Fwd: Re: understanding 15 EDO

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
--- In tuning@yahoogroups.com, "Stephen Szpak" <stephen_szpak@h...>
wrote:
>
> (If anyone wants to comment to this that's fine. Please try to be
as
>simple as possible.)
>
>
> Is it common to alter 15 EDO to make it fifths significatly less
than 720
>cents?

I wouldn't call anything involving 15 common, but I have proposed 15
out of 27-et, or tripletone[15], as a scale.

STEPHEN SZPAK WRITES:

What? Is this a 15 note scale or a 27 note scale?
====================================================
> Also, since a comma is about 25 cents, and the fifths are about
20 cents
>sharp,
> are these 2 things related? What are "vanishing commas"? Does
12 EDO
>have them
> and why? Does 15 EDO have them and why?

Any equal temperament has them. Both 12 and 15 have 400 cent major
thirds, so they share 128/125 as a vanishing comma.

STEPHEN SZPAK WRITES:

Not that it is important but 128/125 is about 41 cents. What has 41 cents got to
do with 400 cent thirds? What actually vanishes?

--- End forwarded message ---
=======================================================
Stephen Szpak

_________________________________________________________________
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--- In tuning@yahoogroups.com, "Stephen Szpak" <stephen_szpak@h...>
wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> I wouldn't call anything involving 15 common, but I have proposed 15
> out of 27-et, or tripletone[15], as a scale.
>
> STEPHEN SZPAK WRITES:
>
> What? Is this a 15 note scale or a 27 note scale?

A 15-note scale, consisting of three chains of fifths separated by
400 cent thirds, where the fifths in each chain are tuned to the
27-et fifth of 711.111 cents, or something close to that.

> Any equal temperament has them. Both 12 and 15 have 400 cent major
> thirds, so they share 128/125 as a vanishing comma.
>
> STEPHEN SZPAK WRITES:
>
> Not that it is important but 128/125 is about 41 cents. What
has 41
> cents got to
> do with 400 cent thirds? What actually vanishes?

What vanishes is that three major thirds do not lead to something a
bit flat from an octave, but are exactly an octave.

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

>I wouldn't call anything involving 15 common, but I have proposed 15
>out of 27-et, or tripletone[15], as a scale.
>
>STEPHEN SZPAK WRITES:
>
> What? Is this a 15 note scale or a 27 note scale?

A 15-note scale, consisting of three chains of fifths separated by
400 cent thirds, where the fifths in each chain are tuned to the
27-et fifth of 711.111 cents, or something close to that.

STEPHEN SZPAK WRITES:

As I try to write this out in a cent series of a scale (0-80-160-240 etc)
or whatever the
real terminology is, perhaps you could send it. The 15 EDO scale has 3
circles of
interlaced fifths with 400 cent thirds already, so, I must be missing
something. The
fifth of 711.111 cents looks good. My concern with altering 15 EDO is
losing the
560 so it becomes sharper AND/OR losing the 960 so it becomes flatter.
Someone wrote to me a while back about using MOS I think. I remember
something
about altering the 15 EDO in a ABC-ABC-ABC-ABC-ABC fashion (I'm sure this
is not the
proper terminology either). Thanks for helping.

FLASH FLASH FLASH: STEPHEN SZPAK WRITES:

Since you haven't had the time to respond (which is O.K.) I have come up with a
new scale. I took 15 EDO and trashed it. I created (hope this is right) 5 tonics with
300 cent minor thirds, 400 cent major thirds and 700 cent fifths. If one uses the tonic
of C you get a subminor third as well. Plus a sub 4th (461 cents) a semi-augmented 4th
(561 cents) and a harmonic 7 th at 961 cents. The goal of all this was 1) to reduce the
fifths from 720 cents to 700 and 2) to create tonics that are not even close to those in
12 EDO. If anyone is interested I might post it. After all it's NOT 15 EDO anymore. It's
a different scale from 12 EDO AND 15EDO with the advantage of actually being playable
by someone with average size hands.

Looking forward to a response to all this at your convenience. Comments from others
are welcome as well.

>Any equal temperament has them. Both 12 and 15 have 400 cent major
>thirds, so they share 128/125 as a vanishing comma.
>
> STEPHEN SZPAK WRITES:
>
> Not that it is important but 128/125 is about 41 cents. What
has 41
>cents got to
> do with 400 cent thirds? What actually vanishes?

