Unequal temperament MP3's and 9:8 (was: 6 pieces...)

On Sat, 13 May 2006 MakeMicroMusic@yahoogroups.com wrote:

> From: "Carl Lumma" ekin@...
> Date: Fri May 12, 2006 2:00pm(PDT)
> Subject: Re: 6 pieces from 'Clash by Night'
>
> > I only wish it had a larger whole tone at times, but one could go for
> >the 8/7s if one wished
>
> Yeah Kraig, I hope you're happy, you ruined 31 for me. :)
> 9:8 is one of my favorite intervals. Actually, I like 193.5 cents
> better melodically in the diatonic scale (both major and minor),
> but harmonically it's a huge difference.

Hi, there, Kraig and Carl, and to keep the spotlight of the
'Clash by Night' thread on Aaron's music, I've decided to take
these asides as an excuse for a new thread in which I might
mention one solution to the "9:8 vs. meantone" dilemma: unequal
temperament, as illustrated by this piece.

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

This is actually in a modified meantone which has become one
of my favorite temperaments. First let's look at the specific
modality of this piece, Bb Dorian, with the minor sixth Gb
above the final sometimes used as an inflection (especially in
descending) like Bb in untransposed D Dorian. Here I'll use
rounded cents:

779
Gb
Bb C Db Eb F G Ab Bb
0 204 274 492 708 900 983 1200
204 71 217 217 192 83 217

Note that there are three kinds of whole tones. The 217-cent
step often featured here is close to half of a 9:7 major
third, as can be seen at Db-Eb-F, where a 434-cent third is
built from two such steps.

The regular 192-cent tone of Zarlino's 2/7-comma temperament occurs
at F-G, and is close to half of 5:4; in nearer transpositions,
this is, of course, the usual tone, two of them forming Zarlino's
meantone major third at 383 cents.

However, there's also another variety of whole tone, around 204
cents, and almost identical to a just 9:8 -- occurring at two
locations, Bb-C and Gb-Ab.

The Bb-C interval plays a very important role in this piece as
the second or ninth above the final, one I use a lot in this
kind of style given my love for 6:8:9 or 4:6:9 as a vertical
sonority.

Note that this modality also features the best septimal
approximations of the temperament, a feature it has in common
with both 22-EDO and 31-EDO, although not quite as accurate as
the former and rather less accurate than the latter.

The basic idea is to tune eight fifths, F-C#, as in Zarlino's
2/7-comma (about 6.145 cents narrow), and the others equally
wide (about 6.424 cents larger than pure), thus "balancing out"
the circle, and getting a 12-note circulating system which
supports a meantone style in the nearer modalities and a
Pythagorean-to-septimal style in more remote modalities.

In a sonority like Bb-F-C, what happens is that the differences
from 3:2 of Bb-F (about 6 cents wide) and F-C (about 6 cents
narrow) "cancel out," making Bb-C almost identical to a just 9:4.
I put "cancel out" in quotes, because the fifths themselves
remain decidedly impure, of course.

While the above piece illustrates a Pythagorean-to-septimal
color, here's a piece mainly focusing on the nearer meantone
part of the circle:

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

By the way, although the idea of eight fifths in Zarlino's
temperament and the rest equally wide was my theoretical
model for this tuning, what happens on a synthesizer in
1024-EDO very closely approximates a slightly different
and more complex model based on fractions of a Pythagorean
comma.

In 1024-EDO, a step is almost exactly 1/20 of a Pythagorean
comma. Here is my actual synthesizer tuning, with the
tempering of the fifths shown both in 1024-EDO steps and in
fractions of a Pythagorean comma (with negative numbers
showing narrow fifths, and positive numbers showing wide ones):

+6 +5 -5 -5 -6 -5 -5 -5 -6 -5 +5 +6
Eb Bb F C G D A E B F# C# G# D#
+.30 .25 -.25 -.25 -.30 -.25 -.25 -.25 -.30 -.25 +.25 +.30

Anyway, in 1024-EDO the 9:8 approximations are virtually just, and
they would be precisely just in the version based on fractions of
a Pythagorean comma.

