Yahoo Tuning Groups Ultimate Backup tuning a problem with 72-tET (??)

I think we've pretty much come to a conclusion that 72-tET is
the "Long Island Princess" of tunings. It pretty much has
everything...

However, I am finding one problem. My synthesizer only has 60 keys!
If I am to use the traditional "Halberstadt" keyboard, I can't even
reach one octave.

Of course, I can get a larger set of MIDI notes by entering note data
directly in a sequencer...

However, it is still severely limited by the number of MIDI notes...
how many are in the total?? Surely it's more than 128 (??)

ANYWAY, I'm thinking for the "keyboard set" there either needs to be
either a different kind of keyboard... a Bosenquet?... Dunno...
Otherwise the reachable pitch spectrum is severely limited.

Likewise, the MIDI limitations... Anybody done some creative
thinking about this "problem??"

Is it possible that 72-tET works better for instruments such as
strings and woodwinds where "discrete" steps don't always have to be
present??

Any thoughts from anybody??

_______ __________ ______ ____
Joseph Pehrson

🔗Afmmjr@aol.com

🔗David Beardsley <xouoxno@virtulink.com>

🔗Rick Tagawa <ricktagawa@earthlink.net>

I've been using 6 synthesizers each globally detuned 17ï¿½. The way I have them arranged is top:
+17ï¿½; 2nd: Concert Pitch; 3rd: -17ï¿½; 4th: -33ï¿½; 5th: -50ï¿½; and 6th: -67ï¿½. I'm using a DX7II; 2
DX7wE!; 1 DX7 and 2 Proteus 1 with Dummy keyboards as has been suggested. It's working out for
pitch reference. All of the pitches is a bit daunting. I've just spent the last couple of days
working out usable notes for each of the 72 pitches.

My latest email to Dave Canright goes as follows:

Subject:
pitch mapping
Date:
Sun, 22 Apr 2001 10:13:00 -0700
From:
Rick Tagawa <ricktagawa@earthlink.net>
To:
Dave <dcanright@nps.navy.mil>

Dear Dave,
This pitch mapping of "usable notes in 72" is really the ticket. As I
study each one of the 72 as I do them I realize a major hurdle is
overcome by figuring out how each transposition affects which notes.
For instance bringing the 1/1 pitch down 1/3 of a semi tone means that
all 67ï¿½ notes have new names. So it's a transposition down to a new
note name and then shifting that note over one column (say from Db to
the C column.)

Finale is excellent for this exact task.

Composing at the 72 keyboard can a little daunting trying to keep track
of these transpositions but with these little pitch maps things should
go more smoothly.

The only other chords I can think of are the equal temperaments.

In C, as a model, I have the following pitch map of notes derived from
overtones and undertones.

+17ï¿½ = C#(15u); F(43o); f#(45u); Ab(5u);B(61o) (small case = error >5ï¿½)

CP = C(1/1); C#(34o);D(9o); D#(38o); e(51u); F(3u); G(3o); ab(51o);
a(54o); Bb(9u); B(34u)

-17ï¿½ = D(61u); E(5o); f#(45o); G(43u); B(15o)

-33ï¿½ = c(63o); C#(58u); D(29u); d#(52u); e(13u); F(21o); F#(23u);
G(47o); Ab(25o); a(53o); Bb(7o); b(59o)

-50ï¿½ = C(31o); C#(33o); D(35o); D#(37o); e(39o); F#(11o); G(11u);
a(39u); Bb(37u); B(35u)

-67ï¿½ = C#(63u); d(59u); D#(7u); e(53u); F(41o); F#(47u); G(23o);
Ab(49o); a(13o); Bb(55o); B(29o)

The way I've spelled it, chords like the minor and dominant seventh are
clear. And what is also clear is the lower ratio versions of each
degree.

I'm eventually going to add all this stuff to my website at:

http://sites.netscape.net/masanoritagawa/homepage

jpehrson@rcn.com wrote:

> I think we've pretty much come to a conclusion that 72-tET is
> the "Long Island Princess" of tunings. It pretty much has
> everything...
>
> However, I am finding one problem. My synthesizer only has 60 keys!
> If I am to use the traditional "Halberstadt" keyboard, I can't even
> reach one octave.
>
> Of course, I can get a larger set of MIDI notes by entering note data
> directly in a sequencer...
>
> However, it is still severely limited by the number of MIDI notes...
> how many are in the total?? Surely it's more than 128 (??)
>
> ANYWAY, I'm thinking for the "keyboard set" there either needs to be
> either a different kind of keyboard... a Bosenquet?... Dunno...
> Otherwise the reachable pitch spectrum is severely limited.
>
> Likewise, the MIDI limitations... Anybody done some creative
> thinking about this "problem??"
>
> Is it possible that 72-tET works better for instruments such as
> strings and woodwinds where "discrete" steps don't always have to be
> present??
>
> Any thoughts from anybody??
>
> _______ __________ ______ ____
> Joseph Pehrson
>
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🔗PERLICH@ACADIAN-ASSET.COM

--- In tuning@y..., jpehrson@r... wrote:

>
> However, I am finding one problem. My synthesizer only has 60
keys!
> If I am to use the traditional "Halberstadt" keyboard, I can't even
> reach one octave.

I thought you were going to use 72-tET for notation, not to actually
map all the notes to a keyboard!
>
> Of course, I can get a larger set of MIDI notes by entering note
data
> directly in a sequencer...
>
> However, it is still severely limited by the number of MIDI
notes...

Just use pitch bends, like John deLaubenfels (who gets an "infinite"
number of pitches from MIDI)
>
> ANYWAY, I'm thinking for the "keyboard set" there either needs to
be
> either a different kind of keyboard... a Bosenquet?... Dunno...

A StarrLabs MicroZone . . .

>
> Is it possible that 72-tET works better for instruments such as
> strings and woodwinds where "discrete" steps don't always have to
be
> present??

Yes, absolutely!
>
> Any thoughts from anybody??
>
Well, if you want to talk about pitch sets that you can put on a
standard keyboard, let's talk about that . . . remember, 72-tET came
up when you were talking about Ben Johnston's notation . . . try
putting all of Ben Johnston's pitches on a keyboard! Ben Johnston did
write a sonata for microtonal piano . . . most of the "octaves" were
not 2/1 . . . so creative subsetting of 72-tET would be one approach
to your keyboard "problem" . . . for example, we could explore 19-
tone subsets of 72-tET, making use of the fact that the 225:224
disappears in 72-tET . . .

🔗paul@stretch-music.com

On Thu, 26 April 2001, Rick Tagawa wrote:
> Dear Dave,
> This pitch mapping of "usable notes in 72" is really the ticket.

I'm having trouble understanding what you mean by usable notes in 72.

> In C, as a model, I have the following pitch map of notes derived from
> overtones and undertones.
> +17¢ = C#(15u); F(43o); f#(45u); Ab(5u);B(61o) (small case = error
> >5¢)

Again, I'm not quite seeing what you're trying to do hear, but I really doubt that appealing to the 43rd overtone is really going to be useful.

You can map all of 72-tET to a 5-prime-limit lattice;
72-tET uniquely articulates all the 11-limit consonances (11-limit Tonality Diamond);
72-tET is only consistent through the 17-limit (ratios of odd numbers 17 or less).

So, even though I'm not sure what you're trying to do, I'm skeptical that 43-limit ratios are going to help you. Perhaps you could explain what you're trying to do in a little more detail, and I'll do my best to help you?
As we've seen, it's particularly easy to conceptualize all the 72-tET pitches in terms of the 11-limit.

🔗Kraig Grady <kraiggrady@anaphoria.com>

🔗Rick Tagawa <ricktagawa@earthlink.net>

I'm also reminded that David Doty said he uses 2 TX-802s and that just
one would output 6 different tunings each outputted on 6 different midi
channels. Of course you would need 6 keyboard controllers but they can
be fairly small and sit on top of each other.
RT

Kraig Grady wrote:

> Joseph!
> One of the things that really got Erv excited about Hanson' s
> keyboard is that it handles 72 where as the Bosanquet does not. I will
> put it up in a day or two or look at his keyboard and you might just
> break the code yourself. hint 53+19 =72
> 19 being number of ranks
>
> jpehrson@rcn.com wrote:
>
>>
>>
>> ANYWAY, I'm thinking for the "keyboard set" there either needs to be
>>
>> either a different kind of keyboard... a Bosenquet?... Dunno...
>> Otherwise the reachable pitch spectrum is severely limited.
>>
>> Any thoughts from anybody??
>>
>> _______ __________ ______ ____
>> Joseph Pehrson
>>
>
>
> -- Kraig Grady
> North American Embassy of Anaphoria island
> http://www.anaphoria.com
>
> The Wandering Medicine Show
> Wed. 8-9 KXLU 88.9 fm
>
>
>
> Yahoo! Groups Sponsor
[www.debticated.com]

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🔗Alexandros Papadopoulos <Alexmoog@otenet.gr>

🔗jpehrson@rcn.com

--- In tuning@y..., Afmmjr@a... wrote:

/tuning/topicId_21636.html#21638

> Joseph, why not get a full-length electronic dummy keyboard to
attach to your synth. Then you'll have full range.
>
> Johnny Reinhard

"Dummy" keyboard?! What are you implying, Johnny? Something about
*ME* or about the keyboard??

I guess you mean an 88-note keyboard, yes, since you're obviously not
talking about a Bosenquet... Well, that's only 28 more keys than I
*already* have... That's not much help, really.

72-tET really "gobbles up" fixed keyboards in a hurry!

By the way, does anybody remember what the maximum number of MIDI
notes available in ENTIRE is??

72 STILL would "gobble up" those pretty fast, too...

________ _______ ___ _____
Joseph Pehrson

🔗jpehrson@rcn.com

--- In tuning@y..., PERLICH@A... wrote:
> --- In tuning@y..., jpehrson@r... wrote:

> > ANYWAY, I'm thinking for the "keyboard set" there either needs to
> be either a different kind of keyboard... a Bosenquet?... Dunno...
>
> A StarrLabs MicroZone . . .
>

Well, of course, that's an excellent idea, and Greg Schiemer just
bought one... They're on sale now for only $12,000...

> > Is it possible that 72-tET works better for instruments such as
> > strings and woodwinds where "discrete" steps don't always have to
> be present??
>
> Yes, absolutely!
> >
> > Any thoughts from anybody??
> >

> Well, if you want to talk about pitch sets that you can put on a
> standard keyboard, let's talk about that . . . remember, 72-tET
came up when you were talking about Ben Johnston's notation . . . try
> putting all of Ben Johnston's pitches on a keyboard!

Forgetaboutit!

>Ben Johnston did write a sonata for microtonal piano . . . most of
the "octaves" were not 2/1 . . . so creative subsetting of 72-tET
would be one approach to your keyboard "problem" . . . for example,
we could explore 19-tone subsets of 72-tET, making use of the fact
that the 225:224 disappears in 72-tET . . .

Yes! This, Paul, is EXACTLY what I need to know now... I need to
know some good 19-tone subsets of 72-tET.

Where would we begin??

I still think I'm inclined to go with the great 19-tone just scales
that you got strictly from "outer space" using your Periodicity Block
method... but I need to have a repertoire of 19-tone 72-tET subsets...

This will "set me up" for quite a bit of composing, I can assure you!

Thanks!

_______ _____ __ _____
Joseph Pehrson

🔗jpehrson@rcn.com

🔗PERLICH@ACADIAN-ASSET.COM

--- In tuning@y..., jpehrson@r... wrote:>
>
> Yes! This, Paul, is EXACTLY what I need to know now... I need to
> know some good 19-tone subsets of 72-tET.
>
> Where would we begin??

Dave Keenan would be the best person to ask . . .
>
> I still think I'm inclined to go with the great 19-tone just scales
> that you got strictly from "outer space" using your Periodicity
Block
> method... but I need to have a repertoire of 19-tone 72-tET
subsets...

Well, first of all, all of those periodicity blocks would of course
work perfectly well in 72-tET . . . just translate the consonant
generating intervals from JI to 72-tET.

