still pretty spooky

Paul Erlich's 19-note JI Scale #3 (the one with spectacular symmetry
and two embedded hexanies) when compared with 19-tET yields the
following results (in cents):

PITCH 19-tET 19-JI#3 difference
C 0 0 0
C# 63 49 -14
Db 126 119 -7
D natural 189 182 -7
D# 253 267 +14
Eb 316 316 0
E natural 379 386 +7
E# 442 435 -7
F 505 498 -7
F# 568 568 0
Gb 632 632 0
G natural 695 702 +7
G# 758 765 +7
Ab 821 814 -7
A natural 884 884 0
A# 947 933 -14
Bb 1011 1018 +7
B natural 1074 1081 +7
B# 1137 1151 +14
C 1200 1200 0

Note that the difference is either +or- 14, +or- 7, or 0.

What is the reason for this again??

Anyway, with all these 7's, this is sure to be a "lucky" scale...

I think I'll go for it...

________ _______ ______ _____
Joseph Pehrson

Joseph wrote,

>Paul Erlich's 19-note JI Scale #3 (the one with spectacular symmetry
>and two embedded hexanies)

In case your friends will let you borrow their car if you use
superparticular step sizes, note that this scale has the following step
sizes:

36/35
25/24
28/27
21/20
36/35
25/24
36/35
28/27
25/24
648/625
25/24
28/27
36/35
25/24
36/35
21/20
28/27
25/24
36/35

So, for the sake of those friends, you might want to look at the other
scales.

:)

>Note that the difference is either +or- 14, +or- 7, or 0.

>What is the reason for this again??

I think you're ready to figure this out for yourself -- look at my
explanation of why "Sparky" only had 0s and ±7s -- and construct a similar
one for this scale.

--- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:

/tuning/topicId_19241.html#19243

> Joseph wrote,
>
> >Paul Erlich's 19-note JI Scale #3 (the one with spectacular
symmetry and two embedded hexanies)
>
> In case your friends will let you borrow their car if you use
> superparticular step sizes, note that this scale has the following
step sizes:
>
> 36/35
> 25/24
> 28/27
> 21/20
> 36/35
> 25/24
> 36/35
> 28/27
> 25/24
> 648/625
> 25/24
> 28/27
> 36/35
> 25/24
> 36/35
> 21/20
> 28/27
> 25/24
> 36/35
>
> So, for the sake of those friends, you might want to look at the
other scales.
>
> :)
>

Oh, absolutely not! I'm a VERY "superparticular" person and so are
my friends! What an amazing property of this scale! I would say the
scale verges on the "supernatural" as well... :)

I wonder how the 648/625 got in there.... one "rotten apple."

There must be some kind of mathematical reason behind this very
particular "superparticularity" but it would certainly be out of *MY*
grasp! Any theory as to why this is happening??

> >Note that the difference is either +or- 14, +or- 7, or 0.
>
> >What is the reason for this again??
>
> I think you're ready to figure this out for yourself -- look at my
> explanation of why "Sparky" only had 0s and ±7s -- and construct
a
similar one for this scale.

Paul... it's always fun to work with you, since I have to do part of
the work... It's a great learning experience!

Anyway, it's clear that some things about the "spooky anomaly" make
sense. Every minor third deviates "0" from the 19-tET scale, which
is as it should be.

However, I'm having more problems when it comes to finding the cents
deviation for the other intervals.

Let's take an A#, for example, since we used it in the "Sparky robot"
example:

The ratio for A# in the 5-limit lattice was 125/72.

Our present 7-limit 19-tone scale has an A# at 12/7.

So, we should get the difference as (12/7)/(125/72) = 864/875

OK. Now I need to make the "cents" out of that...

1200*log(.9874)/log(2)=

1200*(-5.494)/.3010=

-6593/.3010=

-21903

What does *that* mean... 21 cents?? It should be 14, yes?

I'm lost.

