46-equal

Is 46-EDO a worthy temperament for 13-limit? I am considering it as a
low-res substitute for 79/80 MOS 159-tET.

Oz.

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> Is 46-EDO a worthy temperament for 13-limit? I am considering it as a
> low-res substitute for 79/80 MOS 159-tET.
>
> Oz.

It is one of the best equal temperamets of all, given the number
of tones, in the 13-limit. Another choice is its nearby
neighbor, 41.

-Carl

I am glad you approve. It might make an excellent Maqam temperament. Gene
might tell if it is consistent in 5, 7, 11 and 13 limits.

Oz.

----- Original Message -----
From: "Carl Lumma" <clumma@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 14 Eyl�l 2007 Cuma 20:15
Subject: [tuning] Re: 46-equal

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> >
> > Is 46-EDO a worthy temperament for 13-limit? I am considering it as a
> > low-res substitute for 79/80 MOS 159-tET.
> >
> > Oz.
>
> It is one of the best equal temperamets of all, given the number
> of tones, in the 13-limit. Another choice is its nearby
> neighbor, 41.
>
> -Carl
>
>

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> I am glad you approve. It might make an excellent Maqam
> temperament. Gene might tell if it is consistent in
> 5, 7, 11 and 13 limits.
>
> Oz.

46 is 13-limit consistent, and 41 is 15-limit consistent.

-Carl

Carl Lumma wrote:
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>> I am glad you approve. It might make an excellent Maqam
>> temperament. Gene might tell if it is consistent in
>> 5, 7, 11 and 13 limits.
>>
>> Oz.
> > 46 is 13-limit consistent, and 41 is 15-limit consistent.
> > -Carl

http://library.wustl.edu/~manynote/consist.txt has a list of consistent ET's in odd-limits up to 31, up to 1200-ET. (Some 11- and 13-prime-limit consistent temperaments may not be on that list if they're not 9-odd-limit consistent.)

Since 41 and 46 are both 13-limit consistent, it could be interesting to try a 41&46 regular temperament (41&46-RT). The TOP period for this temperament is very close to a just octave at 1200.188030 cents, and the TOP generator is 234.428646 cents. This would be a 13-limit version of "rodan" temperament.

Tuning map:
[<1, 1, -1, 3, 6, 8], <0, 3, 17, -1, -13, -22]]

There are lots more 13-limit temperaments related to 46-ET (I think the 9&46-RT, apparently called "twothirdtonic", looks somewhat interesting), but I'll have to look into those when I get more time.

Herman Miller wrote:

> http://library.wustl.edu/~manynote/consist.txt has a list of consistent > ET's in odd-limits up to 31, up to 1200-ET. (Some 11- and 13-prime-limit > consistent temperaments may not be on that list if they're not > 9-odd-limit consistent.)
> > Since 41 and 46 are both 13-limit consistent, it could be interesting to > try a 41&46 regular temperament (41&46-RT). The TOP period for this > temperament is very close to a just octave at 1200.188030 cents, and the > TOP generator is 234.428646 cents. This would be a 13-limit version of > "rodan" temperament.
> > Tuning map:
> [<1, 1, -1, 3, 6, 8], <0, 3, 17, -1, -13, -22]]

Um, wasn't the tuning map the size of each prime (in cents)?

> There are lots more 13-limit temperaments related to 46-ET (I think the > 9&46-RT, apparently called "twothirdtonic", looks somewhat interesting), > but I'll have to look into those when I get more time.

Indeed there are! I ran a very quick search so I don't claim to be exhaustive. I'm also not sure which of the equal temperament seeds are ambiguous so I'll give you the mappings to check.

At the top of the arbitrary badness ordering is something generated by 31-equal, canonical mapping

<1, 1, 2, 3, 3, 3]
<0, 9, 5, -3, 7, 11]

Complexity 6.7 (I think that's Kees-max), TOP-RMS error 2.0 cents/octave. Looks like 31 is ambiguous here, so the mapping is

<31, 49, 72, 87, 107, 115]

Next, a kind of 19&27. Canonical mapping

<1, -1, -1, -2, -8, 0]
<0, 7, 9, 13, 31, 10]

Complexity 9.0, TOP-RMS 1.4 cents/octave.

Then 12&46, I think this is the standard diaschismic extension, so also involving the unambiguous mapping of 58. Canonical mapping

<2, 3, 5, 7, 9, 10]
<0, 1, -2, -8, -12, -15]

Complexity 9.4, TOP-RMS 0.83 cents/octave.

Then 46&26. Canonical mapping
<2, 5, 8, 5, 6, 8]
<0, -6, -11, 2, 3, -2]
Complexity 11.2, TOP-RMS 0.56 cents/octave.
From what I remember of the numbers scrolling past, this is important in higher limits.

Now -- hurrah! -- your 41&46 comes up. Complexity 13.3, TOP-RMS 0.63 cents/octave.

Then, two more variations on 31&46, a 46&29, a 46&53, and *then* your 46&9. Canonical mapping
<1, 3, 2, 4, 4, 5]
<0, -13, 3, -11, -5, -12]
Complexity 9.5, TOP-RMS 1.4 cents/octave.

Graham

Since 79/80 MOS 159-tET is too complicated a temperament, I think 46-EDO
will make a wonderful substitute as the basic tuning of a new Maqam Music
treatise. It is considerably smaller than 53-EDO too.

But I wonder if a lot may be gained by ditching 53. People are so much used
to thinking in terms of 9-commas per whole tone, 53 commas to the octave.

Oz.

----- Original Message -----
From: "Graham Breed" <gbreed@gmail.com>
To: <tuning@yahoogroups.com>
Sent: 15 Eyl�l 2007 Cumartesi 9:12
Subject: Re: [tuning] Re: 46-equal

> Herman Miller wrote:
>
> > http://library.wustl.edu/~manynote/consist.txt has a list of consistent
> > ET's in odd-limits up to 31, up to 1200-ET. (Some 11- and 13-prime-limit
> > consistent temperaments may not be on that list if they're not
> > 9-odd-limit consistent.)
> >
> > Since 41 and 46 are both 13-limit consistent, it could be interesting to
> > try a 41&46 regular temperament (41&46-RT). The TOP period for this
> > temperament is very close to a just octave at 1200.188030 cents, and the
> > TOP generator is 234.428646 cents. This would be a 13-limit version of
> > "rodan" temperament.
> >
> > Tuning map:
> > [<1, 1, -1, 3, 6, 8], <0, 3, 17, -1, -13, -22]]
>
> Um, wasn't the tuning map the size of each prime (in cents)?
>
> > There are lots more 13-limit temperaments related to 46-ET (I think the
> > 9&46-RT, apparently called "twothirdtonic", looks somewhat interesting),
> > but I'll have to look into those when I get more time.
>
> Indeed there are! I ran a very quick search so I don't
> claim to be exhaustive. I'm also not sure which of the
> equal temperament seeds are ambiguous so I'll give you the
> mappings to check.
>
>
> At the top of the arbitrary badness ordering is something
> generated by 31-equal, canonical mapping
>
> <1, 1, 2, 3, 3, 3]
> <0, 9, 5, -3, 7, 11]
>
> Complexity 6.7 (I think that's Kees-max), TOP-RMS error 2.0
> cents/octave. Looks like 31 is ambiguous here, so the
> mapping is
>
> <31, 49, 72, 87, 107, 115]
>
>
> Next, a kind of 19&27. Canonical mapping
>
> <1, -1, -1, -2, -8, 0]
> <0, 7, 9, 13, 31, 10]
>
> Complexity 9.0, TOP-RMS 1.4 cents/octave.
>
>
> Then 12&46, I think this is the standard diaschismic
> extension, so also involving the unambiguous mapping of 58.
> Canonical mapping
>
> <2, 3, 5, 7, 9, 10]
> <0, 1, -2, -8, -12, -15]
>
> Complexity 9.4, TOP-RMS 0.83 cents/octave.
>
>
> Then 46&26. Canonical mapping
> <2, 5, 8, 5, 6, 8]
> <0, -6, -11, 2, 3, -2]
> Complexity 11.2, TOP-RMS 0.56 cents/octave.
> From what I remember of the numbers scrolling past, this is
> important in higher limits.
>
>
> Now -- hurrah! -- your 41&46 comes up. Complexity 13.3,
> TOP-RMS 0.63 cents/octave.
>
>
> Then, two more variations on 31&46, a 46&29, a 46&53, and
> *then* your 46&9. Canonical mapping
> <1, 3, 2, 4, 4, 5]
> <0, -13, 3, -11, -5, -12]
> Complexity 9.5, TOP-RMS 1.4 cents/octave.
>
>
> Graham
>

Come to think of it, 41 seems way better.

Back to the drawing board.

Oz.

----- Original Message -----
From: "Herman Miller" <hmiller@IO.COM>
To: <tuning@yahoogroups.com>
Sent: 15 Eyl�l 2007 Cumartesi 6:43
Subject: Re: [tuning] Re: 46-equal

> Carl Lumma wrote:
> > --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> >> I am glad you approve. It might make an excellent Maqam
> >> temperament. Gene might tell if it is consistent in
> >> 5, 7, 11 and 13 limits.
> >>
> >> Oz.
> >
> > 46 is 13-limit consistent, and 41 is 15-limit consistent.
> >
> > -Carl
>
> http://library.wustl.edu/~manynote/consist.txt has a list of consistent
> ET's in odd-limits up to 31, up to 1200-ET. (Some 11- and 13-prime-limit
> consistent temperaments may not be on that list if they're not
> 9-odd-limit consistent.)
>
> Since 41 and 46 are both 13-limit consistent, it could be interesting to
> try a 41&46 regular temperament (41&46-RT). The TOP period for this
> temperament is very close to a just octave at 1200.188030 cents, and the
> TOP generator is 234.428646 cents. This would be a 13-limit version of
> "rodan" temperament.
>
> Tuning map:
> [<1, 1, -1, 3, 6, 8], <0, 3, 17, -1, -13, -22]]
>
> There are lots more 13-limit temperaments related to 46-ET (I think the
> 9&46-RT, apparently called "twothirdtonic", looks somewhat interesting),
> but I'll have to look into those when I get more time.
>
>
>
> You can configure your subscription by sending an empty email to one
> of these addresses (from the address at which you receive the list):
> tuning-subscribe@yahoogroups.com - join the tuning group.
> tuning-unsubscribe@yahoogroups.com - leave the group.
> tuning-nomail@yahoogroups.com - turn off mail from the group.
> tuning-digest@yahoogroups.com - set group to send daily digests.
> tuning-normal@yahoogroups.com - set group to send individual emails.
> tuning-help@yahoogroups.com - receive general help information.
>
> Yahoo! Groups Links
>
>
>

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> Since 79/80 MOS 159-tET is too complicated a temperament, I
> think 46-EDO will make a wonderful substitute as the basic
> tuning of a new Maqam Music treatise. It is considerably
> smaller than 53-EDO too.
>
> But I wonder if a lot may be gained by ditching 53. People
> are so much used to thinking in terms of 9-commas per whole
> tone, 53 commas to the octave.
>
> Oz.

Who are you and what have you done with Ozan?

-Carl

Now would be an excellent time for anyone and everyone
to come forth with the names of organ tuners doing stuff
like this, or who would even have the slightest idea
what any of it means.

-Carl

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
//
> Indeed there are! I ran a very quick search so I don't
> claim to be exhaustive. I'm also not sure which of the
> equal temperament seeds are ambiguous so I'll give you the
> mappings to check.
>
>
> At the top of the arbitrary badness ordering is something
> generated by 31-equal, canonical mapping
>
> <1, 1, 2, 3, 3, 3]
> <0, 9, 5, -3, 7, 11]
>
> Complexity 6.7 (I think that's Kees-max), TOP-RMS error 2.0
> cents/octave. Looks like 31 is ambiguous here, so the
> mapping is
>
> <31, 49, 72, 87, 107, 115]
>
>
> Next, a kind of 19&27. Canonical mapping
>
> <1, -1, -1, -2, -8, 0]
> <0, 7, 9, 13, 31, 10]
>
> Complexity 9.0, TOP-RMS 1.4 cents/octave.
>
>
> Then 12&46, I think this is the standard diaschismic
> extension, so also involving the unambiguous mapping of 58.
> Canonical mapping
>
> <2, 3, 5, 7, 9, 10]
> <0, 1, -2, -8, -12, -15]
>
> Complexity 9.4, TOP-RMS 0.83 cents/octave.
>
>
> Then 46&26. Canonical mapping
> <2, 5, 8, 5, 6, 8]
> <0, -6, -11, 2, 3, -2]
> Complexity 11.2, TOP-RMS 0.56 cents/octave.
> From what I remember of the numbers scrolling past, this is
> important in higher limits.
>
>
> Now -- hurrah! -- your 41&46 comes up. Complexity 13.3,
> TOP-RMS 0.63 cents/octave.
>
>
> Then, two more variations on 31&46, a 46&29, a 46&53, and
> *then* your 46&9. Canonical mapping
> <1, 3, 2, 4, 4, 5]
> <0, -13, 3, -11, -5, -12]
> Complexity 9.5, TOP-RMS 1.4 cents/octave.
>
>
> Graham

I am afraid he is indisposed at the moment. I am his long lost fraternal
twin. LOL!

----- Original Message -----
From: "Carl Lumma" <clumma@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 15 Eyl�l 2007 Cumartesi 22:28
Subject: [tuning] Re: 46-equal

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> >
> > Since 79/80 MOS 159-tET is too complicated a temperament, I
> > think 46-EDO will make a wonderful substitute as the basic
> > tuning of a new Maqam Music treatise. It is considerably
> > smaller than 53-EDO too.
> >
> > But I wonder if a lot may be gained by ditching 53. People
> > are so much used to thinking in terms of 9-commas per whole
> > tone, 53 commas to the octave.
> >
> > Oz.
>
> Who are you and what have you done with Ozan?
>
> -Carl
>

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> I am afraid he is indisposed at the moment. I am his long lost
> fraternal twin. LOL!

I mean, isn't he the one who so reviled the 53-comma system
of contemporary Turkey? And wasn't he completely satisfied
with his 79-tone system for years on end?

-Carl

Carl Lumma wrote:
> Now would be an excellent time for anyone and everyone
> to come forth with the names of organ tuners doing stuff
> like this, or who would even have the slightest idea
> what any of it means.
> The almost-reverse question is also interesting---how many people, besides Gene, are doing work in this stuff and actually taking the time to also create some music in it?

-A.

