-comma meantone.
>
> Nope -- everything's wider than in 1/4-comma meantone by a fixed
> percentage,

Ah ok, the maj/min seconds were exact to the nearest cent and I didn't look
any closer (or farther).

> as discussed on tuning-math . . . note that the octave,
> major third, and minor sixth are all just in 1/4-comma meantone but
> wider than just here . . .

Yes, notably the octave. Of course!

Is it at all possible to do this same kind of thing while restricting things
to 2:1 octaves? I take it the whole approach falls apart unless all the
generators are "free"?

>> But how the intervals are
>> sorted in the octave doesn't look like what I initially would have
>> expected.
>
> As I mentioned, I extended the chain of generators alterately in
> the "up" and "down" directions. More typical is to extend always in
> one direction, but then there would be even less correspondence with
> the rotations you're used to.

Yes, see my comment at the bottom.

> Basically, this 12-note meantone range that you're familiar with
> allows for the following key signatures:
>
> 2b = Bb major, G minor
> 1b = F major, D minor
> 0b/# = C major, A minor
> 1# = G major, E minor
> 2# = D major, B minor
> 3# = A major, F# minor
>
> Since modulations to the sharp keys were slightly more common than
> modulations to the flat keys, the configuration you're familiar with
> corresponds to a 'best center' of C major / A minor, or 'C' for
> short. That's all there is to it, and it carries no more meaning than
> that. Trying to determine the most 'tonal' (major?) mode for one of
> the rings in each of the horagrams, in order to extend this kind of
> thinking to them, seems to me like 'speculational overkill' right
> now . . .

Probably so. I didn't mention this but I was thinking already in terms of
software implementations. For that, an agreeable way of specifying the key
would need to be worked out. Probably canonicality is best observed by
keeping it simple and using the convention of of extending the generators
only in one direction, and letting the user be responsible for understanding
the consequences.

Thanks.

-Kurt

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
> /tuning/topicId_51352.html#51824
>
> > Basically, this 12-note meantone range that you're familiar with
> > allows for the following key signatures:
> >
> > 2b = Bb major, G minor
> > 1b = F major, D minor
> > 0b/# = C major, A minor
> > 1# = G major, E minor
> > 2# = D major, B minor
> > 3# = A major, F# minor
> >
> > Since modulations to the sharp keys were slightly more common
than
> > modulations to the flat keys, the configuration you're familiar
> with
> > corresponds to a 'best center' of C major / A minor, or 'C' for
> > short. That's all there is to it, and it carries no more meaning
> than
> > that.
>
> ***This became clear to me when, armed with Kyle Gann's "good key"
> 1/4 comma meantone chart, I orchestrated several pieces from the
> Elizabethan period (most from the Fitzwilliam Virginal Book) for a
> production of The Tempest. The pieces were all, of course, in
these
> keys...
>
> J. Pehrson

Don't forget, though, that Renaissance and most Baroque keyboardists
were expected to *retune*, for example, all the G#s to Abs and all
the C#s to Dbs, when faced with a piece in what today would be known
as an Ab major or F minor key signature. Except, of course, those
lucky enough to have split keys on their keyboards, which was
actually quite a few.

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> on 1/15/04 1:45 PM, wallyesterpaulrus <paul@s...> wrote:
>
> > --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
> >
> >>> Why don't we start by looking at the *meantone* example, which
> > should
> >>> be fairly clear and familiar. In particular, you should
recognize
> > the
> >>> 5-, 7-, and 12-tone rings as *very* familiar scales . . . no?
> >>
> >> The intervals are exactly 1/4-comma meantone.
> >
> > Nope -- everything's wider than in 1/4-comma meantone by a fixed
> > percentage,
>
> Ah ok, the maj/min seconds were exact to the nearest cent and I
didn't look
> any closer (or farther).
>
> > as discussed on tuning-math . . . note that the octave,
> > major third, and minor sixth are all just in 1/4-comma meantone
but
> > wider than just here . . .
>
> Yes, notably the octave. Of course!
>
> Is it at all possible to do this same kind of thing while
restricting things
> to 2:1 octaves?

We haven't come to a consensus on that yet.

> I take it the whole approach falls apart unless all the
> generators are "free"?

When you assume octave-equivalence, you move from a Tenney complexity
measure to a "ratio of" measure (you can look up "ratio of"). Then
there's never any benefit to tempering octaves.

> Probably canonicality is best observed by
> keeping it simple and using the convention of of extending the
generators
> only in one direction, and letting the user be responsible for
understanding
> the consequences.

So would the canonical 12-tone meantone have 5 sharps or 5 flats
or . . . ?

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:

> > Is it at all possible to do this same kind of thing while
> restricting things
> > to 2:1 octaves?
>
> We haven't come to a consensus on that yet.

It clearly can be done. We could simply look for the point nearest
the JIP on the intersection of the temperament subpace and the pure
octaves subspace. Alternatively, we could find a norm on odd ratios
which we like.