_______ ____ ___ ___ _
Joseph Pehrson

The 271 in the 3rd is equivalent to 171 TET (EDO).
This very good 7 limit scale is based on the '7-limit comma' 2401/2400, and
that's exactly the 7 cent interval popping up here.

Kees

> -----Original Message-----
> From: Bohlen, Heinz [mailto:HEINZ.BOHLEN@mpp.cpii.com]
> Sent: Thursday, February 22, 2001 8:49 AM
> To: 'tuning@yahoogroups.com'
> Cc: 'hpbohlen@aol.com'
> Subject: RE: [tuning] still pretty spooky
>
>
> There are more examples around for the peculiar role of roughly 7 cents
> intervals in music. To me they seem to represent a kind of tuning "quarks"
> (the physicists among us will hopefully forgive me). If you are
> interested,
> look up
> http://members.aol.com/bpsite/271tones.html
>
> Heinz Bohlen
>
>
> -----Original Message-----
> From: jpehrson@rcn.com [mailto:jpehrson@rcn.com]
> Sent: Wednesday, February 21, 2001 9:10 PM
> To: tuning@yahoogroups.com
> Subject: [tuning] still pretty spooky
>
>
> Paul Erlich's 19-note JI Scale #3 (the one with spectacular symmetry
> and two embedded hexanies) when compared with 19-tET yields the
> following results (in cents):
>
>
> PITCH 19-tET 19-JI#3 difference
> C 0 0 0
> C# 63 49 -14
> Db 126 119 -7
> D natural 189 182 -7
> D# 253 267 +14
> Eb 316 316 0
> E natural 379 386 +7
> E# 442 435 -7
> F 505 498 -7
> F# 568 568 0
> Gb 632 632 0
> G natural 695 702 +7
> G# 758 765 +7
> Ab 821 814 -7
> A natural 884 884 0
> A# 947 933 -14
> Bb 1011 1018 +7
> B natural 1074 1081 +7
> B# 1137 1151 +14
> C 1200 1200 0
>
>
> Note that the difference is either +or- 14, +or- 7, or 0.
>
> What is the reason for this again??
>
> Anyway, with all these 7's, this is sure to be a "lucky" scale...
>
> I think I'll go for it...
>
> ________ _______ ______ _____
> Joseph Pehrson
>
>
>
>
> You do not need web access to participate. You may subscribe through
> email. Send an empty email to one of these addresses:
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> emails.
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>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
>
>
>

I didn't see how any of this was going to address Joseph's question
(as expressed in the subject line) until I realized that 171 = 19*9.

So Joseph, basically, 7-limit JI happens to coincide, to within 1
cent, with 171-tET, which is a 9-fold subdivision of 19-tET. Hence,
any 7-limit JI scale, when compared with 19-tET, will have deviations
which are an integer number of 171-tET steps. Since 171-tET is a
scale of 7-cent steps, all the deviations will be 0, +7, -7, +14, -
14, +21, -21, etc.

Thanks to all for helping me realize this!