> -Carl
>
> --- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> //
> >> Indeed there are! I ran a very quick search so I don't >> claim to be exhaustive. I'm also not sure which of the >> equal temperament seeds are ambiguous so I'll give you the >> mappings to check.
>>
>>
>> At the top of the arbitrary badness ordering is something >> generated by 31-equal, canonical mapping
>>
>> <1, 1, 2, 3, 3, 3]
>> <0, 9, 5, -3, 7, 11]
>>
>> Complexity 6.7 (I think that's Kees-max), TOP-RMS error 2.0 >> cents/octave. Looks like 31 is ambiguous here, so the >> mapping is
>>
>> <31, 49, 72, 87, 107, 115]
>>
>>
>> Next, a kind of 19&27. Canonical mapping
>>
>> <1, -1, -1, -2, -8, 0]
>> <0, 7, 9, 13, 31, 10]
>>
>> Complexity 9.0, TOP-RMS 1.4 cents/octave.
>>
>>
>> Then 12&46, I think this is the standard diaschismic >> extension, so also involving the unambiguous mapping of 58. >> Canonical mapping
>>
>> <2, 3, 5, 7, 9, 10]
>> <0, 1, -2, -8, -12, -15]
>>
>> Complexity 9.4, TOP-RMS 0.83 cents/octave.
>>
>>
>> Then 46&26. Canonical mapping
>> <2, 5, 8, 5, 6, 8]
>> <0, -6, -11, 2, 3, -2]
>> Complexity 11.2, TOP-RMS 0.56 cents/octave.
>> From what I remember of the numbers scrolling past, this is >> important in higher limits.
>>
>>
>> Now -- hurrah! -- your 41&46 comes up. Complexity 13.3, >> TOP-RMS 0.63 cents/octave.
>>
>>
>> Then, two more variations on 31&46, a 46&29, a 46&53, and >> *then* your 46&9. Canonical mapping
>> <1, 3, 2, 4, 4, 5]
>> <0, -13, 3, -11, -5, -12]
>> Complexity 9.5, TOP-RMS 1.4 cents/octave.
>>
>>
>>

Graham Breed wrote:
> Herman Miller wrote:
> >> http://library.wustl.edu/~manynote/consist.txt has a list of consistent >> ET's in odd-limits up to 31, up to 1200-ET. (Some 11- and 13-prime-limit >> consistent temperaments may not be on that list if they're not >> 9-odd-limit consistent.)
>>
>> Since 41 and 46 are both 13-limit consistent, it could be interesting to >> try a 41&46 regular temperament (41&46-RT). The TOP period for this >> temperament is very close to a just octave at 1200.188030 cents, and the >> TOP generator is 234.428646 cents. This would be a 13-limit version of >> "rodan" temperament.
>>
>> Tuning map:
>> [<1, 1, -1, 3, 6, 8], <0, 3, 17, -1, -13, -22]]
> > Um, wasn't the tuning map the size of each prime (in cents)?

Ah, whatever. I'm way too tired to look up whatever ended up as the term for this thing.

>> There are lots more 13-limit temperaments related to 46-ET (I think the >> 9&46-RT, apparently called "twothirdtonic", looks somewhat interesting), >> but I'll have to look into those when I get more time.
> > Indeed there are! I ran a very quick search so I don't > claim to be exhaustive. I'm also not sure which of the > equal temperament seeds are ambiguous so I'll give you the > mappings to check.

My preference is to check the largest and smallest deviation from the prime numbers up to the limit (13 in this case) and see if the difference is less than half a step. So I include 9-ET even though it's not 9-(odd)-limit consistent, but it is (2,3,5,7,11,13)-consistent.

> At the top of the arbitrary badness ordering is something > generated by 31-equal, canonical mapping
> > <1, 1, 2, 3, 3, 3]
> <0, 9, 5, -3, 7, 11]
> > Complexity 6.7 (I think that's Kees-max), TOP-RMS error 2.0 > cents/octave. Looks like 31 is ambiguous here, so the > mapping is
> > <31, 49, 72, 87, 107, 115]

Hmm, I get something a bit different with this mapping.

<1, 1, 2, 3, 3, 5]
<0, 9, 5, -3, 7, -20]

> Next, a kind of 19&27. Canonical mapping
...
> Then 12&46, I think this is the standard diaschismic > extension, so also involving the unambiguous mapping of 58. I'll have to check these out when I get the time. I haven't put much thought into trying to use ambiguous ET's, even though I know of one or two useful regular temperaments that use them. Mainly because there's such a flood of temperaments built from consistent ET's already that I don't even have time to explore more than a tiny fraction of those!

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <aaron@...> wrote:
>
> Carl Lumma wrote:
> > Now would be an excellent time for anyone and everyone
> > to come forth with the names of organ tuners doing stuff
> > like this, or who would even have the slightest idea
> > what any of it means.
> >
> The almost-reverse question is also interesting---how many people,
> besides Gene, are doing work in this stuff and actually taking
> the time to also create some music in it?
>
> -A.

Music may be your goal (in which case, why are you posting
this?) but it isn't necessarily Graham's.

You think that because the people doing the theory don't
write music in it that invalidates it?

Gene's music, by the way, is some of my favorite I've ever
heard on these lists.

-Carl

Carl Lumma wrote:

> Music may be your goal (in which case, why are you posting
> this?) but it isn't necessarily Graham's.

Heh. Well, as it happens, I'm just back from buying an input keyboard. But certainly writing up the theory takes precedence over making music now. I think it's a good theory and worth spending the time on. At this stage there's no need for me to be actively involved in music to develop it.

Right now, though, my priority's to have a lie down...

Graham

Sadly, yes.

It appears, the details of the 79-tone tuning are too much to bear for my
countrymen.

79/80 MOS 159-tET could remain as an experimental model nonetheless.

Oz.

----- Original Message -----
From: "Carl Lumma" <clumma@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 16 Eyl�l 2007 Pazar 4:01
Subject: [tuning] Re: 46-equal

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> > I am afraid he is indisposed at the moment. I am his long lost
> > fraternal twin. LOL!
>
> I mean, isn't he the one who so reviled the 53-comma system
> of contemporary Turkey? And wasn't he completely satisfied
> with his 79-tone system for years on end?
>
> -Carl
>

Herman Miller wrote:
> Graham Breed wrote:
> >>Herman Miller wrote:
>>
>>
>>>http://library.wustl.edu/~manynote/consist.txt has a list of consistent >>>ET's in odd-limits up to 31, up to 1200-ET. (Some 11- and 13-prime-limit >>>consistent temperaments may not be on that list if they're not >>>9-odd-limit consistent.)
>>>
>>>Since 41 and 46 are both 13-limit consistent, it could be interesting to >>>try a 41&46 regular temperament (41&46-RT). The TOP period for this >>>temperament is very close to a just octave at 1200.188030 cents, and the >>>TOP generator is 234.428646 cents. This would be a 13-limit version of >>>"rodan" temperament.
>>>
>>>Tuning map:
>>>[<1, 1, -1, 3, 6, 8], <0, 3, 17, -1, -13, -22]]
>>
>>Um, wasn't the tuning map the size of each prime (in cents)?
> > > Ah, whatever. I'm way too tired to look up whatever ended up as the term > for this thing.
> > >>>There are lots more 13-limit temperaments related to 46-ET (I think the >>>9&46-RT, apparently called "twothirdtonic", looks somewhat interesting), >>>but I'll have to look into those when I get more time.
>>
>>Indeed there are! I ran a very quick search so I don't >>claim to be exhaustive. I'm also not sure which of the >>equal temperament seeds are ambiguous so I'll give you the >>mappings to check.
> > > My preference is to check the largest and smallest deviation from the > prime numbers up to the limit (13 in this case) and see if the > difference is less than half a step. So I include 9-ET even though it's > not 9-(odd)-limit consistent, but it is (2,3,5,7,11,13)-consistent.
> > >>At the top of the arbitrary badness ordering is something >>generated by 31-equal, canonical mapping
>>
>><1, 1, 2, 3, 3, 3]
>><0, 9, 5, -3, 7, 11]
>>
>>Complexity 6.7 (I think that's Kees-max), TOP-RMS error 2.0 >>cents/octave. Looks like 31 is ambiguous here, so the >>mapping is
>>
>><31, 49, 72, 87, 107, 115]
> > > Hmm, I get something a bit different with this mapping.
> > <1, 1, 2, 3, 3, 5]
> <0, 9, 5, -3, 7, -20]

Um, yes. The equal temperament mappings are

<31, 49, 72, 87, 107, 115]
<15, 24, 35, 42, 52, 56]

so it implies a non-standard mapping of 46 as well. This must be the old badness where simple temperaments float to the top. So this one's simple but the error's even worse than neat 46-equal.

>>Next, a kind of 19&27. Canonical mapping
> > ...
> >>Then 12&46, I think this is the standard diaschismic >>extension, so also involving the unambiguous mapping of 58. > > I'll have to check these out when I get the time. I haven't put much > thought into trying to use ambiguous ET's, even though I know of one or > two useful regular temperaments that use them. Mainly because there's > such a flood of temperaments built from consistent ET's already that I > don't even have time to explore more than a tiny fraction of those!

You have to be selective, of course. But if you filter for the worst error it doesn't really matter if the original ETs are ambiguous or not.

Graham

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> Sadly, yes.
>
> It appears, the details of the 79-tone tuning are too much to
> bear for my countrymen.
>
> 79/80 MOS 159-tET could remain as an experimental model nonetheless.
>
> Oz.

Don't be too disappointed. Even if you suggest the simpler
(than 53) 41-tone tuning, you may run into trouble. In the West,
many have tried and failed to promote systems like 19 or 31
as new standards. :(

Now, back to theory. Has it ever been suggested to you that
you try to describe the maqamat as periodicity blocks? I
think that would be a very interesting paper.

-Carl

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> Since 79/80 MOS 159-tET is too complicated a temperament, I think 46-
EDO
> will make a wonderful substitute as the basic tuning of a new Maqam
Music
> treatise. It is considerably smaller than 53-EDO too.

There's a lot to be said in favor of 46. The list of 13-limit
consistent ets goes 26, 29, 41, 46, 58, 72, 80, 87, 94... The first to
give really decent results is 41. While I see you've concluded 41 is
better, I've tried both and I actually prefer 46.

Of course, your reasons for liking things probably bear little
relationship to mine. For instance, I like the mmms chord, meaning
three minor thirds and one subminor third, of 46; it's an in-tune
version of a diminished seventh. I doubt there is much application to
maqam music.

--- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:

> Since 41 and 46 are both 13-limit consistent, it could be interesting
to
> try a 41&46 regular temperament (41&46-RT).

Good old Rodan the Flying Monster temperament (clueing you that an
approximate 8/7 generates.) It's on my list of things to try, but with
all the medical problems I'm having, I can't seem to get any work done.

It would be a good excuse to try 87-et, which Barbour mentions but
which I don't know if anyone has tried. That Barbour mentions it tips
you off that it is a good 5-limit system, but it's also got this higher
limit thing going, including a nice 15-limit rodan.

> It's on my list of things to try, but with
> all the medical problems I'm having, I can't seem to get
> any work done.

Hope you're O-K. For the first time in my life, I'm having
a healthcare experience. Just got started last week. Nothing
too serious, yet. Having some bloodwork done this week.
Have no idea what it's going to tell them. They seem obsessed
with blood. :)

-Carl

----- Original Message -----
From: "Carl Lumma" <carl@lumma.org>
To: <tuning@yahoogroups.com>
Sent: 16 Eyl�l 2007 Pazar 19:17
Subject: [tuning] Re: 46-equal

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> >
> > Sadly, yes.
> >
> > It appears, the details of the 79-tone tuning are too much to
> > bear for my countrymen.
> >
> > 79/80 MOS 159-tET could remain as an experimental model nonetheless.
> >
> > Oz.
>
> Don't be too disappointed. Even if you suggest the simpler
> (than 53) 41-tone tuning, you may run into trouble. In the West,
> many have tried and failed to promote systems like 19 or 31
> as new standards. :(
>

The West seems to be burdened by the limitations of the Halberstadt keyboard
design, whereas there are no keyboard instruments in authentic Maqam Music.
Thus, it is viable to set 41-equal on tanburs and qanuns. The rest becomes
simple.

> Now, back to theory. Has it ever been suggested to you that
> you try to describe the maqamat as periodicity blocks? I
> think that would be a very interesting paper.
>

What are periodicity blocks in layman's terms?

> -Carl
>
>
>

Oz.

----- Original Message -----
From: "Gene Ward Smith" <genewardsmith@sbcglobal.net>
To: <tuning@yahoogroups.com>
Sent: 17 Eyl�l 2007 Pazartesi 0:47
Subject: [tuning] Re: 46-equal

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> >
> > Since 79/80 MOS 159-tET is too complicated a temperament, I think 46-
> EDO
> > will make a wonderful substitute as the basic tuning of a new Maqam
> Music
> > treatise. It is considerably smaller than 53-EDO too.
>
> There's a lot to be said in favor of 46. The list of 13-limit
> consistent ets goes 26, 29, 41, 46, 58, 72, 80, 87, 94... The first to
> give really decent results is 41. While I see you've concluded 41 is
> better, I've tried both and I actually prefer 46.
>

On second thought, I find 46-equal too radical a temperament for the
traditionalists who will likely opt for something more "Pythagorean".
Fortunately, 41-EDO saves the day.

> Of course, your reasons for liking things probably bear little
> relationship to mine. For instance, I like the mmms chord, meaning
> three minor thirds and one subminor third, of 46; it's an in-tune
> version of a diminished seventh. I doubt there is much application to
> maqam music.
>

A "maqam polyphony" trend might find good use for it.

Oz.

> > Now, back to theory. Has it ever been suggested to you that
> > you try to describe the maqamat as periodicity blocks? I
> > think that would be a very interesting paper.
>
> What are periodicity blocks in layman's terms?

They are scales in just intonation. Actually they are more
like scale archetypes, as usually a number of different but
related scales can be said to come from the same periodicity
block. They are also archetypes of temperaments in a way,
as each periodicity block suggests by its nature ways it may
be tempered. In my view, they are the foundation of the
theory of regular temperaments.

What they actually are: sections ("blocks") of a JI lattice
that are bounded by commas (small intervals). By bounded,
we mean these intervals do not occur between any two notes
in the block. For example, 81/80 does not occur in the
diatonic scale 1/1 9/8/ 5/4 4/3 3/2 5/3 15/8 2/1. However,
if we add 10/9, which is adjacent to 5/3 and 4/3 on the
lattice, then 9/8 - 10/9 = 81/80, and 81/80 will occur as
a scale step ("second") in our scale.

Not only are the blocks bounded in this way, it is usually
the case that no seconds of the scale are smaller than these
bounding intervals. If the diatonic scale had an interval
smaller than 81/80, why not include 81/80 too?

Scala actually has built-in tools for dealing with periodicity
blocks, but I haven't tried to learn how to use them yet.

For more information, consult Paul Erlich's excellent
tutorial:
http://www.tonalsoft.com/enc/f/fokker-gentle-1.aspx

Generally it is a good setup for the kind of paper I suggest
if someone who knows the maqamat but not periodicity blocks
supplies a list of maqamat in just intonation. And then
the other person attempts to describe them in terms of
periodicity blocks. :) But the maqamat are mysterious
things, and you may have an easier time learning about
periodicity blocks than I will about maqamat.

