desire for meantone with an 11-limit interval on piano

Hi, All,

I'm thinking of putting a new tuning on my piano, which would be something
vaguely near a quarter-comma meantone with the wolf between Bb and F instead
of the usual G# to Eb. This has the interesting quality of making the key
of C very consonant both otonally and utonally(somewhat). Specifically you
get the near-pure 6:7 on C-Eb and G-Bb, which gives you the near-pure
subminor-seventh in C, besides giving you the pure 4:5:6:7 in C (and also in
F).

So if that were the only goal, I'd probably temper it for a good compromise
among the near-pure 2:3, 4:5, and 6:7, perhaps with the best beat ratios.

However, I want to make one more tweak to this scale, which is to flatten
the F# to make it form a near-pure 8:11 with C. I might (depending on how
the rest of the tuning works) do this only in the upper octaves, leaving
good D-F# thirds in the lower octaves. But still I'd like to tweak the
whole temperament to make the best of this.

A primary goal would be to have good audible implied fundamentals hanging
around the 4:5:6:7:8:9:10:11 chords and subsets thereof. This usually
requires the intervals involved to be within something like 7 cents of pure,
(maybe closer in case of the intervals involving 11?). But I'm wondering
how far I can stretch this 7-cent limit in order to achieve even other
purposes with the temperament. So I was wondering: suppose for example the
7:11 is within 7 cents, and the 4:7 is within 7 cents but the 4:11 is not?
This is just for example, there may be some better way to structure the
compromise, depending on what else could be achieved. I'm wondering if
people have had any experience with how this works particularly with piano
tone and inharmonicity, and if anyone has found any good near-meantone
tunings with 2 good 6:7's and a good ...:7:...:11.

Then with regard to piano, I'm wondering as usual about the octaves. My
piano tuner tells me not to compromise the octaves too much because you
basically want to stretch to achieve a certain overall resonance with the
octaves. So narrowing the octaves ends up being bad for the overall sound.

Then with regard to other things that could be achieved with the tuning, the
main question is, once I mess with that F# to give it 11-limit utility in C,
what else could I do with it at the same time? For example, if I flattened
the F# a little less and flattened the Eb a little more then I might get
another decent subminor triad at Eb-F#-Bb (excuse my bad enharmonic notation
habits). I think that's really pushing it, but it seems conceivable that
there might be something interesting in this "vicinity", something with
perhaps some useful circulating key-color qualities that somehow integrates
the 11-limite F# and the 7-limit Eb and Bb into some kind of a temperament
that has some kind of meantone roudness to it. And although I mentioned
quarter-comma as a kind of starting point, I'd probably rather have
something that has a bit more variation in the intervals rather than having
most of the major 3rds being the same.

Problem is that getting the piano retunred to try something like this is a
lot of trouble, so I'm looking for some perhaps piano-specific advice in
advance. However, any non-piano-oriented advice in relation to a meantonish
scale with an 11-limit otonal feature would be great. Having the implied
fundamental "thing" happening but perhaps happening right at the edge of
perceptual possibility might be interesting, especially if this permits more
flexibility in the overall temperament.

If there can be any kind of integration of the 11-limit F#, then I'd
probably use it across the whole keyboard instead of just the upper end, as
I originally suggested.

Thanks in advance for any thoughts. I'm probably asking for too much from a
12-tone octave, sort of an impossible holy grail, but it is worth a try.

-Kurt

On 5/13/06, Kurt Bigler <kkb@breathsense.com> wrote:
[...]
> However, I want to make one more tweak to this scale, which is to flatten
> the F# to make it form a near-pure 8:11 with C. I might (depending on how
> the rest of the tuning works) do this only in the upper octaves, leaving
> good D-F# thirds in the lower octaves. But still I'd like to tweak the
> whole temperament to make the best of this.

Well, if you want C-F# to be an acceptable 8:11, then the average of
the sixth fifths C-G, G-D, D-A, A-E, E-B, and B-F# has to be at most
694 cents. That's even less than in 1/3-comma meantone, so if you want
most of the fifths close to 1/4-comma, you'll have to have another
wolf in there.

[...]
> Thanks in advance for any thoughts. I'm probably asking for too much from a
> 12-tone octave, sort of an impossible holy grail, but it is worth a try.

Exactly what I was thinking. I guess it all depends how much
modulation you want to do.

Keenan

on 5/13/06 5:52 PM, Keenan Pepper <keenanpepper@gmail.com> wrote:

> On 5/13/06, Kurt Bigler <kkb@breathsense.com> wrote:
>
> [...]
>> However, I want to make one more tweak to this scale, which is to flatten
>> the F# to make it form a near-pure 8:11 with C. I might (depending on how
>> the rest of the tuning works) do this only in the upper octaves, leaving
>> good D-F# thirds in the lower octaves. But still I'd like to tweak the
>> whole temperament to make the best of this.
>
> Well, if you want C-F# to be an acceptable 8:11, then the average of
> the sixth fifths C-G, G-D, D-A, A-E, E-B, and B-F# has to be at most
> 694 cents. That's even less than in 1/3-comma meantone, so if you want
> most of the fifths close to 1/4-comma, you'll have to have another
> wolf in there.

Yes, I've kind of accepted that already. The rough plan is to only move the
F# and let the additional wolves fall where they may, unless there turns out
to be some other useful tempering I hadn't anticipated.

>
> [...]
>
>> Thanks in advance for any thoughts. I'm probably asking for too much from a
>> 12-tone octave, sort of an impossible holy grail, but it is worth a try.
>
> Exactly what I was thinking. I guess it all depends how much
> modulation you want to do.

I tried a related scale on my harpsichord, which is close to 1/4-comma (with
the unusual key center) since I can tune that myself by ear, and just the F#
modified to be 11-limit.

I find I can do a lot of "normal" things with that scale.

It does add some extra "color" in abnormal places, but I'm wondering if the
most extreme issues can be subdued a little by some tempering and still have
some meaningful 11-limit functioning around that F#.

-Kurt

Kurt,

I usually think that with only 12 notes, one should either avoid the
11- (or even at times the 7-limit), *or*, go for something that you
would tend to do more spacious, non-modulating JI with, and do
something like a Ben Johnston harmonic tuning.

If you plan to modulate, that F# will get pretty annoying (IMO).

I think Kraig Grady's centaur is about as far as one can take the
7-limit while having meantone style limited-modulation, and that's
probably because it avoids the 11-limit...in my experience with
12-note tunings, once you introduce the 11-limit, you are stuck with
only a few usable roots (traditionally speaking)

But maybe you're going for something entirely different, or entirely
un-traditional! (just my two cents)

Good to hear from you!

Best,
Aaron.

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@...> wrote:
>
> on 5/13/06 5:52 PM, Keenan Pepper <keenanpepper@...> wrote:
>
> > On 5/13/06, Kurt Bigler <kkb@...> wrote:
> >
> > [...]
> >> However, I want to make one more tweak to this scale, which is to
flatten
> >> the F# to make it form a near-pure 8:11 with C. I might
(depending on how
> >> the rest of the tuning works) do this only in the upper octaves,
leaving
> >> good D-F# thirds in the lower octaves. But still I'd like to
tweak the
> >> whole temperament to make the best of this.
> >
> > Well, if you want C-F# to be an acceptable 8:11, then the average of
> > the sixth fifths C-G, G-D, D-A, A-E, E-B, and B-F# has to be at most
> > 694 cents. That's even less than in 1/3-comma meantone, so if you want
> > most of the fifths close to 1/4-comma, you'll have to have another
> > wolf in there.
>
> Yes, I've kind of accepted that already. The rough plan is to only
move the
> F# and let the additional wolves fall where they may, unless there
turns out
> to be some other useful tempering I hadn't anticipated.
>
> >
> > [...]
> >
> >> Thanks in advance for any thoughts. I'm probably asking for too
much from a
> >> 12-tone octave, sort of an impossible holy grail, but it is worth
a try.
> >
> > Exactly what I was thinking. I guess it all depends how much
> > modulation you want to do.
>
> I tried a related scale on my harpsichord, which is close to
1/4-comma (with
> the unusual key center) since I can tune that myself by ear, and
just the F#
> modified to be 11-limit.
>
> I find I can do a lot of "normal" things with that scale.
>
> It does add some extra "color" in abnormal places, but I'm wondering
if the
> most extreme issues can be subdued a little by some tempering and
still have
> some meaningful 11-limit functioning around that F#.
>
> -Kurt
>

Aaron,

on 5/14/06 7:25 PM, Aaron Krister Johnson <aaron@akjmusic.com> wrote:

> Kurt,
>
> I usually think that with only 12 notes, one should either avoid the
> 11- (or even at times the 7-limit), *or*, go for something that you
> would tend to do more spacious, non-modulating JI with, and do
> something like a Ben Johnston harmonic tuning.
>
> If you plan to modulate, that F# will get pretty annoying (IMO).
>
> I think Kraig Grady's centaur is about as far as one can take the
> 7-limit

I'll definitely need to refresh my memory on that one.

> while having meantone style limited-modulation, and that's
> probably because it avoids the 11-limit...in my experience with
> 12-note tunings, once you introduce the 11-limit, you are stuck with
> only a few usable roots (traditionally speaking)
>
> But maybe you're going for something entirely different, or entirely
> un-traditional! (just my two cents)

Yes, probably it ends up being untraditional usage I'm going for, but still
somewhat influenced by traditional usages. So that means either improv or
severe adaptation. But I'm just looking for a road somewhere between a
quasi-meantone that I have on the piano and the dual-harmonic (modified)
that I have on the harpsichord:

http://k.breathsense.com/music/

So I've been in the severely-limited boat of the dual-harmonic, and the
totally-open-to-classical/romantic quasi-meantone, and I want something
in-between. With meantone as a reference, I want a more usable 7-limit, and
I'm just asking the question whether I can squeeze the single 11-limit into
the picture. I think I can, even for limited classically-informed
improvisation, based on my limited experience with the modified
quarter-comma that I tuned up on the harpsichord a couple days ago. I can
still play a little Bach on it, for example. The availability of the
7-limit is sweet and something I've been dying for in a context that has
traditional western stuff still somewhat within reach. The 11-limit is
admittedly a question-mark, and its presence clearly destroys any normal key
progression and requires a fair amount of what will seem like avoidance at
first. But it will seem like total freedom compared to having a total of 4
11- and 13-limit notes in the octave.

