Yahoo Tuning Groups Ultimate Backup tuning Proportional beating of various types and new Secor

I think it was H.A. Kellner who first broached the 1/5 *ditonic*
comma meantone as a way to produce the same beat rates in the major
third and fifth of the triad. But if one wants to be exact, the
meantone which does this is 5/23 syntonic comma. George's new tuning
is built on units of 1/23 syntonic comma, with the schisma
incorporated into the fifth between G# and Eb. (I remain ignorant
about what metameantone is...)

There is a whole family of meantones with proportional beating: thez
are fractions 5/26, 5/29, 5/32, 5/35 = 1/7 (etc.) of a syntonic
comma, where the major third beats 2x, 3x, 4x, 5x (etc.) as fast as
the fifth.

If one wants to extend a regular core of such tuning to produce a
temp. ordinaire or true circular temperament, it is handy to note
that including fifths which are tempered by -5 units, -2 units, +1
units and +4 units (where the unit is 1/23 or whatever fraction of a
syntonic comma) will end up producing nice ratios of beat rates.

I wasn't sure what George was doing with the position of F sharp in
his D major triad - maybe I made a mistake in arithmetic, not having
a spreadsheet program handy. I imagined that one would want D-F# to
beat twice as fast as D-A, in which case F# should be up around 586.4
or 586.5 cent, and B-F#-C# would be regular.

I'll end by exhibiting (round the cycle of fifths) a rather mild,
truly circular temperament built on units of 1/32 comma. Cent values
and beat rate ratios (mostly integers) are left as an exercise...

C -5 -5 -5 A -5 -2 -2 F# -2 -2 X Eb -2 -2 -2 C

Where X is the leftover after 34/32 of a syntonic comma have been
taken out of the ditonic comma. It turns out that X is almost equal
to -1 unit, since the ditonic comma is close to 35/32 syntonic commas.

