Annotated Dave Keenan file

I've taken this file

http://dkeenan.com/Music/7LimitGenerators.htm

and annotated it by adding names when available, and descriptions when
not, of the 7-limit linear temperaments it discusses. While I think
the value judgments are a little eccentric, it surely is worth looking
at the temperaments here which don't have names. Maybe Dave wants to
suggest something?

The nameless ones are these:

<3 7 -1 4 -10 -22| {36/35, 1029/1024}
TOP: [1205.820043, 1890.417958, 2803.215176, 3389.260823]
rms gens: [1200., 227.4263791]

<2 -6 -6 -14 -15 3| {50/49, 135/128}
TOP: [1206.548264, 1891.576247, 2771.109113, 3374.383246]
rms gens: [600.0000000, 79.05132740]

<6 -2 -2 -17 -20 1| {50/49, 525/512}
TOP: [1203.400986, 1896.025764, 2777.627538, 3379.328030]
rms gens: [600.0000000, 231.2978354]

Dave's article:

This appeared in Tuning Digest 245, 10 Jul 1999.

[A typo has been corrected where the MA err for 227c was given as
22.4c]

Message: 10
Date: Sat, 10 Jul 1999 10:39:56 +1000
From: David C Keenan [address removed]
Subject: Good 7-limit generators

I've been doing some serious number crunching and can now announce
that there
are only 3 single-chain generators of good (octave-based) 7-limit
scales, and
only 6 double-chain (with a half-octave) generators.

To qualify, all they had to do was to have _more_ complete tetrads in
a chain
of 10 notes, than meantone using augmented sixths (2), and have lower
errors
(either RMS or max-absolute) than the best chain of fourths/fifths
where the
dominant 7th chord is the 4:5:6:7 approximation. i.e. around a 702.5c
fifth
(= 497.5c fourth).

Here's the info on these not-quite-good-enough generators for
comparison:

Not good enough: No. generators in
Min Min interval
Generator No. tetrads 7-limit 7-limit 2 4 5 4 5 6
(+-0.5c) in 10 notes RMS error MA err. 3 5 6 7 7 7
-------------------------------------------------------------
503.4c 2 3.6c 5.4c -1 -4 3-10 -6 -9 meantone
497.5c 8 20.2c 25.4c -1 -4 3 2 6 3 dominant7

Single chain: No. generators in
Min Min interval
Generator No. tetrads 7-limit 7-limit 2 4 5 4 5 6
(+-0.5c) in 10 notes RMS error MA err. 3 5 6 7 7 7
-------------------------------------------------------------
125c 6 12.2c 17.9c -4 3 -7 -2 -5 2
tertiathirds
227c 4 16.5c 24.4c 3 7 -4 -1 -8 -4 <3 7 -1 4
-10 -22| {36/35, 1029/1024}
317c 8 12.3c 17.9c 6 5 1 3 -2 -3 kleismic

The generator sizes are only given to +-0.5c because the exact value
will
depend on whether RMS error or Max-Absolute error or
Max-otonal-beat-rate
(not shown) is the measure to be optimised.

Note that the minor-third generator, that we've been discussing
recently, is
the best possible for a single chain. Note also that 227c is
equivalent to
1200-227 = 973c, a 4:7 generator.

Double chain: No. generators in
Min Min interval
Generator No. tetrads 7-limit 7-limit 2 4 5 4 5 6
(+-0.5c) in 10 notes RMS error MA err. 3 5 6 7 7 7
-------------------------------------------------------------
71c 4 12.5c 18.2c 1 -3 -4 -3 0 4 <2 -6 -6
-14 -15 3| {50/49, 135/128}
230c 4 11.8c 17.5c 3 -1 4 -1 0 -4 <6 -2 -2
-17 -20 1| {50/49, 525/512}
380.5c 4 10.3c 17.5c -3 1 4 1 0 -4 <6 -2 -2
-17 -20 1| {50/49, 525/512}
491c 8 10.9c 17.5c -1 2 -3 2 0 3 pajara
506.5c 4 11.2c 17.5c -1 -4 3 -4 0 -3 injera
521c 4 16.9c 23.1c -1 3 -4 3 0 4 <2 -6 -6
-14 -15 3| {50/49, 135/128}

In the table above, I haven't shown whether the half octave is
included in an
interval or not, but that's not difficult to figure out.

Note that 230c is equivalent to 970c, an approximate 4:7. 380.5c is a
major
third. The last three are fourths that correspond to fifths of 709c,
693.5c
and 679c. The first two of these last three were recognised by Paul
Erlich as
being in the vicinity of the 22-tET and 26-tET fifths, and the last
one is not
really good enough, with a 23.1c error in its fifths.

So far, 491c with a half-octave (22-tET) is the winner, with the
(recently
discovered?) 317c minor-third a close second, and 125c in third place
(based
on treating number-of-tetrads as more important than accuracy).

I wonder which of these have been discovered before? I wonder if I've
missed
any? I'm 99% sure I haven't, but I'll be rechecking. Any journals
likely to be
interested in this?

Does somone want to work out what ET's these embed in with sufficient
accuracy? Or how many notes in the largest 125c MOS with 12 notes or
less?

