Meantone Piano

Fellow MicroMusicMakers,

A while ago I tuned my mother's piano to quarter comma meantone. It took a day or two for me since I had no good tools nor a good algorithm, but it was worth it :). I highly encourage others on this list to try the same, or why not some other tuning. I got very positive comments from my family, to the effect that it sounds very harmonious and relaxing.

My wife helped me to record a little improvisation kick-started by a predecided theme. Both sound quality and playing could be better, but it might give you an idea of what it sounds like. I've uploaded the recording in mp3 format here:

http://magnus.smartelectronix.com/music/

(Click on "magnus_jonsson_-_mea..>")

Hope this will inspire someone.

Cheers,
Magnus Jonsson

On 8/18/06, Magnus Jonsson <magnus@...> wrote:
> Fellow MicroMusicMakers,
>
> A while ago I tuned my mother's piano to quarter comma meantone. It took a
> day or two for me since I had no good tools nor a good algorithm, but it
> was worth it :). I highly encourage others on this list to try the same,
> or why not some other tuning. I got very positive comments from my family,
> to the effect that it sounds very harmonious and relaxing.

Well, as long as you avoid the wolf! Personally, I prefer having seven
or eight quarter-comma fifths and four or five wider fifths to split
the difference. That was the standard tuning for a while before equal
temperament became popular, right?

> My wife helped me to record a little improvisation kick-started by a
> predecided theme. Both sound quality and playing could be better, but it
> might give you an idea of what it sounds like. I've uploaded the recording
> in mp3 format here:
>
> http://magnus.smartelectronix.com/music/
>
> (Click on "magnus_jonsson_-_mea..>")
>
> Hope this will inspire someone.

I'll remember to listen to it when I have some speakers.

Keenan

On Sat, 19 Aug 2006, Keenan Pepper wrote:

> Well, as long as you avoid the wolf! Personally, I prefer having seven
> or eight quarter-comma fifths and four or five wider fifths to split
> the difference. That was the standard tuning for a while before equal
> temperament became popular, right?

That is probably a good idea. I wasn't aware this is how people did it. On the other hand, I really enjoy playing with the 7-limit tetrads on Bb and Eb. I don't use them in the mp3 though. Maybe I'll make a recording with more 7-limit stuff later.

Hi Magnus,

I enjoyed your quarter-comma meantone improv - thanks for sharing it
with us!

--- In MakeMicroMusic@yahoogroups.com, Magnus Jonsson wrote:
>
> On Sat, 19 Aug 2006, Keenan Pepper wrote:
>
> > Well, as long as you avoid the wolf! Personally, I prefer having
seven or eight quarter-comma fifths and four or five wider fifths to
split the difference. That was the standard tuning for a while
before equal temperament became popular, right?
>
> That is probably a good idea. I wasn't aware this is how people
did it. On the other hand, I really enjoy playing with the 7-limit
tetrads on Bb and Eb. I don't use them in the mp3 though. Maybe I'll
make a recording with more 7-limit stuff later.
>

That would be interesting.

Not to create difficulties for you, but if you can retune a piano,
have you thought of trying the one-fifth comma meantone?

Wouldn't it be wonderful to have half-a-dozen pianos at hand in
different tunings? ;-) Luckily we have software to help out our
budgets ...

Regards,
Yahya

I also enjoyed what I percieved to be a 'down home' feeling.
Dave Hill, Kurt Bigler, and Paul Bailey are all artists that
have both done work with the meantone piano (and new music
for the meantone harpsichord). Kurt has a webpage now...

http://voxdigitalis.com/

-Carl

At 07:21 PM 8/19/2006, you wrote:
>Hi Magnus,
>
>I enjoyed your quarter-comma meantone improv - thanks for sharing it
>with us!