But you can do things in 72-tET that you can't do in JI . . . James
Tenney has exploited this fact in his compositions . . . the 225:224
vanishing means you can get more consonant intervals in a 72-tET than
in a JI scale with the same number of notes . . . Dave Keenan should
be able to give you some excellent ideas along these lines.

🔗monz <joemonz@yahoo.com>

--- In tuning@y..., jpehrson@r... wrote:

/tuning/topicId_21636.html#21636

Joe, many of us here who have advocated 72-tET like it not
especially as a *tuning*, but as a *notation*.

It was my understanding that you were originally looking for
a good way to notate JI. 72-tET fits the bill.

So why not just program your keyboard to play the JI pitches
you need, then write it all down in 72-tET?

I realize that you would prefer to have all 72 pitches before
you, but there's no way around the limitations of Halberstadt
keyboard, other than Rick Tagawa's idea of using several at once.
Otherwise, save up your $$$ for the MicroZone...

-monz
http://www.monz.org
"All roads lead to n^0"

🔗PERLICH@ACADIAN-ASSET.COM

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> --- In tuning@y..., jpehrson@r... wrote:
>
> /tuning/topicId_21636.html#21636
>
> > I think we've pretty much come to a conclusion that 72-tET is
> > the "Long Island Princess" of tunings. It pretty much has
> > everything...

May I ask what "Long Island Princess" means? I'm from Queens . . .
>
>
> Joe, many of us here who have advocated 72-tET like it not
> especially as a *tuning*, but as a *notation*.
>
> It was my understanding that you were originally looking for
> a good way to notate JI. 72-tET fits the bill.
>
> So why not just program your keyboard to play the JI pitches
> you need, then write it all down in 72-tET?
>
Monz, these were my thoughts exactly. Joseph seemed to have chosen a
JI approach, then got interested in notation for JI, and settled on
72-based notation. Now he's reconsidering his basic approach, and
talking as if he wants to compose in 72-tET. Nothing wrong with that
(in fact I encourage exploiting the fact that 19-out-of-72-tET can
have more consonances than 19-out-of-JI), and no offense to Joseph,
but it does seem like Joseph is proceeding in his thinking
by "jumping around" and "taking tangents" rather than a sequential,
logical inquiry. That's OK -- in fact he may produce a better body of
musical work if he keeps "jumping around" rather than adopting a
consistent philosophy and sticking to it -- who knows?

🔗monz <joemonz@yahoo.com>

--- In tuning@y..., jpehrson@r... wrote:

/tuning/topicId_21636.html#21667

> By the way, does anybody remember what the maximum number of
> MIDI notes available in ENTIRE is??

MIDI gives you 128 notes, period.

Of course, as Paul pointed out in connection with John
deLaubenfels's work, using pitch-bend in MIDI gives you
essentially an infinite number of pitches to work with.
The MIDI pitch-bend spec of 4096 [= 2^12] units per semitone
amounts to 49152-EDO. That's far more precise than anything
that anyone's ears can hear.

But it's a pain in the ass to work that way. Take it from
me - I know. I do all of my microtonal music in MIDI.

So whether it's MIDI, CSound, or keyboards, they all have
their problems and limitations in regard to tuning.

I suppose the best solution is _a cappella_ singing...

-monz
http://www.monz.org
"All roads lead to n^0"

🔗jpehrson@rcn.com

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

/tuning/topicId_21636.html#21691

> --- In tuning@y..., jpehrson@r... wrote:>
>
> Well, first of all, all of those periodicity blocks would of course
> work perfectly well in 72-tET . . . just translate the consonant
> generating intervals from JI to 72-tET.
>

Well, that would make sense but, of course, 72-tET wouldn't
accomodate ALL the just intervals in the 19-tone periodicity blocks
we created... or WOULD it (??)

> But you can do things in 72-tET that you can't do in JI . . . James
> Tenney has exploited this fact in his compositions . . . the
225:224 vanishing means you can get more consonant intervals in a 72-
tET than in a JI scale with the same number of notes . . . Dave
Keenan should be able to give you some excellent ideas along these
lines.

Well, Tenney is a pretty smart "dude"... I've been reading about
him... I can't say that I exactly understand why I would get more
consonant intervals, except for the vague notion that 72-tET is
more "transposible..."

Dave Keenan hasn't been around much recently... I guess I'll have to
contact him privately...

_______ ______ _____ ___
Joseph Pehrson

🔗jpehrson@rcn.com

--- In tuning@y..., "monz" <joemonz@y...> wrote:

/tuning/topicId_21636.html#21693

>
> > I think we've pretty much come to a conclusion that 72-tET is
> > the "Long Island Princess" of tunings. It pretty much has
> > everything...
> >
> > However, I am finding one problem. My synthesizer only has 60
> > keys! If I am to use the traditional "Halberstadt" keyboard,
> > I can't even reach one octave.
>
>
> Joe, many of us here who have advocated 72-tET like it not
> especially as a *tuning*, but as a *notation*.
>

Whoopie, doopie... I hadn't even THOUGHT about that duality! What
would I do without "youse guys..."

> It was my understanding that you were originally looking for
> a good way to notate JI. 72-tET fits the bill.
>
> So why not just program your keyboard to play the JI pitches
> you need, then write it all down in 72-tET?
>

Of course! That makes a big distinction! Thanks so much for this
intelligence, Joe. You're a wizard...

>
> I realize that you would prefer to have all 72 pitches before
> you, but there's no way around the limitations of Halberstadt
> keyboard, other than Rick Tagawa's idea of using several at once.

I can't imagine anyone who isn't an octopus playing this. I'd like
to see it in "realtime," especially when lots of just sonorities are
required, jumping intervals with 6 keyboards!

Maybe that would be good "theatre" for Johnny Reinhard... I'll
discuss it with him!

> Otherwise, save up your $$$ for the MicroZone...
>

Somebody said that the MicroZone really wasn't that great for 72-
tET... Who said that, why did they say it, and what does that mean??

_________ ______ _____ ___
Joseph Pehrson

🔗paul@stretch-music.com

--- In tuning@y..., jpehrson@r... wrote:
> --- In tuning@y..., PERLICH@A... wrote:
>
> /tuning/topicId_21636.html#21691
>
> > --- In tuning@y..., jpehrson@r... wrote:>
> >
> > Well, first of all, all of those periodicity blocks would of
course
> > work perfectly well in 72-tET . . . just translate the consonant
> > generating intervals from JI to 72-tET.
> >
>
> Well, that would make sense but, of course, 72-tET wouldn't
> accomodate ALL the just intervals in the 19-tone periodicity blocks
> we created... or WOULD it (??)

Joseph . . . those blocks were constructed out of only 7-limit
consonant intervals . . . 72-tET can represent not only all 7-limit
consonant intervals, but also all 11-limit consonant intervals
uniquely, and even all 17-limit consonant intervals consistently (but
not necessarily uniquely). These were only 7-limit periodicity
blocks, but I would generally have little hesitation translating any
17-limit periodicity block to 72-tET.
>
> > But you can do things in 72-tET that you can't do in JI . . .
James
> > Tenney has exploited this fact in his compositions . . . the
> 225:224 vanishing means you can get more consonant intervals in a
72-
> tET than in a JI scale with the same number of notes . . . Dave
> Keenan should be able to give you some excellent ideas along these
> lines.
>
> Well, Tenney is a pretty smart "dude"... I've been reading about
> him... I can't say that I exactly understand why I would get more
> consonant intervals, except for the vague notion that 72-tET is
> more "transposible..."

OK, Joseph, let's go back to diatonic scales, shall we. My claim is
that diatonic scales can have more consonant intervals in meantone
temperament than in JI. Are you clear on that, or do we need to go
over this? If you're clear on it, you'll know that the reason is that
the 81:80 vanishes in meantone. And any time a comma vanishes, you
get extra consonant intervals connecting notes on one end of the
periodicity block with those on the opposite end. So . . . if you
have a 19-tone periodicity block in which 225:224 was one of the
consonant intervals, and you play it in 72-tET (where the 225:224
vanishes), you'll end up with some extra consonant intervals
connecting pitches on opposite sides of the periodicity block.

For each of the periodicity blocks I posted for you, I posted all
three unison vectors. So in any of the blocks where one of the unison
vectors was 225:224 (and even some of those where one wasn't), you'll
get a larger number of consonant intervals in 72-tET than you would
in JI.
>
> Dave Keenan hasn't been around much recently... I guess I'll have
to
> contact him privately...

I just did . . . he's probably sleeping (in Australia).

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

🔗jpehrson@rcn.com

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

/tuning/topicId_21636.html#21697

> --- In tuning@y..., "monz" <joemonz@y...> wrote:
> >
> > --- In tuning@y..., jpehrson@r... wrote:
> >
> > /tuning/topicId_21636.html#21636
> >
> > > I think we've pretty much come to a conclusion that 72-tET is
> > > the "Long Island Princess" of tunings. It pretty much has
> > > everything...
>
> May I ask what "Long Island Princess" means? I'm from Queens . . .
> >

Oh... well just a silly analogy. If you go out on Long Island there
are certain "princesses" who "have everything." They're probably
doing their nails right now as we speak. Surely, this is no
criticism of them, directly, but just an analogy that 72-tET is
"stacked," as it were, in the direction of multiple tuning
fulfillments...

Everybody out there has 15-speed bikes as well!

> > Joe, many of us here who have advocated 72-tET like it not
> > especially as a *tuning*, but as a *notation*.
> >
> > It was my understanding that you were originally looking for
> > a good way to notate JI. 72-tET fits the bill.
> >
> > So why not just program your keyboard to play the JI pitches
> > you need, then write it all down in 72-tET?
> >

> Monz, these were my thoughts exactly. Joseph seemed to have chosen
a JI approach, then got interested in notation for JI, and settled
on 72-based notation. Now he's reconsidering his basic approach, and
> talking as if he wants to compose in 72-tET. Nothing wrong with
that (in fact I encourage exploiting the fact that 19-out-of-72-tET
can have more consonances than 19-out-of-JI), and no offense to
Joseph, but it does seem like Joseph is proceeding in his thinking
> by "jumping around" and "taking tangents" rather than a sequential,
> logical inquiry. That's OK -- in fact he may produce a better body
of musical work if he keeps "jumping around" rather than adopting a
> consistent philosophy and sticking to it -- who knows?

Answer: I doubt it...

Frankly, I should probably clarify my process of "thought" or "anti-
thought."

Actually, I was ALWAYS thinking about the notion of COMPOSING in Just
Intonation. It never occured to me to think of a NOTATION SEPARATE
from composing...or one tuning system functioning ONLY AS A NOTATION
for another different one... I just never thought about it. But,
clearly, a person can set out to compose in a certain way, and
then use a convenient notation to convey it...

When we started with the Ben Johnston notation there was no question
that the COMPOSING in Just Intonation and the NOTATION were identical
(if, however, flawed.)

The same for "Monzowolfellholtz."

HOWEVER, it is only when we get into 72-tET that there is a
possibility of a notation... EQUAL TEMPERED that DIFFERS from the
COMPOSING PROCESS of Just Intonation.

And, in my "defense," and there is a *slight* one... I was ALWAYS AT
FIRST thinking about using 72-tET as a way to approximate JUST
Intonation... even if I had all the pitches available on keyboards, I
was still BASICALLY thinking about using the just intervals that you
outline on the Tagawa webpage chart.

HOWEVER, would I *really* compose that way with it?? Answer:
probably not... Once I got all the pitches and started "hearing
things" I would probably forget all about the just systems and start
just USING 71-tET in one way or another.

So, from that standpoint you're right... I would have "evolved" more
into the system than I had intended originally. And I _know_ Paul
that you're interested in equal temperaments for some of the
transpositions and consonances that they imply... and you're right, I
probably would have started working that way.

Whether that is being "inconsistent," "jumping around" or just
"evolving" depends on ones point of view...

Also, many of these ideas are still rather new to me and I'm a little
like a "kid in a candy store" taking a bit of this here and a bit of
this there. Some of it I "like" and some I don't "like," but I
haven't even TRIED everything yet!

After all, Paul... I've been studying this stuff for about a year and
a half, and I would guess you have been thinking about it for 10
years...

Is that right??

