Steve,

       It was a rounding error.





       Cell # 356
      1.49661827761514














       M
      1.00112915039062
      (Schisma)













       J
      1.00113137110290
      (Progression comma factor)





       P
      1.05707299111353
      (Piagui scales semitone factor)





       F#(PM)
      1.49661495400000
      The rounded G.







      (For the three Justharm scales)




















       F#
      1.41421356237310
      Equals to 2^(0.5)













       F#(PM)
      1.49661495781239
      Fourteen decimals (G)







      (The accurate G frequency)













       J/M
      1.00000221820759
      This ratio gives the associated
      consonant cell.












       F#(PM)(J/M)
      1.49661827761505
      The accurate frequency of Cell
      # 356












       The M and J comma factors of the progression are the two options that
any cell can take for establishing the following cell and the J/M ratio
applied to any cell determines another consonant frequency that could be
called "associated cell".  The associated cell (J/M) (Cell # 356) that is
equal to the accurate G.



       Regards,



       Mario Pizarro



       Lima, November 25

       -------------------------







----- Original Message -----
From: "martinsj013" <martinsj@...>
To: <tuning@yahoogroups.com>
Sent: Wednesday, November 25, 2009 6:01 AM
Subject: [tuning] Re: IMO


>I have realised I should be able to answer my own question:
>
> Well first of all I can't find 1.496614954 precisely in the system.
> Cell 356 is 1.4966182776 (this is 3/2/M^2)
>
> No point looking in any other cell since M ("schisma"), J, U and PC^(1/12)
> ("grad") are all about the same size approx 2 cents.
>
> 3/2/(MxJ) is 1.496614958 - very close indeed, perhaps the difference from
> your value is a rounding error?  This can be found as the ratio between
> Cell 359 and Cell 3, for example.  But I may have misunderstood something
> about the system.
>
> By the way, Cell 357 is 3/2/M    = 2^14/3^7/5  (schisma tempered 5th)
>
> Steve M.
>
>
>
> ------------------------------------

Steve, 1.496614954  is a rounded figure. The attachment explains, starting
from the G frequency of F#(PM), how cell # 356 is linked to the accurate G
frequency. Ratio J/M always establishes an associated consonant cell. I see
that you bought my book otherwise you wouldn´t know the value of cell # 356.
isn´t?

       Cell # 356
      1.49661827761514














       M
      1.00112915039062
      (Schisma)













       J
      1.00113137110290
      (Progression comma factor)





       P
      1.05707299111353
      (Piagui scales semitone factor)





       F#(PM)
      1.49661495400000
      The rounded G.







      (For the three Justharm scales)




















       F#
      1.41421356237310
      Equals to 2^(0.5)













       F#(PM)
      1.49661495781239
      Fourteen decimals (G)







      (The accurate G frequency)













       J/M
      1.00000221820759
      This ratio gives the associated
      consonant cell.












       F#(PM)(J/M)
      1.49661827761505
      The accurate frequency of Cell
      # 356












       The M and J comma factors of the progression are the two options that
any cell can take for establishing the following cell and the J/M ratio
applied to any cell determines another consonant frequency that could be
called "associated cell".

       The  associated
      cell (J/M)(Cell # 356)
      equals to the "accurate G".






-------------------------------------------------------------------------------
IT IS A ROUNDING ERROR, ABOVE YOU HAVE THE EXPLANATION.

REGARDS,

Mario Pizarro
piagui@...
Lima, November 25, 2009
-------------------------------------------------------------------------------
----- Original Message -----
From: "martinsj013" <martinsj@...>
To: <tuning@yahoogroups.com>
Sent: Wednesday, November 25, 2009 6:01 AM
Subject: [tuning] Re: IMO


>I have realised I should be able to answer my own question:
>
> Well first of all I can't find 1.496614954 precisely in the system.
> Cell 356 is 1.4966182776 (this is 3/2/M^2)
>
> No point looking in any other cell since M ("schisma"), J, U and PC^(1/12)
> ("grad") are all about the same size approx 2 cents.
>
> 3/2/(MxJ) is 1.496614958 - very close indeed, perhaps the difference from
> your value is a rounding error?  This can be found as the ratio between
> Cell 359 and Cell 3, for example.  But I may have misunderstood something
> about the system.
> YES STEVE, IT IS A ROUNDING ERROR, THIS TIME I USED EXCEL.
> By the way, Cell 357 is 3/2/M    = 2^14/3^7/5  (schisma tempered 5th)
>
> Steve M.
>
>
>
> ------------------------------------
>
>

--- On Wed, 11/25/09, Mike Battaglia <battaglia01@...> wrote:

> From: Mike Battaglia <battaglia01@...>

> > The sound is a combination of:
> low string tension, hammer
> > material, and the lack of tripled unisons. Regarding
> the
> > first point -- what I was trying to say -- I strongly
> suspect
> > his tuning bender could not function at piano
> tension.

My issue might just be with the hammer material or something simple like that. I
just really like the low harpsichord-like buzz of bass strings on a
Classical-era fortepiano:

http://www.youtube.com/watch?v=ZMUb9nqnk8o (Mozart's Phantasy in C minor)

But since string tension, while much less than that of a modern piano, still has
to be greater than in a harpsichord, the tuning sliders might have to apply the
level principle... tuning machines of some sort, for instance. That would
require more time and effort to retune a string, among other inconveniences.
Obviously, you'd have to compromise.

