T[n] revisited

>Fourththirds[5]

Name[size]

>[16/15, 28/27, 77/75]

The commas.

>[1, -1, 3, -4, -4, 2, -10, 10, -6, -22]

Map?

>[[1, 2, 2, 4, 2], [0, -1, 1, -3, 4]]

Za?

>bad 6476.838089 comp 20.25383770 rms 43.03787612
>graham 7 scale size 5

"graham" is Graham complexity, I assume? What's "comp"?

>Heptadec[9] [36/35, 56/55, 77/75]
>[5, 3, 7, 4, -7, -3, -11, 8, -1, -13]
>[[1, 1, 2, 2, 3], [0, 5, 3, 7, 4]]
>bad 6400.766110 comp 32.19555159 rms 19.64440328
>graham 7 scale size 9
>
>
>Blackwood[10]
>[0, 5, 0, 8, 0, -14]
>[[5, 8, 12, 14], [0, 0, -1, 0]]
>bad 1662.988586 comp 10.25428060 rms 15.81535241
>graham 5 scale size 10 ratio 2.000000
>
>
>Pajara[10]
>[2, -4, -4, -11, -12, 2]
>[[2, 3, 5, 6], [0, 1, -2, -2]]
>bad 1550.521632 comp 11.92510946 rms 10.90317748
>graham 6 scale size 10 ratio 1.666667
>
>
>Pajarous[10] [50/49, 55/54, 64/63]
>[2, -4, -4, 10, -11, -12, 9, 2, 37, 42]
>[[2, 3, 5, 6, 6], [0, 1, -2, -2, 5]]
>bad 6667.906202 comp 43.76707564 rms 12.26714784
>graham 14 scale size 10
>
>
>Check this out Carl[10]
>[2, 1, -4, -3, -12, -12]
>[[1, 1, 2, 4], [0, 2, 1, -4]]
>bad 2431.810368 comp 9.849243627 rms 25.06824586
>graham 6 scale size 10 ratio 1.666667
>
>
>Here's another Carl[10]
>[2, 6, 6, 5, 4, -3]
>[[2, 3, 4, 5], [0, 1, 3, 3]]
>bad 2682.600306 comp 11.92510946 rms 18.86388854
>graham 6 scale size 10 ratio 1.666667
>
>
>NB Carl
>[2, 6, 6, 5, 4, -3]
>[[2, 3, 4, 5], [0, 1, 3, 3]]
>bad 2682.600306 comp 11.92510946 rms 18.86388854
>graham 6 scale size 10 ratio 1.666667

I assume your maple can turn these into generator/ie pairs?

>Dominant sevenths[7]
>rms error 20.163 cents
>
>
>Hemifourths[9]
>rms error 12.690
>
>
>Tertiathirds[9]
>rms error 12.189 cents
>
>
>Hexadecimal[9]
>rms error 18.585

Where, pray tell, is the Official Source of data on these?

-Carl

Gene? You said NB, and NB I did.

-C.