If you prefer the A flat major triad to beat proportionally (and
therefore not the E major one), you can simply shift G# up a tiny bit
so that it is now C#-G# that is tempered by X and Ab-Eb is tempered
by -2/32 comma.

~~~T~~~

🔗Gene Ward Smith <gwsmith@svpal.org>

To be really exact, this gives a brat of almost 4, but not precisely.
Using the formula that a brat of b is a root of the polynomial
(3-2b)x^4 - 10x + 10b, a brat of 4 is from the positive real root of
x^4+2x-8=0.

> There is a whole family of meantones with proportional beating: thez
> are fractions 5/26, 5/29, 5/32, 5/35 = 1/7 (etc.) of a syntonic
> comma, where the major third beats 2x, 3x, 4x, 5x (etc.) as fast as
> the fifth.

Corresponding to brats of 11/4, 7/3, 17/8, 2. It's the 2 which is
interesting.

You can't go just by fifths in isolation, since you need four of them
to get a major third.

An scl file would be nice.

🔗Kraig Grady <kraiggrady@anaphoria.com>

Hello Tom
If you look at the following link you will find the solution to equal beating triads (root position).
Secor has since confirmed this.
You will have to ignore the info on meta mavila at the moment asit is outside any western music application.
http://anaphoria.com/meantone-mavila.PDF

Metameantone can either be constructed either by a recurrent sequence of ratio or one can just take the point at which these converge as the generator.
Secors works better at 12 tones, although a different seeding of metameantone should produce a similar results. On the other hand one can continually add onto metameantone giving higher number scales, not that i am sure that Secor could not come up with good meantones for 19 and 31 etc.
I have been enjoying Secors tuning on a keyboard here

> From: "Tom Dent" <stringph@gmail.com>
>Subject: >
>
>I think it was H.A. Kellner who first broached the 1/5 *ditonic* >comma meantone as a way to produce the same beat rates in the major >third and fifth of the triad. But if one wants to be exact, the >meantone which does this is 5/23 syntonic comma. George's new tuning >is built on units of 1/23 syntonic comma, with the schisma >incorporated into the fifth between G# and Eb. (I remain ignorant >about what metameantone is...)
>
>There is a whole family of meantones with proportional beating: thez >are fractions 5/26, 5/29, 5/32, 5/35 = 1/7 (etc.) of a syntonic >comma, where the major third beats 2x, 3x, 4x, 5x (etc.) as fast as >the fifth. >
>If one wants to extend a regular core of such tuning to produce a >temp. ordinaire or true circular temperament, it is handy to note >that including fifths which are tempered by -5 units, -2 units, +1 >units and +4 units (where the unit is 1/23 or whatever fraction of a >syntonic comma) will end up producing nice ratios of beat rates.
>
>I wasn't sure what George was doing with the position of F sharp in >his D major triad - maybe I made a mistake in arithmetic, not having >a spreadsheet program handy. I imagined that one would want D-F# to >beat twice as fast as D-A, in which case F# should be up around 586.4 >or 586.5 cent, and B-F#-C# would be regular.
>
>I'll end by exhibiting (round the cycle of fifths) a rather mild, >truly circular temperament built on units of 1/32 comma. Cent values >and beat rate ratios (mostly integers) are left as an exercise...
>
>C -5 -5 -5 A -5 -2 -2 F# -2 -2 X Eb -2 -2 -2 C
>
>Where X is the leftover after 34/32 of a syntonic comma have been >taken out of the ditonic comma. It turns out that X is almost equal >to -1 unit, since the ditonic comma is close to 35/32 syntonic commas.
>
>If you prefer the A flat major triad to beat proportionally (and >therefore not the E major one), you can simply shift G# up a tiny bit >so that it is now C#-G# that is tempered by X and Ab-Eb is tempered >by -2/32 comma.
>
>~~~T~
> >
--
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

🔗Tom Dent <stringph@gmail.com>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
> > (...) produce the same beat rates in the major
> > third and fifth of the triad. But if one wants to be exact, the
> > meantone which does this is 5/23 syntonic comma.
>
> To be really exact, this gives a brat of almost 4, but not
precisely.
> Using the formula that a brat of b is a root of the polynomial
> (3-2b)x^4 - 10x + 10b, a brat of 4 is from the positive real root of
> x^4+2x-8=0.

If I understand correctly, this concerns the major third and the
minor third. But I never made a claim about the minor third. Are you
saying that 5/23 comma doesn't really give a 1:1 ratio for the fifth
and the major third? In any case it's close enough for me...

> > (...) fractions 5/26, 5/29, 5/32, 5/35 = 1/7 (etc.) of a syntonic
> > comma, where the major third beats 2x, 3x, 4x, 5x (etc.) as fast
as
> > the fifth.
>
> Corresponding to brats of 11/4, 7/3, 17/8, 2. It's the 2 which is
> interesting.

If you do like 1/7 comma meantone (for m3-related reasons), probably
one can play fairly well with it without any modification, because
the wolf is only half a comma and the thirds going across it are not
too horrible.

> > If one wants to extend a regular core of such tuning to produce a
> > temp. ordinaire or true circular temperament, it is handy to note
> > that including fifths which are tempered by -5 units, -2 units,
+1
> > units and +4 units (where the unit is 1/23 or whatever fraction
of a
> > syntonic comma) will end up producing nice ratios of beat rates.
>
> You can't go just by fifths in isolation, since you need four of
them
> to get a major third.

Er, I do know that. It should be clear what I was talking about, you
start with four or more fifths of a given size (the regular core)
then put fifths of *different* size on each end to hopefully close
the circle. If you do a bit of tinkering with these fractions of a
comma you find, when you do select any of these fifths and put them
on the end, the resulting major third beat rate is guaranteed to have
some integer or half-integer ratio to the fifth beat rate in the
triad. It's just an generalization of what George was doing for the
5/23 comma meantone. I make no claims for the minor 3rd, though...

> > (...) circular temperament built on units of 1/32 comma. Cent
values
> > and beat rate ratios (mostly integers) are left as an exercise...
> >
> > C -5 -5 -5 A -5 -2 -2 F# -2 -2 X Eb -2 -2 -2 C
>
> An scl file would be nice.

Besides not having the relevant program, I'm not sure I can do it
with enough precision for the decimal place fiends here. (I only have
a pocket calculator...) Comma fractions are exact.

Anyway, since there are only three sizes of fifth in the tuning, I
could do rough calcs easily enough. The four fifths written as -5 are
of size:

701.955 - 5/32 * 21.49 = 698.597

the seven fifths written as -2 are of size

701.955 - 2/32 * 21.49 = 700.612

and the remainder is of size

701.955 - 34/32 * 21.49 + 23.46 = 701.328

I expect people may try this and say 'oh, it sounds like equal'...
but that is how I prefer my piano, just a little bit unequal.

~~~T~~~

🔗monz <monz@tonalsoft.com>

Hi Tom,

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:

> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> > --- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
> > >
> > > C -5 -5 -5 A -5 -2 -2 F# -2 -2 X Eb -2 -2 -2 C
> >
> > An scl file would be nice.
>
> Besides not having the relevant program, I'm not sure I
> can do it with enough precision for the decimal place
> fiends here. (I only have a pocket calculator...) Comma
> fractions are exact.

1)
Scala is a free program, easily downloaded and installed,
and works on Windows, Mac, and Linux:

2)
You're posting here from a computer and you're using
a pocket calculator to do your calculations?
?????????