> Not to mention that doing this with tripled unisons would
> likely turn
> an already complicated piece of engineering into a complete
> nightmare.
>
> What would be awesome, however, is if an electroacoustic
> version were
> made. I would buy one of those in a heartbeat, no joke.

I'm with you there--two strings a note is enough.

An electric tunable piano might be something like the old electric grands and
uprights: http://en.wikipedia.org/wiki/Electric_Grand_Piano . They never sounded
anything like a concert grand, of course, but they did have a fortepiano-like
sound, so having pickups under strings would work well for this instrument, I'd
imagine.

And I hope there are eventual plans for an upright version of this thing. I'd
like one with the middle pedal lowering a felt shield for a muffled sound, like
the Yamaha uprights have had...

~D.

I have realised I should be able to answer my own question:

Well first of all I can't find 1.496614954 precisely in the system.
Cell 356 is 1.4966182776 (this is 3/2/M^2)

No point looking in any other cell since M ("schisma"), J, U and PC^(1/12)
("grad") are all about the same size approx 2 cents.

3/2/(MxJ) is 1.496614958 - very close indeed, perhaps the difference from your
value is a rounding error?  This can be found as the ratio between Cell 359 and
Cell 3, for example.  But I may have misunderstood something about the system.

By the way, Cell 357 is 3/2/M    = 2^14/3^7/5  (schisma tempered 5th)

Steve M.

Custom Scale Editor (CSE) and Tonal Plexus Editor (TPXE) are both
free software applications for Mac and Windows which give you control
over microtonal pitch. CSE is for standard MIDI keyboards, while TPXE
gives the experience of having a virtual Tonal Plexus keyboard at
your fingertips.

	 ... a few features of these updates:
	 - numerous bug fixes, including compatability with Mac OSX 10.6
	 - New AnaMark .MSF Multiple Scale File exporting options
	 - TPXE Live Input support for U-PLEX keyboards

DETAILS:
<http://www.h-pi.com/CSEsoftware.html>
<http://www.h-pi.com/TPXEsoftware.html>

DOWNLOAD:
<http://www.h-pi.com/downloads.html>

Cheers,
Aaron
=====
Aaron Andrew Hunt
H-Pi Instruments
Blog: <http://www.h-pi.com/wordpress/>

> The sound is a combination of: low string tension, hammer
> material, and the lack of tripled unisons. Regarding the
> first point -- what I was trying to say -- I strongly suspect
> his tuning bender could not function at piano tension.
>
> -Carl

Not to mention that doing this with tripled unisons would likely turn
an already complicated piece of engineering into a complete nightmare.

What would be awesome, however, is if an electroacoustic version were
made. I would buy one of those in a heartbeat, no joke.

-Mike

>
> > > Of course the design and sound needs work,
> >
> > How do you mean?  I suspect it sounds the way it does
> > for quite practical reasons.  Or to put it another way,
> > what he's trying to do is very hard.
>
> Oh I didn't mean "needs work" as in "it sucks", and I know
> it's a prototype (I couldn't build anything like this myself);
> I just wonder if it could sound a little more piano-like. I've
> always liked the sound of the old fortepianos. It might be the
> brightness of the hammers or something though.

The sound is a combination of: low string tension, hammer
material, and the lack of tripled unisons.  Regarding the
first point -- what I was trying to say -- I strongly suspect
his tuning bender could not function at piano tension.

-Carl

--- On Tue, 11/24/09, Carl Lumma <carl@...> wrote:

> From: Carl Lumma <carl@...>
> Subject: [tuning] Re: Microtonal piano - UK
> To: tuning@yahoogroups.com
> Date: Tuesday, November 24, 2009, 7:43 PM
>
> --- In tuning@yahoogroups.com,
> Danny Wier <dawiertx@...> wrote:
>
> > Of course the design and sound needs work,
>
> How do you mean?  I suspect it sounds the way it does
> for
> quite practical reasons.  Or to put it another way,
> what
> he's trying to do is very hard.

Oh I didn't mean "needs work" as in "it sucks", and I know it's a prototype (I
couldn't build anything like this myself); I just wonder if it could sound a
little more piano-like. I've always liked the sound of the old fortepianos. It
might be the brightness of the hammers or something though.

I still like the santur-like sound on its own merits though. ~D.

--- In tuning@yahoogroups.com, Danny Wier <dawiertx@...> wrote:

> Of course the design and sound needs work,

How do you mean?  I suspect it sounds the way it does for
quite practical reasons.  Or to put it another way, what
he's trying to do is very hard.

-Carl

--- On Tue, 11/24/09, Charles Lucy <lucy@...> wrote:

> From: Charles Lucy <lucy@...>
> Subject: [tuning] Microtonal piano - UK
> To: tuning@yahoogroups.com
> Date: Tuesday, November 24, 2009, 8:17 AM
> http://www.guardian.co.uk/music/video/2009/nov/22/fluid-piano-classical-music

I'm going to be wanting one of these, not just for the obvious reason, but
because I like that combination fortepiano-santur sound. You might have to
amplify it in a full orchestral setting. (I also like the idea of sliding tuners
for qanuns.)

Of course the design and sound needs work, and I'd really like to see a model
with nineteen notes per octave, using split black keys and tiny black keys for
B#/Cb and E#/Fb.

~D.