What vanishes is that three major thirds do not lead to something a
bit flat from an octave, but are exactly an octave.

STEPHEN SZPAK WRITES:

I understand now. (At least as well as I can.) I would never have come up
with this. Thanks.

Stephen Szpak

_________________________________________________________________
Have fun customizing MSN Messenger � learn how here! http://www.msnmessenger-download.com/tracking/reach_customize

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
--- In tuning@yahoogroups.com, "Stephen Szpak" <stephen_szpak@h...>
wrote:
>
> (If anyone wants to comment to this that's fine. Please try to be
as
>simple as possible.)
>
>
> Is it common to alter 15 EDO to make it fifths significatly less
than 720
>cents?

You can view it as altering 15-equal, or perhaps better is to view 15-
equal as the alteration of such scales. Gene gave you a couple of
great examples. Another important one, at least for 5-limit, would be
Kleismic-15:

0
68.319
180.44
248.76
317.08
385.4
497.52
565.84
634.16
702.48
814.6
882.92
951.24
1019.6
1131.7

Lots of quite pure major and minor triads here.

> are these 2 things related? What are "vanishing commas"? Does 12 EDO
>have them
> and why? Does 15 EDO have them and why?

I'm glad you finally (and pretty quickly) understood Gene's comment
on this. More could be said. See my reply to Mark, which I hope
you'll read and try to follow. In that post I brought up the 'maximal
diesis', which can be expressed as 250;243 -- it vanishes in 15-equal
but not in 12-equal. This implies certain 'porcupine' scale
formations and chord progressions which work in 15-equal but not in
12-equal -- Herman Miller has explored these in his music. Meanwhile,
perhaps the most important comma of all, the 'syntonic comma' or
simply 'the comma' or 81;80, vanishes in 12-equal but not in 15-
equal; thus most conventional common-practice triadic diatonic music
will work in 12-equal but not in 15-equal.

STEPHEN SZPAK WRITES:::::::::::::::::

Just writting to let you know I read this. I hope to analyze your scale if I can get caught up
(and learn how to use Scala 18). Thanks. As for understanding people's comments,
that is problematic.

Stephen Szpak

_________________________________________________________________
Expand your wine savvy � and get some great new recipes � at MSN Wine. http://wine.msn.com

--- In tuning@yahoogroups.com, "Stephen Szpak" <stephen_szpak@h...> wrote:
--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
--- In tuning@yahoogroups.com, "Stephen Szpak" <stephen_szpak@h...>
wrote:
>
> (If anyone wants to comment to this that's fine. Please try to be
as
>simple as possible.)
>
>
> Is it common to alter 15 EDO to make it fifths significatly less
than 720
>cents?

You can view it as altering 15-equal, or perhaps better is to view 15-
equal as the alteration of such scales. Gene gave you a couple of
great examples. Another important one, at least for 5-limit, would be
Kleismic-15:

0
68.319
180.44
248.76
317.08
385.4
497.52
565.84
634.16
702.48
814.6
882.92
951.24
1019.6
1131.7

Lots of quite pure major and minor triads here.

> are these 2 things related? What are "vanishing commas"? Does 12 EDO
>have them
> and why? Does 15 EDO have them and why?

I'm glad you finally (and pretty quickly) understood Gene's comment
on this. More could be said. See my reply to Mark, which I hope
you'll read and try to follow. In that post I brought up the 'maximal
diesis', which can be expressed as 250;243 -- it vanishes in 15-equal
but not in 12-equal. This implies certain 'porcupine' scale
formations and chord progressions which work in 15-equal but not in
12-equal -- Herman Miller has explored these in his music. Meanwhile,
perhaps the most important comma of all, the 'syntonic comma' or
simply 'the comma' or 81;80, vanishes in 12-equal but not in 15-
equal; thus most conventional common-practice triadic diatonic music
will work in 12-equal but not in 15-equal.

STEPHEN SZPAK WRITES:::::::::::::::::

Just writting to let you know I read this. I hope to analyze your scale
if I can get caught up
(and learn how to use Scala 18). Thanks. As for understanding people's
comments,
that is problematic.

STEPHEN SZPAK WRITES::::::::::::::::::

After getting caught up with things I can now reply.
If 3 major 3rds lead exactly to an octave we have a vanishing comma. A comma
is 128/125 which is about equal to 41 cents. What 41 cents has to do with anything I
don't know.
As for the 250/243 of 49.166 cents...I don't know what that's got to do with
either.
Please state the message number of "my reply to Mark" sited above.