I'll probably try it, record some improvs and report back with them.

It looks like 13/60 (syntonic) comma meantone makes C-Eb and Eb-F# (the 11/8
F# so to speak) both 8.8 cents wide of pure 7/6, putting 3 subminor triads
and some interesting diminished chords within the realm. So I have to put
that on the organ/synth and hear whether it works (whether 8.8 cents error
is too much). If so then I need to somehow equal-beat-ify the thing,
something I'm not yet experienced with. But I want to take the equality out
of the equal intervals and get more different intervals sizes, and
equal-beating seems to be a good way to do that.

Thanks, I'll keep you posted.

Yes 12-notes is a bear. Dying for that affordable 2-d keyboard. The wait
is grueling with the expectations seeded for what seems like so many years
now. Well, there's still plenty to do in the meantime, actually. ;)

Congrats on your family expansion!

-Kurt

--- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...> wrote:
>
>
> Kurt,
>
> I usually think that with only 12 notes, one should either avoid the
> 11- (or even at times the 7-limit), *or*, go for something that you
> would tend to do more spacious, non-modulating JI with, and do
> something like a Ben Johnston harmonic tuning.

I think there are interesting 11-limit possibilities with just twelve
notes. Here is the bihexany, which I think is quite interesting:

! bihexany.scl
Hole around [0, 1/2, 1/2, 1/2]
12
!
35/33
7/6
5/4
14/11
15/11
3/2
35/22
5/3
7/4
20/11
21/11
2

Tempering out 441/440 and 540/539, for instance by 161-et, is a
retuning plan.

> If you plan to modulate, that F# will get pretty annoying (IMO).
>
> I think Kraig Grady's centaur is about as far as one can take the
> 7-limit while having meantone style limited-modulation, and that's
> probably because it avoids the 11-limit...in my experience with
> 12-note tunings, once you introduce the 11-limit, you are stuck with
> only a few usable roots (traditionally speaking)

The bihexany above has a considerable quantity of 11-limit chords.

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@...> wrote:

> So I've been in the severely-limited boat of the dual-harmonic, and the
> totally-open-to-classical/romantic quasi-meantone, and I want something
> in-between. With meantone as a reference, I want a more usable
7-limit, and
> I'm just asking the question whether I can squeeze the single
11-limit into
> the picture.

It's hard to do very much for 11 with just twelve meantone notes. You
have two reasonable mappings: either 18 fifths up (31&43), C-Ex, or 13
fifths down, (31&50), C-Gbb. Of course 19 notes of meantone, tuned to
50-et, would give you a lot of 11s to play with.

Gene wrote:

> It's hard to do very much for 11 with just twelve meantone notes. You
> have two reasonable mappings: either 18 fifths up (31&43), C-Ex, or 13
> fifths down, (31&50), C-Gbb. Of course 19 notes of meantone, tuned to
> 50-et, would give you a lot of 11s to play with.

Even more, a 24-tone chain allows good combinations of both 7-limit and
11-limit, like "B-D#-F#-Gx-C#-Fb".
And if you are willing to accept a 29-tone chain, you can try this
experiment:

C1 C2 G2 C3
E3 G3 A#3 C4
D4 E4 Gbb4 G4
G##4 A#4 B4 C5

You see what I'm trying to imitate?

Petr

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>

> I think there are interesting 11-limit possibilities with just twelve
> notes. Here is the bihexany, which I think is quite interesting:
>
> ! bihexany.scl
> Hole around [0, 1/2, 1/2, 1/2]
> 12
> !
> 35/33
> 7/6
> 5/4
> 14/11
> 15/11
> 3/2
> 35/22
> 5/3
> 7/4
> 20/11
> 21/11
> 2
>
> Tempering out 441/440 and 540/539, for instance by 161-et, is a
> retuning plan.
> only a few usable roots (traditionally speaking)
>
> The bihexany above has a considerable quantity of 11-limit chords.
>

In my experience, these kinds of scales are cool
harmonically/motivically, but very frustrating if one wants to think
melodically (which of course is not always the only game in town)

-Aaron.

on 5/16/06 1:18 PM, Gene Ward Smith <genewardsmith@coolgoose.com> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@...> wrote:
>
>> So I've been in the severely-limited boat of the dual-harmonic, and the
>> totally-open-to-classical/romantic quasi-meantone, and I want something
>> in-between. With meantone as a reference, I want a more usable
> 7-limit, and
>> I'm just asking the question whether I can squeeze the single
> 11-limit into
>> the picture.
>
> It's hard to do very much for 11 with just twelve meantone notes. You
> have two reasonable mappings: either 18 fifths up (31&43), C-Ex, or 13
> fifths down, (31&50), C-Gbb. Of course 19 notes of meantone, tuned to
> 50-et, would give you a lot of 11s to play with.

Oh, well. Tell that to my piano. ;)

Maybe if I pray and chant and play pure just recordings for it for a few
months it will sprout some new 7- and 11-limit keys.

Good to know these things, of course, but it is a solution to a different
problem. And in anticipation of some future 2-dimensional keyboard of my
dreams, I *am* thinking about those things too.

-Kurt

on 5/16/06 8:02 PM, Aaron Krister Johnson <aaron@akjmusic.com> wrote:

> --- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
>
>> I think there are interesting 11-limit possibilities with just twelve
>> notes. Here is the bihexany, which I think is quite interesting:
>>
>> ! bihexany.scl
>> Hole around [0, 1/2, 1/2, 1/2]
>> 12
>> !
>> 35/33
>> 7/6
>> 5/4
>> 14/11
>> 15/11
>> 3/2
>> 35/22
>> 5/3
>> 7/4
>> 20/11
>> 21/11
>> 2
> In my experience, these kinds of scales are cool
> harmonically/motivically, but very frustrating if one wants to think
> melodically (which of course is not always the only game in town)
>
> -Aaron.

I don't mind uneven steps, but it is nice if they stay within a
half-semitone of ET.

Also for retuning a piano it becomes a bit testy for old strings to be
retuned that far (and back).

-Kurt

on 5/16/06 1:11 PM, Gene Ward Smith <genewardsmith@coolgoose.com> wrote:

> --- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...> wrote:
>>
>> I usually think that with only 12 notes, one should either avoid the
>> 11- (or even at times the 7-limit), *or*, go for something that you
>> would tend to do more spacious, non-modulating JI with, and do
>> something like a Ben Johnston harmonic tuning.
>
> I think there are interesting 11-limit possibilities with just twelve
> notes. Here is the bihexany, which I think is quite interesting:
>
> ! bihexany.scl
> Hole around [0, 1/2, 1/2, 1/2]
> 12
> !
> 35/33
> 7/6
> 5/4
> 14/11
> 15/11
> 3/2
> 35/22
> 5/3
> 7/4
> 20/11
> 21/11
> 2
>
> Tempering out 441/440 and 540/539, for instance by 161-et, is a
> retuning plan.
>
>> If you plan to modulate, that F# will get pretty annoying (IMO).
>>
>> I think Kraig Grady's centaur is about as far as one can take the
>> 7-limit while having meantone style limited-modulation, and that's
>> probably because it avoids the 11-limit...in my experience with
>> 12-note tunings, once you introduce the 11-limit, you are stuck with
>> only a few usable roots (traditionally speaking)
>
> The bihexany above has a considerable quantity of 11-limit chords.

Yes, and without skipping the 7-limit either.

Something to consider for another piano-year. This time I think I want only
a dash of 11-limit, a bit of 7, and a good range of 5-limit possibilities.

-Kurt

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@...> wrote:
>
> on 5/16/06 1:18 PM, Gene Ward Smith <genewardsmith@...> wrote:
>
> > --- In tuning@yahoogroups.com, Kurt Bigler <kkb@> wrote:
> >
> >> So I've been in the severely-limited boat of the dual-harmonic,
and the
> >> totally-open-to-classical/romantic quasi-meantone, and I want
something
> >> in-between. With meantone as a reference, I want a more usable
> > 7-limit, and
> >> I'm just asking the question whether I can squeeze the single
> > 11-limit into
> >> the picture.
> >
> > It's hard to do very much for 11 with just twelve meantone notes.
You
> > have two reasonable mappings: either 18 fifths up (31&43), C-Ex,
or 13
> > fifths down, (31&50), C-Gbb. Of course 19 notes of meantone, tuned
to
> > 50-et, would give you a lot of 11s to play with.
>
> Oh, well. Tell that to my piano. ;)
>
> Maybe if I pray and chant and play pure just recordings for it for a
few
> months it will sprout some new 7- and 11-limit keys.
>
> Good to know these things, of course, but it is a solution to a
different
> problem. And in anticipation of some future 2-dimensional keyboard
of my
> dreams, I *am* thinking about those things too.
>
> -Kurt
>

Another way, depending on your piano, if you are handy you might be
able to make an old fashioned harmonic damper, something simple like
aftermarket muting rails with bichord wedge damper felt applied to the
bass strings, with a pedal at the end of a bicycle cable held onto the
struts with clamps. This one shows the general idea