> 1b. Re: Meantone Piano
> Posted by: "Magnus Jonsson" magnus@... zealmange
> Date: Sat Aug 19, 2006 4:29 am (PDT)
>
> On Sat, 19 Aug 2006, Keenan Pepper wrote:
>
> > Well, as long as you avoid the wolf! Personally, I prefer having seven
> > or eight quarter-comma fifths and four or five wider fifths to split
> > the difference. That was the standard tuning for a while before equal
> > temperament became popular, right?
>
> That is probably a good idea. I wasn't aware this is how people did it. On
> the other hand, I really enjoy playing with the 7-limit tetrads on Bb and
> Eb. I don't use them in the mp3 though. Maybe I'll make a recording with
> more 7-limit stuff later.

Hello, there, Magnus and Keenan.

What you are describing is one variety of temperament ordinaire, with a
good number (often a majority) of meantone fifths and the remainder
tempered wide so as, in a number of versions, to avoid any Wolf fifth. As
your discussion is bringing out, this kind of temperament does have its
advantages and compromises.

With 1/4-comma, let's suppose we place eight regular meantone fifths at
F-C#. Thus this portion of the temperament will be identical to regular
meantone; and the diminished fourth C#/Db-F will also be identical at
32/25 or about 427 cents -- great for special effects in a meantone style,
or for other purposes I'll explain below, but not likely for representing
a 5:4 third! As in regular meantone, and unlike a usual well-temperament,
we'll have some major thirds dramatically larger than a Pythagorean 81:64
(about 408 cents), and minor thirds smaller than 32:27 (about 294 cents).

How about those septimal minor sevenths -- and thirds -- of meantone? We
have another compromise, with ratios closer than in a usual well-tempered
system but considerably less accurate than in regular meantone. Let's see
why this is.

In regular meantone, a near-septimal minor seventh is formed from two
unequal fourths: a regular meantone fourth at around 503.42 cents, and a
Wolf fourth at about 462.36 cents. This adds to about 965.78 cents, very
close to 7:4 (968.83 cents).

In our 1/4-comma temperament ordinaire with eight regular meantone fifths,
and the other four equally wide, the wide fifths will each be about 4.89
cents larger than pure -- and the narrow fourths equally narrow of pure.
Thus two such fourths at about 493.16 cents will produce a minor seventh
of about 986.32 cents (Scala says about 986.314 cents), rather narrower
than 16:9 but still more than 17 cents wider than 7:4. This is better than
a usual well-temperament where all fourths are pure or wider, and vice
versa for fifths -- but much less accurate than regular 1/4-comma.

Also, if you are specifically interested in approximating 4:5:6:7, let's
consider first the situation with regular 1/4-comma, where we have two
such good approximations, in the locations with a meantone augmented
sixth equivalent to a 7:4 minor seventh, Bb-G# and Eb-C#:

Bb D F G#
0 386 697 966

Eb G Bb C#
0 386 697 966

With an 8/4 temperament, based on the eight regular fifths at F-C#, we
have a more complicated situation, with greater compromises in some of the
5:4 major thirds as well as 7:4 minor sevenths:

F A C Eb/D#
0 386 697 986

Bb D F G#/Ab
0 397 707 986

While F-A-C-Eb yields the best approximation of 4:5:6:7 from the viewpoint
of ratios above the lowest note, note that C-Eb/D# is considerably wider
than 7:6, almost 290 cents (and very close to 13:11).

Now we find that Bb-D-F-Ab/G# has the same accuracy for 7:4, and greater
accuracy for 7:6 (about 279 cents, still considerably further than a
regular 1/4-comma), since here we have a wide fifth Bb-F, by the way a tad
closer to 3:2 than a regular meantone fifth (about 4.89 cents wide versus
5.38 cents narrow). However, Bb-D in this tuning is not a pure major third
at 5:4, but one compromised by 1/4 of a 128:125 diesis, about 10.265
cents, in the wide direction. This third is formed by three meantone
fifths plus a wide fifth: Bb-F-C-G-D (with Bb-F wide). Thus from a 4:5:6:7
perspective, we need to consider the impurity not only of the 986-cent
minor seventh but the 397-cent major third.