_________ _______ ____ ______
Joseph Pehrson

🔗jpehrson@rcn.com

--- In tuning@y..., "monz" <joemonz@y...> wrote:

/tuning/topicId_21636.html#21698
>
> --- In tuning@y..., jpehrson@r... wrote:
>
> /tuning/topicId_21636.html#21667
>
> > By the way, does anybody remember what the maximum number of
> > MIDI notes available in ENTIRE is??
>
>
> MIDI gives you 128 notes, period.
>
> Of course, as Paul pointed out in connection with John
> deLaubenfels's work, using pitch-bend in MIDI gives you
> essentially an infinite number of pitches to work with.
> The MIDI pitch-bend spec of 4096 [= 2^12] units per semitone
> amounts to 49152-EDO. That's far more precise than anything
> that anyone's ears can hear.
>
> But it's a pain in the ass to work that way. Take it from
> me - I know. I do all of my microtonal music in MIDI.
>

And the pitch bends were the way you did the great Johnston example,
yes?? Man, I'd LOVE to hear more of his music done by you like
that!!! Amazing.

_________ _______ _____ _
Joseph Pehrson

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

🔗jpehrson@rcn.com

--- In tuning@y..., paul@s... wrote:

/tuning/topicId_21636.html#21707

> >
> > Well, that would make sense but, of course, 72-tET wouldn't
> > accomodate ALL the just intervals in the 19-tone periodicity
blocks we created... or WOULD it (??)
>
> Joseph . . . those blocks were constructed out of only 7-limit
> consonant intervals . . . 72-tET can represent not only all 7-limit
> consonant intervals, but also all 11-limit consonant intervals
> uniquely, and even all 17-limit consonant intervals consistently
(but not necessarily uniquely). These were only 7-limit periodicity
> blocks, but I would generally have little hesitation translating
any 17-limit periodicity block to 72-tET.
> >

Wow... I've really been "underestimating" this scale. This is really
"coming together" now... I can see LOTS of even "higher limit" just
in this scale! Of course, the basic interval is so "teeny-tiny," I
suppose I shouldn't be so surprised it could do things like this
somehow...
>

> OK, Joseph, let's go back to diatonic scales, shall we.

By the way... this was a great refresher for me... The implications
really projected for me onto 72... Thanks again for the "reprise..."

>
> For each of the periodicity blocks I posted for you, I posted all
> three unison vectors.

Finally, I'm beginning to understand what that all was about. I
never *did* get an explanation, Paul of how those vectors worked..

They seem, in fact, to work a little like the "exponential" vectors
that Joe Monzo was using... or am I totally off base on that??

> > Dave Keenan hasn't been around much recently... I guess I'll have
> to contact him privately...
>
> I just did . . . he's probably sleeping (in Australia).

He just woke up! I saw his post!

_______ _______ ______ __
Joseph Pehrson

🔗monz <joemonz@yahoo.com>

--- In tuning@y..., jpehrson@r... wrote:

/tuning/topicId_21636.html#21723

> And the pitch bends were the way you did the great Johnston
> example, yes??

Yup. Actually, I entered the pitches using my handy little
Cakewalk auxilliary program "micro.CAL"
http://www.ixpres.com/interval/monzo/micro/micro.htm

With Micro, I can simply enter each note one at a time,
specifying the duration the way musicians are used to thinking
about it (which is *really* nice, and took me a year of hard
thinking to figure out!), and the program calculates the
correct duration and pitch-bend (within ~2 cents). All I have
to edit is the "octave", because Micro puts all the notes
within one "octave".

Then I can simply open the MIDI "event list" window and
fine-tune each pitch-bend if I want 100% accuracy, using,
for example (for ratios up to 7-limit), this lattice
http://www.ixpres.com/interval/monzo/lattices/pitch-bend-lattice.htm

> Man, I'd LOVE to hear more of his music done by you like
> that!!! Amazing.

Thanks, Joe, glad you liked it so much.

I realized after I posted that excerpt from his 8th quartet
that I didn't work on any of the dynamics! So it would actually
be quite easy to make the example *you* were so impressed with
even a whole lot better!

Of course, I wrote in the post you quoted here that working
like this in MIDI is a pain in the ass. Well, add to that all
the complications of figuring out Johnston's notation, and
having to have *his* lattices in front of me as well as mine,
and... well, that's why I've done so little of it. I have only
a few short examples of about 4 of his pieces. The one I posted
was the longest, and the best-sounding.

But hopefully, someday, I'll get around to doing more.
I'd like to finish an entire piece, or at least a whole
movement, and find out what Ben himself thinks of it.

-monz
http://www.monz.org
"All roads lead to n^0"

🔗monz <joemonz@yahoo.com>

--- In tuning@y..., jpehrson@r... wrote:

/tuning/topicId_21636.html#21727

> Wow... I've really been "underestimating" this scale [72-EDO].
> This is really "coming together" now... I can see LOTS of
> even "higher limit" just in this scale! Of course, the basic
> interval is so "teeny-tiny," I suppose I shouldn't be so
> surprised it could do things like this somehow...

But Joe, it's not really because the basic interval is
"teeny-tiny". It's really because of the fortuitous
happenstance that the larger steps which are built out of
this "teeny-tiny" one happen to coincide so closely with
the basic prime intervals:

The 12-tET notes are close to 3-, 17- and 19-limit ratios.
The 1/12-tone =/- gives the 5-limit correction from 12-tET.
The 1/6-tone </> gives the 7-limit correction from 12-tET.
The 1/4-tone ^/v gives the 11-limit correction from 12-tET.
(and to some extent the 13-limit, but not as good)

> They seem, in fact, to work a little like the "exponential" vectors
> that Joe Monzo was using... or am I totally off base on that??

Joe, clarify for me what you mean by the "exponential" vectors
I was using, and maybe I can help answer.

-monz
http://www.monz.org
"All roads lead to n^0"

🔗monz <joemonz@yahoo.com>

--- In tuning@y..., "monz" <joemonz@y...> wrote:

/tuning/topicId_21636.html#21731

> Then I can simply open the MIDI "event list" window and
> fine-tune each pitch-bend if I want 100% accuracy, using,
> for example (for ratios up to 7-limit), this lattice
> http://www.ixpres.com/interval/monzo/lattices/pitch-bend-lattice.htm

Hmmm... I just took a look at that lattice, and I see that
it's very pertinent to our recent discussion of 72-EDO notation.

I indicated pitches on this lattice using the "combined"
method I elaborated on in a previous post: 72-EDO accidentals
with each note-name, and pitch-bend values for the actual JI
pitches that the 72-EDO approximately represents.

Just thought I should point that out to you, Joe, in case you
glance at that lattice.

Hmmm again... in case you ultimately decide that you prefer
Johnny Reinhard-style 1200-EDO notation, maybe you'd instead
like to "upgrade" to 49152-EDO and get your performers thinking
in terms of MIDI pitch-bend.

Just kidding... but then again, if you make computer mock-ups
of your pieces to help your performers get the intonation correct
(as Ezra Sims does), 49152-EDO would give you ease in switching
back and forth between MIDI and human performers. (But still,
this was supposed to be a joke... I recommend staying with
72-EDO.)

-monz
http://www.monz.org
"All roads lead to n^0"

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

Hi Joseph,

Here's the best 11-limit 19-tone scale that has no steps smaller than
3 steps of 72-tET.

I've given everything using 72-tET degree numbers. It would be a lot
easier to understand if it was rewritten using standard 72-tET
notation. Could someone else please do that and post the result?

Pitches Step
sizes
(1/72nds of an octave)
--------------
0 3
3 4
7 3
10 6
16 3
19 4
23 3
26 4
30 3
33 6
39 3
42 4
46 3
49 4
53 3
56 6
62 3
65 4
69 3

I think it is a periodicity block with the following unison vectors.

1 3 0 1
2 2 -1 0
-1 5 0 0
3 4 0 0

Here's a lattice for it.
62
/
/
30 /
69------39
/ \ /
/ \53 /
/ 7 \ /
5 46------16
/ \ / \ /
/ 7 \ 39 / \30 /
/ 11 \ / 56 \ /
4-------6-------9 53------23------65
otonal hexad / \ 62 / \ /
legend /16 \ / \ 7 /
/ \ / 33 \ /
30-------0------42
/ \ 39 / \ /
/65 \ / \56 / utonal hexad
/ \ / 10 \ / legend
7------49------19 1/9-----1/6-----1/4
/ \ 16 / \ 1/11/
/42 \ / 33 \1/7/
/ \ / \ /
56------26 1/5
/ \ 65 /
/19 \ /
/ \ /
33-------3
/ 42
/
/
10

Notice that it doesn't have any hexads. To get hexads in 19 notes you
have to allow some steps to be 2/72nds of an octave (a much less even
and probably improper scale). Do you want one of those?

-- Dave Keenan

🔗monz <joemonz@yahoo.com>

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

/tuning/topicId_21636.html#21734

> Here's the best 11-limit 19-tone scale that has no steps
> smaller than 3 steps of 72-tET.
>
> I've given everything using 72-tET degree numbers. It would
> be a lot easier to understand if it was rewritten using
> standard 72-tET notation. Could someone else please do that
> and post the result?

Done.

Pitches are notated in my adaptation of 72-EDO:

+ - 1/12-tone
< > 1/6-tone
^ v 1/4-tone

72-EDO 72-EDO
note degree step
size
--------------------
B^ 69 3
B- 65 4
Bb> 62 3
A> 56 6
A- 53 3
Ab+ 49 4
Ab< 46 3
G 42 4
F#^ 39 3
F^ 33 6
F 30 3
E> 26 4
E- 23 3
Eb+ 19 4
Eb< 16 3
D< 10 6
C#+ 7 3
C^ 3 4
C 0 3

Notice that I inverted Dave's listing. He's not a musician,
but I am, and to me it makes a lot more sense to have the lowest
pitch at the bottom of the list and the highest pitch at the top.

> I think it is a periodicity block with the following unison
> vectors.
>
> 1 3 0 1
> 2 2 -1 0
> -1 5 0 0
> 3 4 0 0
>
> Here's a lattice for it.

Bb>
/
/
F /
B^......F#^
/ \ /
/ \A- /
/ C#+ \ /
5 Ab<.....Eb<
/ \ / \ /
/ 7 \ F#^ / \ F /
/ 11 \ / A> \ /
4-------6-------9 A-......E-......B-
otonal hexad / \ Bb> / \ /
legend /Eb<\ / \C#+/
/ \ / F^ \ /
F.......C.......G
/ \ F#^ / \ /
/B- \ / \A> / utonal hexad
/ \ / D< \ / legend
C#+.....Ab+.....Eb+ 1:9-----1:6-----1:4
/ \ Eb< / \ 1:11/
/G \ / F^ \1:7/
/ \ / \ /
A>.......E> 1:5
/ \ B- /
/Eb+\ /
/ \ /
F^.......C^
/ G
/
/
D<

Notice that I used periods instead of hyphens for the horizontal
lines, to avoid confusion with the minus signs: cf. the line
A-......E-......B- . I used colons instead of slashes in the
utonal hexad legend for the same reason.

-monz
http://www.monz.org
"All roads lead to n^0"

🔗monz <joemonz@yahoo.com>

🔗paul@stretch-music.com

--- In tuning@y..., jpehrson@r... wrote:

> Also, many of these ideas are still rather new to me and I'm a little
> like a "kid in a candy store" taking a bit of this here and a bit of
> this there. Some of it I "like" and some I don't "like," but I
> haven't even TRIED everything yet!

That's exactly the impression I was getting . . . and it's been great fun serving as one of the
many "guides" to this big "chocolate factory tour"...
>
> After all, Paul... I've been studying this stuff for about a year and
> a half, and I would guess you have been thinking about it for 10
> years...

Yeah . . . must've been about 15 years ago that I finally understood 12-tone equal
temperament (after years of puzzlement), then I went on to study other ETs with my
Commodore 64 (remember those?) by programming ear-training exercises for myself,
calculating degree of approximation to harmonics, etc. . . . 10 years ago, holed up in a summer
internship, I discovered the decatonic scale in 22-tET as well as the idea of consistency . . .
really, my viewpoints are kind of an outgrowth of my early interest in Bach and the Beatles, with
simple functional or non-functional triadic harmony and diatonic melody . . . it works so well . . . so
mine was a search for how to do something new which will work similarly well.