Stephen Szpak

_________________________________________________________________
Make your home warm and cozy this winter with tips from MSN House & Home. http://special.msn.com/home/warmhome.armx

--- In tuning@yahoogroups.com, "Stephen Szpak" <stephen_szpak@h...>
wrote:

> After getting caught up with things I can now reply.
> If 3 major 3rds lead exactly to an octave we have a
vanishing
> comma. A comma
> is 128/125 which is about equal to 41 cents.

That particular "comma" is called a diesis, not a comma. Confusing,
but there it is. "Comma" means both some small interval, often one
which is being tempered out, and certain particular such intervals,
81/80 and 3^12/2^19.

--- In tuning@yahoogroups.com, "Stephen Szpak" <stephen_szpak@h...>
wrote:
>
> --- In tuning@yahoogroups.com, "Stephen Szpak" <stephen_szpak@h...>
wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, "Stephen Szpak" <stephen_szpak@h...>
> wrote:
> >
> > (If anyone wants to comment to this that's fine. Please try to
be
> as
> >simple as possible.)
> >
> >
> > Is it common to alter 15 EDO to make it fifths significatly
less
> than 720
> >cents?
>
> You can view it as altering 15-equal, or perhaps better is to view
15-
> equal as the alteration of such scales. Gene gave you a couple of
> great examples. Another important one, at least for 5-limit, would
be
> Kleismic-15:
>
> 0
> 68.319
> 180.44
> 248.76
> 317.08
> 385.4
> 497.52
> 565.84
> 634.16
> 702.48
> 814.6
> 882.92
> 951.24
> 1019.6
> 1131.7
>
> Lots of quite pure major and minor triads here.
>
> > are these 2 things related? What are "vanishing commas"? Does
12 EDO
> >have them
> > and why? Does 15 EDO have them and why?
>
> I'm glad you finally (and pretty quickly) understood Gene's comment
> on this. More could be said. See my reply to Mark, which I hope
> you'll read and try to follow. In that post I brought up
the 'maximal
> diesis', which can be expressed as 250;243 -- it vanishes in 15-
equal
> but not in 12-equal. This implies certain 'porcupine' scale
> formations and chord progressions which work in 15-equal but not in
> 12-equal -- Herman Miller has explored these in his music.
Meanwhile,
> perhaps the most important comma of all, the 'syntonic comma' or
> simply 'the comma' or 81;80, vanishes in 12-equal but not in 15-
> equal; thus most conventional common-practice triadic diatonic music
> will work in 12-equal but not in 15-equal.
>
> STEPHEN SZPAK WRITES:::::::::::::::::
>
> Just writting to let you know I read this. I hope to analyze
your scale
> if I can get caught up
> (and learn how to use Scala 18). Thanks. As for understanding
people's
> comments,
> that is problematic.
>
> STEPHEN SZPAK WRITES::::::::::::::::::
>
> After getting caught up with things I can now reply.
> If 3 major 3rds lead exactly to an octave we have a
vanishing
> comma. A comma
> is 128/125 which is about equal to 41 cents. What 41 cents has
to do with
> anything I
> don't know.
> As for the 250/243 of 49.166 cents...I don't know what
that's got
> to do with
> either.
> Please state the message number of "my reply to Mark"
sited
> above.
>
>
>
> Stephen Szpak
>
> _________________________________________________________________
> Make your home warm and cozy this winter with tips from MSN House &
Home.
> http://special.msn.com/home/warmhome.armx

Hi Stephen,

Thanks for your patience. I know this material and the language
associated with it can be rough going at first, and you're doing
impressively well considering your short time here.

When I wrote:

> I'm glad you finally (and pretty quickly) understood Gene's comment
> on this.

I was referring to

/tuning/topicId_50568.html#50582

But I guess I misinterpreted your reply there. It's true that 128:125
is about 41 cents. When we speak of 128;125 vanishing, though, we are
no longer speaking about Just Intonation. We are speaking of some
temperament where all the consonant intervals are adjusted by some
small amount. In Just Intonation, 128:125 is the difference between
three major thirds and an octave (for example). So if we say 128;125
vanishes in some temperament, we mean that the difference between
three major thirds and an octave is zero in that temperament.