--- In tuning@yahoogroups.com, "Mario Pizarro" <piagui@...> wrote:
> As I said, when I knew that Piagui D major chord wave peak graph showed
> unexpected peaks, started the complementary analysis which uses K and P
> semitone factors. ... K for instance, is also given by
> (M^32)*(J^18)*(U^2)= 1.06066017178...-- 32 + 18 + 2 = 52, so cell # 52
> equals K. Similarly, P = 1.0570729911.... = (M^30)*(J^17)*(U^2) = Cell # 49.
> Neither K nor P involves powers of 3 and of 2, as you can see.

Well, yes they do:
K = 2^(-3/2) x 3^1
P = 2^(13/4) x 3^(-2)

Your formulae for M, J, U are missing some ^ and () I think

M = [(3^8 x 5) / 2^15] = [(32805 / 32768)] = 1.00112915039062

J = [(2^25 x 2^(1/4))/(3^13 x 5^2)] = [(33554432x21/4)/39858075]=1.001131371103

U = [(2^12 x 5^2 x 3^(1/2)) / 3^11] = [(102400 x 3^(1/2))/177147] =
1.0012136965066

> ¿Would you explain me how can I calculate "1/6 PC tempered 5th"?. This way I
> could answer your question. Is "tempered 5th" related to the equal tempered
> scale ?-----

PC is 3^12 / 2^19, and "1/6 PC tempered 5th" means "3/2 / PC^(1/6)"
i.e. 2^(13/6) / 3
It is used in a meantone and in several circulating temperaments.

Equal tempered scale has "1/12 PC tempered 5th", this means "3/2 / PC^(1/12)"
i.e. 2^(7/12)

>
> Last week, after many years the progression was deduced, I noticed the
> interesting ratio:
> [(K^4)/(P^4)] = 1.01364326477 = Pythagorean comma which is also given by the
> twelfth cell of the progression

Yes, starting with:
K = 2^(-3/2) x 3^1
P = 2^(13/4) x 3^-2

we can see that (K^4)/(P^4) = 2^(-19) x 3^12 QED.

Steve M.

--- In tuning@yahoogroups.com, Klaus Schmirler <KSchmir@...> wrote:

> > 36:42:54:63 -> 36:42:48:60 - > 32:40:48:64
> >
> > Billy
>
> Wow!

These are the same chords I suggested, except it starts
with D=9/8 instead of 8/7.  That means here, the two Cs
are a 64/63 apart

C B C
A G G
F F E
D D C

vs. the final two Gs the way I had it

1/1 40/21 1/1
12/7 32/21 3/2
4/3 4/3 5/4
8/7 8/7 1/1

As mentioned, these two progressions are the same in
pajara, since 64/63 vanishes.  One can use an adaptive
solution based on pajara if one desires just intonation.

-Carl

http://www.guardian.co.uk/music/video/2009/nov/22/fluid-piano-classical-music


Charles Lucy
lucy@...

-- Promoting global harmony through LucyTuning --

For more information on LucyTuning go to:

http://www.lucytune.com

LucyTuned Lullabies (from around the world) can found at:

http://www.lullabies.co.uk

duckfeetbilly schrieb:
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>> I've posted about other frustrating comma pumps before, but this one
>> has me at my wit's end. This might be the most common chord
>> progression in all of music and it's also the most annoying to deal
>> with in JI.
>>
>> Some things that seem to not work (let's assume we're in C):
>> - Rooting the D on 9/8 and building up 6/5's and 5/4's from there.
>> This makes the C in the iim7 a comma sharp of the finishing I, which
>> sounds awful.
>> - Rooting the D on 10/9 and building up 6/5's and 5/4's from there.
>> This makes the D in the iim7 a comma flat of the D in the V7, which
>> destroys the voice leading and also sounds awful.
>> - Rooting the D on 9/8 and building up 32/27's and 81/64's from there.
>> This sounds completely out of place if your goal is to have beatless
>> "just" voicings, and since ii-V's are so common in almost everything,
>> it almost sounds like the piece isn't in JI at all. On the other hand,
>> this one avoids weird comma shifts.
>>
>
> This is something that I have found to be the most curious of the JI dilemmas,
especially considering it is a simple chord set as well as one natural sounding
to the ear.
>
> When I try to come up with a solution to this I base it on the idea that I am
discovering the real arithmetic behind the ii-V-I that explains its musical
quality, i.e. what is the ear hearing about it that has made it stand the test
of time?
>
> One of my ideas is that the particular minor 7th chord that appears in this
progression could actually be tuned differently, as an 11th chord extension of
the following dominant. The tuning I have in mind is the sub-minor 7th, or
septimal-minor 7th. 12:14:18:21. As such, it really would share a fundamental
with the following dominant. Using this math, you can use the following
harmonics, relative to the tonic as the fundamental, to complete the whole chord
progresssion:
>
> 36:42:54:63 -> 36:42:48:60 - > 32:40:48:64
>
> Billy

Wow!

--- In tuning@yahoogroups.com, "duckfeetbilly" <billygard@...> wrote:

> The tuning I have in mind is the sub-minor 7th,
> or septimal-minor 7th. 12:14:18:21.
> As such, it really would share a fundamental
> with the following dominant. Using this math,
> you can use the following harmonics,
> relative to the tonic as the fundamental,
> to complete the whole chord progresssion:
>
> 36:42:54:63 -> 36:42:48:60 - > 32:40:48:64 = 8*(4:5:6:7:8)
>
Hi Billy, how about to extend that
to Hindemith's more resonant progession

iim9 -> V9 -> I9

20:24:30:35:45 -> 12:15:18:21:27 -> 4:5:6:7:9

within generalized 7-limit JI?