🔗paul@stretch-music.com

🔗jpehrson@rcn.com

--- In tuning@y..., "monz" <joemonz@y...> wrote:

/tuning/topicId_21636.html#21731
>
> > And the pitch bends were the way you did the great Johnston
> > example, yes??
>
>
> Yup. Actually, I entered the pitches using my handy little
> Cakewalk auxilliary program "micro.CAL"
> http://www.ixpres.com/interval/monzo/micro/micro.htm
>

This is truly incredible! Where have you been hiding this stuff?? I
don't believe I've ever seen this before!

I'm sorry I don't use Cakewalk... maybe I should start sometime...

>
> Of course, I wrote in the post you quoted here that working
> like this in MIDI is a pain in the ass. Well, add to that all
> the complications of figuring out Johnston's notation, and
> having to have *his* lattices in front of me as well as mine,
> and... well, that's why I've done so little of it. I have only
> a few short examples of about 4 of his pieces. The one I posted
> was the longest, and the best-sounding.
>
> But hopefully, someday, I'll get around to doing more.
> I'd like to finish an entire piece, or at least a whole
> movement, and find out what Ben himself thinks of it.
>

This would be very much worth your time, in my opinion, since it is
the most incredible xenharmonic music I HAVE EVER HEARD!!!

(I've been talking about it for DAYS!)
________ ______ ______
Joseph Pehrson

🔗jpehrson@rcn.com

--- In tuning@y..., "monz" <joemonz@y...> wrote:

/tuning/topicId_21636.html#21732

>
> --- In tuning@y..., jpehrson@r... wrote:
>
> /tuning/topicId_21636.html#21727
>
> > Wow... I've really been "underestimating" this scale [72-EDO].
> > This is really "coming together" now... I can see LOTS of
> > even "higher limit" just in this scale! Of course, the basic
> > interval is so "teeny-tiny," I suppose I shouldn't be so
> > surprised it could do things like this somehow...
>
>
> But Joe, it's not really because the basic interval is
> "teeny-tiny". It's really because of the fortuitous
> happenstance that the larger steps which are built out of
> this "teeny-tiny" one happen to coincide so closely with
> the basic prime intervals:
>
> The 12-tET notes are close to 3-, 17- and 19-limit ratios.
> The 1/12-tone =/- gives the 5-limit correction from 12-tET.
> The 1/6-tone </> gives the 7-limit correction from 12-tET.
> The 1/4-tone ^/v gives the 11-limit correction from 12-tET.
> (and to some extent the 13-limit, but not as good)
>

But, of course, the reason this works is that the closest ratios in
12-tET to just are multiplied by prime factors so as to produce the
parallel "bike chains" we have been discussing, correct?

However, the "fortuitous" part is the fact that all these intervals
would approximate closely 72 equal steps per octave, right??

"Fortuitous" implies chance... so that is indeed just (no pun
intended) chance, correct??

That seems peculiar. Spooky... Whooooo

> > They seem, in fact, to work a little like the "exponential"
vectors that Joe Monzo was using... or am I totally off base on that??
>
>
> Joe, clarify for me what you mean by the "exponential" vectors
> I was using, and maybe I can help answer.

On Paul's 19-tone just "leadsheets" he has the following unison
vectors, for example:

Unison vectors:

2 -3 1
-5 1 2
-3 3 1

And, since Paul never really had time to explain what these meant, I
was wondering if they were related to the vectors you use in you
JustMusic notation...

For example, on the Johnston page of your book (p. 141):

A 3^-1 5^1

E 5^1

B 3^1 5^1

etc...

_________ ______ _____
Joseph Pehrson

🔗jpehrson@rcn.com

--- In tuning@y..., "monz" <joemonz@y...> wrote:

/tuning/topicId_21636.html#21733

>
> > Then I can simply open the MIDI "event list" window and
> > fine-tune each pitch-bend if I want 100% accuracy, using,
> > for example (for ratios up to 7-limit), this lattice
> > http://www.ixpres.com/interval/monzo/lattices/pitch-bend-
lattice.htm
>
>
> Hmmm... I just took a look at that lattice, and I see that
> it's very pertinent to our recent discussion of 72-EDO notation.
>
> I indicated pitches on this lattice using the "combined"
> method I elaborated on in a previous post: 72-EDO accidentals
> with each note-name, and pitch-bend values for the actual JI
> pitches that the 72-EDO approximately represents.
>
> Just thought I should point that out to you, Joe, in case you
> glance at that lattice.
>

THAT'S FOR SURE THIS IS IMPORTANT! OF COURSE I LOOKED CAREFULLY AT
THIS!

This sums it all up, and shows why you like to use 72-tET as a
NOTATION as well!

Monz, I don't want to be peevish... but where do you HIDE all this
stuff??

I suggested at one time that you really make some kind of INDEX for
all your pages by topic, right from your HOME page! You have SO MUCH
stuff... and lots is just hidden away someplace. I thought I'd seen
all your Webpages by now (!!)

>
> Hmmm again... in case you ultimately decide that you prefer
> Johnny Reinhard-style 1200-EDO notation, maybe you'd instead
> like to "upgrade" to 49152-EDO and get your performers thinking
> in terms of MIDI pitch-bend.
>

I think we should get Greg Schiemer to design headsets, Johnny
Reinhard will conduct and we will send 49152-EDO pitch bend data
directly to the BRAIN of the performer!

_______ _________ ____
Joseph Pehrson

🔗paul@stretch-music.com

--- In tuning@y..., jpehrson@r... wrote:
>
> However, the "fortuitous" part is the fact that all these intervals
> would approximate closely 72 equal steps per octave, right??
>
> "Fortuitous" implies chance... so that is indeed just (no pun
> intended) chance, correct??
>
> That seems peculiar. Spooky... Whooooo

72-tET is pretty spooky . . . first of all, if I took my famous "figure 1" of how well various ETs
approximate various JI intervals, and extended it beyond 34-tET . . . all the way into the
hundreds . . . you'd see just a few "spikes", and 72-tET is one of them. In fact, there isn't a
bigger one until 118-tET . . .

So 72-tET is an "unusually good" ET for the number of notes it contains. But what makes it
spooky is that it's a multiple of 12! And that 12-tET gives you the ratios of 3 so accurately, hence
making all the higher odd factors (through 17) correspond to a particular shift between 12-tET
"gears".
>
>
> On Paul's 19-tone just "leadsheets" he has the following unison
> vectors, for example:
>
> Unison vectors:
>
> 2 -3 1
> -5 1 2
> -3 3 1
>
> And, since Paul never really had time to explain what these meant, I
> was wondering if they were related to the vectors you use in you
> JustMusic notation...

Yes, they are.

"2 -3 1" means

3^2 * 5^-3 * 7^1

which, putting in enough factors of 2 to bring you within 1 octave, equals 126:125

"-5 1 2" means

3^-5 * 5^1 * 7^2 . . . or 245:243 . . . and

"-3 3 1" means

3^-3 * 5^3 * 7^1 . . . can you figure out what ratio that is?

>
> For example, on the Johnston page of your book (p. 141):
>
> A 3^-1 5^1
>
> E 5^1
>
> B 3^1 5^1
>
> etc...
>
> ??

The only difference is that Monz is here using the notation to refer to pitches . . . while I was using
it to refer to intervals . . .

🔗jpehrson@rcn.com

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

/tuning/topicId_21636.html#21734

> Hi Joseph,
>
> Here's the best 11-limit 19-tone scale that has no steps smaller
than 3 steps of 72-tET.
>

Thanks so much, Dave... I appreciate this post.

How on earth (or anyplace else) can you calculate such a thing??

> I've given everything using 72-tET degree numbers. It would be a
lot easier to understand if it was rewritten using standard 72-tET
> notation. Could someone else please do that and post the result?
>

Actually, using the Maneri book, I probably "should" be able to
figure this out for myself... However, probably somebody could do it
much "faster" than _I_ could...

> Notice that it doesn't have any hexads.

Sorry... but why do I *want* those again??

To get hexads in 19 notes you have to allow some steps to be 2/72nds
of an octave (a much less even and probably improper scale). Do you
want one of those?
>

Sure! I'll try ANYTHING once! But, why is it you wanted to keep the
scale steps large... well, relatively large?

It seems as though the larger scale steps means that there is a
better chance that the scale is "proper..."

Could somebody please give me a "refresher" on why that is again??

______ _____ _______ ___
Joseph Pehrson

🔗jpehrson@rcn.com

--- In tuning@y..., "monz" <joemonz@y...> wrote:

/tuning/topicId_21636.html#21735
> > I've given everything using 72-tET degree numbers. It would
> > be a lot easier to understand if it was rewritten using
> > standard 72-tET notation. Could someone else please do that
> > and post the result?
>
>
> Done.
>

Wow... That was FAST...

>
> Pitches are notated in my adaptation of 72-EDO:
>
> + - 1/12-tone
> < > 1/6-tone
> ^ v 1/4-tone
>

You know, I wish Maneri himself had used ASCII symbols rather than
the ink symbols he uses in his book. Has this been discussed before
(most things have!)? It seems it would have been much more
applicable in the "computer age..." and now it's kind of late, since
you have all these instrumentalists in Boston knowing the other
symbols... (??)

>
> 72-EDO 72-EDO
> note degree step
> size
> --------------------
> B^ 69 3
> B- 65 4
> Bb> 62 3
> A> 56 6
> A- 53 3
> Ab+ 49 4
> Ab< 46 3
> G 42 4
> F#^ 39 3
> F^ 33 6
> F 30 3
> E> 26 4
> E- 23 3
> Eb+ 19 4
> Eb< 16 3
> D< 10 6
> C#+ 7 3
> C^ 3 4
> C 0 3
>

So I guess I just multiply the steps by 16.6 to put cents into SCALA
and that's it!

Thanks so much!

________ ______ ________
Joseph Pehrson

🔗jpehrson@rcn.com

--- In tuning@y..., paul@s... wrote:

/tuning/topicId_21636.html#21764

> > Also, many of these ideas are still rather new to me and I'm a
little like a "kid in a candy store" taking a bit of this here and a
bit of this there. Some of it I "like" and some I don't "like," but
I haven't even TRIED everything yet!
>

> That's exactly the impression I was getting . . . and it's been
great fun serving as one of the many "guides" to this big "chocolate
factory tour"...

And I, also, have been VERY much enjoying this learning process!!!!

>then I went on to study other ETs with my
> Commodore 64 (remember those?)

I had more fun and excitement with the Commodore 64 than with ANY
aspect of computing since that time...

Of course... it was all so NEW. My first one was hooked up to a
television set! :)

_________ ______ _____ __
Joseph Pehrson

🔗jpehrson@rcn.com

--- In tuning@y..., paul@s... wrote:

/tuning/topicId_21636.html#21774

>
> 72-tET is pretty spooky . . . first of all, if I took my famous
"figure 1" of how well various ETs
> approximate various JI intervals, and extended it beyond 34-tET . .
. all the way into the hundreds . . . you'd see just a few "spikes",
and 72-tET is one of them. In fact, there isn't a bigger one until
118-tET . . .
>

That would be great if it went out to 72 at least....

> So 72-tET is an "unusually good" ET for the number of notes it
contains. But what makes it spooky is that it's a multiple of 12! And
that 12-tET gives you the ratios of 3 so accurately, hence
> making all the higher odd factors (through 17) correspond to a
particular shift between 12-tET "gears".

That really is amazing... and the way it also fits into our
"traditional" system.
>
> "2 -3 1" means
>
> 3^2 * 5^-3 * 7^1
>
> which, putting in enough factors of 2 to bring you within 1 octave,
equals 126:125
>
> "-5 1 2" means
>
> 3^-5 * 5^1 * 7^2 . . . or 245:243 . . . and
>
> "-3 3 1" means
>
> 3^-3 * 5^3 * 7^1 . . . can you figure out what ratio that is?
>

I enjoy it when you leave me a little bit of the puzzle to do!

Well, these numbers are getting rather large, but how about 875:864
(??)