-Carl

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> Sadly, yes.
>
> It appears, the details of the 79-tone tuning are too much to bear
for my
> countrymen.
>
> 79/80 MOS 159-tET could remain as an experimental model nonetheless.
>
> Oz.

Wouldn't it be possible to treat the tones of 159-ET as 53 three-tone
clusters, whereby the tones of 53-ET would be altered up or down by
1deg159 to fine-tune intervals containing primes 7, 11, and 17?

--George

regarding 46-equal, anyone know of any mp3s/oggs anywhere in this group or in the larger 'net?

Hi Oz and Carl,

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> What are periodicity blocks in layman's terms?
>
> <snip>
>
> What they actually are: sections ("blocks") of a JI lattice
> that are bounded by commas (small intervals). By bounded,
> we mean these intervals do not occur between any two notes
> in the block. For example, 81/80 does not occur in the
> diatonic scale 1/1 9/8/ 5/4 4/3 3/2 5/3 15/8 2/1. However,
> if we add 10/9, which is adjacent to 5/3 and 4/3 on the
> lattice, then 9/8 - 10/9 = 81/80, and 81/80 will occur as
> a scale step ("second") in our scale.

Along with Paul's excellent tutorial, i also have
entries for "periodicity-block" and the related
"unison-vector" in the Encyclopedia:

http://tonalsoft.com/enc/p/periodicity-block.aspx

http://tonalsoft.com/enc/u/unison-vector.aspx

Tonescape utilizes the periodicity-block concept
at its core. Its "Tonespace" files in fact simply
*are* periodicity-blocks, from which "Tuning" files
are generated.

A periodicity-block is essentially a unit-cell,
which tiles the whole tonespace, similar to the
way that in a crystal structure there is a unit-cell
which is repeated over and over again in all three
dimensions. However, a periodicity-block can have
as many boundaries as there are dimensions in the
tonespace, it's not restricted to 3.

Near the bottom of my 12-edo page, i have a graphic
showing a bingo-card lattice of 12-edo with many
of the different possible periodicity-blocks outlined
in green:

http://tonalsoft.com/enc/number/12edo.aspx

Note that in order to create a periodicity-block,
the number of unison-vectors must equal the number
of dimensions in the tonespace. If there are less
unison-vectors (but at least one), there will still
be periodicity in the tonespace, but some dimensions
will be open-ended (infinite) -- thus, for example:

In classic 5-limit meantone, the full tonespace has
3 dimensions, representing prime-factors 2, 3, and 5.
Assuming the normal identity-interval of ratio 2/1,
we may ignore 2 and use a 2-dimensional prime-space
representing factors 3 and 5. If there is only one
unison-vector and it is the syntonic-comma, whose
ratio is 81/80 and 2,3,5-monzo is [-4 4, -1>, then
one boundary of periodicity will be a vector which
can be expressed in classic vector form as <4i-j>
where i is the 3-axis and j is the 5-axis -- thus
the vector extends to a point which is 4 positive
steps along the 3-axis and one negative step along
the 5-axis. This single unison-vector thus creates
a slightly angled "periodicity-slice" across the
2-dimensional prime-space. If 3 is the horizontal
axis and 5 is the vertical, the periodicity-slice
will go from the upper-right to the lower-left, and
all pitches further to the left or right are considered
equivalent to those within the slice. But the slice
will extend infinitely on the top and bottom.

If a second unison-vector is used, then a periodicity-block
will be formed, with the top and bottom boundaries being
the vector of the syntonic-comma itself, and the left
and right boundaries being the second unison-vector.

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

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> On second thought, I find 46-equal too radical a temperament for the
> traditionalists who will likely opt for something more
> "Pythagorean". Fortunately, 41-EDO saves the day.
>

Well, from my point of view: the less tones, the better :-)
Anyway, if you want to play all notes on standard pianos, you would
need four in any case, so it's not that big a difference... What would
good keyboard layouts be?

BTW, it's interesting that 41 and 46 appear here - since exactly those
two were also proposed for indian music (
/tuning/topicId_63593.html#63593 and
/tuning/topicId_63594.html#63594 ). The same
reason perhaps?
--
Hans Straub

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <aaron@...> wrote:
>
> regarding 46-equal, anyone know of any mp3s/oggs anywhere in this
group
> or in the larger 'net?

How about a very short midi file (along with a score) that you can
compare with a bunch of other divisions (as well as JI)?

http://dkeenan.com/sagittal/exmp/index.htm

--George

----- Original Message -----
From: "hstraub64" <hstraub64@telesonique.net>
To: <tuning@yahoogroups.com>
Sent: 17 Eyl�l 2007 Pazartesi 23:56
Subject: [tuning] Re: 46-equal

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> >
> > On second thought, I find 46-equal too radical a temperament for the
> > traditionalists who will likely opt for something more
> > "Pythagorean". Fortunately, 41-EDO saves the day.
> >
>
> Well, from my point of view: the less tones, the better :-)
> Anyway, if you want to play all notes on standard pianos, you would
> need four in any case, so it's not that big a difference... What would
> good keyboard layouts be?
>

I wouldn't know for sure. Let us design a 41-EDO keyboard.

> BTW, it's interesting that 41 and 46 appear here - since exactly those
> two were also proposed for indian music (
> /tuning/topicId_63593.html#63593 and
> /tuning/topicId_63594.html#63594 ). The same
> reason perhaps?

Possibly. Great minds think alike. ;)

Oz.

> --
> Hans Straub
>
>

I am all opposed to the conception of 53-EDO as having anything to do with
Maqam Music. It is a very crude tuning for explaining middle seconds. 41-EDO
can do much the same better with far less. If more detail is 123-EDO may be
employed in a similar manner to what you say.

Oz.

----- Original Message -----
From: "George D. Secor" <gdsecor@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 17 Eyl�l 2007 Pazartesi 20:41
Subject: [tuning] Re: 46-equal

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> >
> > Sadly, yes.
> >
> > It appears, the details of the 79-tone tuning are too much to bear
> for my
> > countrymen.
> >
> > 79/80 MOS 159-tET could remain as an experimental model nonetheless.
> >
> > Oz.
>
> Wouldn't it be possible to treat the tones of 159-ET as 53 three-tone
> clusters, whereby the tones of 53-ET would be altered up or down by
> 1deg159 to fine-tune intervals containing primes 7, 11, and 17?
>
> --George
>
>

41 tone keyboards ( tune how you wish)
http://anaphoria.com/xen3a.PDF
page 6
http://anaphoria.com/xen3b.PDF
page 6, 10, 14

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/index.html>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles

are not periodicity blocks constant structures? The latter is a lot easier to say if they are!
--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/index.html>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles

Ozan Yarman wrote:
> > ----- Original Message -----
> From: "hstraub64" <hstraub64@telesonique.net>
> To: <tuning@yahoogroups.com>
> Sent: 17 Eyl�l 2007 Pazartesi 23:56
> Subject: [tuning] Re: 46-equal
> > >> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>>> On second thought, I find 46-equal too radical a temperament for the
>>> traditionalists who will likely opt for something more
>>> "Pythagorean". Fortunately, 41-EDO saves the day.
>>>
>> Well, from my point of view: the less tones, the better :-)
>> Anyway, if you want to play all notes on standard pianos, you would
>> need four in any case, so it's not that big a difference... What would
>> good keyboard layouts be?
>>
> > > I wouldn't know for sure. Let us design a 41-EDO keyboard.

http://www.anaphoria.com/xen3b.PDF

See page 6, "Positive Linear Mapping Template to modulus 41", and page 10, "Keyboarding Genus 41". Lots of good info packed into these diagrams.

Graham Breed wrote:

> You have to be selective, of course. But if you filter for > the worst error it doesn't really matter if the original ETs > are ambiguous or not.

You know, I don't know why it hasn't occurred to me to compare different ET mappings and take the one with the least error. So take for instance my favorite inconsistent 5-limit ET, 64-ET. The possibilities for combining with 12-ET include:

<4, 6, 9], <0, 1, 1] P = 299.1603 G = 101.6696 TOP: 3.358740
<4, 7, 9], <0, -2, 1] P = 299.4908 G = 95.6261 TOP: 2.036857
<4, 6, 8], <0, 1, 4] P = 300.4246 G = 96.7151 TOP: 1.698520

If I don't care that it has four "bicycle chains", the last one has the smallest TOP error. (If the number looks familiar, it's exactly the same as meantone). The middle one is better if you want a single bicycle chain. But there's always the possibility that some other mapping combination would result in a smaller error. If you've got any further ideas along this line, that could be a good topic for tuning-math.

In any case, what interests me about these "inconsistent" ET's is that they can have two usable fifths (or thirds, or some other interval), which could provide another source of expressive variation or unique chord progressions like the comma pumps of regular temperaments. I'm wondering if temperaments based on these inconsistent ET's might have similar properties.

Herman Miller wrote:
> Graham Breed wrote:
> >>You have to be selective, of course. But if you filter for >>the worst error it doesn't really matter if the original ETs >>are ambiguous or not.
> > > You know, I don't know why it hasn't occurred to me to compare different > ET mappings and take the one with the least error. So take for instance > my favorite inconsistent 5-limit ET, 64-ET. The possibilities for > combining with 12-ET include:
> > <4, 6, 9], <0, 1, 1] P = 299.1603 G = 101.6696 TOP: 3.358740
> <4, 7, 9], <0, -2, 1] P = 299.4908 G = 95.6261 TOP: 2.036857
> <4, 6, 8], <0, 1, 4] P = 300.4246 G = 96.7151 TOP: 1.698520
> > If I don't care that it has four "bicycle chains", the last one has the > smallest TOP error. (If the number looks familiar, it's exactly the same > as meantone). The middle one is better if you want a single bicycle > chain. But there's always the possibility that some other mapping > combination would result in a smaller error. If you've got any further > ideas along this line, that could be a good topic for tuning-math.

It's certainly a tuning-math topic, and has been discussed on tuning-math before. With TOP-max you're fairly safe that no other mapping will give a smaller error, but there's always the chance that another mapping will give the same error.

> In any case, what interests me about these "inconsistent" ET's is that > they can have two usable fifths (or thirds, or some other interval), > which could provide another source of expressive variation or unique > chord progressions like the comma pumps of regular temperaments. I'm > wondering if temperaments based on these inconsistent ET's might have > similar properties.

That's an interesting compositional approach that I don't think has been explored. Certainly you can't stop the listeners hearing more than one mapping when it's ambiguous.

Graham

George wrote...

>> regarding 46-equal, anyone know of any mp3s/oggs anywhere
>> in this group or in the larger 'net?
>
> How about a very short midi file (along with a score)
> that you can compare with a bunch of other divisions (as
> well as JI)?
>
> http://dkeenan.com/sagittal/exmp/index.htm

I think I'd been to this page before, but man, it's so rock.

After listening to the complete example in JI, the first beat
of 12 (only a triad!) nearly sent me through the roof.

To my ear, 46 outshines 41, most immediately audibly on the
fourth chord (min 7th). The difference isn't huge, but to
me is probably worth 5 notes.
53 sounds better than 46 in the first half of the example,
and about the same in the second half. Is it worth 7 more
notes?

Whereas 12 and 19 just garble this example, 22 seems usable.
It delivers about what I'd expect of 12 in a triadic
example -- the underlying structure and meaning of the music
is there, but it's fuzzy.

31 sounds good until the final trine exposes the beating
of the 5th. Urg.

I'd like to see 99 on this list.

I do feel like the 21/16 at the end scotches the ability
of my ear to use this as a JI approximation test, to some
extent. It's like I have to come back inside for protection
for a second... I know it's coming, so I can't be fully
out there listening.

I would be very interested to hear others' reactions.

-Carl

> are not periodicity blocks constant structures? The latter is a lot
> easier to say if they are!

They are related, but a periodicity block is a little more.

-Carl

--- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:
> Graham Breed wrote:
>
> > You have to be selective, of course. But if you filter for
> > the worst error it doesn't really matter if the original ETs
> > are ambiguous or not.
>
> You know, I don't know why it hasn't occurred to me to compare
> different ET mappings and take the one with the least error.

It's precisely Paul's suggestion (and action too, I think) to
take test mappings and use the one with the lowest TOP error.
And 64 was always his favorite example.

-Carl

I haven't listened tothe page yet, but---

re:53 vs. 46, wouldn't there be some advantage to having a prime numbered n-tet (e.g. complete circulation)

I suppose one can get off, musically on the "switching gears" effect of 46=2x23, but what else is virtuous.

Regarding numbers of notes, when they are that high to begin with, piling a few extra doesn't make a difference, except from the point of view of live acoustic instuments with fixed pitch set (guitar frets, keyboard keys). electronics and trombone and voice, and so on, pose no limit, and I gather that these edos are just theory at that point to such instruments, unless you are performing with a synth backing track like Toby Twining does.

One of the attractions of 19 in spite of its less than stellar approximations is that it's easily done with current resources, and its different enough in a very obvioous way.

53 on acoustic instruments really deosn't exist---it just becomes adaptive JI, let's face it, with all the error in cents that acoustic instruments will bring to it. I would say that without synth backing tracks, 31 or 34 might be the upper limit of what acoustic instruments could do with any accuracy or meaning.

-A.

Carl Lumma wrote:
> George wrote...
>
> >>> regarding 46-equal, anyone know of any mp3s/oggs anywhere
>>> in this group or in the larger 'net?
>>> >> How about a very short midi file (along with a score)
>> that you can compare with a bunch of other divisions (as
>> well as JI)?
>>
>> http://dkeenan.com/sagittal/exmp/index.htm
>> >
> I think I'd been to this page before, but man, it's so rock.
>
> After listening to the complete example in JI, the first beat
> of 12 (only a triad!) nearly sent me through the roof.
>
> To my ear, 46 outshines 41, most immediately audibly on the
> fourth chord (min 7th). The difference isn't huge, but to
> me is probably worth 5 notes.
> 53 sounds better than 46 in the first half of the example,
> and about the same in the second half. Is it worth 7 more
> notes?
>
> Whereas 12 and 19 just garble this example, 22 seems usable.
> It delivers about what I'd expect of 12 in a triadic
> example -- the underlying structure and meaning of the music
> is there, but it's fuzzy.
>
> 31 sounds good until the final trine exposes the beating
> of the 5th. Urg.
>
> I'd like to see 99 on this list.
>
> I do feel like the 21/16 at the end scotches the ability
> of my ear to use this as a JI approximation test, to some
> extent. It's like I have to come back inside for protection
> for a second... I know it's coming, so I can't be fully
> out there listening.
>
> I would be very interested to hear others' reactions.
>
>

"Aaron K. Johnson" <aaron@...> wrote:
>
> I haven't listened tothe page yet, but---
>
> re:53 vs. 46, wouldn't there be some advantage to having
> a prime numbered n-tet (e.g. complete circulation)

12-ET is the opposite of complete circlation, and it seems
to do pretty well.