-- In tuning@yahoogroups.com, "hpiinstruments" <aaronhunt@...> wrote:

> Start this video at about 1:10 for the a version
> of the smoother septimal ii7 chord with added
> harmonic series extensions:
> http://www.youtube.com/watch?v=b-YqbdQtmvI

You can also do

http://www.youtube.com/watch?v=b-YqbdQtmvI#t=1m10s

-C.

There's a ii-V-i in minor at the bridge (about 1/3 way through)
on this MIDI file too, along with some other shifting extensions:
<http://www.h-pi.com/midi/Dream2.mid>


--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > Great topic. This is one of my favorite flavors of ii7 V7 I.
> > Supertonic 7 with the small minor thirds sounds really dark
> > and leads perfectly to a septimal dominant 7, and that small
> > minor seventh sounds so perfect moving to a small major third.
> > I'm with duckfeetbilly on this one. But there are so many ways
> > to do it, and they all have unique affect. Like, I love the tortured
> > sound of comma shifting from a ii7 tuned with large minor
> > third and seventh into the septimal dominant. It pulls
> > everything down and just ... kills. It's all just awesome.
> > MUSIC IS AWESOME.
>
> You're saying something like 10/9 4/3 5/3 2/1 -> 9/8 21/16 3/2 15/8 ->
> 1/1 5/4 3/2 2/1 for the second one?
>
> -Mike
>
> > --- In tuning@yahoogroups.com, "duckfeetbilly" <billygard@> wrote:
> > >
> > > --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> > > >
> > > > I've posted about other frustrating comma pumps before, but this one
> > > > has me at my wit's end. This might be the most common chord
> > > > progression in all of music and it's also the most annoying to deal
> > > > with in JI.
> > > >
> > > > Some things that seem to not work (let's assume we're in C):
> > > > - Rooting the D on 9/8 and building up 6/5's and 5/4's from there.
> > > > This makes the C in the iim7 a comma sharp of the finishing I, which
> > > > sounds awful.
> > > > - Rooting the D on 10/9 and building up 6/5's and 5/4's from there.
> > > > This makes the D in the iim7 a comma flat of the D in the V7, which
> > > > destroys the voice leading and also sounds awful.
> > > > - Rooting the D on 9/8 and building up 32/27's and 81/64's from there.
> > > > This sounds completely out of place if your goal is to have beatless
> > > > "just" voicings, and since ii-V's are so common in almost everything,
> > > > it almost sounds like the piece isn't in JI at all. On the other hand,
> > > > this one avoids weird comma shifts.
> > > >
> > >
> > > This is something that I have found to be the most curious of the JI
dilemmas, especially considering it is a simple chord set as well as one natural
sounding to the ear.
> > >
> > > When I try to come up with a solution to this I base it on the idea that I
am discovering the real arithmetic behind the ii-V-I that explains its musical
quality, i.e. what is the ear hearing about it that has made it stand the test
of time?
> > >
> > > One of my ideas is that the particular minor 7th chord that appears in
this progression could actually be tuned differently, as an 11th chord extension
of the following dominant. The tuning I have in mind is the sub-minor 7th, or
septimal-minor 7th. 12:14:18:21. As such, it really would share a fundamental
with the following dominant. Using this math, you can use the following
harmonics, relative to the tonic as the fundamental, to complete the whole chord
progresssion:
> > >
> > > 36:42:54:63 -> 36:42:48:60 - > 32:40:48:64
> > >
> > > Billy
> > >
> >
> >
>

Well, to simplify, spelling local harmonic series
on roots 9/8, 3/2 and 1/1, I mean the chords:

ii7 on 9/8 = 10:12:15:18
V7 on 3/2 = 4:5:6:7:(9)
I on 1/1 = 8:10:12:(15)

In C it would be the third 6/5 from 9/8 of the ii
chord becoming the seventh 7/4 from 3/2 of
the dominant that is the comma shift motion I
find so appealing. You could use the root of ii
on 10/9 instead, but to me it doesn't sound
as good - just an opinion.

Start this video at about 1:10 for the a version
of the smoother septimal ii7 chord with added
harmonic series extensions:
<http://www.youtube.com/watch?v=b-YqbdQtmvI>