Similarly for 250:243. It's the difference between three minor thirds
and two fourths (for example). In Just Intonation it happens to be 49
cents. In 12-equal it's 100 cents. But in 15-equal and some other
temperaments (called "porcupine" temperaments), it's zero, it
vanishes. This fact leads to new scales and chord progressions that
are, some say, as "natural" in 15-equal (and other porcupine
temperaments) as the diatonic scale and its typical chord
progressions are in 12-equal (and other meantone temperaments).

The reply to Mark I was referring to was

/tuning/topicId_21432.html#50652

I'm sure you'll have more questions, and I'll do my best to answer
them!

-Paul

>--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
>--- In tuning@yahoogroups.com, "Stephen Szpak" <stephen_szpak@h...>
>wrote:
> >
> > (If anyone wants to comment to this that's fine. Please try to
be
>as
> >simple as possible.)
> >
> >
> > Is it common to alter 15 EDO to make it fifths significatly
less
>than 720
> >cents?
>
>You can view it as altering 15-equal, or perhaps better is to view
15-
>equal as the alteration of such scales. Gene gave you a couple of
>great examples. Another important one, at least for 5-limit, would
be
>Kleismic-15:
>
> 0
> 68.319
> 180.44
> 248.76
> 317.08
> 385.4
> 497.52
> 565.84
> 634.16
> 702.48
> 814.6
> 882.92
> 951.24
> 1019.6
> 1131.7
>
>Lots of quite pure major and minor triads here.
>
> > are these 2 things related? What are "vanishing commas"? Does
12 EDO
> >have them
> > and why? Does 15 EDO have them and why?
>
>I'm glad you finally (and pretty quickly) understood Gene's comment
>on this. More could be said. See my reply to Mark, which I hope
>you'll read and try to follow. In that post I brought up
the 'maximal
>diesis', which can be expressed as 250;243 -- it vanishes in 15-
equal
>but not in 12-equal. This implies certain 'porcupine' scale
>formations and chord progressions which work in 15-equal but not in
>12-equal -- Herman Miller has explored these in his music.
Meanwhile,
>perhaps the most important comma of all, the 'syntonic comma' or
>simply 'the comma' or 81;80, vanishes in 12-equal but not in 15-
>equal; thus most conventional common-practice triadic diatonic music
>will work in 12-equal but not in 15-equal.
>
>

Hi Stephen,

When I wrote:

>I'm glad you finally (and pretty quickly) understood Gene's comment
>on this.

I was referring to

/tuning/topicId_50568.html#50582

But I guess I misinterpreted your reply there. It's true that 128:125
is about 41 cents. When we speak of 128;125 vanishing, though, we are
no longer speaking about Just Intonation. We are speaking of some
temperament where all the consonant intervals are adjusted by some
small amount. In Just Intonation, 128:125 is the difference between
three major thirds and an octave (for example). So if we say 128;125
vanishes in some temperament, we mean that the difference between
three major thirds and an octave is zero in that temperament.

Similarly for 250:243. It's the difference between three minor thirds
and two fourths (for example). In Just Intonation it happens to be 49
cents. In 12-equal it's 100 cents. But in 15-equal and some other
temperaments (called "porcupine" temperaments), it's zero, it
vanishes. This fact leads to new scales and chord progressions that
are, some say, as "natural" in 15-equal (and other porcupine
temperaments) as the diatonic scale and its typical chord
progressions are in 12-equal (and other meantone temperaments).

The reply to Mark I was referring to was

/tuning/topicId_21432.html#50652

I'm sure you'll have more questions, and I'll do my best to answer
them!

-Paul
--- End forwarded message ---

STEPHEN SZPAK WRITES:::::::::::::::::::

Thanks Paul (Erlich) for your help. I don't know how well I'm going to understand
message 50652, but to make sure what I've got is right so far, please see below.

maximal diesis 250;243 is the difference between 3 minor thirds and 2 fourths

12 EDO's maximal diesis is 3 x 300 = 900 minus 2 x 500 = 1000
it is 100 cents, it does NOT vanish

15 Porcupine maximal diesis is 3 x 325 = 975 minus 2 x 488 = 976
it is about 0 cents, it DOES vanish

great diesis 128;125 is the difference between 3 major thirds and an octave

12 EDO's maximal diesis is 3 x 400 = 1200 minus 1200
it is 0 cents , it DOES vanish

15 Porcupine maximal diesis is 3 x 400 = 1200 minus 1220
it is 0 cents, it DOES vanish

syntonic comma 81;80 is the difference between? the major3rd and the major 3rd?