/tuning/files/claviharpe.PDF

Clark

back in the mid 80's when i only had a 12 tone marimba to retune to go along with my set of dallesandro tubes with the full 1-3-5-7-9-11 CPS, i settled on tuning it to the 2 5-7-9-11- hexanies a 3/2 apart which worked for quite a bit of music. Not good for diatonic music at all, but it didn't seem to matter
>
> Message 1 > From: "Gene Ward Smith" > Date: Tue May 16, 2006 1:11pm(PDT) > Subject: Re: desire for meantone with an 11-limit interval on piano
>
> --- In tuning@yahoogroups.com, "Aaron Krister Johnson" <aaron@...> wrote:
> >> Kurt,
>>
>> I usually think that with only 12 notes, one should either avoid the
>> 11- (or even at times the 7-limit), *or*, go for something that you
>> would tend to do more spacious, non-modulating JI with, and do
>> something like a Ben Johnston harmonic tuning.
>> >
> I think there are interesting 11-limit possibilities with just twelve
> notes. Here is the bihexany, which I think is quite interesting:
>
> ! bihexany.scl
> Hole around [0, 1/2, 1/2, 1/2]
> 12
> !
> 35/33
> 7/6
> 5/4
> 14/11
> 15/11
> 3/2
> 35/22
> 5/3
> 7/4
> 20/11
> 21/11
> 2
>
> Tempering out 441/440 and 540/539, for instance by 161-et, is a
> retuning plan.
>
> >> If you plan to modulate, that F# will get pretty annoying (IMO).
>>
>> I think Kraig Grady's centaur is about as far as one can take the
>> 7-limit while having meantone style limited-modulation, and that's
>> probably because it avoids the 11-limit...in my experience with
>> 12-note tunings, once you introduce the 11-limit, you are stuck with
>> only a few usable roots (traditionally speaking)
>> >
> The bihexany above has a considerable quantity of 11-limit chords.
>
>
>
> -- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

> back in the mid 80's when i only had a 12 tone marimba to
> retune to go along with my set of dallesandro tubes with
> the full 1-3-5-7-9-11 CPS, i settled on tuning it to the
> 2 5-7-9-11 hexanies a 3/2 apart which worked for quite a bit
> of music. Not good for diatonic music at all, but it didn't
> seem to matter

Hiya Kraig- did I get this right?

!
Kraig Grady's dual [5 7 9 11] hexany scale.
12
!
35/33
10/9
7/6
14/11
15/11
81/56
3/2
45/28
135/77
81/44
27/14
2
!

Gene - is the scale you gave constructed anything like this?

-Carl

Hi all,

Carl Lumma wrote on Thu May 18, 2006:
>
> > back in the mid 80's when i only had a 12 tone marimba to
> > retune to go along with my set of dallesandro tubes with
> > the full 1-3-5-7-9-11 CPS, i settled on tuning it to the
> > 2 5-7-9-11 hexanies a 3/2 apart which worked for quite a bit
> > of music. Not good for diatonic music at all, but it didn't
> > seem to matter
>
> Hiya Kraig- did I get this right?
>
> !
> Kraig Grady's dual [5 7 9 11] hexany scale.
> 12
> !
> 35/33
> 10/9
> 7/6
> 14/11
> 15/11
> 81/56
> 3/2
> 45/28
> 135/77
> 81/44
> 27/14
> 2
> !
>
> Gene - is the scale you gave constructed anything like this?

Carl,

The ratios are clearly different; those you cite
are more complex, but the step sizes are less
extreme and overall, their average deviation
from 12-EDO is smaller. I can tune them both
up on my Roland E-28 keyboard, but only by dint
of moving the reference tone A440 by a few
cents.

(Gene's) Bihexany Tuning plan for Roland E-28
Note Ratio Cents Semitone Adjust Step Semitone Adjust
C 1/1 0.00 0 0 0 -52
Db 35/33 101.87 1 2 101.87 1 -50
Eb- 7/6 266.87 3 -33 165.00 2 15
E- 5/4 386.31 4 -14 119.44 3 34
E+ 14/11 417.51 4 18 31.19 4 -34
F+ 15/11 536.95 5 37 119.44 5 -15
G 3/2 701.96 7 2 165.00 6 50
Ab 35/22 803.82 8 4 101.87 7 52
A- 5/3 884.36 9 -16 80.54 8 32
Bb- 7/4 968.83 10 -31 84.47 9 17
Bb+ 20/11 1035.00 10 35 66.17 10 -17
B+ 21/11 1119.46 11 19 84.47 11 -33
C 2/1 1200.00 80.54 average 0
ave dev 33

Carl Lumma's interpretation of
Kraig Grady's dual [5 7 9 11] hexany scale Tuning plan for Roland E-28
Note Ratio Cents Semitone Adjust Step Semitone Adjust
C 1/1 0.00 0 0 0 -19
Db 35/33 101.87 1 2 101.87 1 -17
D- 10/9 182.40 2 -18 80.54 2 -37
Eb- 7/6 266.87 3 -33 84.47 3 -52
E+ 14/11 417.51 4 18 150.64 4 -1
F+ 15/11 536.95 5 37 119.44 5 18
Gb- 81/56 638.99 6 39 102.04 6 20
G 3/2 701.96 7 2 62.96 7 -17
Ab+ 45/28 821.40 8 21 119.44 8 2
A+ 135/77 972.03 10 -28 150.64 9 53
Bb+ 81/44 1056.50 11 -43 84.47 10 38
B+ 27/14 1137.04 11 37 80.54 11 18
C 2/1 1200.00 62.96 average 0
ave dev 24

I've played a bit with Gene's tuning; my favourite
melodic inflection is between the notes I've marked
E- and E+ above, a tiny step of ~31 cents.

Harmonically, I find it works better if I treat the
E+ as the root of an "F minor" - a subdominant minor
- chord where the Eb- belongs to the C or tonic minor.
And G, B+, Db, F+ makes a usable dominant seventh.
Overall, I find it a bit weird in terms of harmonic
progressions - none are particularly satisfying. But
it *is* fun melodically.

I haven't tried your alternative yet, since I haven't
yet finished fiddling with Gene's, but it looks pro-
mising.

Regards,
Yahya

--
No virus found in this outgoing message.
Checked by AVG Free Edition.
Version: 7.1.392 / Virus Database: 268.6.0/341 - Release Date: 16/5/06

i am going to assume this is right as i don't have time to figure it out right now. Gene's looks like it has a similar construction cause it is outwardly mirroring at two points something i have noticed in CPS structures the idea was to get opposite hexanies in the eikosany so that no matter where you are, you might have something usable. From: "Carl Lumma" Date: Thu May 18, 2006 0:15am(PDT)

!
Kraig Grady's dual [5 7 9 11] hexany scale.
12
!
35/33
10/9
7/6
14/11
15/11
81/56
3/2
45/28
135/77
81/44
27/14
2

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

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

> Gene - is the scale you gave constructed anything like this?

No; mine is constructed in a way analogous to the construction of a
hexany as a lattice hole. This is a lattice hole in an 11-limit lattice.

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@...> wrote:
>>You asked for yours piano:
>pure 4:5:6:7 in C?

Try out the following circle of a dozen 5ths in absolute frequencies:
C 33 cps or Hz, "the fundamental base of the chord
G 99 :=33*3 "just pure 5th
D (37,74,148,296)297 := 99*3 = 33*9 "just pure 9th
A (55,110)111:= 37*3 yields as normal pitch A4=444 Hz := 111Hz*4
E (41,82,164)165 := 55*3 "just pure 3rd & 10th
B ´(61,122)123 := 41*3
F# 33*11:=363(366,183 :=61*3)"pure 11th-partial,attend:366/363=133/132
C# (273,546)545,1090(1089 := 363*3)
G# (205,410,820)819 := 273*3
Eb (77,154,308,616)615 := 205*3
Bb (29,58,116,232)231 := 77*3 = 33*7 hence just pure 7th
F (11,22,44,88)87 := 29*3
C 33 again := 11*3 "cycle of synchron-beating 5ths closed ready done

PC=3^12/2^19=531441/524288= subpartition
(297/296)(111/110)(165/164)(123/122)(133/132)(1089/1090)(819/820)
(615/616)(231/232)(87/88)

that tuning satisfies even yours further demand:
>A primary goal would be to have good audible implied fundamentals
>hanging around the 4:5:6:7:8:9:10:11 chords and subsets thereof.
if played on keys C:E:G:Bb:C':D':E':F#'

have a lot of fun
with that amazing sounding 8-fold chord
perhaps maybe soon on yours piano too
A.S.

On 5/19/06, a_sparschuh <a_sparschuh@yahoo.com> wrote:
> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@...> wrote:
> >>You asked for yours piano:
> >pure 4:5:6:7 in C?
>
> Try out the following circle of a dozen 5ths in absolute frequencies:
> C 33 cps or Hz, "the fundamental base of the chord
> G 99 :=33*3 "just pure 5th
> D (37,74,148,296)297 := 99*3 = 33*9 "just pure 9th
> A (55,110)111:= 37*3 yields as normal pitch A4=444 Hz := 111Hz*4
> E (41,82,164)165 := 55*3 "just pure 3rd & 10th
> B ´(61,122)123 := 41*3
> F# 33*11:=363(366,183 :=61*3)"pure 11th-partial,attend:366/363=133/132
> C# (273,546)545,1090(1089 := 363*3)
> G# (205,410,820)819 := 273*3
> Eb (77,154,308,616)615 := 205*3
> Bb (29,58,116,232)231 := 77*3 = 33*7 hence just pure 7th
> F (11,22,44,88)87 := 29*3
> C 33 again := 11*3 "cycle of synchron-beating 5ths closed ready done

I really can't understand what this means. Why did you round
everything to whole numbers of hertz? Are the octaves really supposed
to be stretched by 10-20 cents?

> PC=3^12/2^19=531441/524288= subpartition
> (297/296)(111/110)(165/164)(123/122)(133/132)(1089/1090)(819/820)
> (615/616)(231/232)(87/88)
[...]

This math doesn't work out. One of your problems might be that 366/363
= 122/121, not 133/132.