Still, if you want a circulating (i.e. "all fifths and fourths playable")
temperament with some minor thirds and sevenths within "hailing distance"
of septimal, this is a way to do it -- but less attractive, possibly, for
4:5:6:7 than for things like 4:6:7 or 12:14:18:21 which don't bring into
play the compromise of three major thirds which would pure in a regular
1/4-comma. These are Bb-D and E-G# at around 397 cents -- and, more
dramatically, Eb-G at around 406.84 cents, or close to Pythagorean.

Now for the "other special uses" of the remote thirds near or beyond
Pythagorean. I like to use these in a kind of "neomedieval" style like
that of the 13th-14th centuries in European music based on Pythagorean
tuning, where the near-septimal intervals and narrow semitones (in an 8/4
temperament based on 1/4-comma, at C-C#/Db, F-F#/Gb, with a size of the
same 76.05 cents as regular meantone chromatic semitones) can have a very
pleasing effect.

Again, an important caution is that this kind of temperament avoids any
wolf fifth or fourth while keeping a wide range of third sizes more
characteristic of meantone than of a well-temperament. This can be an
advantage if we're looking for septimal ratios -- but it also means that
C#-F, for example, will be just as unlikely a representation of 5:4 in a
sonority such as Db-F-Ab as if the tuning were regular meantone! If we
want Db-F-Ab to approximate 14:18:21, however, then the arrangement is
more advantageous.

Also, there are times when one might desire the full color and excitement
of C#-F as a special meantone interval, for example, in a rhythm that
places some stress on the first C#:

F E F#
C C# D C# B C# D
A D

If you do experiment with this tuning, an advantage is that you'll only
need to retune three notes in each octave from the regular 1/4-comma
meantone. If your meantone is Eb-G#, then the notes to be retuned are
Bb, Eb, and Ab/G#. The idea is to make the fifths Bb-F, Eb-Bb, Ab/G#-Eb
and C#-G#/Ab equally wide, and a tad less impure than the regular meantone
fifths.

David Hill has done some CD's with a piano in regular 1/4-comma meantone,
and I'd emphasize that the kind of irregular temperament we're discussing
is something I'm familiar with on synthesizer rather than an acoustical
instrument.

Peace and love,

Margo

On Sun, 20 Aug 2006, yahya_melb wrote:
>
> Hi Magnus,
>
> I enjoyed your quarter-comma meantone improv - thanks for sharing it
> with us!

Thanks Yahya!

> --- In MakeMicroMusic@yahoogroups.com, Magnus Jonsson wrote:
>>
>> On Sat, 19 Aug 2006, Keenan Pepper wrote:
>>
>> That is probably a good idea. I wasn't aware this is how people
> did it. On the other hand, I really enjoy playing with the 7-limit
> tetrads on Bb and Eb. I don't use them in the mp3 though. Maybe I'll
> make a recording with more 7-limit stuff later.
>>
>
> That would be interesting.

Okay, I'll see what I can do. I have already recorded something in harmonic G minor with some minimal excursions to 7-limit land, but I should do something overtly 7-limit to really emphasize it.

> Not to create difficulties for you, but if you can retune a piano,
> have you thought of trying the one-fifth comma meantone?

Yes, and actually I was a little surprised at first when I heard how wide the large semitones are in 1/4 comma. At that point I got interested in 1/5 comma, since the large semitone will be exactly 15/16, which should sound ok since it's just. But 1/4 is a lot easier to tune so I settled for 1/4. What happens to the septimal intervals in 1/5? Are they still good?

> Wouldn't it be wonderful to have half-a-dozen pianos at hand in
> different tunings? ;-) Luckily we have software to help out our
> budgets ...

:-D, that would be awesome indeed.

- Magnus

Hi Magnus,

--- In MakeMicroMusic@yahoogroups.com, Magnus Jonsson wrote:
>
> > I enjoyed your quarter-comma meantone improv - thanks for
sharing it with us!
>
> Thanks Yahya!

My pleasure, truly!