While sharing my enthusiasm for this 12-note temperament, I must
also share two prudent cautions. First, while it's a fine 12-note
circulating temperament for me, it might not be so compelling for
others who want either meantone or septimal types of intervals in
more locations, or who aren't as excited as I am about intervals
such as major thirds at around 421 cents (actually 421.875 cents
in the 1024-EDO version), prominently featured in my first piece
in one standard cadence to Bb where a major third contracts to a
unison:

C Bb
Ab Bb

From my stylistic viewpoint, Ab-C at 421 or 422 cents makes a great
"usual" major third in this kind of modality, and also a great meantone
diminished fourth (spelled G#-C) in a 16th-century style. However,
someone looking for major/minor keys rather than modalities might
find this third size not exactly what they intended.

A second caution is simply to note what's not included, especially
neutral steps and intervals, as well as steps smaller than 70 cents.
However, arranged, 12 circulating notes can only do so much, and
Aaron's music has lots of good reasons why often "12 is not enough."

Peace and love,

Margo

Quite compelling little gem here with some quite nice tonal shifts of focus on different tonal areas. Somehow i can picture a satie like character behind such a thng.
Mainly for it use of different tonal areas and harmonies for color nd a boldness one might miss if one wasn't playing attention
Margo Schulter wrote:
>
> <http://www.bestII.com/~mschulter/MMMYear001.mp3>
>
>
>
>
>
>
> -- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

--- In MakeMicroMusic@yahoogroups.com, Kraig Grady <kraiggrady@...>
wrote:
>
> Quite compelling little gem here with some quite nice tonal shifts
of
> focus on different tonal areas. Somehow i can picture a satie like
> character behind such a thng.
> Mainly for it use of different tonal areas and harmonies for
color nd
> a boldness one might miss if one wasn't playing attention

Kraig,

I agree with all of these positive sentiments. I just listened again
to 'Evening Improvistation', and I wanted to write down some
impressions. I so like the piece for it's meditative character, and
so does my little 11-day old Annika-bean, who listened so intently
to every phrase...I think she likes Margo's music alot, as well
as 'temperament extraordinaire'

Margo, I love the blend of near-eastern and raga-like sensibilities
with medieval type cadences and linear thought. The drone at the
beginning struck me as almost like an Indian raga improvisation, and
then it so beautifully enters a descant-like texture a minute or so
into it!

The ending octave resolution was so nicely timed--we knew it was
coming, but the way your performed it played with our expectation so
expressively. Thank you also for you informative exposition on the
uses of modality in your modified 2/7-comma temperament.

I would love to hear what you would do with this style and some
percussion underneath!

Brava!

Best,
Aaron.

>> > I only wish it had a larger whole tone at times, but one could
>> > go for the 8/7s if one wished
>>
>> Yeah Kraig, I hope you're happy, you ruined 31 for me. :)
>> 9:8 is one of my favorite intervals. Actually, I like 193.5 cents
>> better melodically in the diatonic scale (both major and minor),
>> but harmonically it's a huge difference.
>
>Hi, there, Kraig and Carl, and to keep the spotlight of the
>'Clash by Night' thread on Aaron's music, I've decided to take
>these asides as an excuse for a new thread in which I might
>mention one solution to the "9:8 vs. meantone" dilemma: unequal
>temperament, as illustrated by this piece.
>
> < http://www.bestII.com/~mschulter/MMMYear001.mp3 >

That's quite beautiful, Margo.

The synth actually seems to bend down at the end, as if you
let go of a real bellows. Does it use a physical model?

>This is actually in a modified meantone which has become one
>of my favorite temperaments.
//
>The basic idea is to tune eight fifths, F-C#, as in Zarlino's
>2/7-comma (about 6.145 cents narrow), and the others equally
>wide (about 6.424 cents larger than pure),

Why is the second number .3 cents bigger?

>thus "balancing out"
>the circle, and getting a 12-note circulating system which
>supports a meantone style in the nearer modalities and a
>Pythagorean-to-septimal style in more remote modalities.