Certainly you can find a free spreadsheet program
that would do a much better job.

3)
For anyone who wants to make the Scala .scl file,
here are the cents values i calculated from Tom's
x/32nds-of-a-comma ... and that was the pythagorean-comma,
right?

C ..... 0
C# ... 94.62374762
D ... 196.5787485
Eb .. 298.5337494
E ... 393.157497
F ... 499.5112498
F# .. 594.1349974
G ... 698.2893742
G# .. 795.1124978
A ... 894.8681227
Bb .. 999.0224996
B .. 1093.646247

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗monz <monz@tonalsoft.com>

🔗Carl Lumma <clumma@yahoo.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> > > There is a whole family of meantones with proportional beating:
> > > thez are fractions 5/26, 5/29, 5/32, 5/35 = 1/7 (etc.) of a
> > > syntonic comma, where the major third beats 2x, 3x, 4x, 5x
> > > (etc.) as fast as the fifth.
> >
> > Corresponding to brats of 11/4, 7/3, 17/8, 2. It's the 2 which
> > is interesting.
>
> Why only the 2?

Look at the various associated beat ratios, meaning between other
pairs in the closed position major triad:

11/4: 7/4 -10 -2/35

7/3: 7/3 -3 -1/7

17/8: 17/8 -4 -2/17

2: 2 -5 -1/10

Only 2 and 7/3 seem reasonably simple, and perhaps I should have
included 7/3. 2 sticks to integers or inverse integers, but it does
have that -1/10. It's an interesting question which beat ratios, for
which chords, we really most want to look at for this business. For
pure meantones, I kind of like -6 or -1:

-6: -6 1/3 -1/2
-1: -1 1 -1

-1 is hard to beat from a pure beat ratio point of view, but the
tuning of -6, 696.296, or 5/19 comma, has a lot to be said for it.

🔗Tom Dent <stringph@gmail.com>

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> > Hi Tom,
> >
> > --- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
> >
> > There is a whole family of meantones with proportional beating: they
> > are fractions 5/26, 5/29, 5/32, 5/35 = 1/7 (etc.) of a syntonic
> > comma, where the major third beats 2x, 3x, 4x, 5x (etc.) as fast as
> > the fifth.
>
> > > > --- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
> > > > >
> > > > > C -5 -5 -5 A -5 -2 -2 F# -2 -2 X Eb -2 -2 -2 C
>
>
> BTW ... your x/32nds-of-a-comma add up to 34 here, not 32.
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software

Er, they're fractions of a SYNTONIC comma, as clear from my original
post and the requirements to produce proportional beating in the M3
and 5 and to have a good temperament in the remote keys.

35/32 syntonic comma is *nearly* a ditonic comma, but not quite, so I
have to write X instead of -1.

My pocket calc. is a Casio fx-85v with full scientific capability,
don't knock it. But I made a mistake in thinking that the syntonic
comma was 21.49 cents rather than 21.506 etc. One should always use
1200 log_2 (81/80) of course...

I've just moved house from UK to Germany, so downloading a spreadsheet
prog. on the temporary system I'm using here and learning how to use
it is *not* high on my list of priorities.

Anyway, here is my CORRECTED calculation of the sizes of the three
types of fifth.

The four fifths written as -5 are:

701.955 - 5/32 * 21.506 = 698.595 -> 1.405c narrower than

the seven fifths written as -2 are of size

701.955 - 2/32 * 21.506 = 700.611 -> 0.611c wider than ET

and the remainder is of size

701.955 + 34/32 * 21.506 - 23.46 = 701.345 -> 1.345c wider than ET

The deviations are as follows:

Eb -1.833
Bb -1.222
F -0.611
C 0
G -1.405
D -2.81
A -4.215
E -5.62
B -5.009
F# -4.398
C# -3.787
G# -3.176

& that's all the accuracy anyone could possibly want!

~~~T~~~

🔗monz <monz@tonalsoft.com>

Ho Tom,

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

> Er, they're fractions of a SYNTONIC comma, as clear from
> my original post

Sorry ... i should have written in my post that i hadn't
really followed the discussion ... just popped in with my
quick contribution 'cause i thot i could help.

> and the requirements to produce proportional beating in the M3
> and 5 and to have a good temperament in the remote keys.
>
> <snip>
>
> Anyway, here is my CORRECTED calculation of the sizes
> of the three types of fifth.
>
> The four fifths written as -5 are:
>
> 701.955 - 5/32 * 21.506 = 698.595 -> 1.405c narrower than
>
> the seven fifths written as -2 are of size
>
> 701.955 - 2/32 * 21.506 = 700.611 -> 0.611c wider than ET
>
> and the remainder is of size
>
> 701.955 + 34/32 * 21.506 - 23.46 = 701.345 -> 1.345c wider than ET
>
> The deviations are as follows:
>
> Eb -1.833
> Bb -1.222
> F -0.611
> C 0
> G -1.405
> D -2.81
> A -4.215
> E -5.62
> B -5.009
> F# -4.398
> C# -3.787
> G# -3.176
>
> & that's all the accuracy anyone could possibly want!

OK, then here's my revised table of values for your
tuning ... and disclaimer to all: consider my last
post totally incorrect, null, and void. Here's the
real deal, with some extra data added this time:

the intermediate columns show x/32nds of a syntonic-comma
(ratio 81/80)

............ ratio ........ cents

G# ....... 1.619312777 ... 834.4580103
-2
C# ....... 1.078704012 ... 131.1588664
-2
F# ....... 1.437155765 ... 627.8597224
-2
B .........1.914720506 .. 1124.560578
-2
E ........ 1.275489653 ... 421.2614345
-5
A ........ 1.697355073 ... 915.9460758
-5
D ........ 1.12937578 .... 210.6307172
-5
G ........ 1.502914356 ... 705.3153586
-5
C ........ 1.0 ............. 0
-2
F ........ 1.332298525 ... 496.700856
-2
Bb ....... 1.77501936 .... 993.4017121
-2
Eb ....... 1.182427838 ... 290.1025681

-monz
http://tonalsoft.com
Tonescape microtonal music software

🔗monz <monz@tonalsoft.com>

🔗Tom Dent <stringph@gmail.com>

I don't know where you got those numbers from - they are absolutely
crazy. All the fifths are sharp!!

I did actually go through and add up the deviations of each fifth from
ET to get the pitch of each note relative to ET. Should be child's
play to produce the Scala from there.

DEVIATIONS FROM ET IN CENTS (cut-n-paste from previous post)

> > C 0
> > C# -3.787
> > D -2.81
> > Eb -1.833
> > E -5.62
> > F -0.611
> > F# -4.398
> > G -1.405
> > G# -3.176
> > A -4.215
> > Bb -1.222
> > B -5.009

CENT PITCHES

0
96.213
197.19
298.167
394.38
499.389
595.602
698.595
796.824
895.785
998.778
1094.991

Clear???

~~~T~~~

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> Hi Tom,
>
>
> >
> > Anyway, here is my CORRECTED calculation of the sizes
> > of the three types of fifth.
> >
> > The four fifths written as -5 are:
> >
> > 701.955 - 5/32 * 21.506 = 698.595 -> 1.405c narrower than ET
> >
> > the seven fifths written as -2 are of size
> >
> > 701.955 - 2/32 * 21.506 = 700.611 -> 0.611c wider than ET
> >
> > and the remainder is of size
> >
> > 701.955 + 34/32 * 21.506 - 23.46 = 701.345 -> 1.345c wider than ET
> >
> > The deviations are as follows:
> >
> > Eb -1.833
> > Bb -1.222
> > F -0.611
> > C 0
> > G -1.405
> > D -2.81
> > A -4.215
> > E -5.62
> > B -5.009
> > F# -4.398
> > C# -3.787
> > G# -3.176
> >
> > & that's all the accuracy anyone could possibly want!
>
> [monz]:
>
> OK, then here's my revised table of values for your
> tuning ... and disclaimer to all: consider my last
> post totally incorrect, null, and void. Here's the
> real deal, with some extra data added this time:
>
> the intermediate columns show x/32nds of a syntonic-comma
> (ratio 81/80)
>
>
> ............ ratio ........ cents
>
> G# ....... 1.619312777 ... 834.4580103
> -2
> C# ....... 1.078704012 ... 131.1588664
> -2
> F# ....... 1.437155765 ... 627.8597224
> -2
> B .........1.914720506 .. 1124.560578
> -2
> E ........ 1.275489653 ... 421.2614345
> -5
> A ........ 1.697355073 ... 915.9460758
> -5
> D ........ 1.12937578 .... 210.6307172
> -5
> G ........ 1.502914356 ... 705.3153586
> -5
> C ........ 1.0 ............. 0
> -2
> F ........ 1.332298525 ... 496.700856
> -2
> Bb ....... 1.77501936 .... 993.4017121
> -2
> Eb ....... 1.182427838 ... 290.1025681
>
>
> -monz
>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:

> (I remain ignorant
> about what metameantone is...)

Welcome Tom,

Metameantone (named by Erv Wilson) is the meantone where the beat
rates in a root-position, close-voiced major triad are in proportions
of 1:1:1.