> > --- In MakeMicroMusic@yahoogroups.com, Magnus Jonsson wrote:
> >>
> >> On Sat, 19 Aug 2006, Keenan Pepper wrote:
> >>
> >> That is probably a good idea. I wasn't aware this is how people
did it. On the other hand, I really enjoy playing with the 7-limit
tetrads on Bb and Eb. I don't use them in the mp3 though. Maybe I'll
make a recording with more 7-limit stuff later.
> >
> > That would be interesting.
>
> Okay, I'll see what I can do. I have already recorded something in
harmonic G minor with some minimal excursions to 7-limit land, but I
should do something overtly 7-limit to really emphasize it.

Great! Can't wait to hear it!

> > Not to create difficulties for you, but if you can retune a
piano, have you thought of trying the one-fifth comma meantone?
>
> Yes, and actually I was a little surprised at first when I heard
how wide the large semitones are in 1/4 comma. At that point I got
interested in 1/5 comma, since the large semitone will be exactly
15/16, which should sound ok since it's just. But 1/4 is a lot
easier to tune so I settled for 1/4. ...

Understood.

> ... What happens to the septimal intervals in 1/5? Are they still
good?

That's a question best answered by others. Any takers?

Regards,
Yahya

Hi Magnus,

--- In MakeMicroMusic@yahoogroups.com, Magnus Jonsson wrote:
>
> > I enjoyed your quarter-comma meantone improv - thanks for
sharing it with us!
>
> Thanks Yahya!

My pleasure, truly!

> > --- In MakeMicroMusic@yahoogroups.com, Magnus Jonsson wrote:
> >>
> >> On Sat, 19 Aug 2006, Keenan Pepper wrote:
> >>
> >> That is probably a good idea. I wasn't aware this is how people
did it. On the other hand, I really enjoy playing with the 7-limit
tetrads on Bb and Eb. I don't use them in the mp3 though. Maybe I'll
make a recording with more 7-limit stuff later.
> >
> > That would be interesting.
>
> Okay, I'll see what I can do. I have already recorded something in
harmonic G minor with some minimal excursions to 7-limit land, but I
should do something overtly 7-limit to really emphasize it.

Great! Can't wait to hear it!

> > Not to create difficulties for you, but if you can retune a
piano, have you thought of trying the one-fifth comma meantone?
>
> Yes, and actually I was a little surprised at first when I heard
how wide the large semitones are in 1/4 comma. At that point I got
interested in 1/5 comma, since the large semitone will be exactly
15/16, which should sound ok since it's just. But 1/4 is a lot
easier to tune so I settled for 1/4. ...

Understood.

> ... What happens to the septimal intervals in 1/5? Are they still
good?

That's a question best answered by others. Any takers?

Regards,
Yahya

Thanks Carl for listening and for the recommendations. I listened to the music on Kurt's web page (hence the time it took to reply), both his harpsichord in dual harmonic series a fifth apart and the modified meantone piano examples. Both interesting, but I preferred the harpsichord work. And I think I would have liked it even better if the 11th and 13th harmonics were removed and replaced by some lower-limit notes.
- Magnus

On Sat, 19 Aug 2006, Carl Lumma wrote:

> I also enjoyed what I percieved to be a 'down home' feeling.
> Dave Hill, Kurt Bigler, and Paul Bailey are all artists that
> have both done work with the meantone piano (and new music
> for the meantone harpsichord). Kurt has a webpage now...
>
> http://voxdigitalis.com/
>
> -Carl
>
> At 07:21 PM 8/19/2006, you wrote:
>> Hi Magnus,
>>
>> I enjoyed your quarter-comma meantone improv - thanks for sharing it
>> with us!
>
>

> > ... What happens to the septimal intervals in 1/5? Are they still
> good?
>
> That's a question best answered by others. Any takers?
>
>
> Regards,
> Yahya

Hi, Yahya and all.

A quick answer is that in 1/5-comma, each of the two augmented sixths
(Eb-C#, Bb-G#) is around 976.537 cents. This is not as close to 7/4 as
in 1/4-comma, of course, but still seems to me quite reasonable as an
approximation.