Interesting.

>While the above piece illustrates a Pythagorean-to-septimal
>color, here's a piece mainly focusing on the nearer meantone
>part of the circle:
>
> < http://www.bestII.com/~mschulter/IntradaFLydian.mp3 >

In my notes, I this as being in "8/4 well temperament".
Is that what you call it?

Something like this?

!
Margo Schulter's "8/4" well temperament.
12
!
70.67
191.62
287.43
383.24
504.19
574.86
695.81
779.05
887.43
995.81
1079.05
2
!

>While sharing my enthusiasm for this 12-note temperament, I must
>also share two prudent cautions. First, while it's a fine 12-note
>circulating temperament for me, it might not be so compelling for
>others who want either meantone or septimal types of intervals in
>more locations, or who aren't as excited as I am about intervals
>such as major thirds at around 421 cents (actually 421.875 cents
>in the 1024-EDO version), prominently featured in my first piece
>in one standard cadence to Bb where a major third contracts to a
>unison:
>
> C Bb
> Ab Bb
>
>From my stylistic viewpoint, Ab-C at 421 or 422 cents makes a great
>"usual" major third in this kind of modality, and also a great meantone
>diminished fourth (spelled G#-C) in a 16th-century style. However,
>someone looking for major/minor keys rather than modalities might
>find this third size not exactly what they intended.

It's incredible how well it works in your music, and how badly
in mine. :)

Thanks for sharing,

-Carl

Hello, Kraig, Aaron, Annika, Carl, and all, and thank you for your
most generous and encouraging comments about my piece.

For now, why don't I just address a few points, and also have a try at
briefly answering some of your tuning questions, Carl.

Kraig:

> Somehow i can picture a satie like character behind such a thng.

This is a fascinating comparison: when in college, I very much enjoyed
some of Satie's pieces, which were getting performed in various genres.
I'd love to hear them again, and maybe after 35 years or so appreciate
them in a new light.

Thank you also for your more general comments as someone who has,
needless to say, done an immense amount of improvisation and has a
wide perspective on many world musics.

Aaron:

> I so like the piece for it's meditative character, and
> so does my little 11-day old Annika-bean, who listened so intently
> to every phrase...I think she likes Margo's music alot, as well
> as 'temperament extraordinaire'

You've made my day with the response I was especially awaiting, and
this calls for a bit of followup, which I hope to do within a week or
so. Annika is quite an inspiration, isn't she -- as well as a budding
musician of distinction, we might suspect.

> I would love to hear what you would do with this style and some
> percussion underneath!

That's a very creative question. For example, in 14th-century Europe,
there was evidently a tradition of pieces called _Istampitas_ which
could be taken as a form of dance music evolving toward a more
"abstract" or meditative nature, with things such as accidental shifts
that might suggest Near Eastern influences -- and different groups
have handled the percussion in different ways. Studying some of those
recordings -- or more recent ones on CD -- might give some ideas.

Carl:

> The synth actually seems to bend down at the end, as if you
> let go of a real bellows. Does it use a physical model?

Having now learned through Google what a "physical model" is, I can
answer that this is something different -- FM synthesis (Yamaha
TX-802). Since the envelope pattern for the instrument I used here is
defined in a preset voice (A17, "EleCello A"), your comment reinforces
my own impression that John Chowning or whoever programmed this did
very nicely! I really like this timbre, to which I made only a few
small modifications to get more pitch stability (or less randomness).

>> The basic idea is to tune eight fifths, F-C#, as in Zarlino's
>> 2/7-comma (about 6.145 cents narrow), and the others equally
>> wide (about 6.424 cents larger than pure),

> Why is the second number .3 cents bigger?

To explain very quickly, in a 12-note tuning circle, the _average_
size of the 12 fifths must be 700 cents, or 1/12 Pythagorean comma
narrow, so that they add up to an even 8400 cents, or seven 2:1
octaves. (This assumes a 2:1 octave, of course.)