12 EDO is 0 cents?
15 Procupine is 0 cents?

5 major 3rds in 15 Porcupine are at 427 cents
10 major 3rds in Porcupine are at 386 cents
RESULT: I'm confused with syntonic comma.

Stephen Szpak

_________________________________________________________________
Make your home warm and cozy this winter with tips from MSN House & Home. http://special.msn.com/home/warmhome.armx

--- In tuning@yahoogroups.com, "Stephen Szpak" <stephen_szpak@h...>
wrote:

> maximal diesis 250;243 is the difference between 3 minor thirds
and 2
> fourths
>
> 12 EDO's maximal diesis is 3 x 300 = 900 minus 2 x 500 = 1000
> it is 100 cents, it does NOT vanish
>
> 15 Porcupine maximal diesis is 3 x 325 = 975 minus 2 x 488 =
976
> it is about 0 cents, it DOES vanish

It vanishes in 15-equal too.

> great diesis 128;125 is the difference between 3 major thirds
and an
> octave
>
> 12 EDO's maximal diesis

you mean great diesis

> is 3 x 400 = 1200 minus 1200
> it is 0 cents , it DOES vanish
>
> 15 Porcupine maximal diesis is 3 x 400 = 1200 minus 1220

minus 1200 . . . and surely you mean 15-equal great diesis here.

> it is 0 cents, it DOES vanish
>
> syntonic comma 81;80 is the difference between? the major3rd
and the
> major 3rd?

Well, it *is* the difference between the Pythagorean major third and
the Just major third, but the Pythagorean major third is not really
a "just" interval (at least according to Dave Keenan's definition).

You can see what it is by factoring the ratio 81/80.

The numerator factors as 81 = 3*3*3*3.

The denominator factors as 80 = 5*2*2*2*2

So the syntonic comma can be written as

(3*3*3*3)/(5*2*2*2*2)

or

((3/2)*(3/2)*(3/2)*(3/2))
/
((5/4)*2*2))

so it's the difference between four perfect fifths, and a major-third-
plus-two-octaves.

It can also be written as

((3/2)*(3/2)*(3/2))
/
((5/3)*2))

so it's the difference between three perfect fifths, and a major-
sixth-plus-octave.

> 12 EDO is 0 cents?

Yes.

> 15 Procupine is 0 cents?

In 15-equal it's 80 cents. 3 x 720 = 2160 minus 2080 = 80 cents.
In optimal porcupine it's 3 x 711 = 2133 minus 2074 = 59 cents.
The only regular porcupine tuning where the syntonic comma vanishes
is 7-equal, but the errors in the consonant intervals are very large
there.

> 5 major 3rds in 15 Porcupine are at 427 cents

Those are not how porcupine represents the 5/4 ratio, instead those
are the "wolves" resulting from cutting porcupine off at a finite
scale, but treating it as if it were 15-equal.

> 10 major 3rds in Porcupine are at 386 cents

These are how optimal porcupine represents the 5/4 ratio.

Porcupine in general is an infinite tuning, only certain special
cases are closed (like 15-equal). The unequal 15-note scales that
Gene and I gave you have "wolves" -- that is, intervals that are more
out-of-tune (relative to JI) than they would be in 15-equal -- which
is the price you pay for making most of the intervals more pure.

--- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
--- In tuning@yahoogroups.com, "Stephen Szpak" <stephen_szpak@h...>
wrote:

> maximal diesis 250;243 is the difference between 3 minor thirds
and 2
>fourths
>
> 12 EDO's maximal diesis is 3 x 300 = 900 minus 2 x 500 = 1000
> it is 100 cents, it does NOT vanish
>
> 15 Porcupine maximal diesis is 3 x 325 = 975 minus 2 x 488 =
976
> it is about 0 cents, it DOES vanish

It vanishes in 15-equal too.

STEPHEN SZPAK WRITES::::::::::

Yes it vanishes in 15 EDO too. I'm not sure if I saw that before or not.