The fifth of regular metameantone is 695.6304 cents. I say 'regular'
because metameantone can also be understood in terms of an infinite
series of unequal JI ratios of increasing complexity, based off of a
recurrent sequence, as shown in Wilson's document (referenced by
Kraig in his reply). While in the infinite limit, the sequence
converges on the regular metameantone, the early portions of the
sequence still have the exact 1:1:1 beat ratios.

And in case you missed this:

The fifth of 5/17-comma meantone is 695.6296 cents.

A difference of less than a thousandth of a cent.

To do better with a n/d-comma meantone, you have to go all the way to
452/1537-comma meantone!

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:

> You will have to ignore the info on meta mavila at the moment asit is
> outside any western music application.
> http://anaphoria.com/meantone-mavila.PDF

If I may interject, I think it is about time that musicians of the
world (which cannot be reasonably divided into East and West anymore),
especially Aaron Johnston :), have a go at Mavila tunings. While
Meantone is based on a vanishing 81:80, Mavila is based on a vanishing
135:128. As it turns out, both systems can therefore be understood as
having a period of an octave and a generator of a fifth, though the
fifth is much flatter in the case of Mavila. If Western music is mapped
from the chain of meantone fifths to the chain of Mavila fifths, the
musical result is wonderful and fascinating -- it's much as if all the
major intervals and chords become minor, and all the minor intervals
and chords become major. These chords are considerably further from JI
than their meantone enantio-counterparts, so it's best to use
percussive, not-especially-harmonic timbres with this tuning, to
prevent rapid beatings and other unpleasant hallmarks of mistuning from
JI. Or you could use specially designed inharmonic timbres, a la Bill
Sethares.

My 'Middle Path' paper contains a particular optimization, and diagrams
a series of scales, for Mavila and 53 other systems of temperament
(some of 5-limit, and some of 7-limit JI). Anyone who wants a copy of
the current version of this paper, send me your snail-mail address and
I'll mail it to you for free.

🔗Ozan Yarman <ozanyarman@superonline.com>

It would be nice if we could have a list of cent-values for easy copy-paste into Scala Paul. I want to try out this Mavilla myself.

Cordially,
Ozan
----- Original Message -----
From: wallyesterpaulrus
To: tuning@yahoogroups.com
Sent: 06 Eylül 2005 Salı 23:19
Subject: [tuning] Mavila (was: Re: Proportional beating of various types and new Secor)

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:

> You will have to ignore the info on meta mavila at the moment asit is
> outside any western music application.
> http://anaphoria.com/meantone-mavila.PDF

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

Ozan, you may read it off from the second section of the 'Middle
Path' paper I sent you. It's spelled "Mavila". Be sure that your
synth can support an interval of repetition that is not precisely
1200 cents; otherwise, I'll have to propose an alternate tuning.

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> It would be nice if we could have a list of cent-values for easy
copy-paste into Scala Paul. I want to try out this Mavilla myself.
>
> Cordially,
> Ozan
> ----- Original Message -----
> From: wallyesterpaulrus
> To: tuning@yahoogroups.com
> Sent: 06 Eylül 2005 Salý 23:19
> Subject: [tuning] Mavila (was: Re: Proportional beating of
various types and new Secor)
>
>
> --- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...>
wrote:
>
> > You will have to ignore the info on meta mavila at the moment
asit is
> > outside any western music application.
> > http://anaphoria.com/meantone-mavila.PDF
>
> If I may interject, I think it is about time that musicians of
the
> world (which cannot be reasonably divided into East and West
anymore),
> especially Aaron Johnston :), have a go at Mavila tunings. While
> Meantone is based on a vanishing 81:80, Mavila is based on a
vanishing
> 135:128. As it turns out, both systems can therefore be
understood as
> having a period of an octave and a generator of a fifth, though
the
> fifth is much flatter in the case of Mavila. If Western music is
mapped
> from the chain of meantone fifths to the chain of Mavila fifths,
the
> musical result is wonderful and fascinating -- it's much as if
all the
> major intervals and chords become minor, and all the minor
intervals
> and chords become major. These chords are considerably further
from JI
> than their meantone enantio-counterparts, so it's best to use
> percussive, not-especially-harmonic timbres with this tuning, to
> prevent rapid beatings and other unpleasant hallmarks of
mistuning from
> JI. Or you could use specially designed inharmonic timbres, a la
Bill
> Sethares.
>
> My 'Middle Path' paper contains a particular optimization, and
diagrams
> a series of scales, for Mavila and 53 other systems of
temperament
> (some of 5-limit, and some of 7-limit JI). Anyone who wants a
copy of
> the current version of this paper, send me your snail-mail
address and
> I'll mail it to you for free.