The eight narrow meantone fifths taken from Zarlino's 2/7-comma
temperament are each about 695.81 cents, or 4.19 cents smaller than
the 700-cent average, so that each of the four remaining equally wide
fifths must be larger than 700 cents by twice this amount, or 8.38
cents -- 708.38 cents. Then everything balances out. As it happens,
the wide fifths have a slightly greater impurity than the narrow ones,
the difference of about 0.3 cents that you have noted.

If the eight meantone fifths were tempered by precisely 1/4
Pythagorean comma, or about 5.87 cents, then the four wide fifths
would be larger than 3:2 by this exact same amount. Here the small
fifths at about 696.09 cents would be around 3.91 cents smaller than
700 cents, and the wide ones at about 707.82 cents around 7.82 cents
larger, again balancing out nicely. This is an interesting case of an
unequal 12-note temperament where all 12 fifths are impure to an
identical degree -- albeit in opposite directions!

This is only one road to "why," maybe one of the shortest and simplest
ones.

> In my notes, I this as being in "8/4 well temperament". Is that
> what you call it?

For my purposes, that's a nice description -- but "modified meantone"
might be better because the term "well-temperament" is often taken to
imply a tuning that will support some acceptable form of major/minor
tonality in any key, thus with major thirds not larger than around
Pythagorean.

The term I use, and Aaron has quoted, is "temperament extraordinaire,"
a play on the French _temperament ordinaire_, a 17th-18th century
genre which typically does have some fifths wider than pure and major
thirds larger than Pythagorean. Maybe the feature that makes this
tuning based on Zarlino's 2/7-comma "extraordinaire," rather than
merely "ordinaire," is the use of regular meantone major thirds a bit
smaller than a pure 5:4, something I haven't seen documented for an
historical European 12-note circulating temperament.

> Something like this?

> !
> Margo Schulter's "8/4" well temperament.
> 12
>!
> 70.67
> 191.62
> 287.43
> 383.24
> 504.19
> 574.86
> 695.81
> 779.05
> 887.43
> 995.81
> 1079.05
> 2

That's right, and my "official" version simply has more
(in)significant decimal plaes <grin>. Out of curiosity, why don't I
give my approximation in 1024-EDO (assuming that the synthesizer is
perfectly implementing this model):

! zarte84n.scl
!
Zarlino temperament extraordinaire, 1024-tET mapping
12
!
70.31250
191.01562
287.10938
383.20313
503.90625
574.21875
696.09375
778.12500
887.10938
996.09375
1079.29688
2/1

Anyway, thanks for the encouragement, the tuning questions, and that
intriguing observation about how that FM algorithm or whatever came
across -- something I'd like to look into more, with lots of people
here to help.

Peace and love,

Margo

>> The synth actually seems to bend down at the end, as if you
>> let go of a real bellows. Does it use a physical model?
>
>Having now learned through Google what a "physical model" is, I can
>answer that this is something different -- FM synthesis (Yamaha
>TX-802). Since the envelope pattern for the instrument I used here is
>defined in a preset voice (A17, "EleCello A"), your comment reinforces
>my own impression that John Chowning or whoever programmed this did
>very nicely! I really like this timbre, to which I made only a few
>small modifications to get more pitch stability (or less randomness).

Nice work, I like it too.

>>> The basic idea is to tune eight fifths, F-C#, as in Zarlino's
>>> 2/7-comma (about 6.145 cents narrow), and the others equally
>>> wide (about 6.424 cents larger than pure),
>
>> Why is the second number .3 cents bigger?
>
>To explain very quickly, in a 12-note tuning circle, the _average_
>size of the 12 fifths must be 700 cents, or 1/12 Pythagorean comma
>narrow, so that they add up to an even 8400 cents, or seven 2:1
>octaves. (This assumes a 2:1 octave, of course.)
>
>The eight narrow meantone fifths taken from Zarlino's 2/7-comma
>temperament are each about 695.81 cents, or 4.19 cents smaller than
>the 700-cent average, so that each of the four remaining equally wide
>fifths must be larger than 700 cents by twice this amount, or 8.38
>cents -- 708.38 cents. Then everything balances out. As it happens,
>the wide fifths have a slightly greater impurity than the narrow ones,
>the difference of about 0.3 cents that you have noted.