Compare 981.82 cents in 22-EDO, which Paul Erlich uses for its
approximations of 4:5:6:7, or around 985 cents in George Secor's 17-tone
well-temperament for the best approximations of 7/4; in a circulating
system based on eight regular fifths in 2/7-comma and the rest equally
wide, one of my favorite meantone variations, the smallest minor sevenths
(or augmented sixths) are around 983.24 cents.

Also, in 1/5-comma, augmented seconds like Bb-C# are around 278.883 cents,
comparable to the best near-7:6 minor thirds in George Secor's 17-WT
mentioned above.

Something like Bb-D-F-G# in regular 1/5-comma would thus be about
0-391-698-976 cents. If something like 22-EDO is practical for 4:5:6:7
approximations (0-382-709-982), then this should be practical also.

A rule, however useful or otherwise: for a meantone tuning, take the size
of the fifth in cents and multiply by 10 to get the size of the augmented
sixth plus five octaves or 6000 cents which you then subtract (an
excellent approxmation of 7:4 around 1/4-comma, and still not too far at
1/5-comma, as we've seen). For example, a 1/4-comma fifth is around 696.58
cents, so the augmented sixth is around 6965.8 cents less 6000 cents, or
965.8 cents -- just a bit narrow of a just 7:4 at 968.83 cents or so.

Peace and love,

Margo
mschulter@...

P. S. While 1/5-comma is still reasonably close for 7/4 and 7/6, it is not
so accurate for 9/7, where 2/7-comma is close to optimal. A diminished
fourth in 1/5-comma is about 418.770 cents, very close to 14/11 rather
than to 9/7. However, this wouldn't be an issue if you're looking mostly
for a couple of locations with a near-4:5:6:7.

Thanks for shedding light on this Margo!

On Mon, 21 Aug 2006, Margo Schulter wrote:

> Hi, Yahya and all.
>
> A quick answer is that in 1/5-comma, each of the two augmented sixths
> (Eb-C#, Bb-G#) is around 976.537 cents. This is not as close to 7/4 as
> in 1/4-comma, of course, but still seems to me quite reasonable as an
> approximation.
>
> Compare 981.82 cents in 22-EDO, which Paul Erlich uses for its
> approximations of 4:5:6:7, or around 985 cents in George Secor's 17-tone
> well-temperament for the best approximations of 7/4; in a circulating
> system based on eight regular fifths in 2/7-comma and the rest equally
> wide, one of my favorite meantone variations, the smallest minor sevenths
> (or augmented sixths) are around 983.24 cents.
>
> Also, in 1/5-comma, augmented seconds like Bb-C# are around 278.883 cents,
> comparable to the best near-7:6 minor thirds in George Secor's 17-WT
> mentioned above.
>
> Something like Bb-D-F-G# in regular 1/5-comma would thus be about
> 0-391-698-976 cents. If something like 22-EDO is practical for 4:5:6:7
> approximations (0-382-709-982), then this should be practical also.
>
> A rule, however useful or otherwise: for a meantone tuning, take the size
> of the fifth in cents and multiply by 10 to get the size of the augmented
> sixth plus five octaves or 6000 cents which you then subtract (an
> excellent approxmation of 7:4 around 1/4-comma, and still not too far at
> 1/5-comma, as we've seen). For example, a 1/4-comma fifth is around 696.58
> cents, so the augmented sixth is around 6965.8 cents less 6000 cents, or
> 965.8 cents -- just a bit narrow of a just 7:4 at 968.83 cents or so.
>
> Peace and love,
>
> Margo
> mschulter@...
>
> P. S. While 1/5-comma is still reasonably close for 7/4 and 7/6, it is not
> so accurate for 9/7, where 2/7-comma is close to optimal. A diminished
> fourth in 1/5-comma is about 418.770 cents, very close to 14/11 rather
> than to 9/7. However, this wouldn't be an issue if you're looking mostly
> for a couple of locations with a near-4:5:6:7.
>
>

Hi Margo,

Thanks for what seems like a very thorough answer.

May we conclude that for the 7-limit, you prefer your circulating
2/7-comma variation, over other alternatives, including regular 1/5
comma and 1/4-comma (circulating or regular)?