> Regarding numbers of notes, when they are that high to begin
> with, piling a few extra doesn't make a difference, except
> from the point of view of live acoustic instuments with fixed
> pitch set (guitar frets, keyboard keys). electronics and
> trombone and voice, and so on, pose no limit, and I gather
> that these edos are just theory at that point to such
> instruments, unless you are performing with a synth backing
> track like Toby Twining does.

I think they will be theory to free instruments the same way
12-ET is theory to free instruments today, yes.

As for number of notes, I think a jump like 31 vs. 46 makes
a big difference in terms of keyboard tractability, whether
electronic or acoustic. The hands are only so large, and
buttons can only be made so small before fingers hit two.

Another very important consideration is the musical effects
of temperament. 22 can do things that 46 cannot, because it
is a *less accurate* temperament. There is a tradeoff
between accuracy and the kinds of subscales you can support.
For example, for meantone, 31 is about it. You can argue
that 55 or 69 are slightly better meantone tunings, but
basically, if you want to temper out 81/80, you can only
get so accurate.

> 53 on acoustic instruments really deosn't exist---it just
> becomes adaptive JI, let's face it, with all the error in
> cents that acoustic instruments will bring to it.

That's my opinion (though it is disputed by some).
Nevertheless, the *conceptual* differences between
these ETs are important.

-Carl

Carl Lumma schrieb:

> Generally it is a good setup for the kind of paper I suggest
> if someone who knows the maqamat but not periodicity blocks
> supplies a list of maqamat in just intonation. And then
> the other person attempts to describe them in terms of
> periodicity blocks. :) But the maqamat are mysterious
> things, and you may have an easier time learning about
> periodicity blocks than I will about maqamat.

Uh ... It will be interesting ro see what Oz will make out of this suggestion, but given the high primes he uses at least in naming his pitches I expect a multi-dimensional sea urchin rather than anything resembling a "block". But maybe for a single maqam, the picture is entirely different and may suggest a small number of "easy" ETs for the most common ones.

klaus

I can glimpse what periodicity blocks are. But, how am I supposed to find 1
9/8 5/4 4/3 3/2 27/16 15/8 in a periodicity block?

Oz.

----- Original Message -----
From: "Carl Lumma" <carl@lumma.org>
To: <tuning@yahoogroups.com>
Sent: 17 Eyl�l 2007 Pazartesi 19:59
Subject: [tuning] Re: 46-equal

> > > Now, back to theory. Has it ever been suggested to you that
> > > you try to describe the maqamat as periodicity blocks? I
> > > think that would be a very interesting paper.
> >
> > What are periodicity blocks in layman's terms?
>
> They are scales in just intonation. Actually they are more
> like scale archetypes, as usually a number of different but
> related scales can be said to come from the same periodicity
> block. They are also archetypes of temperaments in a way,
> as each periodicity block suggests by its nature ways it may
> be tempered. In my view, they are the foundation of the
> theory of regular temperaments.
>
> What they actually are: sections ("blocks") of a JI lattice
> that are bounded by commas (small intervals). By bounded,
> we mean these intervals do not occur between any two notes
> in the block. For example, 81/80 does not occur in the
> diatonic scale 1/1 9/8/ 5/4 4/3 3/2 5/3 15/8 2/1. However,
> if we add 10/9, which is adjacent to 5/3 and 4/3 on the
> lattice, then 9/8 - 10/9 = 81/80, and 81/80 will occur as
> a scale step ("second") in our scale.
>
> Not only are the blocks bounded in this way, it is usually
> the case that no seconds of the scale are smaller than these
> bounding intervals. If the diatonic scale had an interval
> smaller than 81/80, why not include 81/80 too?
>
> Scala actually has built-in tools for dealing with periodicity
> blocks, but I haven't tried to learn how to use them yet.
>
> For more information, consult Paul Erlich's excellent
> tutorial:
> http://www.tonalsoft.com/enc/f/fokker-gentle-1.aspx
>
> Generally it is a good setup for the kind of paper I suggest
> if someone who knows the maqamat but not periodicity blocks
> supplies a list of maqamat in just intonation. And then
> the other person attempts to describe them in terms of
> periodicity blocks. :) But the maqamat are mysterious
> things, and you may have an easier time learning about
> periodicity blocks than I will about maqamat.
>
> -Carl
>

I concur with Klaus. Still, there might be room for these periodicity
blocks.

Oz.

----- Original Message -----
From: "Klaus Schmirler" <KSchmir@online.de>
To: <tuning@yahoogroups.com>
Sent: 17 Eyl�l 2007 Pazartesi 22:17
Subject: Re: [tuning] Re: 46-equal

> Carl Lumma schrieb:
>
> > Generally it is a good setup for the kind of paper I suggest
> > if someone who knows the maqamat but not periodicity blocks
> > supplies a list of maqamat in just intonation. And then
> > the other person attempts to describe them in terms of
> > periodicity blocks. :) But the maqamat are mysterious
> > things, and you may have an easier time learning about
> > periodicity blocks than I will about maqamat.
>
> Uh ... It will be interesting ro see what Oz will make out of this
> suggestion, but given the high primes he uses at least in naming his
> pitches I expect a multi-dimensional sea urchin rather than anything
> resembling a "block". But maybe for a single maqam, the picture is
> entirely different and may suggest a small number of "easy" ETs for
> the most common ones.
>
> klaus
>

Ozan Yarman schrieb:
> I can glimpse what periodicity blocks are. But, how am I supposed to find 1
> 9/8 5/4 4/3 3/2 27/16 15/8 in a periodicity block?

By ordering them:

/25/16/
/10/9/ /5/3/ 5/4 15/8 /45/32/
/32/27/ /16/9/ 4/3 1/1 3/2 9/8 27/16 /81/64/

The numbers that are hopefully in italics are in the tone space but outside the (central) periodicity block.

klaus

> > Oz.
> > ----- Original Message -----
> From: "Carl Lumma" <carl@lumma.org>
> To: <tuning@yahoogroups.com>
> Sent: 17 Eyl�l 2007 Pazartesi 19:59
> Subject: [tuning] Re: 46-equal
> > >>>> Now, back to theory. Has it ever been suggested to you that
>>>> you try to describe the maqamat as periodicity blocks? I
>>>> think that would be a very interesting paper.
>>> What are periodicity blocks in layman's terms?
>> They are scales in just intonation. Actually they are more
>> like scale archetypes, as usually a number of different but
>> related scales can be said to come from the same periodicity
>> block. They are also archetypes of temperaments in a way,
>> as each periodicity block suggests by its nature ways it may
>> be tempered. In my view, they are the foundation of the
>> theory of regular temperaments.
>>
>> What they actually are: sections ("blocks") of a JI lattice
>> that are bounded by commas (small intervals). By bounded,
>> we mean these intervals do not occur between any two notes
>> in the block. For example, 81/80 does not occur in the
>> diatonic scale 1/1 9/8/ 5/4 4/3 3/2 5/3 15/8 2/1. However,
>> if we add 10/9, which is adjacent to 5/3 and 4/3 on the
>> lattice, then 9/8 - 10/9 = 81/80, and 81/80 will occur as
>> a scale step ("second") in our scale.
>>
>> Not only are the blocks bounded in this way, it is usually
>> the case that no seconds of the scale are smaller than these
>> bounding intervals. If the diatonic scale had an interval
>> smaller than 81/80, why not include 81/80 too?
>>
>> Scala actually has built-in tools for dealing with periodicity
>> blocks, but I haven't tried to learn how to use them yet.
>>
>> For more information, consult Paul Erlich's excellent
>> tutorial:
>> http://www.tonalsoft.com/enc/f/fokker-gentle-1.aspx
>>
>> Generally it is a good setup for the kind of paper I suggest
>> if someone who knows the maqamat but not periodicity blocks
>> supplies a list of maqamat in just intonation. And then
>> the other person attempts to describe them in terms of
>> periodicity blocks. :) But the maqamat are mysterious
>> things, and you may have an easier time learning about
>> periodicity blocks than I will about maqamat.
>>
>> -Carl
>>
> > > > You can configure your subscription by sending an empty email to one
> of these addresses (from the address at which you receive the list):
> tuning-subscribe@yahoogroups.com - join the tuning group.
> tuning-unsubscribe@yahoogroups.com - leave the group.
> tuning-nomail@yahoogroups.com - turn off mail from the group.
> tuning-digest@yahoogroups.com - set group to send daily digests.
> tuning-normal@yahoogroups.com - set group to send individual emails.
> tuning-help@yahoogroups.com - receive general help information.
> > Yahoo! Groups Links
> > >

Hi klaus,

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

> Uh ... It will be interesting ro see what Oz will
> make out of this suggestion, but given the high primes
> he uses at least in naming his pitches I expect a
> multi-dimensional sea urchin rather than anything
> resembling a "block".

Wow, that description really struck a chord of recognition
with me!

By any chance, have you tried Tonescape? Its Lattice
window has a "Geometry" menu which offers the
"Closed Curved" option for temperaments

When you start with a 1-dimensional JI Lattice and then
temper it, the "Closed Curved" option turns it into a
circle (big surprise, i know). And when you start with
a 2-dimensional JI Lattice, the "Closed Curved" version
becomes a helix if you temper out only one unison-vector,
and a torus if you temper out both unison-vectors.

But when you start with a JI Lattice of 3 or more
dimensions, the "Closed Curved" option gives you
something that still looks like a crystal structure
if you temper out one unison-vector, something that
looks like one helix embedded inside another if you
temper out two, and something that *really* looks
like a multi-dimensional sea-urchin if you temper
out 3 or more!

Gosh, my Encyclopedia is in sore need of updating:
now that Tonescape can make all these different Lattices,
it's time to start illustrating the tuning concepts
in the Encyclopedia with graphics of them.

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

Hi Oz,

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> I can glimpse what periodicity blocks are. But, how
> am I supposed to find 1 9/8 5/4 4/3 3/2 27/16 15/8 in
> a periodicity block?

Paul's tutorial explains how a 2-dimensional
periodicity-block, in the shape of a parallelogram,
can be created, on this page:

http://tonalsoft.com/enc/f/fokker-gentle-2.aspx

And the next installment, the "excursion", explains
how you can "chop off" corners of the parallelogram
and move them the distance of a unison-vector, to
another part of the lattice, to create shapes other
than the parallelogram:

http://tonalsoft.com/enc/f/fokker-gentle-ex.aspx

Now ... if you look again at the second graphic on
the "gentle-2" page, you'll see a central parallelogram
outlined in green which delimits the "usual" JI
diatonic major scale: 1/1 9/8 5/4 4/3 3/2 5/3 15/8.

Using the principles explained on the "excursion" page,
all you have to do to get your scale, is to insert
a new boundary-line running southwest-to-northeast,
which creates a triangle or wedge on the upper-left
side of the periodicity-block which contains only
the 5/3 ratio, and then move that wedge to the right
side of the periodicity-block so that it encompasses
27/16 instead of 5/3. Voila, there's your scale.

Note that the new shape of the periodicity-block
will still be a parallelogram, but now the left
and right borders will run southwest-to-northeast
instead of southeast-to-northwest as in the original.
Here's an ASCII diagram of it:

(if viewing on the stupid Yahoo web interface, click
the "Option" link and then "Use Fixed Width Font".)

. . . . . . .
. . . . . . .
| | | * | | | |
| | | * | | | |
| | | | * | | |
| | | | * | | |
...--40/27---10/9-----5/3--*--5/4----15/8*---45/32--135/128-...
| | | | | * | |
| | | | | | * |
| | | * | | | * |
| | | | | | | *
| | | | | | | *
| | * | | | |
| | | | | | |
...--32/27---16/9*----4/3-----1/1-----3/2-----9/8----27/16-*...
| | * | | | | |
| | | * | | | |
| | | * | | | | *
| | | | * | | |
| | | | * | | |
| | | | | * | *
| | | | | * | |
...-256/135--64/45---16/15----8/5-----6/5-----9/5*---27/20--...
| | | | | | * |
. . . . . . .
. . . . . . .

See how the new parallelogram (delimited by asterisks)
contains only the ratios you specified?

Paul's "excursion" goes into more sophisticated
territory than this, because he moves *two* wedges,
changing the parallelogram into a hexagon.

None of this matters. The point of a periodicity-block
is that any ratio outside the periodicity-block is
considered to be equivalent to a ratio inside the
block which lies at the distance of a unison-vector
from the ratio outside the block. Thus, in this case,
5/3 and 27/16 are considered to be the same note.

What's going on here is obvious in the case of temperaments,
because in a meantone temperament (for example), 5/3 and
27/16 really *are* the same note. It might not seem to make
as much sense for JI, but my opinion is that the
periodicity-block concept is important for our conception
of scales and other tuning structures. Some JI scales will
recognize the syntonic-comma and others will not -- those
that do not will follow the principles embedded in the
example periodicity-blocks we're discussing here.

Some time ago, i had updated my Encyclopedia page about
"pythagorean", and included new graphics near the beginning
which illustrate how the age-old pythagorean scales are
derived from unison-vectors and periodicity-blocks:

http://tonalsoft.com/enc/p/pythagorean.aspx

(In this case, they're not really blocks, but just
line-segments, but "periodicity-block" has been taken,
by most tuning theorists who bother with the concept, to
be a generic term representing any closed periodic shape.)

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

> > Generally it is a good setup for the kind of paper I suggest
> > if someone who knows the maqamat but not periodicity blocks
> > supplies a list of maqamat in just intonation. And then
> > the other person attempts to describe them in terms of
> > periodicity blocks. :) But the maqamat are mysterious
> > things, and you may have an easier time learning about
> > periodicity blocks than I will about maqamat.
>
> Uh ... It will be interesting ro see what Oz will make out of
> this suggestion, but given the high primes he uses at least in
> naming his pitches I expect a multi-dimensional sea urchin
> rather than anything resembling a "block". But maybe for a
> single maqam, the picture is entirely different and may suggest
> a small number of "easy" ETs for the most common ones.
>
> klaus

Yes, there's a possibility of that. Perhaps Ozan will
agree to work in the 5-limit for a time.

On the other hand, perhaps you are right. Perhaps maqamat
are ultimately Pythagorean with altered notes, not
periodicity blocks (well, they would be 3-limit blocks with
altered notes). After all, periodicity blocks more
naturally arise in polyphonic settings.

-Carl

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> I can glimpse what periodicity blocks are. But, how am I supposed
> to find 1 9/8 5/4 4/3 3/2 27/16 15/8 in a periodicity block?

If you plot these pitches on a 5-limit triangular lattice, you
will see they form a convex lump which is bounded by 81/80
and 135/128.

-Carl

I do not think it possible to constrain myself with 5-limit, however I am
reconsidering 46-equal for better 5-limit support.

Oz.