At 01:41 AM 10/13/2003, I wrote:
>>Fourththirds[5]
>
>Name[size]
>
>>[16/15, 28/27, 77/75]
>
>The commas.
>
>>[1, -1, 3, -4, -4, 2, -10, 10, -6, -22]
>
>Map?
>
>>[[1, 2, 2, 4, 2], [0, -1, 1, -3, 4]]
>
>Za?
>
>>bad 6476.838089 comp 20.25383770 rms 43.03787612
>>graham 7 scale size 5
>
>"graham" is Graham complexity, I assume? What's "comp"?
>
>>Heptadec[9] [36/35, 56/55, 77/75]
>>[5, 3, 7, 4, -7, -3, -11, 8, -1, -13]
>>[[1, 1, 2, 2, 3], [0, 5, 3, 7, 4]]
>>bad 6400.766110 comp 32.19555159 rms 19.64440328
>>graham 7 scale size 9
>>
>>
>>Blackwood[10]
>>[0, 5, 0, 8, 0, -14]
>>[[5, 8, 12, 14], [0, 0, -1, 0]]
>>bad 1662.988586 comp 10.25428060 rms 15.81535241
>>graham 5 scale size 10 ratio 2.000000
>>
>>
>>Pajara[10]
>>[2, -4, -4, -11, -12, 2]
>>[[2, 3, 5, 6], [0, 1, -2, -2]]
>>bad 1550.521632 comp 11.92510946 rms 10.90317748
>>graham 6 scale size 10 ratio 1.666667
>>
>>
>>Pajarous[10] [50/49, 55/54, 64/63]
>>[2, -4, -4, 10, -11, -12, 9, 2, 37, 42]
>>[[2, 3, 5, 6, 6], [0, 1, -2, -2, 5]]
>>bad 6667.906202 comp 43.76707564 rms 12.26714784
>>graham 14 scale size 10
>>
>>
>>Check this out Carl[10]
>>[2, 1, -4, -3, -12, -12]
>>[[1, 1, 2, 4], [0, 2, 1, -4]]
>>bad 2431.810368 comp 9.849243627 rms 25.06824586
>>graham 6 scale size 10 ratio 1.666667
>>
>>
>>Here's another Carl[10]
>>[2, 6, 6, 5, 4, -3]
>>[[2, 3, 4, 5], [0, 1, 3, 3]]
>>bad 2682.600306 comp 11.92510946 rms 18.86388854
>>graham 6 scale size 10 ratio 1.666667
>>
>>
>>NB Carl
>>[2, 6, 6, 5, 4, -3]
>>[[2, 3, 4, 5], [0, 1, 3, 3]]
>>bad 2682.600306 comp 11.92510946 rms 18.86388854
>>graham 6 scale size 10 ratio 1.666667
>
>I assume your maple can turn these into generator/ie pairs?
>
>>Dominant sevenths[7]
>>rms error 20.163 cents
>>
>>
>>Hemifourths[9]
>>rms error 12.690
>>
>>
>>Tertiathirds[9]
>>rms error 12.189 cents
>>
>>
>>Hexadecimal[9]
>>rms error 18.585
>
>Where, pray tell, is the Official Source of data on these?
>
>-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >Fourththirds[5]
>
> Name[size]
>
> >[16/15, 28/27, 77/75]
>
> The commas.
>
> >[1, -1, 3, -4, -4, 2, -10, 10, -6, -22]
>
> Map?

The wedgie

> >[[1, 2, 2, 4, 2], [0, -1, 1, -3, 4]]
>
> Za?

This is the prime mapping.

> >bad 6476.838089 comp 20.25383770 rms 43.03787612
> >graham 7 scale size 5
>
> "graham" is Graham complexity, I assume? What's "comp"?

Geometric complexity--using the old-style natural log defintion.

Generator: 455.25

> >Heptadec[9] [36/35, 56/55, 77/75]
> >[5, 3, 7, 4, -7, -3, -11, 8, -1, -13]
> >[[1, 1, 2, 2, 3], [0, 5, 3, 7, 4]]
> >bad 6400.766110 comp 32.19555159 rms 19.64440328
> >graham 7 scale size 9

Generator 141.16

> >
> >Blackwood[10]
> >[0, 5, 0, 8, 0, -14]
> >[[5, 8, 12, 14], [0, 0, -1, 0]]
> >bad 1662.988586 comp 10.25428060 rms 15.81535241
> >graham 5 scale size 10 ratio 2.000000

Generators: [240, 90.61]

> >
> >Pajara[10]
> >[2, -4, -4, -11, -12, 2]
> >[[2, 3, 5, 6], [0, 1, -2, -2]]
> >bad 1550.521632 comp 11.92510946 rms 10.90317748
> >graham 6 scale size 10 ratio 1.666667

Generators: [600, 108.81]

> >Pajarous[10] [50/49, 55/54, 64/63]
> >[2, -4, -4, 10, -11, -12, 9, 2, 37, 42]
> >[[2, 3, 5, 6, 6], [0, 1, -2, -2, 5]]
> >bad 6667.906202 comp 43.76707564 rms 12.26714784
> >graham 14 scale size 10

Generators: [600, 109.88]
> >
> >Check this out Carl[10]
> >[2, 1, -4, -3, -12, -12]
> >[[1, 1, 2, 4], [0, 2, 1, -4]]
> >bad 2431.810368 comp 9.849243627 rms 25.06824586
> >graham 6 scale size 10 ratio 1.666667

Generator: 358.03

> >
> >Here's another Carl[10]
> >[2, 6, 6, 5, 4, -3]
> >[[2, 3, 4, 5], [0, 1, 3, 3]]
> >bad 2682.600306 comp 11.92510946 rms 18.86388854
> >graham 6 scale size 10 ratio 1.666667

Generators: [600, 128.51]

> Where, pray tell, is the Official Source of data on these?