Or is it more complex than that; eg, would you choose a temperament
based on whether you plan to include particular intervals, like 9/7?

Regards,
Yahya

--- In MakeMicroMusic@yahoogroups.com, Margo Schulter wrote:
>
> > > ... What happens to the septimal intervals in 1/5? Are they
still good?
> >
> > That's a question best answered by others. Any takers?
> >
> Hi, Yahya and all.
>
> A quick answer is that in 1/5-comma, each of the two augmented
sixths (Eb-C#, Bb-G#) is around 976.537 cents. This is not as close
to 7/4 as in 1/4-comma, of course, but still seems to me quite
reasonable as an approximation.
>
> Compare 981.82 cents in 22-EDO, which Paul Erlich uses for its
approximations of 4:5:6:7, or around 985 cents in George Secor's 17-
tone well-temperament for the best approximations of 7/4; in a
circulating system based on eight regular fifths in 2/7-comma and
the rest equally wide, one of my favorite meantone variations, the
smallest minor sevenths (or augmented sixths) are around 983.24
cents.
>
> Also, in 1/5-comma, augmented seconds like Bb-C# are around
278.883 cents, comparable to the best near-7:6 minor thirds in
George Secor's 17-WT mentioned above.
>
> Something like Bb-D-F-G# in regular 1/5-comma would thus be about
0-391-698-976 cents. If something like 22-EDO is practical for
4:5:6:7 approximations (0-382-709-982), then this should be
practical also.
>
> A rule, however useful or otherwise: for a meantone tuning, take
the size of the fifth in cents and multiply by 10 to get the size of
the augmented sixth plus five octaves or 6000 cents which you then
subtract (an excellent approxmation of 7:4 around 1/4-comma, and
still not too far at 1/5-comma, as we've seen). For example, a 1/4-
comma fifth is around 696.58 cents, so the augmented sixth is around
6965.8 cents less 6000 cents, or 965.8 cents -- just a bit narrow of
a just 7:4 at 968.83 cents or so.
>
> Peace and love,
>
> Margo
> mschulter@...
>
> P. S. While 1/5-comma is still reasonably close for 7/4 and 7/6,
it is not so accurate for 9/7, where 2/7-comma is close to optimal.
A diminished fourth in 1/5-comma is about 418.770 cents, very close
to 14/11 rather than to 9/7. However, this wouldn't be an issue if
you're looking mostly for a couple of locations with a near-4:5:6:7.

> Hi Margo,
>
> Thanks for what seems like a very thorough answer.
>
> May we conclude that for the 7-limit, you prefer your circulating
> 2/7-comma variation, over other alternatives, including regular 1/5
> comma and 1/4-comma (circulating or regular)?

Dear Yahya,

Please let me quickly explain, before appending a longer answer, that
what I'm typically going after might be rather different than "7-limit"
as very reasonably interpreted by many people. Thus I'm concerned that
a statement of preference could be misleading as applied to this thread,
unless people like Magnus are especially seeking to play a piano in a
Renaissance 5-limit style in the nearer transpositions and a kind of
(neo-)medieval style with intervals ranging from around Pythagorean to
near-septimal in the remote region.

While I can give a fuller explanation of what I do on a synthesizer
often emulating an organ or harpsichord, etc., it isn't necessarily
what others should or would do on a piano in different styles.

> Or is it more complex than that; eg, would you choose a temperament
> based on whether you plan to include particular intervals, like 9/7?

For me, it's more a matter of mood or fine shading. If I want to stick
close to an historical 16th-century sound, I'd go with a regular
meantone, for me either 1/4-comma or 2/7-comma (although there's period
evidence for something around 1/5-comma or 1/6-comma as well).

A circulating variation on either shade of meantone is more "modernistic"
in concept: I like it, but am aware that playing C-Eb around 287 or 290
cents could be my taste rather than period practice, although I could
come up with a scenario drawing on Praetorius or Werckmeister as to how
Eb/D# _might_ have been "compromised" to produce such a result.