🔗Herman Miller <hmiller@IO.COM>

Ozan Yarman wrote:
> It would be nice if we could have a list of cent-values for easy > copy-paste into Scala Paul. I want to try out this Mavilla myself.
> I usually just use the PYTH function and type in the generator and period values (since it depends on how many notes you want to use, what pitch to center on, how many fifths up or down, and so on).

Here for instance is a 9-note mavila scale:

! mavila9.scl
!
9-note scale of mavila temperament (TOP tuning)
9
!
163.50770
327.01540
358.01258
521.52028
685.02798
848.53568
879.53287
1043.04057
1206.54826

And a 12-note version for warping major to minor / minor to major:

! mavila12.scl
!
A 12-note mavila scale (for warping meantone-based music)
12
!
-30.99719
163.50770
358.01258
327.01540
521.52028
490.52310
685.02798
654.03080
848.53568
1043.04057
1012.04338
1206.54826

But this is one of many possible mavila tunings, and 23-ET is another. (Parts of Easley Blackwood's 23-ET microtonal etude might be analyzed as using a mavila scale.) I have a kalimba which I used to tune to a 7-note subset of 16-ET, which is also in the range of mavila temperament. In some ways I think 16-ET is better for warping major to minor and vice versa, since the small steps aren't so small as they are in the TOP tuning.

On a somewhat related note, I recently discovered that the Virtual Sound Canvas DXi has a lot more than the regular General MIDI sounds, and I've been adapting some of my MIDI files to use these "new" timbres. Here's an example in TOP mavila tuning that I did the other day:

http://home.comcast.net/~teamouse/kenet-vsc.mp3

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Herman Miller <hmiller@IO.COM>

Gene Ward Smith wrote:

> Can this be used as a sound font? How do you use it?

This is a DXi plugin which was bundled with the version of Cakewalk that I have, but is also available separately from Edirol (with both DXi and VST plugins). It's an emulation of the old Roland Sound Canvas in software.

http://www.edirol.com/products/info/vscmp1.html

This version apparently can take MIDI files and make them into WAV files directly, according to the web site. What I've got is just the DXi plugin, which runs inside of the Cakewalk application. DXi plugins act similar to MIDI devices, except that their parameter settings show up directly on the Cakewalk menus. It turns out that the VSC has controls for things like brightness, vibrato speed, attack time, and things like that in addition to the normal MIDI controls like expression and modulation. As far as I can tell it doesn't have any way to retune individual notes aside from MIDI pitch bends, but it does respond well to pitch bends on multiple channels (which is how I'm using it).

🔗monz <monz@tonalsoft.com>

🔗Kraig Grady <kraiggrady@anaphoria.com>

I agree with most of what you point out here , but say i have not had any problem using normal timbre with this tuning.

On my Interiors CD done in the late 90's i used it on both the organ and the hammer dulcimer piece. One reviewer thought i was a using a synth on the organ with sometype of phase shifter and despite what i had sent out, people refused to believe that this was being done by acoustical means. nor truly understand what it was they were hearing.

My latest CD Without R&R likewise uses this scale on hammer dulcimer.

Unlike meantone which is the third is determined by going up 4 fifths, you go the opposite direction down 3 fifths.

Like Meta Slendro, there is much to be gained by also looking at the converging sequence that spawned the scale in the first place.
if seeded and approached in such a way, one can have all type of subtle variations in the scale that are extremely musical. and is a wide open door for an individual scale determined strictly by ones own personal taste. Which will vary from Paul's version, so i recommend one try these versions also found here.
I have used the scale starting at 11 and 37 ( my more recent favorite)
http://anaphoria.com/meantone-mavila.PDF
Page 5 shows a possible keyboard layout as well as some frequencies Also there is another recommended seed at the top of page 2.
While it does not, in my opinion really make the best Pelogs, it does a good job of duplicating those pelog-like ones that are we find in Thailand and Burma, not to mention the Chopi scales in Mozambique ( which contains the village of Mavila ). Although it does not appear that there is an ensemble there any more.
One reason i understand that the decline in marimba ensembles in this area of the world is the wood wenge, had become so valuable due to demand, that those there can no longer afford it.