Keenan

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...>
wrote on:
>> C 33
>> G 99
>> D (37,74,148,296)297
>> A (55,110)111
>> E (41,82,164)165
>> B (61,122)123
>> F# 363(366,183)
>> C# (273,546)545,1090(1089)
>> G# (205,410,820)819
>> Eb (77,154,308,616)615
>> Bb (29,58,116,232)231
>> F (11,22,44,88)87
>> C 33
>
> I really can't understand what this means.
Perhaps you will be able to comprehend the numbers a little bit more,
if you get the same series represented in ascending order,
resorted upwards in seize afferently.
The pitches result then in the middle octave as frequencies too:

C4 264 Hz middle C
C# 272.5
D4 297
Eb 307.5
E4 330
F4 348
F# 363
G4 396
G# 409.5
A4 444 Hz, that's 4Hz sharper above standard normal-pitch 440Hz
Bb 462
B4 492
C5 524=264*2=C4*2

or if you prefer it rather in scala-file convention:

! sync_beat_11-limit.scl
!
synchronous beating 11-limit scale for C4=264Hz or A4=444Hz
12
!
545/524
9/8
615/524
5/4
348/264 ! =(4/3)(87/88)
11/8
3/2
273/176
37/22 ! =(5/3)(111/110)
7/4
41/22 ! =(15/8)(164/165)
2/1

Attend: The absolute pitches went lost in the *.scl case, because:
Scala demands normalisation of the middle C=264Hz to the ratio 1/1
proportion.

> Why did you round
> everything to whole numbers of hertz?
There is no rounding here needed.
The round brackets indicate for the tempering steps in 5hts:
How far should the according 5ths be flattened or widened.

> Are the octaves really supposed
> to be stretched by 10-20 cents?
not at all, because the temperament setting can stay completely
within the middle octave in order to define the 12 base keys.
After that it's up to your taste how much you want to stretch
the octaves alongside in treble and bass, depending also on
the string-inharmonicity found in yours piano.
That turns out individual different from instrument to instrument.

>
> > PC=3^12/2^19=531441/524288= subpartition
> > (297/296)(111/110)(165/164)(123/122)(133/132)(1089/1090)(819/820)
> > (615/616)(231/232)(87/88)
366/363=122/121, hence:
the formerly wrong subfactor 133/132
has to be replaced now by the correct one: 122/121

> One of your problems might be that 366/363
> = 122/121, not 133/132.
That miscalulation had no effect on the tuning.
> [...]
>
> This math doesn't work out.
but sounds fine
A.S.

> > Gene - is the scale you gave constructed anything like this?
>
> No; mine is constructed in a way analogous to the construction of
> a hexany as a lattice hole. This is a lattice hole in an 11-limit
> lattice.

I'd love to hear more about the difference.

-Carl

> I haven't tried your alternative yet, since I haven't
> yet finished fiddling with Gene's, but it looks pro-
> mising.

Great, Yahya!

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> > > Gene - is the scale you gave constructed anything like this?
> >
> > No; mine is constructed in a way analogous to the construction of
> > a hexany as a lattice hole. This is a lattice hole in an 11-limit
> > lattice.
>
> I'd love to hear more about the difference.

Here are some relevant articles:

/tuning-math/message/11813

/tuning-math/message/11833

/tuning-math/message/11834

/tuning-math/message/11835

/tuning-math/message/11839

/tuning-math/message/11843

Anyone wishing to avoid tuning-math stuff could just click on the last
message, which gets to to actual music.

> > > > Gene - is the scale you gave constructed anything like this?
> > >
> > > No; mine is constructed in a way analogous to the
> > > construction of a hexany as a lattice hole. This is a
> > > lattice hole in an 11-limit lattice.
> >
> > I'd love to hear more about the difference.
>
> Here are some relevant articles:
>
> /tuning-math/message/11813
>
> /tuning-math/message/11833
>
> /tuning-math/message/11834
>
> /tuning-math/message/11835
>
> /tuning-math/message/11839

Yes, well I think I grok all that. The scale I gave was
supposed to be two hexanies (as you point out, in the
11-limit, hexanies are consonant chords) a 3:2 apart. It
looks like the same is true of your scale. But I don't
follow your paragraph at the top (what "this" is referring
to -- my scale or yours). Are you saying the scale I
gave isn't as deep of a hole in the 11-limit lattice?
Incidentally, I didn't think there *were* any 12-note
holes in the 11-limit lattice.

-Carl

> --- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@>
> wrote:
Dear Keenan,
here comes additional the conversion-step into the middle octave too:

C4 264:= 33*8 > > C 33
G4 396:= 99*4 > > G 99
D4 297_______ > > D (37,74,148,296)297
A4 444:=111*4 > > A (55,110)111
E4 330:=165*2 > > E (41,82,164)165
B4 462:=123*4 > > B (61,122)123
F# 363________ >> F# 363(366,183)
C# 272.5=545/2 >> C# (273,546)545,1090(1089)
G# 409.5=819/2 >> G# (205,410,820)819
Eb 307.5=615/2 >> Eb (77,154,308,616)615
Bb 462:=231*2 > > Bb (29,58,116,232)231
F4 348:= 87*4 > > F (11,22,44,88)87
C5 524:=33*16 > > C 33
> >
> > I really can't understand what this means.
Reordering that 5ths-circle-pitches
into an arithmetical ascending series
yields chromatical:

> C4 264 Hz middle C
> C# 272.5
> D4 297
> Eb 307.5
> E4 330
> F4 348
> F# 363
> G4 396
> G# 409.5
> A4 444 Hz, that's 4Hz sharper above standard normal-pitch 440Hz
> Bb 462
> B4 492
> C5 524=264*2=C4*2
>
Dividing all that 12 pitches by C=264Hz normalizes
the frequencies it into
dimensinonless scala-file ratios with C=1/1:
>
> ! sync_beat_11-limit.scl
> !
> synchronous beating 11-limit scale for C4=264Hz or A4=444Hz
> 12
> !
> 545/524
> 9/8
> 615/524
> 5/4
> 348/264 ! =(4/3)(87/88)
> 11/8
> 3/2
> 273/176
> 37/22 ! =(5/3)(111/110)
> 7/4
> 41/22 ! =(15/8)(164/165)
> 2/1

> > Why did you round
> > everything to whole numbers of hertz?
> There is no rounding here.
It's a kind of
http://tonalsoft.com/enc/b/bridging.aspx
by using the superparticular (epimoric) bridges
as tempering steps inbetween the 5ths.
The round brackets indicate the tempering steps in 5hts:
about how far should the according 5ths be flattened or widened.
>
> > Are the octaves really supposed
> > to be stretched by 10-20 cents?
> not at all, because.....that turns out individual different from
> instrument to instrument.
http://en.wikipedia.org/wiki/Inharmonicity

Above procedere divides the PC into
> > > PC=3^12/2^19=531441/524288= subpartition
> > > (297/296)(111/110)(165/164)(123/122)(122/121)(1089/1090)(819/820)
> > > (615/616)(231/232)(87/88)

Alreday Andreas Werckmeister in his "Musicalische Temperatur 1691"
http://diapason.xentonic.org/ttl/ttl01.html
used that superparticular-subdivision method successfully
in his #6, the "Septenarius"-tuning:
/tuning/topicId_63465.html#63471

As far as i do know at the moment:
You are the first man since 315 years,
that doubts about W's idea:
> > This math doesn't work out.
Amazing!?

Above modified version is merely adapted for producing an pure
4:5:6:7 :8 :9 :10:11 :12 chord on the keys
C:E:G:Bb:C':D':E':F#':G'.
A.S.

On 5/20/06, a_sparschuh <a_sparschuh@yahoo.com> wrote:
> Dear Keenan,
> here comes additional the conversion-step into the middle octave too:
>
> C4 264:= 33*8 > > C 33
> G4 396:= 99*4 > > G 99
> D4 297_______ > > D (37,74,148,296)297
> A4 444:=111*4 > > A (55,110)111
> E4 330:=165*2 > > E (41,82,164)165
> B4 462:=123*4 > > B (61,122)123
> F# 363________ >> F# 363(366,183)
> C# 272.5=545/2 >> C# (273,546)545,1090(1089)
> G# 409.5=819/2 >> G# (205,410,820)819
> Eb 307.5=615/2 >> Eb (77,154,308,616)615
> Bb 462:=231*2 > > Bb (29,58,116,232)231
> F4 348:= 87*4 > > F (11,22,44,88)87
> C5 524:=33*16 > > C 33

There are still some simple arithmetic errors here. 123*4 = 492, not
462, but I suspect that's just a typo.

I can't understand how you get from B to F#. It seems like you're
trying to make all the fifths differ from 3/2 by a superparticular
ratio, but 363/246 differs from 3/2 by 123/121, which is not
superparticular. It's unclear to me what the parentheses and the
ordering of the numbers mean.

More importantly, what are you trying to achieve with this procedure?

> > ! sync_beat_11-limit.scl
> > !
> > synchronous beating 11-limit scale for C4=264Hz or A4=444Hz
> > 12
> > !
> > 545/524
> > 9/8
> > 615/524
> > 5/4
> > 348/264 ! =(4/3)(87/88)
> > 11/8
> > 3/2
> > 273/176
> > 37/22 ! =(5/3)(111/110)
> > 7/4
> > 41/22 ! =(15/8)(164/165)
> > 2/1

Shouldn't C# be 545/528 rather than 545/524?

> > > Why did you round
> > > everything to whole numbers of hertz?
> > There is no rounding here.
> It's a kind of
> http://tonalsoft.com/enc/b/bridging.aspx
> by using the superparticular (epimoric) bridges
> as tempering steps inbetween the 5ths.
> The round brackets indicate the tempering steps in 5hts:
> about how far should the according 5ths be flattened or widened.

I'm familiar with that, but it seems a little old-fashioned now that
we have a solid mathematical theory of regular temperaments.

> > > Are the octaves really supposed
> > > to be stretched by 10-20 cents?
> > not at all, because.....that turns out individual different from
> > instrument to instrument.
> http://en.wikipedia.org/wiki/Inharmonicity

I'm quite familiar with that, but when you gave 55 and 111 as Hertz
values for the same pitch class I didn't know what to think. Forget
it.

> Above procedere divides the PC into
> > > > PC=3^12/2^19=531441/524288= subpartition
> > > > (297/296)(111/110)(165/164)(123/122)(122/121)(1089/1090)(819/820)
> > > > (615/616)(231/232)(87/88)
>
> Alreday Andreas Werckmeister in his "Musicalische Temperatur 1691"
> http://diapason.xentonic.org/ttl/ttl01.html
> used that superparticular-subdivision method successfully
> in his #6, the "Septenarius"-tuning:
> /tuning/topicId_63465.html#63471
>
> As far as i do know at the moment:
> You are the first man since 315 years,
> that doubts about W's idea:

Um, I never said anything about ideas, I just pointed out that that
mathematical equation is false. Probably because you left out 545/546.