--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > Great topic. This is one of my favorite flavors of ii7 V7 I.
> > Supertonic 7 with the small minor thirds sounds really dark
> > and leads perfectly to a septimal dominant 7, and that small
> > minor seventh sounds so perfect moving to a small major third.
> > I'm with duckfeetbilly on this one. But there are so many ways
> > to do it, and they all have unique affect. Like, I love the tortured
> > sound of comma shifting from a ii7 tuned with large minor
> > third and seventh into the septimal dominant. It pulls
> > everything down and just ... kills. It's all just awesome.
> > MUSIC IS AWESOME.
>
> You're saying something like 10/9 4/3 5/3 2/1 -> 9/8 21/16 3/2 15/8 ->
> 1/1 5/4 3/2 2/1 for the second one?
>
> -Mike
>
> > --- In tuning@yahoogroups.com, "duckfeetbilly" <billygard@> wrote:
> > >
> > > --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> > > >
> > > > I've posted about other frustrating comma pumps before, but this one
> > > > has me at my wit's end. This might be the most common chord
> > > > progression in all of music and it's also the most annoying to deal
> > > > with in JI.
> > > >
> > > > Some things that seem to not work (let's assume we're in C):
> > > > - Rooting the D on 9/8 and building up 6/5's and 5/4's from there.
> > > > This makes the C in the iim7 a comma sharp of the finishing I, which
> > > > sounds awful.
> > > > - Rooting the D on 10/9 and building up 6/5's and 5/4's from there.
> > > > This makes the D in the iim7 a comma flat of the D in the V7, which
> > > > destroys the voice leading and also sounds awful.
> > > > - Rooting the D on 9/8 and building up 32/27's and 81/64's from there.
> > > > This sounds completely out of place if your goal is to have beatless
> > > > "just" voicings, and since ii-V's are so common in almost everything,
> > > > it almost sounds like the piece isn't in JI at all. On the other hand,
> > > > this one avoids weird comma shifts.
> > > >
> > >
> > > This is something that I have found to be the most curious of the JI
dilemmas, especially considering it is a simple chord set as well as one natural
sounding to the ear.
> > >
> > > When I try to come up with a solution to this I base it on the idea that I
am discovering the real arithmetic behind the ii-V-I that explains its musical
quality, i.e. what is the ear hearing about it that has made it stand the test
of time?
> > >
> > > One of my ideas is that the particular minor 7th chord that appears in
this progression could actually be tuned differently, as an 11th chord extension
of the following dominant. The tuning I have in mind is the sub-minor 7th, or
septimal-minor 7th. 12:14:18:21. As such, it really would share a fundamental
with the following dominant. Using this math, you can use the following
harmonics, relative to the tonic as the fundamental, to complete the whole chord
progresssion:
> > >
> > > 36:42:54:63 -> 36:42:48:60 - > 32:40:48:64
> > >
> > > Billy
> > >
> >
> >
>

> Great topic. This is one of my favorite flavors of ii7 V7 I.
> Supertonic 7 with the small minor thirds sounds really dark
> and leads perfectly to a septimal dominant 7, and that small
> minor seventh sounds so perfect moving to a small major third.
> I'm with duckfeetbilly on this one. But there are so many ways
> to do it, and they all have unique affect. Like, I love the tortured
> sound of comma shifting from a ii7 tuned with large minor
> third and seventh into the septimal dominant. It pulls
> everything down and just ... kills. It's all just awesome.
> MUSIC IS AWESOME.

You're saying something like 10/9 4/3 5/3 2/1 -> 9/8 21/16 3/2 15/8 ->
1/1 5/4 3/2 2/1 for the second one?

-Mike

> --- In tuning@yahoogroups.com, "duckfeetbilly" <billygard@...> wrote:
> >
> > --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> > >
> > > I've posted about other frustrating comma pumps before, but this one
> > > has me at my wit's end. This might be the most common chord
> > > progression in all of music and it's also the most annoying to deal
> > > with in JI.
> > >
> > > Some things that seem to not work (let's assume we're in C):
> > > - Rooting the D on 9/8 and building up 6/5's and 5/4's from there.
> > > This makes the C in the iim7 a comma sharp of the finishing I, which
> > > sounds awful.
> > > - Rooting the D on 10/9 and building up 6/5's and 5/4's from there.
> > > This makes the D in the iim7 a comma flat of the D in the V7, which
> > > destroys the voice leading and also sounds awful.
> > > - Rooting the D on 9/8 and building up 32/27's and 81/64's from there.
> > > This sounds completely out of place if your goal is to have beatless
> > > "just" voicings, and since ii-V's are so common in almost everything,
> > > it almost sounds like the piece isn't in JI at all. On the other hand,
> > > this one avoids weird comma shifts.
> > >
> >
> > This is something that I have found to be the most curious of the JI
dilemmas, especially considering it is a simple chord set as well as one natural
sounding to the ear.
> >
> > When I try to come up with a solution to this I base it on the idea that I
am discovering the real arithmetic behind the ii-V-I that explains its musical
quality, i.e. what is the ear hearing about it that has made it stand the test
of time?
> >
> > One of my ideas is that the particular minor 7th chord that appears in this
progression could actually be tuned differently, as an 11th chord extension of
the following dominant. The tuning I have in mind is the sub-minor 7th, or
septimal-minor 7th. 12:14:18:21. As such, it really would share a fundamental
with the following dominant. Using this math, you can use the following
harmonics, relative to the tonic as the fundamental, to complete the whole chord
progresssion:
> >
> > 36:42:54:63 -> 36:42:48:60 - > 32:40:48:64
> >
> > Billy
> >
>
>

Great topic. This is one of my favorite flavors of ii7 V7 I.
Supertonic 7 with the small minor thirds sounds really dark
and leads perfectly to a septimal dominant 7, and that small
minor seventh sounds so perfect moving to a small major third.
I'm with duckfeetbilly on this one. But there are so many ways
to do it, and they all have unique affect. Like, I love the tortured
sound of comma shifting from a ii7 tuned with large minor
third and seventh into the septimal dominant. It pulls
everything down and just ... kills. It's all just awesome.
MUSIC IS AWESOME.