>Message: 1 > Date: Tue, 06 Sep 2005 20:19:01 -0000
> From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
>Subject: Mavila (was: Re: Proportional beating of various types and new Secor)
>
>--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
>
> >
>>You will have to ignore the info on meta mavila at the moment asit is >>outside any western music application.
>>http://anaphoria.com/meantone-mavila.PDF
>> >>
>
>If I may interject, I think it is about time that musicians of the >world (which cannot be reasonably divided into East and West anymore), >especially Aaron Johnston :), have a go at Mavila tunings. While >Meantone is based on a vanishing 81:80, Mavila is based on a vanishing >135:128. As it turns out, both systems can therefore be understood as >having a period of an octave and a generator of a fifth, though the >fifth is much flatter in the case of Mavila. If Western music is mapped >from the chain of meantone fifths to the chain of Mavila fifths, the >musical result is wonderful and fascinating -- it's much as if all the >major intervals and chords become minor, and all the minor intervals >and chords become major. These chords are considerably further from JI >than their meantone enantio-counterparts, so it's best to use >percussive, not-especially-harmonic timbres with this tuning, to >prevent rapid beatings and other unpleasant hallmarks of mistuning from >JI. Or you could use specially designed inharmonic timbres, a la Bill >Sethares.
>
>My 'Middle Path' paper contains a particular optimization, and diagrams >a series of scales, for Mavila and 53 other systems of temperament >(some of 5-limit, and some of 7-limit JI). Anyone who wants a copy of >the current version of this paper, send me your snail-mail address and >I'll mail it to you for free.
>
>
>
> >
>
> >

--
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

🔗Tom Dent <stringph@gmail.com>

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
>
> > (I remain ignorant
> > about what metameantone is...)
>
> Welcome Tom,
>
> Metameantone (named by Erv Wilson) is the meantone where the beat
> rates in a root-position, close-voiced major triad are in proportions
> of 1:1:1.

I didn't know that was possible!

> The fifth of regular metameantone is 695.6304 cents. I say 'regular'
> because metameantone can also be understood in terms of an infinite
> series of unequal JI ratios of increasing complexity, based off of a
> recurrent sequence, as shown in Wilson's document (referenced by
> Kraig in his reply). While in the infinite limit, the sequence
> converges on the regular metameantone, the early portions of the
> sequence still have the exact 1:1:1 beat ratios.

OK, that was the handwritten document that I didn't understand. A case
of a picture that would take a thousand words to explain...

> And in case you missed this:
>
> The fifth of 5/17-comma meantone is 695.6296 cents.
>
> A difference of less than a thousandth of a cent.

So, that's another link in the chain of 5/X comma meantones with X =
17, 20, 23, 26, 29, 32, 35 ... that have very close to proportional
beat relations - 5/20 and 5/35 being the simplest.