> > > This math doesn't work out.
> Amazing!?
>
> Above modified version is merely adapted for producing an pure
> 4:5:6:7 :8 :9 :10:11 :12 chord on the keys
> C:E:G:Bb:C':D':E':F#':G'.
> A.S.

That's quite clear to me, but what about the other notes?

Keenan

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
correction:
> > C4 264:= 33*8 > > C 33
> > G4 396:= 99*4 > > G 99
> > D4 297_______ > > D (37,74,148,296)297
> > A4 444:=111*4 > > A (55,110)111
> > E4 330:=165*2 > > E (41,82,164)165
___ B4 492:=123*4 > > B (61,122)123 "instead former-wrong "typo": 462
!!! F# 363________ >> F# 363(366,183) "366/363=122/121
> > C# 272.5=545/2 >> C# (273,546)545,1090(1089)
> > G# 409.5=819/2 >> G# (205,410,820)819
> > Eb 307.5=615/2 >> Eb (77,154,308,616)615
> > Bb 462:=231*2 > > Bb (29,58,116,232)231
> > F4 348:= 87*4 > > F (11,22,44,88)87
> > C5 528:=33*16 > > C 33
>
> I can't understand how you get from B to F#.
The !!! 5th B>F# has to be flattend down by the product of
(123/122)*(122/121) = 123/121 = 61.5/60.5
because 122 cancels out in nominator versus denominator.
I frankly admit:
Tempering the 5th: B>F# about
(1 200 * ln(61.5 / 60.5)) / ln(2) = ~28.38...Cents
flat sounds a bit harsh even in my ears, due to enforcing the:
http://www.google.de/search?as_q=&num=10&hl=de&btnG=Google-Suche&as_epq=alphorn+fa&as_oq=&as_eq=&lr=lang_en&as_ft=i&as_filetype=&as_qdr=all&as_occt=any&as_dt=i&as_sitesearch=&as_rights=&safe=images
11/8 alphorn-fa on C>F#.

> It seems like you're
> trying to make all the fifths differ from 3/2 by a superparticular
> ratio,
Yes, in deed, in imitating Werckmeister's "Septenarian" way,
See for deeper ananlysis also the later decomposition into prime-factors.

> but 363/246 differs from 3/2 by 123/121, which is not
> superparticular.
but the composite 61.5/60.5:=123/121 satisfies again that proprty,
if we allow additional half-integral superparticulars as valid too.

> It's unclear to me what the parentheses and the
> ordering of the numbers mean.
The value in parentheses versus the bare without the
parenthesis indicate the amount of tempering the 5ths.
Hence:
The values enclosed inbetween the brackets represent
only the virtual pitches, that an just-pure 5th step
(factor 3:2) would have merely thought ,
instead/versus the real tempered pitch-numbers,
without any parentheses barely.
>
> More importantly, what are you trying to achieve with this
> procedure?
Just an circle of a dozen tempered 5ths that includes an
> > 4:5:6:7 :8 :9 :10:11 :12 chord on the keys
> > C:E:G:Bb:C':D':E':F#':G'.

>
> > > ! sync_beat_11-limit.scl
> > > !
> > > synchronous beating 11-limit scale for C4=264Hz or A4=444Hz
> > > 12
> > > !
_____ 545/528 !now corrected instead faulty denominator 524 formerly
> > > 9/8
> > > 615/528
> > > 5/4
> > > 87/66 ! =(4/3)(87/88)
> > > 11/8
> > > 3/2
> > > 273/176
> > > 37/22 ! =(5/3)(111/110)
> > > 7/4
> > > 41/22 ! =(15/8)(164/165)
> > > 2/1
>
> Shouldn't C# be 545/528 rather than 545/524?
The denominator 528 is correct, due to C5 528:=33*16,
hence just another typo error in transferring from
paper to posting. Sorry! Thanx for yours patience.
'hope, that now at least my numbers fit accurate.
>
>
> I'm familiar with that, but it seems a little old-fashioned now that
> we have a solid mathematical theory of regular temperaments.
Antediluvianic integer arithmtics avoids the faultyness of
logarithms in the "regular" theory, in order to get rid of
accumualting logarithmic rounding-errors, that you have inavoidable
alyways inherent included in modern ET "regular" systems.

Consider the Advantages of the traditional way:
1.Everything can executed easily merely by pencil and paper,
without any need of electronic calculators or even slide rulers.
2. All 5ths beatings are synchroneous to 1 Hz or Metronome: 60 beats.
3. You got exaclty all the demanded ratios 4:5:....,11:12 just pure,
instead merelyarbitray incontrolable numerical approximations
whatsoever.
4. The way of calculating represents the procedere in practical tuning
too.
Are that enough convincing arguments in order to prefer W's old method?
> > > > Are the octaves really supposed
> > > > to be stretched by ...the... instruments.
> > http://en.wikipedia.org/wiki/Inharmonicity
>
> I'm quite familiar with that, but when you gave 55 and 111 as Hertz
> values for the same pitch class I didn't know what to think.

On A2,3,4 only 111,222,444 got tuned in practice.
in contrast remain (55,110,220,440 in the brackets)
merely virtual meant, without got tuned real in practice:
A (55,110)111 merely 111 matters to represent the significant pitch
E 165:=55*3
hence the 5th A>E amounts =165/111. It becomes 111/110
(1 200 * ln(111 / 110)) / ln(2) = ~15.67....Cents
flattend down, than if it would be just pure
3/2 = 165/110 =, because =(165/111)(111/110).

In general:
Subtracting any arbitray argument N the difference (-1)
is algebraic equivalent to an multiplication of N times (N-1)/N.
Proof: N * ((N-1)/N) = N-1. q.e.d. done by shortening.

>Forget it.
Why?
But W's old method yields exact the desired result,
instead merely approximating the true ratios by
irrational-act numbers of the "regular" ET theory.
The "regular" ET theory excludes inherently, due to of beeing
resticted only to irrational-numbers, to obtain the correct ratios of the
4:5:6:7...:11:12 chord in an finte amount of numerical steps,
directly correct, neither on the paper nor on the machine,
and must hence refused as inferior,
in applying Occams-razor!
>
> > Above procedere divides the PC into
> > > > > PC=3^12/2^19=531441/524288= subpartition
> > > > > (297/296)(111/110)(165/164)(123/122)(122/121)(1089/1090)
insert here: (545/546) becaue i forgot that factor.
> > > >(819/820)
> > > > > (615/616)(231/232)(87/88)
> Um,... I just pointed out that that
> mathematical equation is false.
ok, let's factorize the 11 terms out into prime decomposition:
__297/296__ __11*3^3/37*2^3
__111/110__ ____37*3/55*2
__165/164__ ____55*3/41*2^2
__123/122__ ____41*3/61*2
__122/121__ ___61*2/11^2
_1089/1090_ 11^2*3^2/545*2
__545/546__ _____545/273*2 "that one went lost last time, sorry
__819/820__ ___273*3/205*2^2
__615/616__ ___205*3/77*2^3
__231/232__ ____77*3/29*8
___87/88___ ____29*3/11*2^3
That results in
total over all : 3^12/2^19,
the collective product over all 11 superparticulares.
Factors not equal to powers of 3 or 2 do cancel out each others in
nominator versus denominator, so that just merely the PC=3^12/2^19
remains:
Simply add the exponents of the 3s and respectively of the 2s.
That yields two sums in the powers of the 3s: =12,
respectively in the 2s: =-19,
makes concluding final: 3^12/2^19=531441/528244. q.e.d.

> Probably because you left out 545/546.
That remark looks already alike,
you got meanwhile be able to understand
a little more about comprehending the concept.
>
> That's quite clear to me, but what about the other notes?
they are chosen in above way inbetween the given specified ratios,
in order to interpolate the cycle of 12 5hts,
as smooth as possible,
under the predetermined restrictions like:
> > pure
> > 4:5:6:7 :8 :9 :10:11 :12 chord on the keys
> > C:E:G:Bb:C':D':E':F#':G'.

Summary in other words:
The values in parenthesises serve merely as
auxiliary-variables, working for the intermediate
5ths-tempering calculation steps inbetween,
but become dispensable for yielding the final result,
hence they do appear carried along in brackets merely virtual,
in order to indicate the priority of the intended 12 bare
pitch-frequencies against the assumed pure ones,
in order to compensate by/on the way the PC,
by its subdivision into superparticular factors.

Is that to grasp really so difficult?
A.S.

On 5/22/06, a_sparschuh <a_sparschuh@yahoo.com> wrote:
> The !!! 5th B>F# has to be flattend down by the product of
> (123/122)*(122/121) = 123/121 = 61.5/60.5
> because 122 cancels out in nominator versus denominator.
> I frankly admit:
> Tempering the 5th: B>F# about
> (1 200 * ln(61.5 / 60.5)) / ln(2) = ~28.38...Cents
> flat sounds a bit harsh even in my ears, due to enforcing the:
> http://www.google.de/search?as_q=&num=10&hl=de&btnG=Google-Suche&as_epq=alphorn+fa&as_oq=&as_eq=&lr=lang_en&as_ft=i&as_filetype=&as_qdr=all&as_occt=any&as_dt=i&as_sitesearch=&as_rights=&safe=images
> 11/8 alphorn-fa on C>F#.

It sounds harsh to my ears too. That's neat about "alphorn fa" though,
maybe I'll work that into a Wikipedia article.

> > but 363/246 differs from 3/2 by 123/121, which is not
> > superparticular.
> but the composite 61.5/60.5:=123/121 satisfies again that proprty,
> if we allow additional half-integral superparticulars as valid too.

LOL, and what if we allow 2.25/1.25 (=9/5) and 6.4/5.4 (=32/27) and so
on? Any ratio can be "superparticular"!