________ _______ _______
Joseph Pehrson

🔗paul@stretch-music.com

--- In tuning@y..., jpehrson@r... wrote:

> >
> > 72-EDO 72-EDO
> > note degree step
> > size
> > --------------------
> > B^ 69 3
> > B- 65 4
> > Bb> 62 3
> > A> 56 6
> > A- 53 3
> > Ab+ 49 4
> > Ab< 46 3
> > G 42 4
> > F#^ 39 3
> > F^ 33 6
> > F 30 3
> > E> 26 4
> > E- 23 3
> > Eb+ 19 4
> > Eb< 16 3
> > D< 10 6
> > C#+ 7 3
> > C^ 3 4
> > C 0 3
> >
>
> So I guess I just multiply the steps by 16.6 to put cents into
SCALA
> and that's it!

I think there's a way to specify a scale as a subset of an ET in
Scala, isn't there?

Anyway, I notice that only three of the steps are full semitones,
while all the other steps are either 1/4 tones or 1/3 tones.
Splitting the semitones into 1/4 tones would lead to a much
more "even" 22-tone scale . . . which makes sense since the 11-limit
lends itself much more readily to 22-tone scales than to 19-tone
scales. Perhaps when I get a chance, I'll see if I can derive this
directly as a 22-tone 11-limit periodicity block . . . but for now,
I'd recommend filling in those "gaps" . . .

Otherwise, if you do want a more "even" 19-tone scale from 72-tET,
I'd recommend taking one of those 19-tone 7-limit periodicity blocks
I made for you, and "translating" it into 72-tET.

🔗paul@stretch-music.com

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

Thanks for that Monz. You wrote:

> Pitches are notated in my adaptation of 72-EDO:
>
> + - 1/12-tone
> < > 1/6-tone
> ^ v 1/4-tone

I assume the 1/6-tone line was a typo and should be

sharpen flatten
+ - 1/12-tone (1 step of 72-EDO)
> < 1/6-tone (2 steps of 72-EDO)
^ v 1/4-tone (3 steps of 72-EDO)

At least that's how you've used them on the lattice, and it makes more
sense to me that a "greater than" symbol should result in sharpening.

We could argue for weeks over which spellings to use. The only way I
can see to get consistent spelling is to use a meantone-ish scheme for
the flats and sharps, where for example all 4:7's are modified
augmented sixths and 8:11's are double-diminished fifths. But this
results in double-flats and double-sharps. Is it a rule in this Boston
notation that there will never be more than one sharp or flat, or
never more than two modifiers in total?

> So I guess I just multiply the steps by 16.6 to put cents into SCALA
> and that's it!

16.7 would be more accurate but you should really multiply
by 16.66666667, or better still, multiply by 1200 and divide by 72.
Here they are.

50
116.6666667
166.6666667
266.6666667
316.6666667
383.3333333
433.3333333
500
550
650
700
766.6666667
816.6666667
883.3333333
933.3333333
1033.333333
1083.333333
1150
2/1

Paul is right about approximating more than 39 11-limit pitches. And
all within about 4 cents. But this isn't a very meaningful way of
comparing a microtemperament with a strictly-just scale. Counting
numbers of various harmonic constructs would be more meaningful.

-- Dave Keenan

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

--- In tuning@y..., paul@s... wrote:
> Anyway, I notice that only three of the steps are full semitones,
> while all the other steps are either 1/4 tones or 1/3 tones.
> Splitting the semitones into 1/4 tones would lead to a much
> more "even" 22-tone scale . . .

Perhaps surprisingly, the three notes you suggest adding, have no
11-limit relationship whatsoever with any other note in the scale. And
between themselves they form only an augmented triad (stacked major
thirds). So one would only do this if there was some pressing melodic
reason. It's hard to see why there would be, since one won't normally
be playing the whole 19 notes as a (melodically intellible) scale
anyway, but will tend to use subsets with variations.

> which makes sense since the 11-limit
> lends itself much more readily to 22-tone scales than to 19-tone
> scales. Perhaps when I get a chance, I'll see if I can derive this
> directly as a 22-tone 11-limit periodicity block . . .

Good luck, but I'm pretty sure it isn't one. Can you confirm that the
19 tones I gave are a periodicity block based on the matrix

3^a * 5^b * 7^d * 11^e
-----------------------------
1 3 0 1
2 2 -1 0
-1 5 0 0
3 4 0 0
-----------------------------

The unison vectors in the first two rows are respectively 11-limit and
7-limit commas/kleismas that vanish in 72-EDO, expressed in a form
that relates the 7 or the 11 to the 5-limit plane and presents 11
independently of 7.

Because the unison vectors in the last two rows don't vanish in 72-EDO
this matrix doesn't uniquely determine the choice of notes. But the
specific notes are chosen so as to maximise 4:7's and 8:11's within a
lattice that is connected in the 5-limit plane. i.e. when projected
onto the 5-limit plane the notes will ideally be inside a
parallellogram whose sides are in the directions (1,3) and (2,2)
obtained from the first two vectors.

Given that the notes will form such a shape in the 5-limit plane, the
last two unison vectors are chosen to prevent the scale containing any
steps smaller than 3 steps of 72-EDO.

> but for now,
> I'd recommend filling in those "gaps" . . .

I don't.

> Otherwise, if you do want a more "even" 19-tone scale from 72-tET,
> I'd recommend taking one of those 19-tone 7-limit periodicity blocks
> I made for you, and "translating" it into 72-tET.

I understood that Joe wanted 11's (and maybe even 13's) in it. Perhaps
I misunderstood? I chose to ignore 13's because I don't think they are
well-enough approximated in 72-tET, at least nowhere near as well as
7's and 11's. It's 7's and 11's are quasi-just, I don't think you can
say the same of its 13's or 17's.

I believe the scale I gave is unique in having the greatest number of
11-limit consonances of any proper 19-tone subset of 72-EDO. In fact I
don't think there are any other proper 19-tone subsets that come
anywhere near it. Adding the 3 notes to it that you suggest, adds only
two more 5-limit consonances, completely disconnected from the rest of
the lattice.

If we are willing to accept steps as small as 2 steps of 72-EDO (most
likely creating an improper scale) then there is much more freedom to
choose a 19-tone scale (from some 22 and 31 note periodicity blocks)
and it can have some actual 11-limit hexads in it. But noone has said
they are interested in these.

Paul, I'd like to see you translate your 19-tone 7-limit periodicity
blocks to steps of 72-EDO, so we can see how even they are and how
many 7-limit consonances they have. Of course I'd like you to respond
off-list re the Shrutar too.

-- Dave Keenan

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

--- In tuning@y..., jpehrson@r... wrote:
> How on earth (or anyplace else) can you calculate such a thing??

Adriaan Fokker found the key (explained to me by Paul Erlich and
Graham Breed) ... Periodicity blocks and the matrix formalism. If you
learn about the these, you may understand the explanation I gave in my
previous message in this thread. But it still isn't something that
I can just plug into a spreadsheet and out pops the answer. I didn't
know how I was going to solve it before I started playing around.

> > Notice that it doesn't have any hexads.
>
> Sorry... but why do I *want* those again??

Hmm. Maybe you'd better just say what you _do_ want from this 19-tone
subset of 72-EDO.

> To get hexads in 19 notes you have to allow some steps to be 2/72nds
> of an octave (a much less even and probably improper scale). Do you
> want one of those?
>
> Sure! I'll try ANYTHING once!

There's a fair bit of time involved, so you'd better _really_ want it
and be able to specify a bit better _what_ you want.

>But, why is it you wanted to keep the
> scale steps large... well, relatively large?

So you'd have more chance of finding melodically useful subsets, even
though they themselves might well be improper.

What I'm doing is essentially a harmonic optimisation but you don't
want to end up with two notes only 17 cents apart, since melodically
speaking, you would have wasted a note.

With the 4D periodicity-block (PB) matrix, plus thinking in terms of
all the notes being connected in the 5-limit plane, I get to use two
rows (unison vectors) to alias these notes into the 7's and 11's
dimensions and then I have two left over which I can use to introduce
this kind of melodic sense into the scale. The first two unison
vectors will correspond to zero steps of 72-EDO (i.e. they are
distributed so they "vanish"). The last two will correspond to 1 or 2
steps of 72-EDO.

The fact that the resulting matrix just happened to have a determinant
of 19 (meaning 19 notes in the PB) is some kind of miracle.

I used Excel (as usual). To find the determinant of a matrix use
MDETERM. In Excel I also made an array representing a large section of
the five limit plane (rectangular not triangular) showing the 72-EDO
degree number in each square. I then worked out from note zero,
colouring the squares blue as I chose specific notes to be in the
scale and colouring yellow (for unavailable) those which were one or
two steps above and below the chosen note. I always tried to grow the
blue patch in the optimum directions (1,3) and (2,2), until I had 19
notes.

-- Dave Keenan

🔗monz <joemonz@yahoo.com>

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

/tuning/topicId_21636.html#21795

> I assume the 1/6-tone line was a typo and should be
>
> sharpen flatten
> + - 1/12-tone (1 step of 72-EDO)
> > < 1/6-tone (2 steps of 72-EDO)
> ^ v 1/4-tone (3 steps of 72-EDO)
>
> At least that's how you've used them on the lattice, and it
> makes more sense to me that a "greater than" symbol should
> result in sharpening.

Yup, I reversed the < > signs. Thanks for catching that,
Dave. Your chart above is correct.

>
> We could argue for weeks over which spellings to use. The
> only way I can see to get consistent spelling is to use a
> meantone-ish scheme for the flats and sharps, where for
> example all 4:7's are modified augmented sixths and 8:11's
> are double-diminished fifths. But this results in double-flats
> and double-sharps. Is it a rule in this Boston notation
> that there will never be more than one sharp or flat, or
> never more than two modifiers in total?

I never seen any "rules" like this from either of the major
published advocates of this notation (Ezra Sims and Joe Maneri).

But I imagine that since the Boston-area 72-tETers think
of the 72-EDO scale *as* a "virtual pitch continuum", they're
looking for the "leanest" and clearest notation possible, and
would therefore avoid double-sharps and double-flats etc.,
similar to what I posted recently about Schoenberg's use
of 12-EDO notation.

-monz
http://www.monz.org
"All roads lead to n^0"

🔗monz <joemonz@yahoo.com>

--- In tuning@y..., jpehrson@r... wrote:

/tuning/topicId_21636.html#21776

> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> /tuning/topicId_21636.html#21735
> > > I've given everything using 72-tET degree numbers. It would
> > > be a lot easier to understand if it was rewritten using
> > > standard 72-tET notation. Could someone else please do that
> > > and post the result?
> >
> >
> > Done.
> >
>
> Wow... That was FAST...

It was easy for me... I have a folder full of 72-EDO stuff
and the list of pitches was in there.

> > Pitches are notated in my adaptation of 72-EDO:
> >
> > + - 1/12-tone
> > > < 1/6-tone
> > ^ v 1/4-tone

[I took the liberty of correcting this: the original was wrong.]

> You know, I wish Maneri himself had used ASCII symbols rather
> than the ink symbols he uses in his book. Has this been
> discussed before (most things have!)? It seems it would have
> been much more applicable in the "computer age..." and now
> it's kind of late, since you have all these instrumentalists
> in Boston knowing the other symbols... (??)

This issue hasn't really been discussed before. I simply
took the initiative to change the Sims/Maneri notation when
discussing 72-EDO in previous Tuning List posts.

Actually, I never liked the "square root" sign Sims/Maneri
use for 1/4-tones, or the fact that Sims/Maneri uses arrows
for 1/12-tones, because I had already been using the up
and down arrows to represent 1/4-tones for a few years
and thought that was a great quarter-tone accidental.

Franz Richter Herf was an Austrian 72-tETer who similarly
used arrows for 1/4-tones, and the same "half arrow" for
1/6-tones that Sims/Maneri use, and Herf used a simple
upward or downward slash for 1/12-tones.

I thought Herf's notation was "cleaner", and it already used
the 1/4-tone notation I liked. So it was easy for me to adapt
ASCII to that, using the caret and lower-case "v" for the arrows,
< > for the "half arrows", and + - instead of the slashes.