Which one you select for building a proportional-beating temperament
depends on how flat you like your fifths and thirds.

~~~T~~~

🔗George D. Secor <gdsecor@yahoo.com>

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
>
> I think it was H.A. Kellner who first broached the 1/5 *ditonic*
> comma meantone as a way to produce the same beat rates in the major
> third and fifth of the triad. But if one wants to be exact, the
> meantone which does this is 5/23 syntonic comma. George's new
tuning
> is built on units of 1/23 syntonic comma, with the schisma
> incorporated into the fifth between G# and Eb. (I remain ignorant
> about what metameantone is...)

I see that Paul explained it briefly:

/tuning/topicId_60165.html#60212

> There is a whole family of meantones with proportional beating:
thez
> are fractions 5/26, 5/29, 5/32, 5/35 = 1/7 (etc.) of a syntonic
> comma, where the major third beats 2x, 3x, 4x, 5x (etc.) as fast as
> the fifth.

And by now I guess you're also familiar with 5/17, 5/14, 5/11, 5/8 as
1x, 2x, 3x., 4x.

> If one wants to extend a regular core of such tuning to produce a
> temp. ordinaire or true circular temperament, it is handy to note
> that including fifths which are tempered by -5 units, -2 units, +1
> units and +4 units (where the unit is 1/23 or whatever fraction of
a
> syntonic comma) will end up producing nice ratios of beat rates.
>
> I wasn't sure what George was doing with the position of F sharp in
> his D major triad - maybe I made a mistake in arithmetic, not
having
> a spreadsheet program handy. I imagined that one would want D-F# to
> beat twice as fast as D-A, in which case F# should be up around
586.4
> or 586.5 cent, and B-F#-C# would be regular.

Since you didn't see the figures in my spreadsheet, I should give
them here for the 8 proportional-beating triads::

Major ----Beat Ratios----
Triad M3/5th m3/5th m3/M3
----- ------ ------ ------
Eb... 15.000 20.000 1.3333
Bb... ------ ------ 1.5000
F.... 5.0000 10.000 2.0000
C.... 1.0000 4.0000 4.0000
G.... 1.0000 4.0000 4.0000
D.... 1.6667 5.0000 3.0000
A.... 3.0000 7.0000 2.3333
E.... 5.0000 10.000 2.0000

I was trying to get whole-number ratios in as many places as
possible. F# at 586.46582 cents would give:

D.... 2.0000 5.5000 2.7500

which, in addition to having only one integer multiple, adds about
1.8 cents of total error (~10% more) to the D major triad and removes
1.8 cents of total error (or < 4% improvement) from B major -- not
worth the trade-off, IMO. (I might add that, when I first heard it,
I was rather impressed by the sound of the D major triad -- much
better than I expected.)

> I'll end by exhibiting (round the cycle of fifths) a rather mild,
> truly circular temperament built on units of 1/32 comma. Cent
values
> and beat rate ratios (mostly integers) are left as an exercise...
>
> C -5 -5 -5 A -5 -2 -2 F# -2 -2 X Eb -2 -2 -2 C
>
> Where X is the leftover after 34/32 of a syntonic comma have been
> taken out of the ditonic comma. It turns out that X is almost equal
> to -1 unit, since the ditonic comma is close to 35/32 syntonic
commas.

From looking at the numbers, I'm quite impressed that you've kept the
maximum total absolute error in *all* of the triads significantly
below that of a pythagorean triad -- your 3 worst triads (B, F#, C#)
are about halfway between 12-ET and pythagorean. AKJ would probably
be interested in this.

The only criticism I could make is that I would prefer to see roughly
equal amounts of total absolute error in F and G, in Bb and D, and in
Eb and A, or possibly an amount in Eb between that in A and E, so as
to make the 8 best major triads in the temperament correspond to
those usable in the Eb-to-G# meantone temperament. (A triad's total
absolute error is the sum of the absolute values of the errors in the
three intervals of the triad.) Like most well-temperaments I've
seen, yours is biased so as to favor the triads in the flat direction.

Anyway, it looks very nice -- I'll have to try this out in Scala.
For anyone interested, here's a file:

! Dent_523WT.scl
!
Tom Dent's 5/32-comma proportional-beating well-temperament
12
!
96.14515
197.18098
298.16743
394.36196
499.38914
595.55308
698.59049
796.89241
895.77147
998.77828
1094.95868
2/1

> If you prefer the A flat major triad to beat proportionally (and
> therefore not the E major one), you can simply shift G# up a tiny
bit
> so that it is now C#-G# that is tempered by X and Ab-Eb is tempered
> by -2/32 comma.

That improves C# major at the expense of E major, which I think is
not very desirable.

--George

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:

> I agree with most of what you point out here , but say i have not
had
> any problem using normal timbre with this tuning.

We probably use it differently. With most timbres, the fifths and
fourths of Mavila tunings are too rough for me, as I'm trying to use
them analogously to the diatonic fifths and fourths, but they're
tempered so severly that they don't sound 'perfect' at all. With
marimba-type timbres, though, these intervals don't have many audible
traces of being "out-of-tune", and work just fine harmonically.
Melodically, you have a new paradigm to get used to, a week or two is
usually enough for my ears.

> One reason i understand that the decline in marimba

Ha!

>ensembles in
> this area of the world is the wood wenge, had become so valuable
>due to
> demand, that those there can no longer afford it.

That's funny -- my bed is wenge. I'd happily have it converted into a
marimba -- don't get much sleep these days anyway!

The decline is tragic, don't mean to make light of it of course . . .

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> > --- In tuning@yahoogroups.com, "Tom Dent" <stringph@g...> wrote:
> >
> > > (I remain ignorant
> > > about what metameantone is...)
> >
> > Welcome Tom,
> >
> > Metameantone (named by Erv Wilson) is the meantone where the beat
> > rates in a root-position, close-voiced major triad are in
proportions
> > of 1:1:1.
>
> I didn't know that was possible!
>
> > The fifth of regular metameantone is 695.6304 cents. I
say 'regular'
> > because metameantone can also be understood in terms of an
infinite
> > series of unequal JI ratios of increasing complexity, based off
of a
> > recurrent sequence, as shown in Wilson's document (referenced by
> > Kraig in his reply). While in the infinite limit, the sequence
> > converges on the regular metameantone, the early portions of the
> > sequence still have the exact 1:1:1 beat ratios.
>
> OK, that was the handwritten document that I didn't understand. A
case
> of a picture that would take a thousand words to explain...
>
> > And in case you missed this:
> >
> > The fifth of 5/17-comma meantone is 695.6296 cents.
> >
> > A difference of less than a thousandth of a cent.
>
>
> So, that's another link in the chain of 5/X comma meantones with X =
> 17, 20, 23, 26, 29, 32, 35 ... that have very close to proportional
> beat relations - 5/20 and 5/35 being the simplest.

Why are those the simplest if ~5/17 gives you 1:1:1, the simplest
possible "tratio"?

> Which one you select for building a proportional-beating temperament
> depends on how flat you like your fifths and thirds.

Or on whether you are looking for, say, a circulating 19-note
temperament . . .