> > I'm familiar with that, but it seems a little old-fashioned now that
> > we have a solid mathematical theory of regular temperaments.
> Antediluvianic integer arithmtics avoids the faultyness of
> logarithms in the "regular" theory, in order to get rid of
> accumualting logarithmic rounding-errors, that you have inavoidable
> alyways inherent included in modern ET "regular" systems.
>
> Consider the Advantages of the traditional way:
> 1.Everything can executed easily merely by pencil and paper,
> without any need of electronic calculators or even slide rulers.

I frequently work out tempered scales with pencil and paper. I just
remember the cents values of the first few prime numbers (3/2 is 702
cents, 5/4 is 386 cents, and so on), and everything else can be
expressed as linear combinations of those.

> 2. All 5ths beatings are synchroneous to 1 Hz or Metronome: 60 beats.

Okay, that makes sense. That's obviously what you meant by
"synchronous beating". Excuse me for not realizing that sooner.

> 3. You got exaclty all the demanded ratios 4:5:....,11:12 just pure,
> instead merelyarbitray incontrolable numerical approximations
> whatsoever.

Well... all the intervals other than those in the hexad C, D, E, F#,
G, Bb are going to be tempered anyway. The resulting system has a
jarring contrast between the pure intervals of the hexad and the
(sometimes drastically) tempered intervals outside it. I say "jarring"
but it could be musically useful - I'd love to hear some music in this
tuning.

> 4. The way of calculating represents the procedere in practical tuning
> too.

Because the beat frequencies are proportional. Got it.

> Are that enough convincing arguments in order to prefer W's old method?

Maybe, but I won't be convinced until I hear some music.

> >Forget it.
> Why?

I meant forget that I said anything about stretched octaves, because I
misinterpreted your numbers.

> But W's old method yields exact the desired result,
> instead merely approximating the true ratios by
> irrational-act numbers of the "regular" ET theory.
> The "regular" ET theory excludes inherently, due to of beeing
> resticted only to irrational-numbers, to obtain the correct ratios of the
> 4:5:6:7...:11:12 chord in an finte amount of numerical steps,
> directly correct, neither on the paper nor on the machine,
> and must hence refused as inferior,
> in applying Occams-razor!

Regular temperaments are not "restricted only to irrational-numbers",
because of something called an "eigenmonzo" (I love that word!). For
example, quarter-comma meantone has an eigenmonzo of 5/4, which means
that although the perfect fifth is tempered to an irrational number,
four of them (less two octaves) make exactly the rational interval
5/4.

Keenan

A.S.,

Thanks, sorry so slow responding...

on 5/19/06 6:40 AM, a_sparschuh <a_sparschuh@yahoo.com> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@...> wrote:
>>> You asked for yours piano:
>> pure 4:5:6:7 in C?
>
> Try out the following circle of a dozen 5ths in absolute frequencies:
> C 33 cps or Hz, "the fundamental base of the chord
> G 99 :=33*3 "just pure 5th
> D (37,74,148,296)297 := 99*3 = 33*9 "just pure 9th
> A (55,110)111:= 37*3 yields as normal pitch A4=444 Hz := 111Hz*4
> E (41,82,164)165 := 55*3 "just pure 3rd & 10th
> B ´(61,122)123 := 41*3
> F# 33*11:=363(366,183 :=61*3)"pure 11th-partial,attend:366/363=133/132
> C# (273,546)545,1090(1089 := 363*3)
> G# (205,410,820)819 := 273*3
> Eb (77,154,308,616)615 := 205*3
> Bb (29,58,116,232)231 := 77*3 = 33*7 hence just pure 7th
> F (11,22,44,88)87 := 29*3
> C 33 again := 11*3 "cycle of synchron-beating 5ths closed ready done
>
> PC=3^12/2^19=531441/524288= subpartition
> (297/296)(111/110)(165/164)(123/122)(133/132)(1089/1090)(819/820)
> (615/616)(231/232)(87/88)
>
> that tuning satisfies even yours further demand:
>> A primary goal would be to have good audible implied fundamentals
>> hanging around the 4:5:6:7:8:9:10:11 chords and subsets thereof.
> if played on keys C:E:G:Bb:C':D':E':F#'
>
> have a lot of fun
> with that amazing sounding 8-fold chord
> perhaps maybe soon on yours piano too
> A.S.

That's interesting. I'll try it on my tunable synth.

-Kurt

Gene and others,

on 5/16/06 1:11 PM, Gene Ward Smith <genewardsmith@coolgoose.com> wrote:

> I think there are interesting 11-limit possibilities with just twelve
> notes. Here is the bihexany, which I think is quite interesting:
>
> ! bihexany.scl
> Hole around [0, 1/2, 1/2, 1/2]
> 12
> !
> 35/33
> 7/6
> 5/4
> 14/11
> 15/11
> 3/2
> 35/22
> 5/3
> 7/4
> 20/11
> 21/11
> 2

I seem to recall that Gene or someone had made some reasonable lattice
representation of this (in 2-d projection), but I can't find it, and I'm not
having any luck figuring it out myself.

I'd appreciate any pointers, if any such useful visual representation
exists.

> Tempering out 441/440 and 540/539, for instance by 161-et, is a
> retuning plan.

And that is probably an even more visualizable result, I'd imagine.

Thanks.

-Kurt

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@...> wrote:
multiply all ratios by factor 264=11*3*2*2 in order to
obtain integral absolute pitch-frequencies
> > ! bihexany.scl
> > Hole around [0, 1/2, 1/2, 1/2]
> > 12
> > !
C4 264 from 1/1 middle C=264Hz
C# 280 > 35/33
(D4 297 > 9/8) inserted instead Cb=B# for smoothening the 5ths
Eb 308 > 7/6
E4 330 > 5/4
F4 336 > 14/11
F# 360 > 15/11
G4 396 > 3/2
G# 420 > 35/22
A4 440 > 5/3 !normal pitch 440Hz
Bb 462 > 7/4
B4 480 > 20/11
(Cb=B# 504 > 21/11)
C5 528 > 2

or arranged in a cycle of a dozen partial tempered 5ths

C 33,..,264 begin, starting @ middle-C
G 99:=33*3,198,396
D (37,74,148,296)297:=99*3
A 55,110(111:=37*3),220,440 ! (111:110)*(297:296)=81:80 the SC
E (41,82,164)165:=55*3,330
B 15,30,60,,(123:=41*3)120,240,480 ! (165:164)*(41:40)=33:32
F# 45:=15*3,..,360
C# 35,70,(135:=45*3)140,280 ! 140:135=28:27
G# (13,26,52,104)105:=35*3,210,420
Eb (39:=13*3,78)77,154,308
!(105:104)*(77:78)=2695:2704=299.4444.../300.4444...
Bb (7,...,228)231:=77*3,462 ! 231:228=77:76
F (11,22)21:=7*3,42,84,168,336
C 33:=11*3 cycle closed, ready done.

i.m.o: heavy harsh tempering, but it works.
A.S.

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@...> wrote:

> I seem to recall that Gene or someone had made some reasonable lattice
> representation of this (in 2-d projection), but I can't find it, and
I'm not
> having any luck figuring it out myself.
>
> I'd appreciate any pointers, if any such useful visual representation
> exists.

The first neat fact to bear in mind is that everything in the 7-limit
can be expressed in terms of 2, 49/40, 10/7, and 2401/2400, and that
last number is really, really small. If we convert the above to monzos
and invert the 4x4 matrix, we find that

[<1 1 1 2|, <0 2 1 1|, <0 0 2 1|, <0 -1 0 0|]

puts the 7-limit into the above system, and we get the projection of
7-limit pitches onto a two-dimensional picture of pitch classes by
ignoring the first (octave) and last (2401/2400) vals. This gives a
lattice picture in the plane of the 7-limit; the hexany turns itself
into a hexagon, which is cute.

Now if we take the 49/40 neutral third to be an 11/9, we have a much
less accurate tuning system, but still one more accurate than (and
compatible with) miracle temperament, so it's not at all bad. If we do
the same trick as befor only with 2, 11/9, 10/7, 243/242, 441/440,
then we can ignore the 2 and the 243/242 and 441/440 vals (obtained as
before by inverting the square matrix) and just use

[<0 2 1 1 5|, <0 0 2 1 0|]

to project 11-limit JI to pitch classes in a planar lattice picture.

This gives the following twelve points when applied to the bihexany:

{[1, 2], [1, 1], [-5, 3], [-4, 1], [-1, 1], [2, 0], [-2, 2], [-1, 2],
[-3, 3], [-2, 1], [-4, 2], [0, 0]}

Plot that and you should get the picture.

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@...> wrote:
> That's interesting. I'll try it on my tunable synth.
if you do understand german,
then try to implement that too in:

http://www.math.tu-dresden.de/~mutabor/

a freeware public-domain musical programming-language.
Mutabor is capable to detect interval-seizes online
and allows instant automatic retuning in realtime,
so that almost every played multi-chord sounds just as pure
as possible. Hence you are no longer resticted
to merely the C-major chord 4:5:6:7:8:9:10:11:12.....
as if in the discussed fix-tuned case.
Caution:
The general keyboard reference-pitch may move around
wandering about in comma steps up and down: 81/80, 64/63 or
yet even in double commatas 33/32:=(66/65)*(65/64) alongside,
depending and following on the actual performed modulations,
while keeping during that all involved intervals
consistent beatless clear and pure;
due to retuning if needed
and demanded by the chord-progression whatsoever you play.
Hence we got rid of any enharmonics:
Mind-boggling!

enjoy that real-time fun
A.S.