(It was interesting to me that < > were the symbols chosen
by Schoenberg to represent quarter-tones, as discussed in
my Microfest presentation. I'd never seen them used in music
before I put them to use in 72-EDO, and wondered why, because
their use seems so logical.)

The use of + - actually grew out of my recognition that the
72-EDO 1/12-tone functioned like a syntonic comma in the
lattice, and I was already using + - in ASCII in discussing
Ben Johnston's notation.

I agree with you, Joe, that it's too bad so many Boston
microtonalists are adopting the Sims/Maneri notation instead
of this one. Same situation as with Johnston/HEWM for JI.
An influential teacher has a school of students who adopt
his notation, and people on this list adopt mine if they
prefer it. It's too bad that there's a divergence.

Hey, since Maneri is putting out a new version of his book,
maybe he can be convinced to change the notation to this
version before publishing it? Someone care to pass it along
to him? Or give me his email address?

-monz
http://www.monz.org
"All roads lead to n^0"

🔗graham@microtonal.co.uk

Joseph Pehrson wrote:

> Somebody said that the MicroZone really wasn't that great for 72-
> tET... Who said that, why did they say it, and what does that mean??

I hope I'm not breaking a conspiracy of silence here.

I think it was Kraig Grady paraphrasing Erv Wilson. The MicroZone uses a
Bosanquet layout, which is designed for tunings with a single chain of
fifths. But 72-equal has 6 different chains (or one chain with 6 gears by
the newer metaphor) so it'd need a different type of keyboard. If you
used it with a MicroZone you'd still get the advantage of having the notes
squashed in to a 2-D arrangement, but you wouldn't get the transpositional
invariance a Bosanquet usually provides.

If you were starting from having a MicroZone in front of you, and wanted
to compose with commas and 11-limit harmony, 41-equal would be the obvious
choice.

Graham

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

Ok Joseph, I couldn't help myself. Here's a 19 tone subset of 72-EDO
that has heaps of 11-limit consonances including two 11-limit hexads,
six 9-limit pentads, six 7-limit tetrads and four 5-limit triads.

It's very uneven but at least it has no comma sized steps, i.e. no
steps which are a single step of 72-EDO.

39
/ \
53 / \23
7 / 49 \
16------58
/ \ /
39 60 30 / \ 0 / 42
14 56 / \ /
53------23------65------35
/ \ / \ / \ /
/16 \37 /58 \ 7 / \49 / 19
/ \ / 33 \ / \ /
60------30-------0------42------12
/ \ / \ 39 / \ /
53 /23 \ /65 \14 /35 \56 /
/ \ / \ / \ /
37-------7------49------19
/ \ / 16 58
30 / 0 \ /42 12 33
/ \ /
14------56
\ 23 / 65
49 \ /19
\ /
33

72-EDO Step sizes
degrees in 72-EDO steps
-----------------------
0 7
7 5
12 2
14 2
16 3
19 4
23 7
30 3
33 2
35 2
37 2
39 3
42 7
49 4
53 3
56 2
58 2
60 5
65 7

It is not a periodicity block. But it is a subset of a 31 tone PB with
the following unison vectors.
Steps of 72-EDO
1 3 0 1 4096:4125 0
2 2 -1 0 224:225 0
4 -1 0 0 80:81 1
-3 -7 0 0 2097152:2109375 1

The full 31 tone PB is very even, as follows.

72-EDO Step sizes
degrees in 72-EDO steps
-----------------------
0 2
2 3
5 2
7 2
9 3
12 2
14 2
16 3
19 2
21 2
23 3
26 2
28 2
30 3
33 2
35 2
37 2
39 3
42 2
44 2
46 3
49 2
51 2
53 3
56 2
58 2
60 3
63 2
65 2
67 3
70 2

-- Dave Keenan

🔗jpehrson@rcn.com

--- In tuning@y..., paul@s... wrote:

/tuning/topicId_21636.html#21789

> --- In tuning@y..., jpehrson@r... wrote:
>
> > >
> > > 72-EDO 72-EDO
> > > note degree step
> > > size
> > > --------------------
> > > B^ 69 3
> > > B- 65 4
> > > Bb> 62 3
> > > A> 56 6
> > > A- 53 3
> > > Ab+ 49 4
> > > Ab< 46 3
> > > G 42 4
> > > F#^ 39 3
> > > F^ 33 6
> > > F 30 3
> > > E> 26 4
> > > E- 23 3
> > > Eb+ 19 4
> > > Eb< 16 3
> > > D< 10 6
> > > C#+ 7 3
> > > C^ 3 4
> > > C 0 3
> > >
> >
> > So I guess I just multiply the steps by 16.6 to put cents into
> SCALA and that's it!
>
> I think there's a way to specify a scale as a subset of an ET in
> Scala, isn't there?
>

I'm seeing both a SELECT command and a DELETE command, but I can't
get EITHER to work properly! If anyone has any suggestions, it would
help!

Let's say I generate the 72-tET scale. Fine.

Then, I go SELECT, and I get:

Enter scale degree:

But when I put anything other than "0" in there, nothing happens!

Additionally, I can't put more than one degree in at the same time.

I must be doing something wrong with my syntax.

ALSO, I can't get DELETE to delete more than one scale degree at a
time.... I try to put a string of numbers in there and I get
"illegal parameters..."

I just wrote to Manul Op de Coul... Most probably there is something
wrong with the syntax I am using...

boo hoo :(

> Anyway, I notice that only three of the steps are full semitones,
> while all the other steps are either 1/4 tones or 1/3 tones.
> Splitting the semitones into 1/4 tones would lead to a much
> more "even" 22-tone scale . . . which makes sense since the 11-
limit lends itself much more readily to 22-tone scales than to 19-
tone
> scales. Perhaps when I get a chance, I'll see if I can derive this
> directly as a 22-tone 11-limit periodicity block . . . but for now,
> I'd recommend filling in those "gaps" . . .
>
> Otherwise, if you do want a more "even" 19-tone scale from 72-tET,
> I'd recommend taking one of those 19-tone 7-limit periodicity
blocks I made for you, and "translating" it into 72-tET.

You mean, obviously, just choosing the "closest" 72-tET pitch to
these scales...

OK... can do.

_________ ______ _____
Joseph Pehrson

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

Paul E.,

I take it all back. You were right. Adding those 3 extra notes to my
proper 19-tone PB _does_ give a 22-tone PB. Here is a set of unison
vectors for it.

1 3 0 1
2 2 -1 0
-3 -7 0 0
-1 5 0 0

So they _are_ connected to the other 19 in the 5-limit plane, albeit
at the very extremities (which is, I guess, why I missed it the first
time).

But it is still true that adding them has no significant effect on the
number of 11-limit consonances.

Here's the 22-tone PB.
/
36
/
/
/
13
/
/
/
62
/
13 /
30 / 0
69------39
36 / \ /
62 / \53 /
/ 7 \ / 49
46------16
13 / \ /
39 / \30 /
/ 56 \ / 26
53------23------65
/ \ 62 / \ /
/16 \ / \ 7 /
/ \ / 33 \ / 3
30-------0------42
69 / \ 39 / \ /
/65 \ / \56 /
/ \ / 10 \ /
7------49------19
46 / \ 16 /
/42 \ / 33
/ \ / 59
56------26
23 / \ 65 /
/19 \ / 10
/ \ / 36
33-------3
0 / 42
/ 59
/
10
/
/
/
59
/

72-EDO Step sizes
degrees in 72-EDO steps
-----------------------
0 3
3 4
7 3
10 3
13 3
16 3
19 4
23 3
26 4
30 3
33 3
36 3
39 3
42 4
46 3
49 4
53 3
56 3
59 3
62 3
65 4
69 3

-- Dave Keenan

🔗jpehrson@rcn.com

🔗Kraig Grady <kraiggrady@anaphoria.com>

Graham!
Erv has pointed out that Hanson keyboard logic allows the 72 to be played on a generalized
keyboard. A loose mock up for the Hebdomekontany can be seen at
http://www.anaphoria.com/images/hebdo.GIF Please notice where the row of natural flats and sharps
are as this is the primary starting point. Erv has filled out The 1-3-5-7-9-11-13-15
Hebdomekontany on this keyboard , filling in the empty keys so if one wanted to one could start
with a Just 72 as a basic set , that would be used in other ways.

graham@microtonal.co.uk wrote:

> I think it was Kraig Grady paraphrasing Erv Wilson. The MicroZone uses a
> Bosanquet layout, which is designed for tunings with a single chain of
> fifths. But 72-equal has 6 different chains (or one chain with 6 gears by
> the newer metaphor) so it'd need a different type of keyboard. If you
> used it with a MicroZone you'd still get the advantage of having the notes
> squashed in to a 2-D arrangement, but you wouldn't get the transpositional
> invariance a Bosanquet usually provides.
>
> If you were starting from having a MicroZone in front of you, and wanted
> to compose with commas and 11-limit harmony, 41-equal would be the obvious
> choice.
>
> Graham

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

🔗paul@stretch-music.com

--- In tuning@y..., paul@s... wrote:
> --- In tuning@y..., jpehrson@r... wrote:
>
> > >
> > > 72-EDO 72-EDO
> > > note degree step
> > > size
> > > --------------------
> > > B^ 69 3
> > > B- 65 4
> > > Bb> 62 3
> > > A> 56 6
> > > A- 53 3
> > > Ab+ 49 4
> > > Ab< 46 3
> > > G 42 4
> > > F#^ 39 3
> > > F^ 33 6
> > > F 30 3
> > > E> 26 4
> > > E- 23 3
> > > Eb+ 19 4
> > > Eb< 16 3
> > > D< 10 6
> > > C#+ 7 3
> > > C^ 3 4
> > > C 0 3
> > >
> >
> > So I guess I just multiply the steps by 16.6 to put cents into
> SCALA
> > and that's it!
>
> I think there's a way to specify a scale as a subset of an ET in
> Scala, isn't there?
>
> Anyway, I notice that only three of the steps are full semitones,
> while all the other steps are either 1/4 tones or 1/3 tones.
> Splitting the semitones into 1/4 tones would lead to a much
> more "even" 22-tone scale . . . which makes sense since the 11-limit
> lends itself much more readily to 22-tone scales than to 19-tone
> scales. Perhaps when I get a chance, I'll see if I can derive this
> directly as a 22-tone 11-limit periodicity block . . . but for now,
> I'd recommend filling in those "gaps" . . .

Actually, I strongly suspect the "correct" "splitting" of those semitones would be into a 1/6 tone
and a 1/3 tone . . . this would make the scale three chains of alternating 1/4 and 1/3 tones,
delimited by three 1/6 tones . . .

As to your question about propriety, Joseph, I actually think it's not too important if a scale is
proper or not . . . but a related property, CS or "constant structures", may be important. It
means that anytime you find a particular interval in the scale, it is always subtended by the same
number of scale degrees . . . Kraig Grady finds this important in composition, and you may too .
. . all the periodicity blocks are CS scales . . .

🔗paul@stretch-music.com

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning@y..., paul@s... wrote:
> > Anyway, I notice that only three of the steps are full semitones,
> > while all the other steps are either 1/4 tones or 1/3 tones.
> > Splitting the semitones into 1/4 tones would lead to a much
> > more "even" 22-tone scale . . .
>
> Perhaps surprisingly, the three notes you suggest adding, have no
> 11-limit relationship whatsoever with any other note in the scale.

Actually, I intuitively guessed that that would be the case after I made that post. Thus I changed
my suggestion to splitting the semitones into a 1/6 tone and a 1/3 tone.
>
> Good luck, but I'm pretty sure it isn't one. Can you confirm that the
> 19 tones I gave are a periodicity block based on the matrix
>
> 3^a * 5^b * 7^d * 11^e
> -----------------------------
> 1 3 0 1
> 2 2 -1 0
> -1 5 0 0
> 3 4 0 0
> -----------------------------

Probably . . . the determinant of that matrix is 19, and Paul Hahn's conjecture that all periodicity
blocks of a certain cardinality and limit are equivalent has yet to be disproved.
>
> Paul, I'd like to see you translate your 19-tone 7-limit periodicity
> blocks to steps of 72-EDO, so we can see how even they are and how
> many 7-limit consonances they have.