Gotcha.

>If the eight meantone fifths were tempered by precisely 1/4
>Pythagorean comma, or about 5.87 cents, then the four wide fifths
>would be larger than 3:2 by this exact same amount. Here the small
>fifths at about 696.09 cents would be around 3.91 cents smaller than
>700 cents, and the wide ones at about 707.82 cents around 7.82 cents
>larger, again balancing out nicely. This is an interesting case of an
>unequal 12-note temperament where all 12 fifths are impure to an
>identical degree -- albeit in opposite directions!

Something like this, I believe

!
Margo Schulter's 1/4-Pythagorean-comma temperament extraordinaire.
12
!
72.63
192.18
276.54
384.36
492.18
576.54
696.09
768.72
888.27
984.36
1080.45
2
!

It's a shame the approximate 9/7 doesn't occur in a mode with
the good 7/4 in these tunings... with a 7 present, I much less
trouble using the 9/7.

>The term I use, and Aaron has quoted, is "temperament extraordinaire,"
>a play on the French _temperament ordinaire_, a 17th-18th century
>genre which typically does have some fifths wider than pure and major
>thirds larger than Pythagorean. Maybe the feature that makes this
>tuning based on Zarlino's 2/7-comma "extraordinaire," rather than
>merely "ordinaire," is the use of regular meantone major thirds a bit
>smaller than a pure 5:4, something I haven't seen documented for an
>historical European 12-note circulating temperament.

I believe George Secor uses that term too. Now I know what he
means!

-Carl

--- In MakeMicroMusic@yahoogroups.com, Margo Schulter <mschulter@...>
wrote:

> The basic idea is to tune eight fifths, F-C#, as in Zarlino's
> 2/7-comma (about 6.145 cents narrow), and the others equally
> wide (about 6.424 cents larger than pure), thus "balancing out"
> the circle, and getting a 12-note circulating system which
> supports a meantone style in the nearer modalities and a
> Pythagorean-to-septimal style in more remote modalities.

Do you have a Scala file for that? Otherwise, I think I'll make one.
If I do, should it be 2/7 comma exactly?

On Wed, 17 May 2006 "Gene Ward Smith" genewardsmith@... wrote:

> --- In MakeMicroMusic@yahoogroups.com, Margo Schulter <mschulter@...>
> wrote:
>
> > The basic idea is to tune eight fifths, F-C#, as in Zarlino's
> > 2/7-comma (about 6.145 cents narrow), and the others equally
> > wide (about 6.424 cents larger than pure), thus "balancing out"
> > the circle, and getting a 12-note circulating system which
> > supports a meantone style in the nearer modalities and a
> > Pythagorean-to-septimal style in more remote modalities.
>
> Do you have a Scala file for that? Otherwise, I think I'll make one.
> If I do, should it be 2/7 comma exactly?

Hi, Gene, and indeed the theoretical version is precisely 2/7 comma for
the eight meantone fifths: it's in the Scala archive under the filename
of schulter_zarte84.scl. Here's essentially the same thing, for which Carl
also gave a fine version:

! zarte84a.scl
!
Temperament extraordinaire, F-C# in Zarlino's 2/7-comma, other 5ths equally wide
12
!
25/24
191.62069
287.43104
383.24139
504.18965
574.86208
695.81035
779.05173
887.43104
995.81035
1079.05173
2/1

The following version shows what happens in one implementation for a
1024-EDO synthesizer (not, at least, part of the Scala archive as far as
I'm aware):

! zarte84n.scl
!
Zarlino temperament extraordinaire, 1024-tET mapping
12
!
70.31250
191.01562
287.10938
383.20313
503.90625
574.21875
696.09375
778.12500
887.10938
996.09375
1079.29688
2/1

Stay tuned for an article I'm posting with MP3 snippets in response to a
query from Carl.

Peace and love,

Margo