--- In tuning@yahoogroups.com, "paulerlich" <paul@s...> wrote:

/tuning/topicId_51352.html#51859
> >
> > ***This became clear to me when, armed with Kyle Gann's "good
key"
> > 1/4 comma meantone chart, I orchestrated several pieces from the
> > Elizabethan period (most from the Fitzwilliam Virginal Book) for
a
> > production of The Tempest. The pieces were all, of course, in
> these
> > keys...
> >
> > J. Pehrson
>
> Don't forget, though, that Renaissance and most Baroque
keyboardists
> were expected to *retune*, for example, all the G#s to Abs and all
> the C#s to Dbs, when faced with a piece in what today would be
known
> as an Ab major or F minor key signature. Except, of course, those
> lucky enough to have split keys on their keyboards, which was
> actually quite a few.

***Thanks Paul. Actually, I *had* forgotten that or, rather, I never
knew it in the first place... :) Well, I knew there was some kind
of "retuning business" associated with this, but I wasn't sure,
specifically, what it was...

Thanks!

JP

on 1/16/04 4:29 PM, wallyesterpaulrus <paul@stretch-music.com> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>> on 1/15/04 1:45 PM, wallyesterpaulrus <paul@s...> wrote:
>>
>>> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:
>>>
>>>>> Why don't we start by looking at the *meantone* example, which
>>> should
>>>>> be fairly clear and familiar. In particular, you should
> recognize
>>> the
>>>>> 5-, 7-, and 12-tone rings as *very* familiar scales . . . no?
>>>>
>>>> The intervals are exactly 1/4-comma meantone.
>>>
>>> Nope -- everything's wider than in 1/4-comma meantone by a fixed
>>> percentage,
>>
>> Ah ok, the maj/min seconds were exact to the nearest cent and I
> didn't look
>> any closer (or farther).
>>
>>> as discussed on tuning-math . . . note that the octave,
>>> major third, and minor sixth are all just in 1/4-comma meantone
> but
>>> wider than just here . . .
>>
>> Yes, notably the octave. Of course!
>>
>> Is it at all possible to do this same kind of thing while
> restricting things
>> to 2:1 octaves?
>
> We haven't come to a consensus on that yet.
>
>> I take it the whole approach falls apart unless all the
>> generators are "free"?
>
> When you assume octave-equivalence, you move from a Tenney complexity
> measure to a "ratio of" measure (you can look up "ratio of"). Then
> there's never any benefit to tempering octaves.
>
>> Probably canonicality is best observed by
>> keeping it simple and using the convention of of extending the
> generators
>> only in one direction, and letting the user be responsible for
> understanding
>> the consequences.
>
> So would the canonical 12-tone meantone have 5 sharps or 5 flats
> or . . . ?

I probably confused things a little by refering to "canonicality" there. I
didn't necessarily want to draw conclusions about the canonical form outside
a software context. However, referring to what you had said:

> I extended the chain of generators alterately in
> the "up" and "down" directions. More typical is to extend always in
> one direction, but then there would be even less correspondence with
> the rotations you're used to.

my impression would be that although the "alternate" approach may be
somewhat "better" on the average it might cause confusion in a software
context because it is less explicit, i.e. contains more assumptions. And
the "alternate" situation is also a little different with odd versus even
numbers of scale degrees, and why should the user have to think through that
complexity ... unless it turns out to be just really useful to do it that
way.

But I think I'd tentatively rather expect the user to know what they want in
terms of sharps and flats. In that case it is easy enough for them to
specify starting at Eb and applying generators upward if they want the
"familiar" C meantone. And this is where it reflects back into
"canonicality". If the software is to be consistent with a pre-fixed set of
diagrams, it might then be better if the diagrams were generated using what
you called the "More typical" approach of "extending always in one
direction". Because then the user could easily relate their choice to the
diagram, with less thinking required to sort out the correspondences.

However, it might just turn out that the "alternating" approach (which I'd
tentatively rather call the "centered" approach) might just do exactly what
is needed most of the time, in which case "why not"? However, if you used
the alternating/centered approach then I am wondering why the "familiar"
meantone did not come out centered on "C" rather than "A". If there are 5
sharps and flats total, I would expect that you would have ended up either
with 2-flats/3-sharps or 3-flats/2-sharps, and you could easily bias it
toward preferrng sharps. So how did the "A" center result? I hope I'm
being clear enough.

-Kurt

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...>
wrote:
>
> > > Is it at all possible to do this same kind of thing while
> > restricting things
> > > to 2:1 octaves?
> >
> > We haven't come to a consensus on that yet.
>
> It clearly can be done. We could simply look for the point nearest
> the JIP on the intersection of the temperament subpace and the pure
> octaves subspace.

Can you give some examples, and find the error function that is being
minimized?

> Alternatively, we could find a norm on odd ratios
> which we like.

Yes, you and I are now discussing this, and whether it's even
possible (given some 'likes', at least), on tuning-math.

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@b...> wrote:

> However, it might just turn out that the "alternating" approach
(which I'd
> tentatively rather call the "centered" approach) might just do
exactly what
> is needed most of the time, in which case "why not"? However, if
you used
> the alternating/centered approach then I am wondering why
the "familiar"
> meantone did not come out centered on "C" rather than "A". If
there are 5
> sharps and flats total, I would expect that you would have ended up
either
> with 2-flats/3-sharps or 3-flats/2-sharps, and you could easily
bias it
> toward preferrng sharps. So how did the "A" center result? I hope
I'm
> being clear enough.
>
> -Kurt

(1) Since the generators was taken as a fourth rather than as a
fifth, we get "A" instead of "D".

(2) The center (in the chain of generators) of the C major scale and
A minor scale -- the null key signature -- is "D", not "C".