Later . . .

>Of course I'd like you to respond
> off-list re the Shrutar too.

Whoops, I keep forgetting to print out those fingering diagrams and bring them home . . .

🔗paul@stretch-music.com

🔗David J. Finnamore <daeron@bellsouth.net>

🔗jpehrson@rcn.com

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

/tuning/topicId_21636.html#21795

>
> We could argue for weeks over which spellings to use. The only way
I can see to get consistent spelling is to use a meantone-ish scheme
for the flats and sharps, where for example all 4:7's are modified
> augmented sixths and 8:11's are double-diminished fifths. But this
> results in double-flats and double-sharps. Is it a rule in this
Boston notation that there will never be more than one sharp or
flat, or never more than two modifiers in total?
>

If I'm understanding this correctly, it seems as though the Maneri
groug is more interested in the continuous LINEAR pitch continuum
than with accuracies of harmonies... Is this possibly why they're
not so taken up with using some kind of modified meantone notation
(??) Just wondering...

>
> > So I guess I just multiply the steps by 16.6 to put cents into
SCALA and that's it!
>
> 16.7 would be more accurate but you should really multiply
> by 16.66666667, or better still, multiply by 1200 and divide by 72.
> Here they are.
>
> 50
> 116.6666667
> 166.6666667
> 266.6666667
> 316.6666667
> 383.3333333
> 433.3333333
> 500
> 550
> 650
> 700
> 766.6666667
> 816.6666667
> 883.3333333
> 933.3333333
> 1033.333333
> 1083.333333
> 1150
> 2/1
>

Thanks, Dave... I will use this. The other is definitely not
accurate enough. Thanks again!

> Paul is right about approximating more than 39 11-limit pitches.
And all within about 4 cents. But this isn't a very meaningful way of
> comparing a microtemperament with a strictly-just scale. Counting
> numbers of various harmonic constructs would be more meaningful.
>

That sounds really interesting! Would it end up looking like a bunch
of different lattices as in Paul Erlich's new paper (??)

________ ______ ____ __
Joseph Pehrson

🔗jpehrson@rcn.com

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

/tuning/topicId_21636.html#21796

>
> Because the unison vectors in the last two rows don't vanish in 72-
EDO this matrix doesn't uniquely determine the choice of notes. But
the specific notes are chosen so as to maximise 4:7's and 8:11's
within a lattice that is connected in the 5-limit plane. i.e. when
projected onto the 5-limit plane the notes will ideally be inside a
> parallellogram whose sides are in the directions (1,3) and (2,2)
> obtained from the first two vectors.
>

Ok... that procedure is actually making sense to me... (

> I understood that Joe wanted 11's (and maybe even 13's) in it.
Perhaps I misunderstood? I chose to ignore 13's because I don't think
they are well-enough approximated in 72-tET, at least nowhere near as
well as 7's and 11's. It's 7's and 11's are quasi-just, I don't think
you can say the same of its 13's or 17's.
>

Yes... I believe I would prefer higher-limit consonances even if it
resulted in a bit more "irregularity" of the step sizes... if this is
what this is all about (!!)

> If we are willing to accept steps as small as 2 steps of 72-EDO
(most likely creating an improper scale) then there is much more
freedom to choose a 19-tone scale (from some 22 and 31 note
periodicity blocks) and it can have some actual 11-limit hexads in
it. But noone has said they are interested in these.
>

That might not be a problem, either, unless the 2-step intervals are
placed in such a way that they seem "out of step" as it were, with
the rest of the scale (??)

> Paul, I'd like to see you translate your 19-tone 7-limit
periodicity blocks to steps of 72-EDO, so we can see how even they
are and how many 7-limit consonances they have.

I'd love to see this too! But, Paul has quite a few of these... He
gave me EIGHT of them, and I've already started composing with them...

But I REALLY would LOVE to see 72-tET approximations of these, if
somebody can do it "easily..."

Thanks again!

Joe

_________ ______ ____ ____
Joseph Pehrson

🔗jpehrson@rcn.com

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

/tuning/topicId_21636.html#21798

> --- In tuning@y..., jpehrson@r... wrote:
> > How on earth (or anyplace else) can you calculate such a thing??
>
> Adriaan Fokker found the key (explained to me by Paul Erlich and
> Graham Breed) ... Periodicity blocks and the matrix formalism. If
you learn about the these, you may understand the explanation I gave
in my previous message in this thread.

Actually, Dave... I think I got a pretty good "glimmer" of what you
were doing. The explanation in that post with the vectors was really
quite clear!

But it still isn't something that
> I can just plug into a spreadsheet and out pops the answer. I
didn't know how I was going to solve it before I started playing
around.

Yes... I can also understand that from your previous description.

> > > Notice that it doesn't have any hexads.
> >
> > Sorry... but why do I *want* those again??
>
> Hmm. Maybe you'd better just say what you _do_ want from this 19-
tone subset of 72-EDO.
>

Sorry, Dave...

OK... well I guess it's time for the Paul Erlich "Hexad Tutorial..."

I learned all about the use of hexaCHORDS, but not so much about
Hexads... They must be related to the Harry Partch hexads that Paul
was talking about...

Well, I don't really see where they are so thoroughly explained in
Partch's book... but I do see that Partch's "Tonality Diamond" has
SIX elements on a side and that, also, it is mentioned that a Kithara
has a "hexad" that contains all six identies from a tonic: 1-3-5-7-9-
11.

Am I on the right track??

Anyway... since I really didn't know anything about them, I wasn't
too concerned about them (yet). Mostly I just wanted a 72-tET scale
with as many consonances to whatever limit was feasible, even if it
had to compromise the step size a bit.

HOWEVER, I do understand the concern that a very small step size
could be a "melodic waste..." so, of course, those should be avoided
as well...

This scale construction process sure can't be easy!

> > To get hexads in 19 notes you have to allow some steps to be
2/72nds of an octave (a much less even and probably improper scale).
Do you want one of those?
> >
> > Sure! I'll try ANYTHING once!
>
> There's a fair bit of time involved, so you'd better _really_ want
it and be able to specify a bit better _what_ you want.
>

I'm sorry Dave... don't go out on a limb for this... However, on the
other hand, it is true that I am saving all this stuff and hope,
possibly, after doing all this research to compose for YEARS with it!

> >But, why is it you wanted to keep the
> > scale steps large... well, relatively large?
>
> So you'd have more chance of finding melodically useful subsets,
even though they themselves might well be improper.
>
> What I'm doing is essentially a harmonic optimisation but you don't
> want to end up with two notes only 17 cents apart, since
melodically speaking, you would have wasted a note.

Got it!

> The fact that the resulting matrix just happened to have a
determinant of 19 (meaning 19 notes in the PB) is some kind of
miracle.
>

Everybody seems to think that 72-tET has "supernatural" qualities...
That isn't very "scientific," but there are lots of peculiar things
here...

> I used Excel (as usual). To find the determinant of a matrix use
> MDETERM. In Excel I also made an array representing a large section
of the five limit plane (rectangular not triangular) showing the 72-
EDO degree number in each square. I then worked out from note zero,
> colouring the squares blue as I chose specific notes to be in the
> scale and colouring yellow (for unavailable) those which were one
or two steps above and below the chosen note. I always tried to grow
the blue patch in the optimum directions (1,3) and (2,2), until I had
19 notes.

This is very cool... Might we see these worksheets??

Actually, maybe they should be on the Web, too. This is fascinating.

________ _____ ____ ____
Joseph Pehrson

🔗jpehrson@rcn.com

--- In tuning@y..., "monz" <joemonz@y...> wrote:

/tuning/topicId_21636.html#21802
>
>
> > > Pitches are notated in my adaptation of 72-EDO:
> > >
> > > + - 1/12-tone
> > > > < 1/6-tone
> > > ^ v 1/4-tone
>
> [I took the liberty of correcting this: the original was wrong.]
>
> > You know, I wish Maneri himself had used ASCII symbols rather
> > than the ink symbols he uses in his book. Has this been
> > discussed before (most things have!)? It seems it would have
> > been much more applicable in the "computer age..." and now
> > it's kind of late, since you have all these instrumentalists
> > in Boston knowing the other symbols... (??)
>
>
> This issue hasn't really been discussed before. I simply
> took the initiative to change the Sims/Maneri notation when
> discussing 72-EDO in previous Tuning List posts.
>

I'm rather surprised that this hasn't come up before... it seems
pretty fundamental...

> Actually, I never liked the "square root" sign Sims/Maneri
> use for 1/4-tones, or the fact that Sims/Maneri uses arrows
> for 1/12-tones, because I had already been using the up
> and down arrows to represent 1/4-tones for a few years
> and thought that was a great quarter-tone accidental.
>

Personally, I think one important "feature" is the fact that as the
"alteration" gets larger, the symbol gets larger as well. This is
one good thing about the Maneri notation...

However, I believe YOUR Ascii representation also serves that
purpose, i.e.: + - seems smaller than > < seems smaller than
^ v.

Well the comparison between the last two pairs might be somewhat
debatable, but that's the "psychological" impression it gives to ME...

>
> I agree with you, Joe, that it's too bad so many Boston
> microtonalists are adopting the Sims/Maneri notation instead
> of this one. Same situation as with Johnston/HEWM for JI.
> An influential teacher has a school of students who adopt
> his notation, and people on this list adopt mine if they
> prefer it. It's too bad that there's a divergence.
>
> Hey, since Maneri is putting out a new version of his book,
> maybe he can be convinced to change the notation to this
> version before publishing it? Someone care to pass it along
> to him? Or give me his email address?
>

Monz... you should get right on this... although I doubt that Maneri
will change at this point, since so many people have studied his
system... but it can't be conveyed into Ascii, which will be a
REQUISITE for the computer age...

My guess is that it will HAVE to evolve at some point, just as it
already has as people are discussing 72-tET music on this list!!

_______ ______ ______ __
Joseph Pehrson

🔗paul@stretch-music.com

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> Can you confirm that the
> 19 tones I gave are a periodicity block based on the matrix
>
> 3^a * 5^b * 7^d * 11^e
> -----------------------------
> 1 3 0 1
> 2 2 -1 0
> -1 5 0 0
> 3 4 0 0
> -----------------------------

For this matrix I get the Fokker perioodicity block (in 72-tET):

62
/
/
/
/
69--------39
/ \ /
/ \ /
/ \ /
/ \ /
46--------16
/ \ /
/ \ /
/ \ /
/ \ /
53--------23--------65
/ \ / \ /
/ \ / \ /
/ \ / \ /
/ \ / \ /
30---------0--------42
/ \ / \ /
/ \ / \ /
/ \ / \ /
/ \ / \ /
7--------49--------19
/ \ /
/ \ /
/ \ /
/ \ /
56--------26
/ \ /
/ \ /
/ \ /
/ \ /
33---------3
/
/
/
/
10

So the scale is 0,3,7,10,16,19,23,26,30,33,39,42,46,49,53,56,62,65,69

or steps of 3,4,3,6,3,4,3,4,3,6,3,4,3,4,3,6,3,4,3

Sure looks like your scale, Dave!

🔗paul@stretch-music.com

--- In tuning@y..., jpehrson@r... wrote:

> Sorry, Dave...
>
> OK... well I guess it's time for the Paul Erlich "Hexad Tutorial..."
>
> I learned all about the use of hexaCHORDS, but not so much about
> Hexads... They must be related to the Harry Partch hexads that
Paul
> was talking about...

Yup.
>
> Well, I don't really see where they are so thoroughly explained in
> Partch's book...

Keep digging . . . they're VERY important to Partch.

> but I do see that Partch's "Tonality Diamond" has
> SIX elements on a side and that, also, it is mentioned that a
Kithara
> has a "hexad" that contains all six identies from a tonic: 1-3-5-7-
9-
> 11.
>
> Am I on the right track??

Yup! A hexad is just a 1:3:5:7:9:11 chord, or a 1/(1:3:5:7:9:11)
chord, with possible octave transpositions. Each hexad has 15
consonant intervals (if you accept the ratios of 11 as consonances).
2 of these 15 are identical -- the 3:1 and the 9:3.
>
> Anyway... since I really didn't know anything about them, I wasn't
> too concerned about them (yet). Mostly I just wanted a 72-tET
scale
> with as many consonances to whatever limit was feasible, even if it
> had to compromise the step size a bit.