--- In tuning@yahoogroups.com, "duckfeetbilly" <billygard@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> >
> > I've posted about other frustrating comma pumps before, but this one
> > has me at my wit's end. This might be the most common chord
> > progression in all of music and it's also the most annoying to deal
> > with in JI.
> >
> > Some things that seem to not work (let's assume we're in C):
> > - Rooting the D on 9/8 and building up 6/5's and 5/4's from there.
> > This makes the C in the iim7 a comma sharp of the finishing I, which
> > sounds awful.
> > - Rooting the D on 10/9 and building up 6/5's and 5/4's from there.
> > This makes the D in the iim7 a comma flat of the D in the V7, which
> > destroys the voice leading and also sounds awful.
> > - Rooting the D on 9/8 and building up 32/27's and 81/64's from there.
> > This sounds completely out of place if your goal is to have beatless
> > "just" voicings, and since ii-V's are so common in almost everything,
> > it almost sounds like the piece isn't in JI at all. On the other hand,
> > this one avoids weird comma shifts.
> >
>
> This is something that I have found to be the most curious of the JI dilemmas,
especially considering it is a simple chord set as well as one natural sounding
to the ear.
>
> When I try to come up with a solution to this I base it on the idea that I am
discovering the real arithmetic behind the ii-V-I that explains its musical
quality, i.e. what is the ear hearing about it that has made it stand the test
of time?
>
> One of my ideas is that the particular minor 7th chord that appears in this
progression could actually be tuned differently, as an 11th chord extension of
the following dominant. The tuning I have in mind is the sub-minor 7th, or
septimal-minor 7th. 12:14:18:21. As such, it really would share a fundamental
with the following dominant. Using this math, you can use the following
harmonics, relative to the tonic as the fundamental, to complete the whole chord
progresssion:
>
> 36:42:54:63 -> 36:42:48:60 - > 32:40:48:64
>
> Billy
>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> I've posted about other frustrating comma pumps before, but this one
> has me at my wit's end. This might be the most common chord
> progression in all of music and it's also the most annoying to deal
> with in JI.
>
> Some things that seem to not work (let's assume we're in C):
> - Rooting the D on 9/8 and building up 6/5's and 5/4's from there.
> This makes the C in the iim7 a comma sharp of the finishing I, which
> sounds awful.
> - Rooting the D on 10/9 and building up 6/5's and 5/4's from there.
> This makes the D in the iim7 a comma flat of the D in the V7, which
> destroys the voice leading and also sounds awful.
> - Rooting the D on 9/8 and building up 32/27's and 81/64's from there.
> This sounds completely out of place if your goal is to have beatless
> "just" voicings, and since ii-V's are so common in almost everything,
> it almost sounds like the piece isn't in JI at all. On the other hand,
> this one avoids weird comma shifts.
>

This is something that I have found to be the most curious of the JI dilemmas,
especially considering it is a simple chord set as well as one natural sounding
to the ear.

When I try to come up with a solution to this I base it on the idea that I am
discovering the real arithmetic behind the ii-V-I that explains its musical
quality, i.e. what is the ear hearing about it that has made it stand the test
of time?

One of my ideas is that the particular minor 7th chord that appears in this
progression could actually be tuned differently, as an 11th chord extension of
the following dominant. The tuning I have in mind is the sub-minor 7th, or
septimal-minor 7th. 12:14:18:21. As such, it really would share a fundamental
with the following dominant. Using this math, you can use the following
harmonics, relative to the tonic as the fundamental, to complete the whole chord
progresssion:

36:42:54:63 -> 36:42:48:60 - > 32:40:48:64

Billy

I forgot to mention: of the 5-limit solutions, I'd go with Hindemith's: ii7 = 9/8 4/3 27/16 3/2.

> I forgot to mention: of the 5-limit solutions, I'd go with Hindemith's: ii7 =
9/8 4/3 27/16 3/2.

Yeah, everyone's suggested that so far... Seems like an interesting
option, but it might not work for sections with a slower harmonic
rhythm. Sometimes a composer will hang onto the ii7 chord for a long
time before switching to the V7 chord, and to have the ii7 chord be
completely dissonant by itself would kind of give away the punchline,
so to speak.

> And how do you get 1003 cents, exactly? I know 25/14 is about 1004...

It's actually 1002 cents, or 1001.612 more precisely. I was
remembering it wrong. It's the metastable interval between 9/5 and
16/9, which is the same thing as the metastable interval between 7/4
and 9/5, which is the same thing as the metastable interval between
7/4 and 2/1.

More info can be found on it at: http://dkeenan.com/Music/NobleMediant.txt

The general idea is that it's a useful and quick approximation to get
"maximally dissonant" intervals, and hence intervals that are not
stable and want to resolve.

> I had 72-edo in mind too. And if you mean the comma between 32/27 and 19/16,
it's Boethius' comma, 513/512.

Eh, I didn't feel like crunching the numbers, but it would actually be
a triad of commas tempered out:
  - Boethius' ridiculously tiny 3.4 cent comma,
  - The irrational interval between 32/27 and 299.657 cents (a perfect
fifth down from the 1002 cent metastable 7th), which turns out to be
5.522 cents,
  - The irrational interval between 19/16 and 299.657 cents, which
turns out to be 2.144 cents.

These are pretty small numbers :D
Although would the irrational intervals still be called "commas" in
any sense, since they aren't rational? Working with metastable
intervals brings stuff like this out into the open.