----- Original Message -----
From: "Carl Lumma" <carl@lumma.org>
To: <tuning@yahoogroups.com>
Sent: 18 Eyl�l 2007 Sal� 21:50
Subject: [tuning] Re: 46-equal

> > > Generally it is a good setup for the kind of paper I suggest
> > > if someone who knows the maqamat but not periodicity blocks
> > > supplies a list of maqamat in just intonation. And then
> > > the other person attempts to describe them in terms of
> > > periodicity blocks. :) But the maqamat are mysterious
> > > things, and you may have an easier time learning about
> > > periodicity blocks than I will about maqamat.
> >
> > Uh ... It will be interesting ro see what Oz will make out of
> > this suggestion, but given the high primes he uses at least in
> > naming his pitches I expect a multi-dimensional sea urchin
> > rather than anything resembling a "block". But maybe for a
> > single maqam, the picture is entirely different and may suggest
> > a small number of "easy" ETs for the most common ones.
> >
> > klaus
>
> Yes, there's a possibility of that. Perhaps Ozan will
> agree to work in the 5-limit for a time.
>
> On the other hand, perhaps you are right. Perhaps maqamat
> are ultimately Pythagorean with altered notes, not
> periodicity blocks (well, they would be 3-limit blocks with
> altered notes). After all, periodicity blocks more
> naturally arise in polyphonic settings.
>
> -Carl
>
>

Aaron K. Johnson schrieb:
> I haven't listened tothe page yet, but---

I have listened, and I think it deserves a link from some prominent place like the group title page or at the beginning of "the FAQ", should it ever exist.

I was surprised - disappointed, rather - that 19 sounded almost as bad 12. 31 was smoother, but still had what I take to be a bias for the 5-limit.

> > > Regarding numbers of notes, when they are that high to begin with, > piling a few extra doesn't make a difference, except from the point of > view of live acoustic instuments with fixed pitch set (guitar frets, > keyboard keys). electronics and trombone and voice, and so on, pose no > limit, and I gather that these edos are just theory at that point to > such instruments, unless you are performing with a synth backing track > like Toby Twining does.
> > One of the attractions of 19 in spite of its less than stellar > approximations is that it's easily done with current resources, and its > different enough in a very obvioous way.
> > 53 on acoustic instruments really deosn't exist---it just becomes > adaptive JI, let's face it, with all the error in cents that acoustic > instruments will bring to it. I would say that without synth backing > tracks, 31 or 34 might be the upper limit of what acoustic instruments > could do with any accuracy or meaning.

I was able to teach myself 29. For months now, I am trying 67 and can't get anywhere. But I think the presence of starting intervals makes a huge difference here: I don't see such an interval with 67, but I could divide the 9/8 into five parts for 29 (omitting refinements). 53, certainly if it's permissible to arrive at 55 or 54, looks feasible to me: dividing 9/8 into 9 _and_ dividing 10/9 into 8 is indeed working with large numbers, but you can break up the divisions (three and another three), and the good major and minor tone and third will help to correct errors. If Leopold Mozart taught it ... What I wanted to say is that high ETs arent't necessarily more difficult than lower ones.

klaus

This has been my experience with 19 and 31. The latter though has some good 7th one can use instead of 5s but still he has what you say

Posted by: "Klaus Schmirler"

I was surprised - disappointed, rather - that 19 sounded almost as bad
12. 31 was smoother, but still had what I take to be a bias for the
5-limit.
--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/index.html>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles

monz schrieb:
> Hi klaus,
> > --- In tuning@yahoogroups.com, Klaus Schmirler <KSchmir@...> wrote:
> >> Uh ... It will be interesting ro see what Oz will
>> make out of this suggestion, but given the high primes
>> he uses at least in naming his pitches I expect a
>> multi-dimensional sea urchin rather than anything >> resembling a "block".
> > > Wow, that description really struck a chord of recognition
> with me!
> > By any chance, have you tried Tonescape? Its Lattice
> window has a "Geometry" menu which offers the > "Closed Curved" option for temperaments

I haven't, but before posting I tried to find the word you used where the prime dimensions were represented, 2-D, by spikes of different lengths and angles. It was easy to interpret them 3-D. It wasn't "crystal", or was it?

klaus

> > When you start with a 1-dimensional JI Lattice and then
> temper it, the "Closed Curved" option turns it into a
> circle (big surprise, i know). And when you start with
> a 2-dimensional JI Lattice, the "Closed Curved" version
> becomes a helix if you temper out only one unison-vector,
> and a torus if you temper out both unison-vectors.
> > But when you start with a JI Lattice of 3 or more
> dimensions, the "Closed Curved" option gives you
> something that still looks like a crystal structure
> if you temper out one unison-vector, something that
> looks like one helix embedded inside another if you
> temper out two, and something that *really* looks
> like a multi-dimensional sea-urchin if you temper
> out 3 or more!
> > Gosh, my Encyclopedia is in sore need of updating:
> now that Tonescape can make all these different Lattices,
> it's time to start illustrating the tuning concepts
> in the Encyclopedia with graphics of them.

Hi klaus,

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

> I haven't, but before posting I tried to find the word
> you used where the prime dimensions were represented,
> 2-D, by spikes of different lengths and angles. It was
> easy to interpret them 3-D. It wasn't "crystal", or was it?

I've often compared harmonic lattice-diagrams to crystal
structures ... but i think what you're referring to is
something that i always called the "Monzo lattice formula",
as explained here:

http://tonalsoft.com/monzo/lattices/lattices.htm

I don't think i ever had a single word for that.

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

Thank you for providing these links monz. I have read as much as I could.
But I wonder, why "unison vector", and not "enharmonic vector"?

Oz.

----- Original Message -----
From: "monz" <monz@tonalsoft.com>
To: <tuning@yahoogroups.com>
Sent: 17 Eyl�l 2007 Pazartesi 23:35
Subject: [tuning] periodicity-blocks (was: 46-equal)

> Hi Oz and Carl,
>
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> > What are periodicity blocks in layman's terms?
> >
> > <snip>
> >
> > What they actually are: sections ("blocks") of a JI lattice
> > that are bounded by commas (small intervals). By bounded,
> > we mean these intervals do not occur between any two notes
> > in the block. For example, 81/80 does not occur in the
> > diatonic scale 1/1 9/8/ 5/4 4/3 3/2 5/3 15/8 2/1. However,
> > if we add 10/9, which is adjacent to 5/3 and 4/3 on the
> > lattice, then 9/8 - 10/9 = 81/80, and 81/80 will occur as
> > a scale step ("second") in our scale.
>
>
> Along with Paul's excellent tutorial, i also have
> entries for "periodicity-block" and the related
> "unison-vector" in the Encyclopedia:
>
> http://tonalsoft.com/enc/p/periodicity-block.aspx
>
> http://tonalsoft.com/enc/u/unison-vector.aspx
>
>
> Tonescape utilizes the periodicity-block concept
> at its core. Its "Tonespace" files in fact simply
> *are* periodicity-blocks, from which "Tuning" files
> are generated.
>
>
> A periodicity-block is essentially a unit-cell,
> which tiles the whole tonespace, similar to the
> way that in a crystal structure there is a unit-cell
> which is repeated over and over again in all three
> dimensions. However, a periodicity-block can have
> as many boundaries as there are dimensions in the
> tonespace, it's not restricted to 3.
>
> Near the bottom of my 12-edo page, i have a graphic
> showing a bingo-card lattice of 12-edo with many
> of the different possible periodicity-blocks outlined
> in green:
>
> http://tonalsoft.com/enc/number/12edo.aspx
>
>
> Note that in order to create a periodicity-block,
> the number of unison-vectors must equal the number
> of dimensions in the tonespace. If there are less
> unison-vectors (but at least one), there will still
> be periodicity in the tonespace, but some dimensions
> will be open-ended (infinite) -- thus, for example:
>
> In classic 5-limit meantone, the full tonespace has
> 3 dimensions, representing prime-factors 2, 3, and 5.
> Assuming the normal identity-interval of ratio 2/1,
> we may ignore 2 and use a 2-dimensional prime-space
> representing factors 3 and 5. If there is only one
> unison-vector and it is the syntonic-comma, whose
> ratio is 81/80 and 2,3,5-monzo is [-4 4, -1>, then
> one boundary of periodicity will be a vector which
> can be expressed in classic vector form as <4i-j>
> where i is the 3-axis and j is the 5-axis -- thus
> the vector extends to a point which is 4 positive
> steps along the 3-axis and one negative step along
> the 5-axis. This single unison-vector thus creates
> a slightly angled "periodicity-slice" across the
> 2-dimensional prime-space. If 3 is the horizontal
> axis and 5 is the vertical, the periodicity-slice
> will go from the upper-right to the lower-left, and
> all pitches further to the left or right are considered
> equivalent to those within the slice. But the slice
> will extend infinitely on the top and bottom.
>
> If a second unison-vector is used, then a periodicity-block
> will be formed, with the top and bottom boundaries being
> the vector of the syntonic-comma itself, and the left
> and right boundaries being the second unison-vector.
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>
>

That is all very well, but what about a Halberstadt-like keyboard as in the
SCALA chromatic clavier?

Oz.

----- Original Message -----
From: "Herman Miller" <hmiller@IO.COM>
To: <tuning@yahoogroups.com>
Sent: 18 Eyl�l 2007 Sal� 4:45
Subject: Re: [tuning] Re: 46-equal

> Ozan Yarman wrote:
> >
> > ----- Original Message -----
> > From: "hstraub64" <hstraub64@telesonique.net>
> > To: <tuning@yahoogroups.com>
> > Sent: 17 Eyl�l 2007 Pazartesi 23:56
> > Subject: [tuning] Re: 46-equal
> >
> >
> >> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> >>> On second thought, I find 46-equal too radical a temperament for the
> >>> traditionalists who will likely opt for something more
> >>> "Pythagorean". Fortunately, 41-EDO saves the day.
> >>>
> >> Well, from my point of view: the less tones, the better :-)
> >> Anyway, if you want to play all notes on standard pianos, you would
> >> need four in any case, so it's not that big a difference... What would
> >> good keyboard layouts be?
> >>
> >
> >
> > I wouldn't know for sure. Let us design a 41-EDO keyboard.
>
> http://www.anaphoria.com/xen3b.PDF
>
> See page 6, "Positive Linear Mapping Template to modulus 41", and page
> 10, "Keyboarding Genus 41". Lots of good info packed into these diagrams.
>
>

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>

> But I wonder, why "unison vector", and not "enharmonic vector"?

That's been a princess' pea for me since I first discovered
this community, but as I already get scoffed upon for proposing
such wild and stupid ideas like "3:1 is not the same frequency as
3:2", I didn't bother to mention it. Glad you brought it up.

Really, thanks!

Oz.

----- Original Message -----
From: "Cameron Bobro" <misterbobro@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 19 Eyl�l 2007 �ar�amba 15:56
Subject: [tuning] Re: periodicity-blocks (was: 46-equal)

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> >
>
> > But I wonder, why "unison vector", and not "enharmonic vector"?
>
>
> That's been a princess' pea for me since I first discovered
> this community, but as I already get scoffed upon for proposing
> such wild and stupid ideas like "3:1 is not the same frequency as
> 3:2", I didn't bother to mention it. Glad you brought it up.
>
>
>

You have outdone yourself with that ASCII graphic. Excellent explanations.
Thank you very much.

Oz.

----- Original Message -----
From: "monz" <monz@tonalsoft.com>
To: <tuning@yahoogroups.com>
Sent: 18 Eyl�l 2007 Sal� 19:33
Subject: [tuning] periodicity-blocks (was: 46-equal)

> Hi Oz,
>
>
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> >
> > I can glimpse what periodicity blocks are. But, how
> > am I supposed to find 1 9/8 5/4 4/3 3/2 27/16 15/8 in
> > a periodicity block?
>
>
> Paul's tutorial explains how a 2-dimensional
> periodicity-block, in the shape of a parallelogram,
> can be created, on this page:
>
> http://tonalsoft.com/enc/f/fokker-gentle-2.aspx
>
>
> And the next installment, the "excursion", explains
> how you can "chop off" corners of the parallelogram
> and move them the distance of a unison-vector, to
> another part of the lattice, to create shapes other
> than the parallelogram:
>
> http://tonalsoft.com/enc/f/fokker-gentle-ex.aspx
>
>
> Now ... if you look again at the second graphic on
> the "gentle-2" page, you'll see a central parallelogram
> outlined in green which delimits the "usual" JI
> diatonic major scale: 1/1 9/8 5/4 4/3 3/2 5/3 15/8.
>
> Using the principles explained on the "excursion" page,
> all you have to do to get your scale, is to insert
> a new boundary-line running southwest-to-northeast,
> which creates a triangle or wedge on the upper-left
> side of the periodicity-block which contains only
> the 5/3 ratio, and then move that wedge to the right
> side of the periodicity-block so that it encompasses
> 27/16 instead of 5/3. Voila, there's your scale.
>
> Note that the new shape of the periodicity-block
> will still be a parallelogram, but now the left
> and right borders will run southwest-to-northeast
> instead of southeast-to-northwest as in the original.
> Here's an ASCII diagram of it:
>
> (if viewing on the stupid Yahoo web interface, click
> the "Option" link and then "Use Fixed Width Font".)
>
> . . . . . . .
> . . . . . . .
> | | | * | | | |
> | | | * | | | |
> | | | | * | | |
> | | | | * | | |
> ...--40/27---10/9-----5/3--*--5/4----15/8*---45/32--135/128-...
> | | | | | * | |
> | | | | | | * |
> | | | * | | | * |
> | | | | | | | *
> | | | | | | | *
> | | * | | | |
> | | | | | | |
> ...--32/27---16/9*----4/3-----1/1-----3/2-----9/8----27/16-*...
> | | * | | | | |
> | | | * | | | |
> | | | * | | | | *
> | | | | * | | |
> | | | | * | | |
> | | | | | * | *
> | | | | | * | |
> ...-256/135--64/45---16/15----8/5-----6/5-----9/5*---27/20--...
> | | | | | | * |
> . . . . . . .
> . . . . . . .
>
> See how the new parallelogram (delimited by asterisks)
> contains only the ratios you specified?
>
>
> Paul's "excursion" goes into more sophisticated
> territory than this, because he moves *two* wedges,
> changing the parallelogram into a hexagon.
>
> None of this matters. The point of a periodicity-block
> is that any ratio outside the periodicity-block is
> considered to be equivalent to a ratio inside the
> block which lies at the distance of a unison-vector
> from the ratio outside the block. Thus, in this case,
> 5/3 and 27/16 are considered to be the same note.
>
> What's going on here is obvious in the case of temperaments,
> because in a meantone temperament (for example), 5/3 and
> 27/16 really *are* the same note. It might not seem to make
> as much sense for JI, but my opinion is that the
> periodicity-block concept is important for our conception
> of scales and other tuning structures. Some JI scales will
> recognize the syntonic-comma and others will not -- those
> that do not will follow the principles embedded in the
> example periodicity-blocks we're discussing here.
>
> Some time ago, i had updated my Encyclopedia page about
> "pythagorean", and included new graphics near the beginning
> which illustrate how the age-old pythagorean scales are
> derived from unison-vectors and periodicity-blocks:
>
> http://tonalsoft.com/enc/p/pythagorean.aspx
>
> (In this case, they're not really blocks, but just
> line-segments, but "periodicity-block" has been taken,
> by most tuning theorists who bother with the concept, to
> be a generic term representing any closed periodic shape.)
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>
>

You mention "teaching yourself 29[edo]"...what...singing? violin? trombone?

just curious...