Gene,

on 5/26/06 2:12 AM, Gene Ward Smith <genewardsmith@coolgoose.com> wrote:

> --- In tuning@yahoogroups.com, Kurt Bigler <kkb@...> wrote:
>> I seem to recall that Gene or someone had made some reasonable lattice
>> representation of this (in 2-d projection), but I can't find it, and
>> I'm not having any luck figuring it out myself.
>
> The first neat fact to bear in mind is that everything in the 7-limit
> can be expressed in terms of 2, 49/40, 10/7, and 2401/2400, and that
> last number is really, really small. If we convert the above to monzos
> and invert the 4x4 matrix, we find that
>
> [<1 1 1 2|, <0 2 1 1|, <0 0 2 1|, <0 -1 0 0|]
>
> puts the 7-limit into the above system, and we get the projection of
> 7-limit pitches onto a two-dimensional picture of pitch classes by
> ignoring the first (octave) and last (2401/2400) vals. This gives a
> lattice picture in the plane of the 7-limit; the hexany turns itself
> into a hexagon, which is cute.
>
> Now if we take the 49/40 neutral third to be an 11/9, we have a much
> less accurate tuning system, but still one more accurate than (and
> compatible with) miracle temperament, so it's not at all bad. If we do
> the same trick as befor only with 2, 11/9, 10/7, 243/242, 441/440,
> then we can ignore the 2 and the 243/242 and 441/440 vals (obtained as
> before by inverting the square matrix) and just use
>
> [<0 2 1 1 5|, <0 0 2 1 0|]
>
> to project 11-limit JI to pitch classes in a planar lattice picture.
>
> This gives the following twelve points when applied to the bihexany:
>
> {[1, 2], [1, 1], [-5, 3], [-4, 1], [-1, 1], [2, 0], [-2, 2], [-1, 2],
> [-3, 3], [-2, 1], [-4, 2], [0, 0]}
>
> Plot that and you should get the picture.

Thanks, I'll take a look at that.

I've got a bit of reviewing to do to understand all this, but it is a good
exercise.

But maybe you can give me a couple of hints.

The originaly scale involved primes 2,3,5,7,11 and is therefore
5-dimensional. Ignoring the 2 makes it 4-dimensional. Tempering out the
2401/2400 should then reduce it to 3-dimensional. I think this is what you
mean by the "plane" of the 7-limit, a 3-plane, right? But: 2401/2400
involves a relation between the prime 7 and the primes 2,3,5. To me
intuitively that is a way of making the prime 7 disappear into the 5-limit,
except oops the 11 still remains. So it confuses me to call this the
7-limit plane. I'd think it would be called 11-limit with 7 missing (is
there a name for that?).

Meanwhile, I think it is the previous projection I was more interested in,
with the nexany turning into a hexagon. Do you have plot coordinates for
that handy? ;)

Thanks.

-Kurt

> > If we do
> > the same trick as befor only with
> > 2, 11/9, 10/7, 243/242, 441/440, then we can ignore
> > the 2 and the 243/242 and 441/440 vals (obtained as
> > before by inverting the square matrix) and just use
> >
> > [<0 2 1 1 5|, <0 0 2 1 0|]
> >
> > to project 11-limit JI to pitch classes in a planar
> > lattice picture.

> > This gives the following twelve points when applied to the
> > bihexany:
> >
> > {[1, 2], [1, 1], [-5, 3], [-4, 1], [-1, 1], [2, 0], [-2, 2],
> > [-1, 2], [-3, 3], [-2, 1], [-4, 2], [0, 0]}
> >
> > Plot that and you should get the picture.
>
> Thanks, I'll take a look at that.

Yeah, if somebody could plot that, I'd love to see it.

> The originaly scale involved primes 2,3,5,7,11 and is therefore
> 5-dimensional. Ignoring the 2 makes it 4-dimensional. Tempering
> out the 2401/2400 should then reduce it to 3-dimensional. I
> think this is what you mean by the "plane" of the 7-limit, a
> 3-plane, right? But: 2401/2400 involves a relation between the
> prime 7 and the primes 2,3,5. To me intuitively that is a way
> of making the prime 7 disappear into the 5-limit, except oops
> the 11 still remains. So it confuses me to call this the
> 7-limit plane. I'd think it would be called 11-limit with 7
> missing (is there a name for that?).

Sorry to butt in, but I think the 2401/2400 thing was just
an example -- a 7-limit one. The actual 11-limit project
ignores 2, 243/242 and 441/440, if I'm reading that right
(above).

There's no simple name for '11-limit with 7 tempered out'
that I know of.

-Carl

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

> Sorry to butt in, but I think the 2401/2400 thing was just
> an example -- a 7-limit one. The actual 11-limit project
> ignores 2, 243/242 and 441/440, if I'm reading that right
> (above).

Since (441/440)^2/(243/242) = 2401/2400, it also tempers that out.
Plus, (243/242)/(441/440) = 540/539.

on 5/26/06 6:58 PM, Carl Lumma <clumma@yahoo.com> wrote:

>>> If we do
>>> the same trick as befor only with
>>> 2, 11/9, 10/7, 243/242, 441/440, then we can ignore
>>> the 2 and the 243/242 and 441/440 vals (obtained as
>>> before by inverting the square matrix) and just use
>>>
>>> [<0 2 1 1 5|, <0 0 2 1 0|]
>>>
>>> to project 11-limit JI to pitch classes in a planar
>>> lattice picture.
>
>>> This gives the following twelve points when applied to the
>>> bihexany:
>>>
>>> {[1, 2], [1, 1], [-5, 3], [-4, 1], [-1, 1], [2, 0], [-2, 2],
>>> [-1, 2], [-3, 3], [-2, 1], [-4, 2], [0, 0]}
>>>
>>> Plot that and you should get the picture.
>>
>> Thanks, I'll take a look at that.
>
> Yeah, if somebody could plot that, I'd love to see it.

I'm working on getting Mathematica to do this. I can probably provide a png
file when I have something that looks decent.

>> The originaly scale involved primes 2,3,5,7,11 and is therefore
>> 5-dimensional. Ignoring the 2 makes it 4-dimensional. Tempering
>> out the 2401/2400 should then reduce it to 3-dimensional. I
>> think this is what you mean by the "plane" of the 7-limit, a
>> 3-plane, right? But: 2401/2400 involves a relation between the
>> prime 7 and the primes 2,3,5. To me intuitively that is a way
>> of making the prime 7 disappear into the 5-limit, except oops
>> the 11 still remains. So it confuses me to call this the
>> 7-limit plane. I'd think it would be called 11-limit with 7
>> missing (is there a name for that?).
>
> Sorry to butt in,

Heck, no problem. Fancy meeting you here. :)

> but I think the 2401/2400 thing was just
> an example -- a 7-limit one. The actual 11-limit project
> ignores 2, 243/242 and 441/440, if I'm reading that right
> (above).

Hmm, ok. Since I haven't yet quite assimilated all the low-level concepts,
it's too easy for me to misconstrue the high-level structure of these
expositions.

> There's no simple name for '11-limit with 7 tempered out'
> that I know of.

Will wonders never cease!

But even "tempered out" remains a slightly elusive concept to me. It seems
to me there are some intuitive (and functional) distinctions that get
glossed over in the fast-and-loose use of the tempering concept that seems
to come so easily to people with years of experience with this stuff.

Here's some things that come to mind along those lines. Maybe somebody
could give me a reality check on these thoughts:

Tempering doesn't make a prime disappear, it just folds the prime space in
some way. And there is probably is a tendency to want to reduce something
to a lower prime limit when possible, for practical reasons, either for
generating lattice diagrams or maybe for aural tuning.

So taking 225/224 as an example, this relates the primes 3,5 to the prime 7
(ignoring 2), and intuitively I've always thought of this as a way of
reducing a 7-limit capability into a 5-limit structure. Or alternately it
is a way of finding 7-limit utility in a 5-limit structure. But I guess you
have to say you are tempering out a comma, not a prime, and I suppose most
usage on this list has been pretty clear about that. It gets even more
obvious with 441/440 which (in the language I've just been using) relates
primes 3,7 to primes 5,11 (ignoring 2 again). Clearly in that case there is
no single prime that can be made to disappear. Rather pairs of primes fold
onto other pairs of primes.

So would you call 2401/2400 (7*7*7*7/(2*2*2*2*2*3*5*5)) a 7-limit comma?

But when you say "The actual 11-limit project ignores 2, 243/242 and
441/440" and you go about creating a lattice for the result, it seems to me
that would usually be called a 5-limit lattice, isn't that right? At the
same time the temperament acknowledges the equivalence of this to an
11-limit functionality. And as I recall, the concept of temperament wants
to remain a little vague about the actual tuning, so you could in this
example make the 7- and 11-limit intervals exact at the expense of others,
or you could make others exact at the expense of 7 and 11. Or you could do
some other optimization.

But what I'm leading to is this question: Isn't it potentially useful to
distinguish:

the minimum limit needed to implement a tuning
the maximum useful functioning limit of that tuning

and also it might be useful for classification to indicate that some primes
are simply not present in any useful way, i.e. you might conceivably have an
11-limit scale that contains no intervals that could conceivably be tempered
to offer any functioning intervals that involved the prime 7. So in this
case you might want to say that 7 is absent. This might be true for 24-et
for example. It is a good 11-limit tuning, but provides no useful 7-limit
capability. Useful is of course relative, so this may technically require
specifying an error tolerance. Nonetheless it could be looked at the other
way, I'd think, i.e. a definition of a temperament might specify what
functionality is intended to be used without specifying a tolerance.

Similarly one might want to specify what limit numbers are considered to be
exact and which are considered to be approximated but useful. For example
Duodene can be tuned exact in the 5-limit but can be construed to provide
useful 7-limit functioning. Yet the 11-limit would not be intended, and
probbly could not be construed in any reasonable way as being present.

This leads me to think of the following kind of classification system, based
like the monzo on an implicit reference to the series of primes, where each
element may be one of the following letters used as abbreviations:

e = exact
a = approximate
- = absent

And you could smash the resulting monzoish entity together into a series of
characters without other punctuation, so that Duodene could be classified as
"eeea" meaning 2,3,5 are exact and 7 is approximate. And 24-et could be
"eaa-a".

Does this seem at all useful?

-Kurt

> I'm working on getting Mathematica to do this. I can probably
> provide a png file when I have something that looks decent.

Great!