Joseph, have you tested, using your own ear, whether you percieve the
ratios of 11 as consonances?

🔗jpehrson@rcn.com

--- In tuning@y..., graham@m... wrote:

/tuning/topicId_21636.html#21803

> Joseph Pehrson wrote:
>
> > Somebody said that the MicroZone really wasn't that great for 72-
> > tET... Who said that, why did they say it, and what does that
mean??
>
> I hope I'm not breaking a conspiracy of silence here.
>
> I think it was Kraig Grady paraphrasing Erv Wilson. The MicroZone
uses a Bosanquet layout, which is designed for tunings with a single
chain of fifths. But 72-equal has 6 different chains (or one chain
with 6 gears by the newer metaphor) so it'd need a different type of
keyboard. If you used it with a MicroZone you'd still get the
advantage of having the notes squashed in to a 2-D arrangement, but
you wouldn't get the transpositional invariance a Bosanquet usually
provides.
>

Hi Graham!

Thank you for your response, but I'm not certain I am entirely
"getting" this...

I just looked at the Microzone Keyboard over at Starr labs again, and
it looks like five Halberstadt keyboards stacked upon one another...

Wouldn't we just need SIX to do the trick??

Kraig Grady sent me the Wilson prototype, and I am very grateful for
this... However, it seems TOTALLY unrelated to the Halberstadt, which
means, for me, that it might be harder for me to find my "way
around." Dunno...

But why won't the Microzone work since it looks so much like stacked
multiples of 12-tET...??

________ ______ _____ _____
Joseph Pehrson

🔗jpehrson@rcn.com

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

/tuning/topicId_21636.html#21805

> Ok Joseph, I couldn't help myself. Here's a 19 tone subset of 72-
EDO that has heaps of 11-limit consonances including two 11-limit
hexads, six 9-limit pentads, six 7-limit tetrads and four 5-limit
triads.
>
> It's very uneven but at least it has no comma sized steps, i.e. no
> steps which are a single step of 72-EDO.

Thanks so much, Dave! I really appreciate this.

I'm filing this stuff carefully away, since I hope to be realizing it
in sound soon. I guess that will be the only way to tell if any
"irregularities" in the scale steps really mean anything.

Frankly, I don't know at this point whether, as far as my OWN music
is concerned, REGULARITY of scale step or CONCORDANCE of intervals is
more important. This is really something that is going to have to be
the subject of "experimentation."

It really a FUNNY question, since it seems to get right to the ROOT,
so to speak (just "regular" root, not squared or anything) of the
duality between the Just Intonation folks and the ET Folks.

On one side we have people who like maximally even scales, and on the
other people who prefer concordances...

Whether maximal evenness or concordance is more important probably
can only be determined by the nature of any given composer's work.

In my pieces right now, the idea of literal transposition and
modulation is a bit "ideosyncratic" at best (!) so I'm tending to go,
at the moment for concordance and JI OVER modulation and ET.

What do the other composers on this list think??

Does your OWN music require regularity of scale step or concordance??

Of course, I bet Paul is going to tell me now that ET's have more
overall concordances than JI scales (??) But they're never PURE...
does that make a difference??

_________ _______ _____ ____
Joseph Pehrson

🔗ligonj@northstate.net

🔗paul@stretch-music.com

--- In tuning@y..., jpehrson@r... wrote:
>
> On one side we have people who like maximally even scales, and on
the
> other people who prefer concordances...
>
> Whether maximal evenness or concordance is more important probably
> can only be determined by the nature of any given composer's work.

Sometimes, the two actually go together, and you have to sacrifice
_both_ for some other goal.

For example, in 22-tET, the maximally even 10-tone scale has 8
consonant 7-limit tetrads. But the "pentachordal" (having identical
pentachords a 3:2 or 4:3 apart in every octave species) 10-tone scale
has only 6 consonant 7-limit tetrads. I find the latter to work
better melodically, though, so often I prefer to sacrifice _both_
maximal evenness _and_ number of consonances.
>
> Of course, I bet Paul is going to tell me now that ET's have more
> overall concordances than JI scales (??)

Often true.

> But they're never PURE...
> does that make a difference??

Not if you can't hear the difference. Ezra Sims said he can't hear
the difference between his music in JI and in 72-tET, and according
to Daniel Wolf, Partch was similarly "fooled" by a 41-tET instrument.

If the music goes really slowly, and rich harmonic timbres are used,
you might hear the difference. But even then, there's no reason to
choose strict JI over adaptive JI, unless you have a limited number
of pitches available (such as on a piano). In general, adaptive JI
will give you more opportunity for consonance than strict JI.

🔗jpehrson@rcn.com

🔗paul@stretch-music.com

🔗jpehrson@rcn.com

--- In tuning@y..., paul@s... wrote:

/tuning/topicId_21636.html#21832

>
> > Well, I don't really see where they are so thoroughly explained
in Partch's book...
>
> Keep digging . . . they're VERY important to Partch.
>

It's about time for a re-read anyway... I read it PRE-Tuning List (!!)

>
> Yup! A hexad is just a 1:3:5:7:9:11 chord, or a 1/(1:3:5:7:9:11)
> chord, with possible octave transpositions. Each hexad has 15
> consonant intervals (if you accept the ratios of 11 as
consonances). 2 of these 15 are identical -- the 3:1 and the 9:3.
> >

> > Anyway... since I really didn't know anything about them, I
wasn't too concerned about them (yet). Mostly I just wanted a 72-tET
> scale with as many consonances to whatever limit was feasible, even
if it had to compromise the step size a bit.
>

> Joseph, have you tested, using your own ear, whether you percieve
the ratios of 11 as consonances?

OK... I'll have to figure out how to do this... but you're already
instilling doubts!

_________ ______ ______
Joseph Pehrson

🔗jpehrson@rcn.com

🔗David C Keenan <D.KEENAN@UQ.NET.AU>

Joseph,

Here's another horribly-uneven but comma-free 19-tone 11-limit job. This
one has four hexads.

Melodically, it wants to be a 22-tone scale, just like the others. And like
the others, the extra three notes would give pitifully few extra consonances.

32
/
46 /
0 /
9
/
32 53 23 /
7 49 /
46------16------58
/ \ /
/ 9 \30 /
14 / 56 \ / 26
53------23------65
/ \ / \ 32 /
/16 \ /58 \ 7 /
/ 63 \ / \ /
30-------0------42
/ \ / \ 9 /
/65 \14 / \56 /
/ 40 \ / \ /
7------49------19
46 / \ 16 / 58
/42 \63 /
/ \ /
14-------56------26
/ 23 65
/ 19 40
/
63
/ 0
/
/
40
49

72-EDO Step sizes
degrees in 72-EDO steps
-----------------------
0 7
7 2
9 5
14 2
16 3
19 4
23 3
26 4
30 2
32 8
40 2
42 4
46 3
49 4
53 3
56 2
58 5
63 2
65 7

Paul mentioned tetrachordality as another criterion which may conflict with
both evenness and number of consonances.

--------
Question
--------

Is there a way to tell, by looking at the shape of the lattice, whether a
scale is tetrachordal or not?
-------------------------------

At first I thought you only had to ensure that every note participates in a
perfect fifth with another note, i.e. is connected to another note by a
horizontal line on the lattice. But I tried the following lattice and soon
found that it wasn't tetrachordal. But it's still an interesting scale,
harmonically.

You'll notice it is a slight modification of the one above. It makes the
scale more even (sort of). We are back to only two hexads, but I think we
gained in consonances overall. This seems like a reasonable compromise.

0
39-------9
/ \ /
53 / \23 /
7 / 49 \ /
46------16------58
/ \ /
39 / 9 \30 /
14 / 56 \ / 26
53------23------65
/ \ / \ /
/16 \ /58 \ 7 /
/ 63 \ / 33 \ /
30-------0------42
/ \ 39 / \ 9 /
/65 \14 / \56 /
/ \ / \ /
7------49------19
46 / \ 16 / 58
/42 \63 / 33
/ \ /
14-------56------26
/ \ 23 / 65
/49 \ /19
/ \ /
63-------33
0

72-EDO Step sizes
degrees in 72-EDO steps
-----------------------
0 7
7 2
9 5
14 2
16 3
19 4
23 3
26 4
30 3
33 6
39 3
42 4
46 3
49 4
53 3
56 2
58 5
63 2
65 7

----------
The method
----------

The more I considered how I was doing this the more I realised it has very
little to do with Fokker's periodicity blocks. I don't think I really
understand how to use them yet.

What I'm doing is the 2-dimensional analog of looking at the patterns that
complete chords make on a chain of fifths (or other generator) and trying
to find a set of notes that gives the most chords for the least notes. It's
a geometric pattern-matching thing.

As requested by Joseph, I've attached a tidied up version of the worksheet
showing the scale above. The finished scale is in blue in the middle of the
72-EDO 5-limit plane.

-----------------------
Detempering from 72-EDO
-----------------------

Note that all these scales can be detempered slightly from 72-EDO in a way
that distributes the 224:225 and 384:385 a bit better, and makes it a
little closer to strictly just. This is what I call "microtempered".

I worked out the tuning that gives the minimum maximum beat rate in the
4:5:6:7:9:11 hexad inversion. My philosophy is to maximise the consonance
of the most consonant chord. Here is a table of all the intervals in that
chord, sorted by interval width, showing the error in cents and the beat
rate on a 130.8 Hz root.

Intvl Error Beat rate when the interval is from a 4:5:6:7:9:11
whose root is C below middle C (130.8 Hz)
(cents) (Hz)
--------------------
4:6 -1.05 1.0
6:9 -1.05 1.4
4:5 -3.18 4.8
5:6 2.13 4.8
4:7 -0.74 1.6
5:7 2.44 6.4
6:7 0.31 1.0
4:9 -2.10 5.7
5:9 1.08 3.7
7:9 -1.35 6.4
4:11 -1.63 5.4
5:11 1.55 6.4
6:11 -0.58 2.9
7:11 -0.89 5.2
9:11 0.47 3.5

Note that equal worst beats occur in the 7:9, 5:7 and 5:11. The maximum
error in any interval is 3.2 c (in the 4:5).

In practice, this might well be indistinguishable from 72-EDO.

-- Dave Keenan

-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

🔗paul@stretch-music.com

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

--- In tuning@y..., paul@s... wrote:
> --- In tuning@y..., David C Keenan <D.KEENAN@U...> wrote:
> > Melodically, it wants to be a 22-tone scale, just like the others.
And like
> > the others, the extra three notes would give pitifully few extra
consonances.
>
> Just like the others? Are you sure that dividing the three "empty
semitones" in your first 19-tone
> scale into a 1/6 tone and a 1/3 tone (rather

This sentence was incomplete in your posting but I guess I know what
you were going to write. You might want to check that other parts of
that post didn't get lost.

Yes dividing the 6/72 oct steps into 2/72 oct and 4/72 oct is better
than 3 and 3. I'd add 14 and 58. If you want it to be symmetrical
you've got no choice but to split the middle one in half by adding
note 36. But adding 35 would be the best choice otherwise.

However, It may be possible to do a better 22-tone 11-limit subset in
ways other than extending this 19-toner.

-- Dave Keenan

🔗jpehrson@rcn.com

--- In tuning@y..., David C Keenan <D.KEENAN@U...> wrote:

/tuning/topicId_21636.html#21871

Hi Dave!

Thanks so much for the additional 19-tone scale from 72-tET. I
appreciate the effort and will explore this one as well...

I was particularly impressed with your Excel chart which you used to
work out this scale! It looks a little like the kind of multi-
colored sketching that Stravinsky used to make in his later years...

For myself, I think I will have to know a bit more about "modular"
arithmetic to get all the details...

However, I believe if one were to project this chart on the wall in
front of a good jazz musician (it looks a little like one of Earle
Brown's compositions!) you would get some nice music, even if they
knew nothing else about it ! :)

Thanks again!

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Joseph Pehrson