-A.

Klaus Schmirler wrote:
> Aaron K. Johnson schrieb:
> >> I haven't listened tothe page yet, but---
>> >
> I have listened, and I think it deserves a link from some prominent > place like the group title page or at the beginning of "the FAQ", > should it ever exist.
>
>
> I was surprised - disappointed, rather - that 19 sounded almost as bad > 12. 31 was smoother, but still had what I take to be a bias for the > 5-limit.
>
> >
> >> Regarding numbers of notes, when they are that high to begin with, >> piling a few extra doesn't make a difference, except from the point of >> view of live acoustic instuments with fixed pitch set (guitar frets, >> keyboard keys). electronics and trombone and voice, and so on, pose no >> limit, and I gather that these edos are just theory at that point to >> such instruments, unless you are performing with a synth backing track >> like Toby Twining does.
>>
>> One of the attractions of 19 in spite of its less than stellar >> approximations is that it's easily done with current resources, and its >> different enough in a very obvioous way.
>>
>> 53 on acoustic instruments really deosn't exist---it just becomes >> adaptive JI, let's face it, with all the error in cents that acoustic >> instruments will bring to it. I would say that without synth backing >> tracks, 31 or 34 might be the upper limit of what acoustic instruments >> could do with any accuracy or meaning.
>> >
> I was able to teach myself 29. For months now, I am trying 67 and > can't get anywhere. But I think the presence of starting intervals > makes a huge difference here: I don't see such an interval with 67, > but I could divide the 9/8 into five parts for 29 (omitting > refinements). 53, certainly if it's permissible to arrive at 55 or 54, > looks feasible to me: dividing 9/8 into 9 _and_ dividing 10/9 into 8 > is indeed working with large numbers, but you can break up the > divisions (three and another three), and the good major and minor tone > and third will help to correct errors. If Leopold Mozart taught it ... > What I wanted to say is that high ETs arent't necessarily more > difficult than lower ones.
>
> klaus
>
>

monz schrieb:
> Hi klaus,
> > > --- In tuning@yahoogroups.com, Klaus Schmirler <KSchmir@...> wrote:
> >> I haven't, but before posting I tried to find the word
>> you used where the prime dimensions were represented,
>> 2-D, by spikes of different lengths and angles. It was
>> easy to interpret them 3-D. It wasn't "crystal", or was it?
> > > I've often compared harmonic lattice-diagrams to crystal
> structures ... but i think what you're referring to is
> something that i always called the "Monzo lattice formula",
> as explained here:
> > http://tonalsoft.com/monzo/lattices/lattices.htm

Yes. The spiky thing near the bottom, intertwangled with one or two other such things of an even higher prime limit -- would be a random selection of tones of Oz's tuning. Again, actual maqam scales may present a very orderly picture, just with prime dimensions of 3, 13, and 23 instead of 3, 5 and 7.

klaus

> > I don't think i ever had a single word for that.

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@> wrote:
> >
>
> > But I wonder, why "unison vector", and not "enharmonic vector"?
>
>
> That's been a princess' pea for me since I first discovered
> this community, but as I already get scoffed upon for proposing
> such wild and stupid ideas like "3:1 is not the same frequency as
> 3:2", I didn't bother to mention it. Glad you brought it up.
>

This is actually an easy one, methinks: "enharmonic" has two
meanings, the first (and one still highly relevant both to the
contemporary tuning community and to musicologists with the recent
explosion in work in classical Greek music) referring to the
classical tetrachordal genus in which a semitone pyknon is divided
into two parts, and the second, as in "enharmonic equivalence", in
which two notated pitched are mapped onto the same physical pitch.
"Unison" is perfectly clear for the second sense, and "enharmonic"
can be reserved for the first, for which it has precedence and for
which there is really no alternative.

djw

Hi Oz,

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> You have outdone yourself with that ASCII graphic.
> Excellent explanations.
> Thank you very much.

Paul Erlich really deserves the credit for the ASCII
graphic. Even tho i made the one in my post, all i
did was copy exactly the format of the ones he made
for his "Gentle Introduction". The only thing i had
to do differently was place a few of the asterisks
in different places.

But anyway, you're welcome. I really do wish (and so
many others) could get Tonescape running, because it
uses all these concepts.

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

Hi Cameron and Oz,

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:
>
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@> wrote:
> >
>
> > But I wonder, why "unison vector", and not
> > "enharmonic vector"?
>
>
> That's been a princess' pea for me since I first
> discovered this community, but as I already get
> scoffed upon for proposing such wild and stupid
> ideas like "3:1 is not the same frequency as
> 3:2", I didn't bother to mention it. Glad you
> brought it up.

There was actually a huge debate about "unison-vector"
on this list a few years back. You should be able to
find it by searching the archives.

Oz, i can understand your problem with it, given the
argument you and i had recently about the "unison"
being an interval (as i hold) or lack thereof
(as you hold).

The terminology comes from Fokker, and makes perfect
sense to me. The whole idea is that the vector which
connects a pitch inside the periodicity-block with
one outside of it represents the concept that both
pitches are a unison.

"Enharmonic" had a much older meaning, from ancient
Greek times. The word "enharmonic" actually means
"properly attuned", but as a genus it implies a
scale with some notes which are approximately a
quarter-tone apart.

The "enharmonic-diesis" arose within the context of
5-limit JI and meantone, and represents the interval
between pairs of notes notated as, for example, Ab
and G#. In the case of JI and 1/4-comma meantone it
is the same: ~41.06 cents. At the wide end of the
meantone spectrum, it is ~62.57 in 1/3-comma meantone
(and similar for 19-edo), and at the other end it
is 0 cents for 12-edo. In the commonly-used 1/6-comma
meantone it is ~19.55 cents (and similar for 55-edo).

So because the enharmonic-diesis vanishes in 12-edo,
folks today are used to thinking of "enharmonic" as
meaning "identical in pitch", but historically that's
not what it meant.

I think "unison-vector" encapsulates perfectly the
mechanics of periodicity-block theory.

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

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> Thank you for providing these links monz. I have read as much as
> I could. But I wonder, why "unison vector", and not "enharmonic
> vector"?
>
> Oz.

That's an interesting suggestion. If you read The Forms of
Tonality, where all of this is explained with pretty diagrams,
it uses the terms *commatic unison vector* and *chromatic
unison vector*....

-Carl

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:
>
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@> wrote:
> >
>
> > But I wonder, why "unison vector", and not "enharmonic vector"?
>
> That's been a princess' pea for me since I first discovered
> this community, but as I already get scoffed upon for proposing
> such wild and stupid ideas like "3:1 is not the same frequency as
> 3:2", I didn't bother to mention it. Glad you brought it up.

Neither 3:1 or 3:2 are frequencies. If you're careful about
it, you might convey that 3/1 and 3/2 are (different) pitches.
(Still not frequencies.)

-Carl

The plot thickens.

----- Original Message -----
From: "Carl Lumma" <carl@lumma.org>
To: <tuning@yahoogroups.com>
Sent: 19 Eyl�l 2007 �ar�amba 20:59
Subject: [tuning] Re: periodicity-blocks (was: 46-equal)

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> > Thank you for providing these links monz. I have read as much as
> > I could. But I wonder, why "unison vector", and not "enharmonic
> > vector"?
> >
> > Oz.
>
> That's an interesting suggestion. If you read The Forms of
> Tonality, where all of this is explained with pretty diagrams,
> it uses the terms *commatic unison vector* and *chromatic
> unison vector*....
>
> -Carl
>
>

You are most kind to divulge these matters monz.

Oz.

----- Original Message -----
From: "monz" <monz@tonalsoft.com>
To: <tuning@yahoogroups.com>
Sent: 19 Eyl�l 2007 �ar�amba 20:55
Subject: [tuning] "unison" vs. "enharmonic" (was: periodicity-blocks)

> Hi Cameron and Oz,
>
>
> --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:
> >
> > --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@> wrote:
> > >
> >
> > > But I wonder, why "unison vector", and not
> > > "enharmonic vector"?
> >
> >
> > That's been a princess' pea for me since I first
> > discovered this community, but as I already get
> > scoffed upon for proposing such wild and stupid
> > ideas like "3:1 is not the same frequency as
> > 3:2", I didn't bother to mention it. Glad you
> > brought it up.
>
>
> There was actually a huge debate about "unison-vector"
> on this list a few years back. You should be able to
> find it by searching the archives.
>
> Oz, i can understand your problem with it, given the
> argument you and i had recently about the "unison"
> being an interval (as i hold) or lack thereof
> (as you hold).
>
> The terminology comes from Fokker, and makes perfect
> sense to me. The whole idea is that the vector which
> connects a pitch inside the periodicity-block with
> one outside of it represents the concept that both
> pitches are a unison.
>
> "Enharmonic" had a much older meaning, from ancient
> Greek times. The word "enharmonic" actually means
> "properly attuned", but as a genus it implies a
> scale with some notes which are approximately a
> quarter-tone apart.
>
> The "enharmonic-diesis" arose within the context of
> 5-limit JI and meantone, and represents the interval
> between pairs of notes notated as, for example, Ab
> and G#. In the case of JI and 1/4-comma meantone it
> is the same: ~41.06 cents. At the wide end of the
> meantone spectrum, it is ~62.57 in 1/3-comma meantone
> (and similar for 19-edo), and at the other end it
> is 0 cents for 12-edo. In the commonly-used 1/6-comma
> meantone it is ~19.55 cents (and similar for 55-edo).
>
> So because the enharmonic-diesis vanishes in 12-edo,
> folks today are used to thinking of "enharmonic" as
> meaning "identical in pitch", but historically that's
> not what it meant.
>
> I think "unison-vector" encapsulates perfectly the
> mechanics of periodicity-block theory.
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>
>

Then kudos for Paul Erlich.

Yes, once I acquire a new laptop, I will explore Tonescape.

Oz.

----- Original Message -----
From: "monz" <monz@tonalsoft.com>
To: <tuning@yahoogroups.com>
Sent: 19 Eyl�l 2007 �ar�amba 18:58
Subject: [tuning] Re: periodicity-blocks (was: 46-equal)

> Hi Oz,
>
>
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> >
> > You have outdone yourself with that ASCII graphic.
> > Excellent explanations.
> > Thank you very much.
>
> Paul Erlich really deserves the credit for the ASCII
> graphic. Even tho i made the one in my post, all i
> did was copy exactly the format of the ones he made
> for his "Gentle Introduction". The only thing i had
> to do differently was place a few of the asterisks
> in different places.
>
> But anyway, you're welcome. I really do wish (and so
> many others) could get Tonescape running, because it
> uses all these concepts.
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>
>
>

Carl Lumma wrote:
> --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:
> >> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@> wrote:
>> >>> But I wonder, why "unison vector", and not "enharmonic vector"?
>>> >> That's been a princess' pea for me since I first discovered
>> this community, but as I already get scoffed upon for proposing >> such wild and stupid ideas like "3:1 is not the same frequency as >> 3:2", I didn't bother to mention it. Glad you brought it up.
>> >
> Neither 3:1 or 3:2 are frequencies. If you're careful about
> it, you might convey that 3/1 and 3/2 are (different) pitches.
> (Still not frequencies.)
> They are neither frequencies nor pitches, but intervals, from where I sit.

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <aaron@...> wrote:

Hi, Aaron. I'm replying to both you and Carl in one swoop.

> Carl Lumma wrote:
> > George wrote...
> >
> >>> regarding 46-equal, anyone know of any mp3s/oggs anywhere
> >>> in this group or in the larger 'net?
> >>>
> >> How about a very short midi file (along with a score)
> >> that you can compare with a bunch of other divisions (as
> >> well as JI)?
> >>
> >> http://dkeenan.com/sagittal/exmp/index.htm
> >
> > I think I'd been to this page before, but man, it's so rock.
> >
> > After listening to the complete example in JI, the first beat
> > of 12 (only a triad!) nearly sent me through the roof.

Yeah, really shocking, isn't it? 8>O

> > To my ear, 46 outshines 41, most immediately audibly on the
> > fourth chord (min 7th). The difference isn't huge, but to
> > me is probably worth 5 notes.
> > 53 sounds better than 46 in the first half of the example,
> > and about the same in the second half. Is it worth 7 more
> > notes?

A lot depends on how much you care about how the pitches are
organized.

> > Whereas 12 and 19 just garble this example, 22 seems usable.
> > It delivers about what I'd expect of 12 in a triadic
> > example -- the underlying structure and meaning of the music
> > is there, but it's fuzzy.
> >
> > 31 sounds good until the final trine exposes the beating
> > of the 5th. Urg.

One glaring omission is 34. I made these files a few years ago,
before I really began to appreciate 34. I'll have to ask Dave K. to
add that one to the web page. In the meantime, you'll find it here:
/tuning/files/secor/examples/
for both the ET and my well-temperament. (All the midi examples on
Dave's website are there, plus even more.)

> > I'd like to see 99 on this list.

I didn't do 99 (not very good above the 9 limit), but you'll find 130
there.

> > I do feel like the 21/16 at the end scotches the ability
> > of my ear to use this as a JI approximation test, to some
> > extent. It's like I have to come back inside for protection
> > for a second... I know it's coming, so I can't be fully
> > out there listening.

For quick relief, try the two 34-tone files. :-)

> > I would be very interested to hear others' reactions.

[Exit Carl; enter Aaron:]

> I haven't listened tothe page yet, but---
>
> re:53 vs. 46, wouldn't there be some advantage to having a prime
> numbered n-tet (e.g. complete circulation)
>
> I suppose one can get off, musically on the "switching gears"
effect of
> 46=2x23, but what else is virtuous.

46 has some nice melodic properties and better consistency: 53 is
only 9-limit consistent, while 46 is 13-limit consistent (strictly
speaking, although its inconsistency at the 17-limit is so slight as
to be practically nonexistent).

Now if you had asked about 34, I'd have plenty to say. It's the best
5-limit division below 46, and it has many of the general
characteristics of 46 as well (wide 5ths, similar melodic properties,
17-limit capability). A big plus is that 34 gives you pajara (with,
arguably, the optimal pajara tuning via well-temperament). 34-ET
isn't quite 17-limit consistent (due to the 7th harmonic), however,
my 34-WT overcomes this deficiency in the best pajara keys. You also
get some other scale structures, such as keemun and sentinel (a non-5
scale subset of 17-ET I discovered a couple decades ago; I saw that
name for it in Scala, although I don't know where Manuel got it).
One big problem with 34 is that you can't map it on a Bosanquet
generalized keyboard -- however, not many people have one of those,
anyway (yet)!