> great diesis 128;125 is the difference between 3 major thirds
and an
>octave
>
> 12 EDO's maximal diesis

you mean great diesis

STEPHEN SZPAK WRITES:::::::::::::::::::::::::::::::::

I'm not sure what I did.

great diesis 128;125 correct??? Difference between 3 major 3rds and an octave.
maximal diesis 250;243 correct????Difference between 3 minor 3rds and two 4ths.

Also,,,does ,let's say: 10;20 mean the same as 10:20 ?

>is 3 x 400 = 1200 minus 1200
> it is 0 cents , it DOES vanish
>
> 15 Porcupine maximal diesis is 3 x 400 = 1200 minus 1220

minus 1200 . . . and surely you mean 15-equal great diesis here.

> it is 0 cents, it DOES vanish
>\=======================================================
========================================================
========================================================
========================================================
=======================================================
STEPHEN SZPAK WRITES:::::::::::::::::::::::::::::::::::

No reply to what's below this time.

> syntonic comma 81;80 is the difference between? the major3rd
and the
>major 3rd?

Well, it *is* the difference between the Pythagorean major third and
the Just major third, but the Pythagorean major third is not really
a "just" interval (at least according to Dave Keenan's definition).

You can see what it is by factoring the ratio 81/80.

The numerator factors as 81 = 3*3*3*3.

The denominator factors as 80 = 5*2*2*2*2

So the syntonic comma can be written as

(3*3*3*3)/(5*2*2*2*2)

or

((3/2)*(3/2)*(3/2)*(3/2))
/
((5/4)*2*2))

so it's the difference between four perfect fifths, and a major-third-
plus-two-octaves.

It can also be written as

((3/2)*(3/2)*(3/2))
/
((5/3)*2))

so it's the difference between three perfect fifths, and a major-
sixth-plus-octave.

> 12 EDO is 0 cents?

Yes.

> 15 Procupine is 0 cents?

In 15-equal it's 80 cents. 3 x 720 = 2160 minus 2080 = 80 cents.
In optimal porcupine it's 3 x 711 = 2133 minus 2074 = 59 cents.
The only regular porcupine tuning where the syntonic comma vanishes
is 7-equal, but the errors in the consonant intervals are very large
there.

> 5 major 3rds in 15 Porcupine are at 427 cents

Those are not how porcupine represents the 5/4 ratio, instead those
are the "wolves" resulting from cutting porcupine off at a finite
scale, but treating it as if it were 15-equal.

> 10 major 3rds in Porcupine are at 386 cents

These are how optimal porcupine represents the 5/4 ratio.

Porcupine in general is an infinite tuning, only certain special
cases are closed (like 15-equal). The unequal 15-note scales that
Gene and I gave you have "wolves" -- that is, intervals that are more
out-of-tune (relative to JI) than they would be in 15-equal -- which
is the price you pay for making most of the intervals more pure.
--- End forwarded message ---

Stephen Szpak

_________________________________________________________________
Make your home warm and cozy this winter with tips from MSN House & Home. http://special.msn.com/home/warmhome.armx

--- In tuning@yahoogroups.com, "Stephen Szpak" <stephen_szpak@h...>
wrote:
> --- In tuning@yahoogroups.com, "Paul Erlich" <paul@s...> wrote:
> --- In tuning@yahoogroups.com, "Stephen Szpak" <stephen_szpak@h...>
> wrote:
>
> > maximal diesis 250;243 is the difference between 3 minor thirds
> and 2
> >fourths
> >
> > 12 EDO's maximal diesis is 3 x 300 = 900 minus 2 x 500 = 1000
> > it is 100 cents, it does NOT vanish
> >
> > 15 Porcupine maximal diesis is 3 x 325 = 975 minus 2 x 488 =
> 976
> > it is about 0 cents, it DOES vanish
>
> It vanishes in 15-equal too.
>
> STEPHEN SZPAK WRITES::::::::::
>
> Yes it vanishes in 15 EDO too. I'm not sure if I saw that before
or not.
>
> > great diesis 128;125 is the difference between 3 major thirds
> and an
> >octave
> >
> > 12 EDO's maximal diesis
>
> you mean great diesis
>
> STEPHEN SZPAK WRITES:::::::::::::::::::::::::::::::::
>
> I'm not sure what I did.
>
> great diesis 128;125 correct??? Difference between 3 major
3rds and an
> octave.
> maximal diesis 250;243 correct????Difference between 3 minor
3rds and two
> 4ths.

correct and correct.