Didn't know there was such a thing.

>> >[1, -1, 3, -4, -4, 2, -10, 10, -6, -22]
>>
>> Map?
>
>The wedgie

Ah.

>> >[[1, 2, 2, 4, 2], [0, -1, 1, -3, 4]]
>>
>> Za?
>
>This is the prime mapping.

Ok, I should have known this. But it never hurts to
label these things in your posts.

//
>> >Check this out Carl[10]
>> >[2, 1, -4, -3, -12, -12]
>> >[[1, 1, 2, 4], [0, 2, 1, -4]]
>> >bad 2431.810368 comp 9.849243627 rms 25.06824586
>> >graham 6 scale size 10 ratio 1.666667
>
>Generator: 358.03
>
>> >
>> >Here's another Carl[10]
>> >[2, 6, 6, 5, 4, -3]
>> >[[2, 3, 4, 5], [0, 1, 3, 3]]
>> >bad 2682.600306 comp 11.92510946 rms 18.86388854
>> >graham 6 scale size 10 ratio 1.666667
>
>Generators: [600, 128.51]

Thanks, dude!

Did you use one of your maple routines to do this?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Did you use one of your maple routines to do this?

Of course--gf7 or gf11 of transpose(mapping to primes).

>> Did you use one of your maple routines to do this?
>
>Of course--gf7 or gf11 of transpose(mapping to primes).

Sweet; this works on my end.

-Carl

I am curious to know who's work you all would cite as the first
serious efforts to explore linear temperaments in depth. Put
another way, who was it that laid the groundwork for all of the
exploration into linear temperaments you are involved with now?
This is not my central interest, and from what little I know, I would
guess that Ellis was really the first to size things up in this area.
How far off is that?

Thanks,
Aaron

>I am curious to know who's work you all would cite as the first
>serious efforts to explore linear temperaments in depth. Put
>another way, who was it that laid the groundwork for all of the
>exploration into linear temperaments you are involved with now?
>This is not my central interest, and from what little I know, I
>would guess that Ellis was really the first to size things up in
>this area. How far off is that?

I'm not familiar with Ellis' work, believe it or not. I've always
thought of Bosanquet as the first to really work with the idea of
linear temperaments (c. 1876). Followed by Fokker, Wilson, Graham
Breed, Dave Keenan, Paul Erlich, and Gene Smith. But Gene's work
may actually come much earlier in this list...

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> I'm not familiar with Ellis' work, believe it or not. I've always
> thought of Bosanquet as the first to really work with the idea of
> linear temperaments (c. 1876). Followed by Fokker, Wilson, Graham
> Breed, Dave Keenan, Paul Erlich, and Gene Smith. But Gene's work
> may actually come much earlier in this list...

Nah. Before finding this list, I worked on JI and equal temperament,
but nothing in between.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> I'm not familiar with Ellis' work, believe it or not. I've always
> thought of Bosanquet as the first to really work with the idea of
> linear temperaments (c. 1876). Followed by Fokker, Wilson, Graham
> Breed, Dave Keenan, Paul Erlich, and Gene Smith. But Gene's work
> may actually come much earlier in this list...

I suspect Paul Erlich should come before me on that list. In any case
Breed-Erlich-Keenan was pretty much a simultaneous cooperative effort
that definitely built on earlier stuff by Wilson.

The answer to the original question really depends on what you mean by
"in depth".

--- In tuning-math@yahoogroups.com, "gooseplex" <cfaah@e...> wrote:
> I am curious to know who's work you all would cite as the first
> serious efforts to explore linear temperaments in depth. Put
> another way, who was it that laid the groundwork for all of the
> exploration into linear temperaments you are involved with now?
> This is not my central interest, and from what little I know, I
would
> guess that Ellis was really the first to size things up in this
area.
> How far off is that?
>
> Thanks,
> Aaron

i don't think ellis did any work in this area at all. erv wilson
mentioned or implied quite a few linear temperaments but seemed to
miss out entirely on the cases where the period turns out to be a
fraction of an octave. bosanquet, if i'm not mistaken, did not even
look outside the period=octave, generator~=fifth cases. if you're not
familiar with erv wilson's work, please spend some time scratching
your head over these:

http://www.anaphoria.com/wilson.html

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
>
> > I'm not familiar with Ellis' work, believe it or not. I've always
> > thought of Bosanquet as the first to really work with the idea of
> > linear temperaments (c. 1876). Followed by Fokker, Wilson, Graham
> > Breed, Dave Keenan, Paul Erlich, and Gene Smith. But Gene's work
> > may actually come much earlier in this list...
>
> Nah. Before finding this list, I worked on JI and equal
temperament,
> but nothing in between.

yet you had already made the keen observation about the properties of
tunings with 81/80 in the kernel differing radically from those
without it as concerns compatibility with western musical thinking
and notation.