> Regarding numbers of notes, when they are that high to begin with,
> piling a few extra doesn't make a difference, except from the point
of
> view of live acoustic instuments with fixed pitch set (guitar
frets,
> keyboard keys). electronics and trombone and voice, and so on, pose
no
> limit, and I gather that these edos are just theory at that point
to
> such instruments, unless you are performing with a synth backing
track
> like Toby Twining does.
>
> One of the attractions of 19 in spite of its less than stellar
> approximations is that it's easily done with current resources, and
its
> different enough in a very obvioous way.

IMO, well-temperaments give you more bang for the buck. Compare the
Exmp19p3.mid file at the above link with the ones for 19- and 31-
equal.

> 53 on acoustic instruments really deosn't exist---it just becomes
> adaptive JI, let's face it, with all the error in cents that
acoustic
> instruments will bring to it. I would say that without synth
backing
> tracks, 31 or 34 might be the upper limit of what acoustic
instruments
> could do with any accuracy or meaning.

Yes, that's pretty much what I thought back in the 1970's, although I
allowed that the extreme upper limit for acoustic instruments might
be 41. (It takes almost that many tones of the miracle temperament
to play my tuning example, which doesn't even modulate. And, yes,
there is a Stud-loco example, ExmpMrcl.mid, out there, using the
minimax generator.)

One thing I was after, back then, was a near-just 13-limit otonal
tuning that allowed more than a minimal amount of modulation with a
reasonable number of tones. As far as I knew it didn't exist, so I
created one (in 29 tones) that, in 6 different keys, is closer to JI
than any EDO below 130 and has some great melodic properties. A midi
example (Exmp29HT.mid) is out there (transposable to 4 other keys
without losing accuracy). You don't get the totally free modulation
of an ET, but you can modulate over all 29 keys if you're willing to
accept (and exploit) variable intonation. I also have a 17-tone
version of this tuning in 3 excellent keys (not a subset of the 29
tones, however), to which I tuned a 2-2/3-octave set of tubulongs;
it's not JI, but it's very close.

--George

> > Neither 3:1 or 3:2 are frequencies. If you're careful about
> > it, you might convey that 3/1 and 3/2 are (different) pitches.
> > (Still not frequencies.)
>
> They are neither frequencies nor pitches, but intervals, from
> where I sit.

3:1 and 3:2 are intervals, 3/1 and 3/2 are pitches, if you
observe one of the weaker list conventions.

-Carl

Hi Oz,

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> You are most kind to divulge these matters monz.
>
> Oz.
>
> ----- Original Message -----
> From: "monz" <monz@...>
> To: <tuning@yahoogroups.com>
> Sent: 19 Eylül 2007 Çarþamba 20:55
> Subject: [tuning] "unison" vs. "enharmonic" (was: periodicity-
blocks)

> > There was actually a huge debate about "unison-vector"
> > on this list a few years back. You should be able to
> > find it by searching the archives.
> >
> > Oz, i can understand your problem with it, given the
> > argument you and i had recently about the "unison"
> > being an interval (as i hold) or lack thereof
> > (as you hold).

Oops ... i got this wrong. In the argument we had a
couple of months ago, i maintained that "prime" is
certainly an interval and that "unison" would be a
good term to use to represent "lack of interval".

So anyway, this only strengthens what i feel is
good about Fokker's terminology. The unison-vector
is indeed meant to represent a "lack of interval"
in the tonespace.

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

Aaron K. Johnson schrieb:
> You mention "teaching yourself 29[edo]"...what...singing? violin? trombone?
> > just curious...
> > -A.

Trombone. I can't imagine dividing an interval by listening or gradings of muscle tension. The trombone also has the advantage of having the harmonic series built in for easy reference.

I'd like to hear more about the singing in EDOs experiment. Was that you? How did you start out?

klaus

Carl Lumma wrote:
>>> Neither 3:1 or 3:2 are frequencies. If you're careful about
>>> it, you might convey that 3/1 and 3/2 are (different) pitches.
>>> (Still not frequencies.)
>>> >> They are neither frequencies nor pitches, but intervals, from
>> where I sit.
>> >
> 3:1 and 3:2 are intervals, 3/1 and 3/2 are pitches, if you
> observe one of the weaker list conventions.
>
> Where was it, but I also read of a convention that 3:1 and 3:2 are melodic intervals, and 3/1 and 3/2 are harmonic (simultaneous) intervals....

IOW, the ':' means melody and the '/' means harmony.

-A.

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:

> > > After listening to the complete example in JI, the first beat
> > > of 12 (only a triad!) nearly sent me through the roof.
>
> Yeah, really shocking, isn't it? 8>O

The first three examples, including 12, won't play for me.

> > 3:1 and 3:2 are intervals, 3/1 and 3/2 are pitches, if you
> > observe one of the weaker list conventions.
> >
> Where was it, but I also read of a convention that 3:1 and 3:2 are
> melodic intervals, and 3/1 and 3/2 are harmonic (simultaneous)
> intervals....
>
> IOW, the ':' means melody and the '/' means harmony.

That's backwards, at best. In the 'American Gamelan' school
of just intonation (and outgrowths like the JI network and
its journal 1/1), slash ratios usually refer to pitches,
relative to some 1/1 pitch. Therefore, melodies can be
spelled out in slash ratios. 1/1 9/8 5/4, etc.

This notation is very limiting and impractical when it comes
to western-like music, which modulates a lot. It was at the
root of many misunderstandings on this list, and in my opinion
limited the scope of the JI Network composers.

Not wanting, however, to walk all over recent history, it
was suggested, and several people agreed (and none dissented)
that colon notation be used for simultaneous dyads and
larger chords, whose roots could be placed in pitch space
only with a detailed look at the history of the piece of
music in question -- i.e. all triads could be written 4:5:6
even though the 4's would be at different pitches.

This is just a cursory overview of the issue. But yeah,
you got it backwards.

-Carl

Carl Lumma wrote:
>>> 3:1 and 3:2 are intervals, 3/1 and 3/2 are pitches, if you
>>> observe one of the weaker list conventions.
>>> >>> >> Where was it, but I also read of a convention that 3:1 and 3:2 are >> melodic intervals, and 3/1 and 3/2 are harmonic (simultaneous)
>> intervals....
>>
>> IOW, the ':' means melody and the '/' means harmony.
>> >
> That's backwards, at best. In the 'American Gamelan' school
> of just intonation (and outgrowths like the JI network and
> its journal 1/1), slash ratios usually refer to pitches,
> relative to some 1/1 pitch. Therefore, melodies can be
> spelled out in slash ratios. 1/1 9/8 5/4, etc.
>
> This notation is very limiting and impractical when it comes
> to western-like music, which modulates a lot. It was at the
> root of many misunderstandings on this list, and in my opinion
> limited the scope of the JI Network composers.
>
> Not wanting, however, to walk all over recent history, it
> was suggested, and several people agreed (and none dissented)
> that colon notation be used for simultaneous dyads and
> larger chords, whose roots could be placed in pitch space
> only with a detailed look at the history of the piece of
> music in question -- i.e. all triads could be written 4:5:6
> even though the 4's would be at different pitches.
>
> This is just a cursory overview of the issue. But yeah,
> you got it backwards.
>
> -Carl

Hey, we agree that they ain't frequencies!

Anyway, the more I think about it, the more I really dont care--you're right, of course, we've seen tetrads like 12:14:17:20 spelled out (someone mentioned this as a good JI dim7th, as I recall), and it wouldn't matter whether is was simultaneous or not. And yes, the whole slash notation thing---oy---that started with Partch, and although the idea is good in theory 9get rid of historical remnants of a system of notation not suited to our new pitches, say the JI folks), man, when you start imagining the monster number of digits needed when you modulate (hence another rationale for thinking of JI as being adaptive on a higher order tempered 'grid' 41-edo? 53-edo?)

I ramble....yes, I'm corrected. Go Carl!

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <aaron@...> wrote:
>
> Carl Lumma wrote:
> >>> 3:1 and 3:2 are intervals, 3/1 and 3/2 are pitches, if you
> >>> observe one of the weaker list conventions.
> >>>
> >>>
> >> Where was it, but I also read of a convention that 3:1 and 3:2
are
> >> melodic intervals, and 3/1 and 3/2 are harmonic (simultaneous)
> >> intervals....
> >>
> >> IOW, the ':' means melody and the '/' means harmony.
> >>
> >
> > That's backwards, at best. In the 'American Gamelan' school
> > of just intonation (and outgrowths like the JI network and
> > its journal 1/1), slash ratios usually refer to pitches,
> > relative to some 1/1 pitch. Therefore, melodies can be
> > spelled out in slash ratios. 1/1 9/8 5/4, etc.
> >
> > This notation is very limiting and impractical when it comes
> > to western-like music, which modulates a lot. It was at the
> > root of many misunderstandings on this list, and in my opinion
> > limited the scope of the JI Network composers.
> >
> > Not wanting, however, to walk all over recent history, it
> > was suggested, and several people agreed (and none dissented)
> > that colon notation be used for simultaneous dyads and
> > larger chords, whose roots could be placed in pitch space
> > only with a detailed look at the history of the piece of
> > music in question -- i.e. all triads could be written 4:5:6
> > even though the 4's would be at different pitches.
> >
> > This is just a cursory overview of the issue. But yeah,
> > you got it backwards.
> >
> > -Carl
>
> Hey, we agree that they ain't frequencies!

No of course not, "refer to different frequencies", not "are".
>
> Anyway, the more I think about it, the more I really dont care--
you're
> right, of course, we've seen tetrads like 12:14:17:20 spelled out
> (someone mentioned this as a good JI dim7th, as I recall), and it
> wouldn't matter whether is was simultaneous or not. And yes, the
whole
> slash notation thing---oy---that started with Partch, and although
the
> idea is good in theory 9get rid of historical remnants of a system
of
> notation not suited to our new pitches, say the JI folks), man,
when you
> start imagining the monster number of digits needed when you
modulate
> (hence another rationale for thinking of JI as being adaptive on a
> higher order tempered 'grid' 41-edo? 53-edo?)
>
> I ramble....yes, I'm corrected. Go Carl!
>

I don't know about all this : and / stuff but I agree it would be
nice to have a convention.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

>
> Not wanting, however, to walk all over recent history, it
> was suggested, and several people agreed (and none dissented)
> that colon notation be used for simultaneous dyads and
> larger chords, whose roots could be placed in pitch space
> only with a detailed look at the history of the piece of
> music in question -- i.e. all triads could be written 4:5:6
> even though the 4's would be at different pitches.

I find it amusing that any terminology/spelling
sloppiness on my part (there's plenty of it, and I always immediately
apologize and correct) never escapes notice, but the
fundamental thinking behind decisions like this one, which
is utterly bogus and a needless burden on the development of
harmony, struts right by eliciting nothing but a few cheerful
waves. Penny wise, pound foolish.

Go ahead and change "none dissented" to "one dissents": :-)
Though of course I'll gain nothing but the entertaining sight
of even more colorful and ebulliant dancing away from issues
of actual importance, LOL.

-Cameron Bobro

Hi Cameron,

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:

> I don't know about all this : and / stuff but I agree
> it would be nice to have a convention.

As Carl explained, it's been a convention on these lists
for several years now to use colon(s) for ratios/proportions
which represent intervals and slashes for ratios of pitches.

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

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> Hi Cameron,
>
> --- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@> wrote:
>
> > I don't know about all this : and / stuff but I agree
> > it would be nice to have a convention.
>
>
> As Carl explained, it's been a convention on these lists
> for several years now to use colon(s) for ratios/proportions
> which represent intervals and slashes for ratios of pitches.

Fine, but what's this business about colons being used in a way
that seems to me to be ratios of pitch classes?

-Cameron Bobro

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> ----- Original Message -----
> From: "hstraub64" <hstraub64@...>
> >
> > Well, from my point of view: the less tones, the better :-)
> > Anyway, if you want to play all notes on standard pianos, you would
> > need four in any case, so it's not that big a difference... What would
> > good keyboard layouts be?
> >
>
> I wouldn't know for sure. Let us design a 41-EDO keyboard.
>

Alright. I have tried to design a mapping of 24 notes out of 41EDO, to
by played on 2 pianos.

Starting point was the observation that approximations for many
important intervals lie one step (29.27 cents) apart - such as:
- two whole tones: 10/9 (6 steps) vs. 9/8 (7 steps)
- pure major third 5/4 (13 steps) vs. pythagorean major third 81/64
(14 steps)
- harmonic seventh 7/4 (33 steps) vs. 3-limit minor seventh (built
from 2 fourths) 16/9 (34 steps)

So I designed the following keyboard mapping, with the difference
between the two keyboard constantly one step, which gives two
identical layouts.

Played on one keyboard gives pythagorean-style major thirds between
C-E and F-A, a 5-limit-style major third between G and B , and a
3-limit minor seventh between C and Bb.

Taking both keyboards, it is possible to play a number of pure major
thirds, a number of harmonic sevenths and also a number of neutral
thirds (12 steps - one step below pure major third and one step above
minor third).

Would that be something suitable for maqams?

41EDO steps:

High 0 4 7 11 14 17 20 24 28 31 34 37
Low 40 3 6 10 13 16 19 23 27 30 33 36

Cents (rounded to whole cents):

High 0 117 205 322 410 498 585 702 820 907 995 1083
Low 1171 88 176 293 380 468 556 673 790 878 966 1053
--
Hans Straub

Dear Hans,

----- Original Message -----
From: "hstraub64" <hstraub64@telesonique.net>
To: <tuning@yahoogroups.com>
Sent: 09 Ekim 2007 Sal� 12:07
Subject: [tuning] 24 out of 41EDO (was: 46-equal)

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> >
> > ----- Original Message -----
> > From: "hstraub64" <hstraub64@...>
> > >
> > > Well, from my point of view: the less tones, the better :-)
> > > Anyway, if you want to play all notes on standard pianos, you would
> > > need four in any case, so it's not that big a difference... What would
> > > good keyboard layouts be?
> > >
> >
> > I wouldn't know for sure. Let us design a 41-EDO keyboard.
> >
>
> Alright. I have tried to design a mapping of 24 notes out of 41EDO, to
> by played on 2 pianos.
>

Very well.

> Starting point was the observation that approximations for many
> important intervals lie one step (29.27 cents) apart - such as:
> - two whole tones: 10/9 (6 steps) vs. 9/8 (7 steps)
> - pure major third 5/4 (13 steps) vs. pythagorean major third 81/64
> (14 steps)
> - harmonic seventh 7/4 (33 steps) vs. 3-limit minor seventh (built
> from 2 fourths) 16/9 (34 steps)
>

Indeed so.