Since "7-based" for me in such a style almost always involves 9:7 as
well as 7:6 and 7:4 -- factors of 2-3-7-9, in other words -- I could go
for either a 2/7-comma or 1/4-comma modified system depending on my
mood. Each has its attractions.

>
> Regards,
> Yahya

Here's my reply with musical examples:

Dear Yahya, Magnus, and all,

Please let me apologize for possibly not keeping my remarks about
regular and modified meantones as closely tied to specific pieces as
is appropriate for MMM. To remedy this fault, I will give a couple of
musical examples, the first in a modified 2/7-comma meantone which was
much inspired by Harold Fortuin, and which like a very recently posted
example of his (Hi, Harold!) uses a 7:4 approximation at one point,
although the near-7:6 is more prominent.

<http://www.bestII.com/~mschulter/IntradaFLydian.mp3>

The second piece shows how the remote portion of the cycle can be used
for a septimal style featuring approximations of 7:6 and 7:4.

<http://www.bestII.com/~mschulter/MMMYear001.mp3>

What might be vital for this thread is to make clear some musical
assumptions that lead me to favor either this tuning or the closely
related modified meantone based on 1/4-comma, which I'd consider as
different musically interchangeable shadings:

(1) In the usual portion of the circle, I'll be playing typically
in a modal style not unlike that of the 16th or early 17th
century, where large diatonic semitones (121 cents in
2/7-comma) may be more congenial than in later major/minor
tonality.

(2) In the remote portion of the circle, while I might use lots of
septimal intervals (e.g. thirds at 275/434 cents), I'm also
happy with sizes such as 300/408 cents or 287/421 cents. People
seeking especially "approximations of small integer ratios"
such as 4:5:6:7 might be happier with a regular 1/4-comma,
if the Wolf fifth isn't a consideration.

My concern is that often it's easier to discuss cents than musical
styles and sensibilities. For me, the modified 1/4-comma or 2/7-comma
scheme is a neat way in a 12-note system to have all 12 fifths
reasonably close to 3:2 and to be able to play in an idiomatic
Renaissance meantone style in the nearer portion of the circle, and an
idiomatic (neo-)medieval style in the remote portion. For pianistic
repertory, the relevant goals and compromises could be quite different!

In peace and love,

Margo

Hallo Margo,

--- In MakeMicroMusic@yahoogroups.com, Margo Schulter wrote:
> > May we conclude that for the 7-limit, you prefer your
circulating 2/7-comma variation, over other alternatives, including
regular 1/5 comma and 1/4-comma (circulating or regular)?
>
> Dear Yahya,
>
> Please let me quickly explain, before appending a longer answer,
that what I'm typically going after might be rather different
than "7-limit" as very reasonably interpreted by many people. Thus
I'm concerned that a statement of preference could be misleading as
applied to this thread, unless people like Magnus are especially
seeking to play a piano in a Renaissance 5-limit style in the nearer
transpositions and a kind of (neo-)medieval style with intervals
ranging from around Pythagorean to near-septimal in the remote
region.
>
> While I can give a fuller explanation of what I do on a synthesizer
often emulating an organ or harpsichord, etc., it isn't necessarily
what others should or would do on a piano in different styles.
>
> > Or is it more complex than that; eg, would you choose a
temperament based on whether you plan to include particular
intervals, like 9/7?
>
> For me, it's more a matter of mood or fine shading. If I want to
stick close to an historical 16th-century sound, I'd go with a
regular meantone, for me either 1/4-comma or 2/7-comma (although
there's period evidence for something around 1/5-comma or 1/6-comma
as well).

[large snip]

Thank you for a full and clear answer. Yes, I have your examples,
but will play them again, perhaps with a little more insight than
formerly.

Somehow I'm reminded that TANSTAAFL (There Ain't No Such Thing As A
Free Lunch); if I'm looking for temperaments near small-ratio JI
including the 7-limit, there's really no shortcut but to tune some
of them up and *play* - which is (lucky people, musicians) our work.

But it's great that we have friends who are willing to set the table
for us! Thanks again.

Regards,
Yahya