--- In tuning@y..., "Kees van Prooijen" <kees@d...> wrote:
>
> The 271 in the 3rd is equivalent to 171 TET (EDO).
> This very good 7 limit scale is based on the '7-limit comma'
2401/2400, and
> that's exactly the 7 cent interval popping up here.
>
> Kees
>
> > -----Original Message-----
> > From: Bohlen, Heinz [mailto:HEINZ.BOHLEN@m...]
> > Sent: Thursday, February 22, 2001 8:49 AM
> > To: 'tuning@y...'
> > Cc: 'hpbohlen@a...'
> > Subject: RE: [tuning] still pretty spooky
> >
> >
> > There are more examples around for the peculiar role of roughly 7
cents
> > intervals in music. To me they seem to represent a kind of
tuning "quarks"
> > (the physicists among us will hopefully forgive me). If you are
> > interested,
> > look up
> > http://members.aol.com/bpsite/271tones.html
> >
> > Heinz Bohlen
> >
> >
> > -----Original Message-----
> > From: jpehrson@r... [mailto:jpehrson@r...]
> > Sent: Wednesday, February 21, 2001 9:10 PM
> > To: tuning@y...
> > Subject: [tuning] still pretty spooky
> >
> >
> > Paul Erlich's 19-note JI Scale #3 (the one with spectacular
symmetry
> > and two embedded hexanies) when compared with 19-tET yields the
> > following results (in cents):
> >
> >
> > PITCH 19-tET 19-JI#3 difference
> > C 0 0 0
> > C# 63 49 -14
> > Db 126 119 -7
> > D natural 189 182 -7
> > D# 253 267 +14
> > Eb 316 316 0
> > E natural 379 386 +7
> > E# 442 435 -7
> > F 505 498 -7
> > F# 568 568 0
> > Gb 632 632 0
> > G natural 695 702 +7
> > G# 758 765 +7
> > Ab 821 814 -7
> > A natural 884 884 0
> > A# 947 933 -14
> > Bb 1011 1018 +7
> > B natural 1074 1081 +7
> > B# 1137 1151 +14
> > C 1200 1200 0
> >
> >
> > Note that the difference is either +or- 14, +or- 7, or 0.
> >
> > What is the reason for this again??
> >
> > Anyway, with all these 7's, this is sure to be a "lucky" scale...
> >
> > I think I'll go for it...
> >
> > ________ _______ ______ _____
> > Joseph Pehrson
> >
> >
> >
> >
> > You do not need web access to participate. You may subscribe
through
> > email. Send an empty email to one of these addresses:
> > tuning-subscribe@y... - join the tuning group.
> > tuning-unsubscribe@y... - unsubscribe from the tuning group.
> > tuning-nomail@y... - put your email message delivery on hold
> > for the tuning group.
> > tuning-digest@y... - change your subscription to daily digest
> > mode.
> > tuning-normal@y... - change your subscription to individual
> > emails.
> > tuning-help@y... - receive general help information.
> >
> >
> > Your use of Yahoo! Groups is subject to
http://docs.yahoo.com/info/terms/
> >
> >
> >

--- In tuning@y..., PERLICH@A... wrote:

/tuning/topicId_19241.html#19498

> I didn't see how any of this was going to address Joseph's question
> (as expressed in the subject line) until I realized that 171 = 19*9.
>
> So Joseph, basically, 7-limit JI happens to coincide, to within 1
> cent, with 171-tET, which is a 9-fold subdivision of 19-tET. Hence,
> any 7-limit JI scale, when compared with 19-tET, will have
deviations which are an integer number of 171-tET steps. Since
171-tET is a scale of 7-cent steps, all the deviations will be 0, +7,
-7,+14, -14, +21, -21, etc.
>
> Thanks to all for helping me realize this!
>
> --- In tuning@y..., "Kees van Prooijen" <kees@d...> wrote:
> >
> > The 271 in the 3rd is equivalent to 171 TET (EDO).
> > This very good 7 limit scale is based on the '7-limit comma'
> 2401/2400, and that's exactly the 7 cent interval popping up here.
> >
> > Kees
> >
>> > >
> > > There are more examples around for the peculiar role of roughly
7 cents intervals in music. To me they seem to represent a kind of
> tuning "quarks" (the physicists among us will hopefully forgive
me). If you are interested,look up
> > > http://members.aol.com/bpsite/271tones.html
> > >
> > > Heinz Bohlen
> > >

Thanks so much, Paul, for coming to the "bottom" of this! (I had
also thanked Mr. Bohlen in a private e-mail...)

Admittedly, I didn't get the "drift" of this from Kees statement, so
I'm sure glad you figured it out!

Well, that drives one more stake in the heart of the "magic-spooky"
irrationals... (I'll stick with "wholesome" whole numbers from here
on out...)

_______ _____ ____ ____
Joseph Pehrson