-Mike

--- On Mon, 11/23/09, Mike Battaglia <battaglia01@...> wrote:

> From: Mike Battaglia <battaglia01@...>
> Subject: Re: [tuning] How do you guys tune iim7-V7-I in JI?
> To: tuning@yahoogroups.com
> Date: Monday, November 23, 2009, 3:59 PM
> > You've already gotten some good
> answers, but I thought of something a bit unorthodox: using
> the 19-limit minor triad. It's similar in sound to the
> Pythagorean minor, and could be a form of adaptive JI, with
> the second shifting by the 513/512 comma (3.378 cents)
> between the first and second chords:
> >
> > ii7 = 64/57 4/3 32/19 2/1
> > V7 = 9/8 4/3 3/2 15/8
> > I7 = 1/1 5/3 3/2 15/8
>
> That is a good idea and it's what I've been doing for the
> moment.
> Actually, to be precise, I've been treating the 7th
> interval over the
> V7 as being ideally placed at the 1003 cent metastable
> interval... And
> then I put the minor seventh in the ii7 a fifth above
> that.

I forgot to mention: of the 5-limit solutions, I'd go with Hindemith's: ii7 =
9/8 4/3 27/16 3/2.

And how do you get 1003 cents, exactly? I know 25/14 is about 1004...

> Since I usually work in 72-equal, this is effectively the
> same thing
> for me as working with the Pythagorean intervals, as is it
> the same as
> working with 19-limit intervals as you suggest... those 3
> types of
> intervals are made to be equivalent, I'm not sure what the
> exact comma
> tempered out is.

I had 72-edo in mind too. And if you mean the comma between 32/27 and 19/16,
it's Boethius' comma, 513/512.

~D.

Steve,

As I said, when I knew that Piagui D major chord wave peak graph showed
unexpected peaks, started the complementary analysis which uses K and P
semitone factors.
K = (9/8)^(1/2) and P = (8/9)*2^(1/4) are cells of a geometric progression
where the first 612 cells cover the octave 1 up to 2. Here, commas M, J and
U are the factors which work in sequence. K for instance, is also given by
(M^32)*(J^18)*(U^2)= 1.06066017178...-- 32 + 18 + 2 = 52, so cell # 52
equals K. Similarly, P = 1.0570729911.... = (M^30)*(J^17)*(U^2) = Cell # 49.

Neither K nor P involves powers of 3 and of 2, as you can see. However,
commas M, J, U do it:

M = [(38 x 5) / 215] = [(32805 / 32768)] = 1.00112915039062

J = [(225x21/4)/(313x52)] = [(33554432x21/4)/39858075]=1.001131371103

U = [(212x 52 x 31/2) / 311] = [(102400 x 31/2)/177147] = 1.0012136965066


<If the mathematical analysis involves powers of 3 and of 2 then I don't
think it <"curious" that the Pythagorean Comma is involved.  The 1.496614954
ratio looks <very close indeed to the "1/6 PC tempered 5th" - is it actually
identical in your <system?
----
¿Would you explain me how can I calculate "1/6 PC tempered 5th"?. This way I
could answer your question. Is "tempered 5th" related to the equal tempered
scale ?-----

Last week, after many years the progression was deduced, I noticed the
interesting ratio:
[(K^4)/(P^4)] = 1.01364326477 = Pythagorean comma which is also given by the
twelft cell of the progression and the sixth cell is equal to the square
root of the PC:
It took me years to derive the progression where the first cell is the
schisma. Cell # 0 is note C. Below I give you the first 20 cells.

THE PROGRESSION OF MUSICAL CELLS

FIRST SEGMENT

        CELL                                RELATIVE FREQUENCY

           No.    COMMA           F(M,J,U)             DECIMAL VALUE

                      -                         -                    1 = C

           1          M                       M
1.00112915039062    ZC

           2          M                       M2
1.00225957576060

           3          J                         M2J
1.00339350328355

           4          J                         M2J2
1.00452871369807

           5          M                       M3J2
1.00566297768753

           6          M                       M4J2
1.00679852243163 (Its square root)

           7          M                       M5J2
1.00793534937651

           8          M                       M6J2
1.00907345996998

SET   9          J                         M6J3
1.01021509652337

   Q     10        J                         M6J4
1.01135802469136    ZC

           11        M                       M7J4               1.0125 ZC

           12        M                       M8J4
1.01364326477051  * Pythag. comma

           13        M                       M9J4
1.01478782045888

           14        M                       M10J4
1.01593366852275

           15        J                         M10J5
1.01708306651785

           16        J                         M10J6
1.01823376490863

           17        M                       M11J6
1.01938350396202

           18        M                       M12J6
1.02053454124372

------------------------------------------

           19        J                         M12J7
1.02168914453326

           20        J                         M12J8
1.02284505410760

           21        M                       M13J8              1.024
ZC

           22        M                       M14J8              1.02515625


Thanks
Mario Pizarro
piagui@...