> Also,,,does ,let's say: 10;20 mean the same as 10:20 ?

I'm using the semicolon in tempered cases just so you know I don't
mean the Just Intonation interval -- in JI, 128:125 is 41 cents and
250:243 is 49 cents. But most people will just use the colon or slash
in all cases.

> syntonic comma 81;80 is the difference between? the major3rd
and the
>major 3rd?

Well, it *is* the difference between the Pythagorean major third and
the Just major third, but the Pythagorean major third is not really
a "just" interval (at least according to Dave Keenan's definition).

You can see what it is by factoring the ratio 81/80.

The numerator factors as 81 = 3*3*3*3.

The denominator factors as 80 = 5*2*2*2*2

So the syntonic comma can be written as

(3*3*3*3)/(5*2*2*2*2)

or

((3/2)*(3/2)*(3/2)*(3/2))
/
((5/4)*2*2))

so it's the difference between four perfect fifths, and a major-third-
plus-two-octaves.

It can also be written as

((3/2)*(3/2)*(3/2))
/
((5/3)*2))

so it's the difference between three perfect fifths, and a major-
sixth-plus-octave.

> 12 EDO is 0 cents?

Yes.

> 15 Procupine is 0 cents?

In 15-equal it's 80 cents. 3 x 720 = 2160 minus 2080 = 80 cents.
In optimal porcupine it's 3 x 711 = 2133 minus 2074 = 59 cents.
The only regular porcupine tuning where the syntonic comma vanishes
is 7-equal, but the errors in the consonant intervals are very large
there.

> 5 major 3rds in 15 Porcupine are at 427 cents

Those are not how porcupine represents the 5/4 ratio, instead those
are the "wolves" resulting from cutting porcupine off at a finite
scale, but treating it as if it were 15-equal.

> 10 major 3rds in Porcupine are at 386 cents

These are how optimal porcupine represents the 5/4 ratio.

Porcupine in general is an infinite tuning, only certain special
cases are closed (like 15-equal). The unequal 15-note scales that
Gene and I gave you have "wolves" -- that is, intervals that are more
out-of-tune (relative to JI) than they would be in 15-equal -- which
is the price you pay for making most of the intervals more pure.
--- End forwarded message ---

STEPHEN SZPAK WRITES::::::::::::::::::::

Paul

So, the syntonic comma vanishes in 12 EDO but not in Porcupine 15 or 15 EDO.

12 EDO math: 3x700=2100 equals 900+1200=2100

I don't understand what this means: "Porcupine in general is an infinite tuning,
only certain special cases are closed"

Also, do you see 15 equal as the same as 15 EDO?

Hate to tell you this but I may need explainations of the relevance of why a ratio
that does or does not vanish in 12 EDO is important to Porcupine 15. I don't know
how to even begin on that as yet. Thanks for helping me out.

Stephen Szpak

_________________________________________________________________
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--- In tuning@yahoogroups.com, "Stephen Szpak" <stephen_szpak@h...>
wrote:

> STEPHEN SZPAK WRITES::::::::::::::::::::
>
> Paul
>
> So, the syntonic comma vanishes in 12 EDO but not in Porcupine
15 or 15
> EDO.
>
> 12 EDO math: 3x700=2100 equals 900+1200=2100

Correct.

> I don't understand what this means: "Porcupine in general is an
infinite
> tuning,
> only certain
special
> cases are closed"

Porcupine in general is generated by repeating an interval of roughly
163 cents, and octave-reducing when necessary. Normally, you can keep
generating more and more notes this way and never get back where you
started. However, certain values of the generator will get you back
where you started after a certain number of iterations. For example,
if you take the generator to be 160 cents, you'll get back where you
started after 15 iterations, and will in the process have generated
15-equal.

> Also, do you see 15 equal as the same as 15 EDO?

Yes.

> Hate to tell you this but I may need explainations of the
relevance of why
> a ratio
> that does or does not vanish in 12 EDO is important to Porcupine
>15.

Certain constructs that are familiar in Western music, such as the
diatonic scale and its common chord progressions, are based on the
syntonic comma vanishing. Such constructs will tend to fall apart in
any tuning where the syntonic comma does not vanish. Meanwhile,
knowing what 'commas' do vanish in a new tuning system may help you
find new scales and chord progressions within that tuning system
which have many of the familiar 'diatonic' properties, but may be
inexpressible in 12-equal.