> So I designed the following keyboard mapping, with the difference
> between the two keyboard constantly one step, which gives two
> identical layouts.
>
> Played on one keyboard gives pythagorean-style major thirds between
> C-E and F-A, a 5-limit-style major third between G and B , and a
> 3-limit minor seventh between C and Bb.
>

By doing so, you have acquired this scale:

0: 1/1 C
1: 117.073 cents C/||\ D!!/
2: 204.878 cents D
3: 321.951 cents D/||\ E!!/
4: 409.756 cents E
5: 497.561 cents F
6: 585.366 cents F||\ G\!!/
7: 702.439 cents G
8: 819.512 cents G/||\ A!!/
9: 907.317 cents A
10: 995.122 cents A||\ B\!!/
11: 1082.927 cents A/|||\ B\!
12: 2/1 C

The most obvious discrepancy is the wolf between E and B\|.

> Taking both keyboards, it is possible to play a number of pure major
> thirds, a number of harmonic sevenths and also a number of neutral
> thirds (12 steps - one step below pure major third and one step above
> minor third).
>
> Would that be something suitable for maqams?
>
>
> 41EDO steps:
>
> High 0 4 7 11 14 17 20 24 28 31 34 37
> Low 40 3 6 10 13 16 19 23 27 30 33 36
>
> Cents (rounded to whole cents):
>
> High 0 117 205 322 410 498 585 702 820 907 995 1083
> Low 1171 88 176 293 380 468 556 673 790 878 966 1053
> --
> Hans Straub
>
>

You have forsaken middle seconds, which makes this an inappropriate setting
for Maqam Music. We need all 41 tones. Here is a 41-tone scale with Yekta-24
at its core:

Yekta-24 extended to 41-quasi equal
|
0: 1/1 0.000 YEGAH/NEVA
1: 26.962 cents -----------------------dik yegah/neva
2: 64.430 cents -----------------------(pest) beyati
3: 256/243 90.225 Nim (Pest) Hisar
4: 2187/2048 113.685 (Pest) Hisar
5: 148.819 cents -----------------------(pest) h�zzam
6: 65536/59049 180.450 Dik (Pest) Hisar
7: 9/8 203.910 H�SEYN�(A��RAN)
8: 233.207 cents ------------------------dik h�seyni(a�iran)
9: 270.675 cents ------------------------nerm acem(a�iran)
10: 32/27 294.135 Acem(a�iran)
11: 19683/16384 317.595 Dik Acem(a�iran)
12: 355.063 cents -----------------------nerm arak/evc
13: 8192/6561 384.360 ARAK/EVC
14: 81/64 407.820 Geve�t/Mahur
15: 439.451 cents -----------------------rehavi/mahurek
16: 2097152/1594323 474.585 Dik Geve�t/Mahur
17: 4/3 498.045 RAST/GERDAN�YE
18: 523.840 cents -----------------------dik rast/gerdaniye
19: 561.308 cents -----------------------nerm zengule/�ehnaz
20: 1024/729 588.270 Nim Zengule/�ehnaz
21: 729/512 611.730 Zengule/�ehnaz
22: 645.696 cents -----------------------suznak/hicazkar
23: 262144/177147 678.495 Dik Zengule/�ehnaz
24: 3/2 701.955 D�GAH/MUHAYYER
25: 730.084 cents -----------------------dik d�gah/muhayyer
26: 767.553 cents -----------------------nerm k�rdi/s�nb�le
27: 128/81 792.180 K�rdi/S�nb�le
28: 6561/4096 815.640 Dik K�rdi/S�nb�le
29: 851.941 cents -----------------------(tiz) nerm segah
30: 32768/19683 882.405 (T�Z) SEGAH
31: 27/16 905.865 (Tiz) Puselik
32: 936.329 cents -----------------------(tiz) ni�ab�r
33: 8388608/4782969 972.630 (Tiz) Dik Puselik
34: 16/9 996.090 (T�Z) �ARGAH
35: 1020.717 cents ------------------------(tiz) dik �argah
36: 1058.186 cents ------------------------(tiz) nerm hicaz
37: 4096/2187 1086.315 (Tiz) Nim Hicaz
38: 243/128 1109.775 (Tiz) Hicaz
39: 1142.574 cents ------------------------(tiz) saba
40: 1048576/531441 1176.540 (Tiz) Dik Hicaz
41: 2/1 1200.000 (T�Z) NEVA

Danny Wier inspired me for the creation of this scale. The SCALA file is
attached. Change key to 17th degree in order to start from perde rast at C.

Oz.

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> You have forsaken middle seconds, which makes this an inappropriate
> setting for Maqam Music. We need all 41 tones. Here is a 41-tone
> scale with Yekta-24 at its core:
>
> Yekta-24 extended to 41-quasi equal
> |
> 0: 1/1 0.000 YEGAH/NEVA
> 1: 26.962 cents -----------------------dik yegah/neva
> 2: 64.430 cents -----------------------(pest) beyati
> 3: 256/243 90.225 Nim (Pest) Hisar
> 4: 2187/2048 113.685 (Pest) Hisar
> 5: 148.819 cents -----------------------(pest) hüzzam
> 6: 65536/59049 180.450 Dik (Pest) Hisar
> 7: 9/8 203.910 HÜSEYNÝ(AÞÝRAN)
> 8: 233.207 cents ------------------------dik hüseyni
(aþiran)
> 9: 270.675 cents ------------------------nerm acem
(aþiran)
> 10: 32/27 294.135 Acem(aþiran)
> 11: 19683/16384 317.595 Dik Acem(aþiran)
> 12: 355.063 cents -----------------------nerm arak/evc
> 13: 8192/6561 384.360 ARAK/EVC
> 14: 81/64 407.820 Geveþt/Mahur
> 15: 439.451 cents -----------------------rehavi/mahurek
> 16: 2097152/1594323 474.585 Dik Geveþt/Mahur
> 17: 4/3 498.045 RAST/GERDANÝYE
> 18: 523.840 cents -----------------------dik
rast/gerdaniye
> 19: 561.308 cents -----------------------nerm
zengule/þehnaz
> 20: 1024/729 588.270 Nim Zengule/Þehnaz
> 21: 729/512 611.730 Zengule/Þehnaz
> 22: 645.696 cents -----------------------suznak/hicazkar
> 23: 262144/177147 678.495 Dik Zengule/Þehnaz
> 24: 3/2 701.955 DÜGAH/MUHAYYER
> 25: 730.084 cents -----------------------dik
dügah/muhayyer
> 26: 767.553 cents -----------------------nerm
kürdi/sünbüle
> 27: 128/81 792.180 Kürdi/Sünbüle
> 28: 6561/4096 815.640 Dik Kürdi/Sünbüle
> 29: 851.941 cents -----------------------(tiz) nerm
segah
> 30: 32768/19683 882.405 (TÝZ) SEGAH
> 31: 27/16 905.865 (Tiz) Puselik
> 32: 936.329 cents -----------------------(tiz) niþabür
> 33: 8388608/4782969 972.630 (Tiz) Dik Puselik
> 34: 16/9 996.090 (TÝZ) ÇARGAH
> 35: 1020.717 cents ------------------------(tiz) dik
çargah
> 36: 1058.186 cents ------------------------(tiz) nerm
hicaz
> 37: 4096/2187 1086.315 (Tiz) Nim Hicaz
> 38: 243/128 1109.775 (Tiz) Hicaz
> 39: 1142.574 cents ------------------------(tiz) saba
> 40: 1048576/531441 1176.540 (Tiz) Dik Hicaz
> 41: 2/1 1200.000 (TÝZ) NEVA
>
> Danny Wier inspired me for the creation of this scale. The SCALA
> file is attached.

Thanks!

> Change key to 17th degree in order to start from
> perde rast at C.
>

I do not well understand the last sentence. 17th degree would be F?

Mapping the above scale to 41EDO, we would get:

41EDO steps:
High 0 4 7 11 14 17 21 24 28 31 34 38
Low 40 3 6 10 13 16 20 23 27 30 33 37

Cents:
High 0 117 205 322 410 498 615 702 820 907 995 1112
Low 1171 88 176 293 380 468 585 673 790 878 966 1083

BTW, I observed this can be extended to play all 41 notes (to be
played on 4 pianos), with the properties:
1) 4 identical layouts, the difference between each one 41EDO step.
2) All notes are not more than 63 cents away from standard 12edo.

The latter is an especially good property for me since 64 cents is
the maximal amount of retuning that my synthesizer supports...

41EDO steps:
1 5 8 12 15 18 22 25 29 32 35 39
0 4 7 11 14 17 21 24 28 31 34 38
40 3 6 10 13 16 20 23 27 30 33 37
39 2 5 9 12 15 19 22 26 29 32 36

Cents:
29 146 234 351 439 527 644 732 849 937 1024 1141
0 117 205 322 410 498 615 702 820 907 995 1112
1171 88 176 293 380 468 585 673 790 878 966 1083
1141 59 146 263 351 439 556 644 761 849 937 1054
--
Hans Straub

----- Original Message -----
From: "hstraub64" <hstraub64@telesonique.net>
To: <tuning@yahoogroups.com>
Sent: 30 Ekim 2007 Sal� 17:16
Subject: [tuning] Re: 24 out of 41EDO (was: 46-equal)

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> You have forsaken middle seconds, which makes this an inappropriate
> setting for Maqam Music. We need all 41 tones. Here is a 41-tone
> scale with Yekta-24 at its core:
>
> Yekta-24 extended to 41-quasi equal
> |
> 0: 1/1 0.000 YEGAH/NEVA
> 1: 26.962 cents -----------------------dik yegah/neva
> 2: 64.430 cents -----------------------(pest) beyati
> 3: 256/243 90.225 Nim (Pest) Hisar
> 4: 2187/2048 113.685 (Pest) Hisar
> 5: 148.819 cents -----------------------(pest) h�zzam
> 6: 65536/59049 180.450 Dik (Pest) Hisar
> 7: 9/8 203.910 H�SEYN�(A��RAN)
> 8: 233.207 cents ------------------------dik h�seyni
(a�iran)
> 9: 270.675 cents ------------------------nerm acem
(a�iran)
> 10: 32/27 294.135 Acem(a�iran)
> 11: 19683/16384 317.595 Dik Acem(a�iran)
> 12: 355.063 cents -----------------------nerm arak/evc
> 13: 8192/6561 384.360 ARAK/EVC
> 14: 81/64 407.820 Geve�t/Mahur
> 15: 439.451 cents -----------------------rehavi/mahurek
> 16: 2097152/1594323 474.585 Dik Geve�t/Mahur
> 17: 4/3 498.045 RAST/GERDAN�YE
> 18: 523.840 cents -----------------------dik
rast/gerdaniye
> 19: 561.308 cents -----------------------nerm
zengule/�ehnaz
> 20: 1024/729 588.270 Nim Zengule/�ehnaz
> 21: 729/512 611.730 Zengule/�ehnaz
> 22: 645.696 cents -----------------------suznak/hicazkar
> 23: 262144/177147 678.495 Dik Zengule/�ehnaz
> 24: 3/2 701.955 D�GAH/MUHAYYER
> 25: 730.084 cents -----------------------dik
d�gah/muhayyer
> 26: 767.553 cents -----------------------nerm
k�rdi/s�nb�le
> 27: 128/81 792.180 K�rdi/S�nb�le
> 28: 6561/4096 815.640 Dik K�rdi/S�nb�le
> 29: 851.941 cents -----------------------(tiz) nerm
segah
> 30: 32768/19683 882.405 (T�Z) SEGAH
> 31: 27/16 905.865 (Tiz) Puselik
> 32: 936.329 cents -----------------------(tiz) ni�ab�r
> 33: 8388608/4782969 972.630 (Tiz) Dik Puselik
> 34: 16/9 996.090 (T�Z) �ARGAH
> 35: 1020.717 cents ------------------------(tiz) dik
�argah
> 36: 1058.186 cents ------------------------(tiz) nerm
hicaz
> 37: 4096/2187 1086.315 (Tiz) Nim Hicaz
> 38: 243/128 1109.775 (Tiz) Hicaz
> 39: 1142.574 cents ------------------------(tiz) saba
> 40: 1048576/531441 1176.540 (Tiz) Dik Hicaz
> 41: 2/1 1200.000 (T�Z) NEVA
>
> Danny Wier inspired me for the creation of this scale. The SCALA
> file is attached.

Thanks!

> Change key to 17th degree in order to start from
> perde rast at C.
>

I do not well understand the last sentence. 17th degree would be F?

[OZ: Yes, If you want to make the starting tone (C) perde Rast, change scale
key in SCALA to 17th degree]

Mapping the above scale to 41EDO, we would get:

41EDO steps:
High 0 4 7 11 14 17 21 24 28 31 34 38
Low 40 3 6 10 13 16 20 23 27 30 33 37

Cents:
High 0 117 205 322 410 498 615 702 820 907 995 1112
Low 1171 88 176 293 380 468 585 673 790 878 966 1083

BTW, I observed this can be extended to play all 41 notes (to be
played on 4 pianos), with the properties:
1) 4 identical layouts, the difference between each one 41EDO step.
2) All notes are not more than 63 cents away from standard 12edo.

The latter is an especially good property for me since 64 cents is
the maximal amount of retuning that my synthesizer supports...

41EDO steps:
1 5 8 12 15 18 22 25 29 32 35 39
0 4 7 11 14 17 21 24 28 31 34 38
40 3 6 10 13 16 20 23 27 30 33 37
39 2 5 9 12 15 19 22 26 29 32 36

Cents:
29 146 234 351 439 527 644 732 849 937 1024 1141
0 117 205 322 410 498 615 702 820 907 995 1112
1171 88 176 293 380 468 585 673 790 878 966 1083
1141 59 146 263 351 439 556 644 761 849 937 1054
--
Hans Straub

[OZ: Is there any possibility of implementing this tuning on a 12-tone
keyboard?]

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> 41EDO steps:
> High 0 4 7 11 14 17 21 24 28 31 34 38
> Low 40 3 6 10 13 16 20 23 27 30 33 37

Garibaldi[24] in 41-et. It's arguably better in 94-et.

> [OZ: Is there any possibility of implementing this tuning on a 12-tone
> keyboard?]
>

Why not?

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
>
> [OZ: Is there any possibility of implementing this tuning on a 12-tone
> keyboard?]
>

Yes - for example on a keyboard that can be run in "split" mode, i.e.
different parts of the keyboard have different midi channels assigned.
My keyboard, e.g., has two split areas (upper half and lower half).
Then I have a multitimbral synthesizer that allows a separate retuning
for each midi channel. I just have to edit the synth's sounds to get
two identical sounds that differ only in the octave range and assign
the higher sounding one to the lower part of the keyboard and the
lower one to the higher part of the keyboard. Then I can play up to 24
tones per octave. This is how I managed to compose a piece for the
midwest microfest and also for the seventeen tone piano project.

For more than 24 tones per octave, the same procedure applies - you
just need another keyboard, such as the Kaway MP5, which supports 4
split areas. Playing this as one person gets a little difficult,
though, since humans have only two hands...
--
Hans Straub