Lima, November 23


----- Original Message -----
From: "martinsj013" <martinsj@...>
To: <tuning@yahoogroups.com>
Sent: Monday, November 23, 2009 12:30 PM
Subject: [tuning] Re: IMO


> --- In tuning@yahoogroups.com, "Mario" <piagui@...> wrote:
>> > I see that some new scale proposals decide on
>> > G = 1.5 = 3/2. Piagui scales also work with this value which was not
>> > selected
>> > when they were derived but was the result of a mathematical process
>> > started
>> > with the Piagui basic equations K^m P^n = 2 and m + n = 12.
> Curiously, the
>> > 1.01364326477 pythagorean comma appeared as a key factor in the
>> > analysis
>> > and finally a new scale system (three variants too) was derived. Here,
>> > the G
>> > frequency is 1.496614954 and this value has a solid connection with the
>> > remaining 11 semitone factors therefore it would be a nonsence to
>> > replace
>> > this narrow fifth by 1.5.
>
> If the mathematical analysis involves powers of 3 and of 2 then I don't
> think it "curious" that the Pythagorean Comma is involved.  The
> 1.496614954 ratio looks very close indeed to the "1/6 PC tempered 5th" -
> is it actually identical in your system?
>
> Steve M.
>
>
>
> ------------------------------------
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>

> You've already gotten some good answers, but I thought of something a bit
unorthodox: using the 19-limit minor triad. It's similar in sound to the
Pythagorean minor, and could be a form of adaptive JI, with the second shifting
by the 513/512 comma (3.378 cents) between the first and second chords:
>
> ii7 = 64/57 4/3 32/19 2/1
> V7 = 9/8 4/3 3/2 15/8
> I7 = 1/1 5/3 3/2 15/8

That is a good idea and it's what I've been doing for the moment.
Actually, to be precise, I've been treating the 7th interval over the
V7 as being ideally placed at the 1003 cent metastable interval... And
then I put the minor seventh in the ii7 a fifth above that.

Since I usually work in 72-equal, this is effectively the same thing
for me as working with the Pythagorean intervals, as is it the same as
working with 19-limit intervals as you suggest... those 3 types of
intervals are made to be equivalent, I'm not sure what the exact comma
tempered out is.

Of course, in 72-equal, it's also the same thing as working in regular
old 12-tet >:| I was hoping there'd be some way to get away from that,
but maybe that's why ii7-V7-I7 sounds so fluid in 12-equal, anyway.

> ~D. ¶¦¬{> http://dannywier.ucoz.com
>

--- On Mon, 11/23/09, Mike Battaglia <battaglia01@...> wrote:

> From: Mike Battaglia <battaglia01@...>
> Subject: [tuning] How do you guys tune iim7-V7-I in JI?
> To: tuning@yahoogroups.com
> Date: Monday, November 23, 2009, 2:59 AM
> I've posted about other frustrating
> comma pumps before, but this one
> has me at my wit's end. This might be the most common
> chord
> progression in all of music and it's also the most annoying
> to deal
> with in JI.
>
> Some things that seem to not work (let's assume we're in
> C):
> - Rooting the D on 9/8 and building up 6/5's and 5/4's from
> there.
> This makes the C in the iim7 a comma sharp of the finishing
> I, which
> sounds awful.
> - Rooting the D on 10/9 and building up 6/5's and 5/4's
> from there.
> This makes the D in the iim7 a comma flat of the D in the
> V7, which
> destroys the voice leading and also sounds awful.
> - Rooting the D on 9/8 and building up 32/27's and 81/64's
> from there.
> This sounds completely out of place if your goal is to have
> beatless
> "just" voicings, and since ii-V's are so common in almost
> everything,
> it almost sounds like the piece isn't in JI at all. On the
> other hand,
> this one avoids weird comma shifts.

You've already gotten some good answers, but I thought of something a bit
unorthodox: using the 19-limit minor triad. It's similar in sound to the
Pythagorean minor, and could be a form of adaptive JI, with the second shifting
by the 513/512 comma (3.378 cents) between the first and second chords:

ii7 = 64/57 4/3 32/19 2/1
V7  = 9/8 4/3 3/2 15/8
I7  = 1/1 5/3 3/2 15/8

~D. ¶¦¬{> http://dannywier.ucoz.com

On 11/23/09, Carl Lumma <carl@...> wrote:
> Different tunings of these chords are suited to different
>  voicings.  Moreover, listening to short chord progressions in
>  isolation doesn't tell much about how they'll serve in a piece
>  of music.  If the progression is used in passing, the tuning
>  might not matter at all.

OK, so let's take a tune like Autumn Leaves:

| Cm7 | F7 | Bb | Eb | Am7 | D7 | Gm | Gm |
| Cm7 | F7 | Bb | Eb | Am7 | D7 | Gm | Gm |
| Am7 | D7 | Gm | Gm | Cm7 | F7 | Bb | Bb |
| Am7 | D7 | Fm | Bb7 | Eb | D7 | Gm | Gm |

That's kind of a simplified version of it, but almost the entire song
can be viewed as a series of ii-V's... just like every other piece of
music from this time period. Not to mention that as a jazz pianist I'd
be throwing in extra ii-V's for harmonic color if I feel like it.

> If the melody has previously been
>  making use of comma shifts, then perhaps you'll want them.  If
>  you just want the 'diatonic' style, meantone-based adaptive JI
>  would probably be best.  If you want 7-limit harmony and no
>  shifts, it can be rooted in pajara instead, e.g. D = '8/7' and
>  the 64/63 between the final two Gs vanishes.

Hey, that pajara idea sounds pretty neat. I'll check that out, thanks
for the reply.

-Mike

Klaus, Marcel, and Andreas - you're all saying the same thing, and
I've never thought of that. I'll check that out. Thanks for the reply.

-Mike