> i don't think ellis did any work in this area at all. erv wilson
> mentioned or implied quite a few linear temperaments but
seemed to
> miss out entirely on the cases where the period turns out to be
a
> fraction of an octave. bosanquet, if i'm not mistaken, did not
even
> look outside the period=octave, generator~=fifth cases. if you're
not
> familiar with erv wilson's work, please spend some time
scratching
> your head over these:
>
> http://www.anaphoria.com/wilson.html

yes, thanks for the link. I've seen Erv's work (and met him at
microfest 2001 as well) and I agree his work is quite cryptic. Ellis
has a few pages in Appendix XX in his translation of Helmholtz's
tome which carry the heading 'linear temperaments'. In this
section, cyclic temperaments fall under this heading, including
31 53, 19, 29, 43, and 55. Granted this is just a catalog of work
done by others.

AH

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> yet you had already made the keen observation about the properties
of
> tunings with 81/80 in the kernel differing radically from those
> without it as concerns compatibility with western musical thinking
> and notation.

Yes, but my attitude was that linear temperaments can always be
expressed in terms of equal temperaments. It hadn't occurred to me
that linear temperaments are crucial for understanding equal
temperaments, however.

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> > I'm not familiar with Ellis' work, believe it or not.
> > I've always thought of Bosanquet as the first to really
> > work with the idea of linear temperaments (c. 1876).
> > Followed by Fokker, Wilson, Graham Breed, Dave Keenan,
> > Paul Erlich, and Gene Smith. But Gene's work
> > may actually come much earlier in this list...
>
> I suspect Paul Erlich should come before me on that list.
> In any case Breed-Erlich-Keenan was pretty much a simultaneous
> cooperative effort that definitely built on earlier stuff by
> Wilson.

i think the history of theoretical speculation on
linear temperaments begins with writings concerning meantone,
doesn't it?

Woolhouse seems to have been the first to give a rigourously
mathematical underpinning to it, c. 1835:

http://sonic-arts.org/monzo/woolhouse/essay.htm

but Zarlino gave a mathematically precise explanation of
/7-comma meantone in 1558:

http://sonic-arts.org/monzo/zarlino/1558/zarlino1558-2.htm

Bosanquet mentioned Woolhouse in the original draft of
his book, but all references to Woolhouse were removed
when it was published. Rasch discusses this in his
foreward to the Diapason Press edition of Bosanquet's book.

REFERENCE
Bosanquet, Robert Halford Macdowall. 1876.
_An Elementary Treatise on Musical Intervals and Temperament_.
MacMillan & Co., London.
- 2nd edition, with introduction by Rudolf Rasch (ed.). 1987.
- Tuning and temperament library vol. 4,
- Diapason Press, Utrecht.

i think that the correct recent chronology is:

Woolhouse-Bosanquet-Wilson-Erlich-Breed-Keenan-Smith

paul?

-monz

oops ...

--- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:

> i think the history of theoretical speculation on
> linear temperaments begins with writings concerning meantone,
> doesn't it?
>
>
> Woolhouse seems to have been the first to give a rigourously
> mathematical underpinning to it, c. 1835:
>
> http://sonic-arts.org/monzo/woolhouse/essay.htm
>
>
> but Zarlino gave a mathematically precise explanation of
> /7-comma meantone in 1558:

that should have been "2/7-comma meantone". sorry.

-monz

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
> wrote:
>
> > yet you had already made the keen observation about the
properties
> of
> > tunings with 81/80 in the kernel differing radically from those
> > without it as concerns compatibility with western musical
thinking
> > and notation.
>
> Yes, but my attitude was that linear temperaments can always be
> expressed in terms of equal temperaments.

More precisely, in terms of the intersection of the kernels of two
equal temperaments.