> But even "tempered out" remains a slightly elusive concept to
> me. It seems to me there are some intuitive (and functional)
> distinctions that get glossed over in the fast-and-loose use
> of the tempering concept that seems to come so easily to people
> with years of experience with this stuff.

You can think in terms of 81/80 reducing the 5-limit to a
linear thing -- and back it up with examples at your piano!

> Here's some things that come to mind along those lines.
> Maybe somebody could give me a reality check on these thoughts:
>
> Tempering doesn't make a prime disappear, it just folds the
> prime space in some way. And there is probably is a tendency
> to want to reduce something to a lower prime limit when
> possible, for practical reasons, either for generating
> lattice diagrams or maybe for aural tuning.

I think this is correct.

It's good not only for diagrams and bearing plans, but also
to reduce the number of distinct pitches needed to convey
chord progressions, melodies, etc.

> So taking 225/224 as an example, this relates the primes
> 3,5 to the prime 7 (ignoring 2), and intuitively I've always
> thought of this as a way of reducing a 7-limit capability
> into a 5-limit structure. Or alternately it is a way of
> finding 7-limit utility in a 5-limit structure.

True. One doesn't have to employ irrational intervals.
One can "temper" simply by treating the 7/4 as a 225/128
or vice versa.

> Clearly in that case there is
> no single prime that can be made to disappear.

You could also think of tempering as causing a prime
to *appear*.

> So would you call 2401/2400 (7*7*7*7/(2*2*2*2*2*3*5*5)) a
> 7-limit comma?

Yes.

> But when you say "The actual 11-limit project ignores 2, 243/242
> and 441/440" and you go about creating a lattice for the result,
> it seems to me that would usually be called a 5-limit lattice,
> isn't that right?

No, I don't think so.

> And as I recall, the concept of temperament wants
> to remain a little vague about the actual tuning, so you could
> in this example make the 7- and 11-limit intervals exact at the
> expense of others, or you could make others exact at the expense
> of 7 and 11. Or you could do some other optimization.

Yes, that's right. While piano tuners might use the terms
scale, tuning, and temperament interchangably, that isn't the
tuning-math-approved way. There are temperaments, which are
abstract objects. One chooses a tuning for them (typically
minimizing something like, say, the max error from JI). And
then one chooses a finite number of notes and arrives at a
scale. Meantone is a temperament, 1/4-comma is a tuning, and
12 notes of 1/4-comma from Ab - D# (or whatever) is a scale.

> Isn't it potentially useful to
> distinguish:
>
> the minimum limit needed to implement a tuning
> the maximum useful functioning limit of that tuning

Not sure what you're asking here.

> and also it might be useful for classification to indicate
> that some primes are simply not present in any useful way,
> i.e. you might conceivably have an 11-limit scale that
> contains no intervals that could conceivably be tempered
> to offer any functioning intervals that involved the prime 7.

I'm not aware of this kind of thing having been undertaken,
but there is the notion of a 'good' periodicity block -- a
section of the JI lattice delimited by commas that contains
no intervals smaller than those commas.

> And you could smash the resulting monzoish entity together into
> a series of characters without other punctuation, so that
> Duodene could be classified as "eeea" meaning 2,3,5 are exact
> and 7 is approximate. And 24-et could be "eaa-a".
>
> Does this seem at all useful?

It seems better to me to just give the errors.

-Carl

Kurt Bigler wrote:

> Tempering doesn't make a prime disappear, it just folds the prime space in
> some way. And there is probably is a tendency to want to reduce something
> to a lower prime limit when possible, for practical reasons, either for
> generating lattice diagrams or maybe for aural tuning.
> > So taking 225/224 as an example, this relates the primes 3,5 to the prime 7
> (ignoring 2), and intuitively I've always thought of this as a way of
> reducing a 7-limit capability into a 5-limit structure. Or alternately it
> is a way of finding 7-limit utility in a 5-limit structure. But I guess you
> have to say you are tempering out a comma, not a prime, and I suppose most
> usage on this list has been pretty clear about that. It gets even more
> obvious with 441/440 which (in the language I've just been using) relates
> primes 3,7 to primes 5,11 (ignoring 2 again). Clearly in that case there is
> no single prime that can be made to disappear. Rather pairs of primes fold
> onto other pairs of primes.
> > So would you call 2401/2400 (7*7*7*7/(2*2*2*2*2*3*5*5)) a 7-limit comma?

Yes. But it doesn't give you a 5-limit structure. One consequence of tempering out 2401:2400 is that you have true neutral thirds: the perfect fifth can be divided into two equal parts. That's why we're talking about a new lattice geometry.

> But when you say "The actual 11-limit project ignores 2, 243/242 and
> 441/440" and you go about creating a lattice for the result, it seems to me
> that would usually be called a 5-limit lattice, isn't that right? At the
> same time the temperament acknowledges the equivalence of this to an
> 11-limit functionality. And as I recall, the concept of temperament wants
> to remain a little vague about the actual tuning, so you could in this
> example make the 7- and 11-limit intervals exact at the expense of others,
> or you could make others exact at the expense of 7 and 11. Or you could do
> some other optimization.
> > But what I'm leading to is this question: Isn't it potentially useful to
> distinguish:
> > the minimum limit needed to implement a tuning
> the maximum useful functioning limit of that tuning
> > and also it might be useful for classification to indicate that some primes
> are simply not present in any useful way, i.e. you might conceivably have an
> 11-limit scale that contains no intervals that could conceivably be tempered
> to offer any functioning intervals that involved the prime 7. So in this
> case you might want to say that 7 is absent. This might be true for 24-et
> for example. It is a good 11-limit tuning, but provides no useful 7-limit
> capability. Useful is of course relative, so this may technically require
> specifying an error tolerance. Nonetheless it could be looked at the other
> way, I'd think, i.e. a definition of a temperament might specify what
> functionality is intended to be used without specifying a tolerance.

Yes, but it's not always a matter of primes. 31-equal works well with 11:9 but not so well with 9:8 or 11:8. So how would you describe it's approximation to the prime 11?

You're touching on the whole idea of searching for temperaments that work well in an arbitrary subset of a chosen odd or prime limit. I think it's an important unsolved theoretical problem. But it's one that should really be discussed on tuning-math.

Graham

on 5/26/06 8:33 PM, Kurt Bigler <kkb@breathsense.com> wrote:

> on 5/26/06 6:58 PM, Carl Lumma <clumma@yahoo.com> wrote:
>
>>>> If we do the same trick as befor only with 2, 11/9, 10/7, 243/242, 441/440,
>>>> then we can ignore the 2 and the 243/242 and 441/440 vals (obtained as
>>>> before by inverting the square matrix) and just use
>>>>
>>>> [<0 2 1 1 5|, <0 0 2 1 0|]
>>>>
>>>> to project 11-limit JI to pitch classes in a planar lattice picture.
>>>> This gives the following twelve points when applied to the bihexany:
>>>>
>>>> {[1, 2], [1, 1], [-5, 3], [-4, 1], [-1, 1], [2, 0], [-2, 2],
>>>> [-1, 2], [-3, 3], [-2, 1], [-4, 2], [0, 0]}
>>>>
>>>> Plot that and you should get the picture.
>>>
>>> Thanks, I'll take a look at that.
>>
>> Yeah, if somebody could plot that, I'd love to see it.
>
> I'm working on getting Mathematica to do this. I can probably provide a png
> file when I have something that looks decent.

Well here's a first cut. It's not very "informational":

http://breathsense.com/published/bihexany1.png

I can add labelling etc. once i understand it better. Meanwhile maybe this
is already a tad useful, at least to Carl. ;)

-Kurt

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@...> wrote:

> Well here's a first cut. It's not very "informational":

It helps if you know that the stick figure with two legs is an ontonal
tetrad, and standing on his head is a utonal tetrad. You can also see
two hexagons with a vertex missing in there, next to each other. Those
would be standard 7-limit hexanies if the hexagons were complete, but
instead they replace one of the vertices with another sort of note.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:

> It helps if you know that the stick figure with two legs is an ontonal
> tetrad, and standing on his head is a utonal tetrad.

You have the origin labled, so that's 1/1. Then the stick figure off
to the right, with 1/1 the left leg, is the 1-5/4-3/2-7/4 hexad. The
3/2 is the right foot, and you get that by taking the
1,7/6,5/4,5/3,7/5,35/24 hexany, and moving 35/24 over to 3/2.
Inverting that and placing them side by side gives the bihexany; of
course there is 11-limit stuff in here, though you can think of it
entirely in the 7-limit if you want. But the neutral thirds can be
taken as 11/9 also.

Other fun stuff to plot would be the 7-limit tonality diamond and the
stellated hexany, I think.

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@...> wrote:

> But when you say "The actual 11-limit project ignores 2, 243/242 and
> 441/440" and you go about creating a lattice for the result, it
seems to me
> that would usually be called a 5-limit lattice, isn't that right?

Not really. It isn't like 225/224, where you can look at it as a way
of mashing the 7s down into the 5-limit lattice. Here the horizontal
lattice relationship is 49/40 (7-limit) and the vertical is 10/7
(again, 7-limit.) So it's clearly a 7-limit lattice, but squished down
to a plane of pitch-classes. The deal is, two 49/40s in a row gives
you your 3/2 (by tempering), and a 49/40 times 10/7 is 7/4 (no
tempering) and times 10/7 again is 5/2 (no tempering.) So it boils
down to using 49/40 and 60/49 for the same thing, and using two of
these neutral thirds to reach the fifth.

In terms of this lattice, the fifth is [2,0], and the major third
[1,2]. The determinant of [[2,0], [1,2]] is 4, so only 1/4 of the
lattice consists of the pure 5-limit.

At the
> same time the temperament acknowledges the equivalence of this to an
> 11-limit functionality. And as I recall, the concept of temperament
wants
> to remain a little vague about the actual tuning, so you could in this
> example make the 7- and 11-limit intervals exact at the expense of
others,
> or you could make others exact at the expense of 7 and 11. Or you
could do
> some other optimization.

Or you could just use 72-et or 130-et (a division I'm exploring these
days.) If you want to favor the 7-limit a bit, 171-et.