I could look at higher multiple chains with the appropriate fraction
of an
octave, if anyone cares. Does anyone want me to change my criteria in
any way?

Regards,
-- Dave Keenan
http://dkeenan.com

>Single chain: No. generators in
> Min Min interval
>Generator No. tetrads 7-limit 7-limit 2 4 5 4 5 6
>(+-0.5c) in 10 notes RMS error MA err. 3 5 6 7 7 7
>-------------------------------------------------------------
>125c 6 12.2c 17.9c -4 3 -7 -2 -5 2
>tertiathirds

Why isn't this negri?

By the way, anybody know names for these...

!
Two pentatonic chains of 7:4's rooted a 5:4 apart, tuned in 31-tet.
10
!
154.839 !.....4
232.258 !.....6
387.097 !....10
464.516 !....12
619.355 !....16
696.774 !....18
851.613 !....22
967.742 !....25
1083.871 !...28
2/1 !........31
!
! Four 5-limit triads on 1-4-7, strictly proper.

!
Two pentatonic chains of 3:2's rooted a 7:4 apart, tuned in 31-tet.
10
!
154.839 !......4
193.548 !......5
387.097 !.....10
464.516 !.....12
658.065 !.....17
696.774 !.....18
890.323 !.....23
967.742 !.....25
1161.290 !....30
2/1 !.........31
!
! Four 5-limit triads on 1-4-7, not proper.

?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >Single chain: No. generators in
> > Min Min interval
> >Generator No. tetrads 7-limit 7-limit 2 4 5 4 5 6
> >(+-0.5c) in 10 notes RMS error MA err. 3 5 6 7 7 7
> >-------------------------------------------------------------
> >125c 6 12.2c 17.9c -4 3 -7 -2 -5 2
> >tertiathirds
>
> Why isn't this negri?

Interesting question. This is the only 7-limit version of negri with a
badness score which is much good, so using the "reasonable tuning"
criterion perhaps it should be. However, <4 -3 -17 -14 -38 -31| is
closer, <4 -3 21 -14 22 57| much closer yet, and <4 -3 40 -14 52 101|
has the identical TOP tuning. What to do?

>> >Single chain: No. generators in
>> > Min Min interval
>> >Generator No. tetrads 7-limit 7-limit 2 4 5 4 5 6
>> >(+-0.5c) in 10 notes RMS error MA err. 3 5 6 7 7 7
>> >-------------------------------------------------------------
>> >125c 6 12.2c 17.9c -4 3 -7 -2 -5 2
>> >tertiathirds
>>
>> Why isn't this negri?
>
>Interesting question. This is the only 7-limit version of negri with
>a badness score which is much good, so using the "reasonable tuning"
>criterion perhaps it should be.
//
>However, <4 -3 -17 -14 -38 -31| is closer, <4 -3 21 -14 22 57| much
>closer yet, and <4 -3 40 -14 52 101| has the identical TOP tuning.

The TOP tuning of what?

>What to do?

I don't get it. Paul's temperament database doesn't list tertiathirds
so I don't know what comma(s) tertiathirds was based on. If it's
previously been a 5-limit linear temperament I don't see how it could
have <-4 3] the same as negri.

-Carl

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> I've taken this file
>
> http://dkeenan.com/Music/7LimitGenerators.htm
>
> and annotated it by adding names when available, and descriptions when
> not, of the 7-limit linear temperaments it discusses. While I think
> the value judgments are a little eccentric, it surely is worth looking
> at the temperaments here which don't have names. Maybe Dave wants to
> suggest something?

I'd like them to be named "", "", and "" respectively (without the
quotes). :-)

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

> >However, <4 -3 -17 -14 -38 -31| is closer, <4 -3 21 -14 22 57| much
> >closer yet, and <4 -3 40 -14 52 101| has the identical TOP tuning.
>
> The TOP tuning of what?

Negri.

> >What to do?
>
> I don't get it. Paul's temperament database doesn't list tertiathirds
> so I don't know what comma(s) tertiathirds was based on. If it's
> previously been a 5-limit linear temperament I don't see how it could
> have <-4 3] the same as negri.

Tertiathirds is a name for the 7-limit temperament with TM comma base
{49/48, 225/224}, wedgie <<4 -3 2 -14 -8 13|| and mapping
[<1 2 2 3|, <0 -4 3 -2|]. There's been a question all along as to
whether it should be called "negri", and I suppose it should be; it's
pretty closely tied to 2/19 as a generator either way.

>> >However, <4 -3 -17 -14 -38 -31| is closer, <4 -3 21 -14 22 57| much
>> >closer yet, and <4 -3 40 -14 52 101| has the identical TOP tuning.
>>
>> The TOP tuning of what?
>
>Negri.
>
>> >What to do?
>>
>> I don't get it. Paul's temperament database doesn't list tertiathirds
>> so I don't know what comma(s) tertiathirds was based on. If it's
>> previously been a 5-limit linear temperament I don't see how it could
>> have <-4 3] the same as negri.
>
>Tertiathirds is a name for the 7-limit temperament with TM comma base
>{49/48, 225/224}, wedgie <<4 -3 2 -14 -8 13|| and mapping
>[<1 2 2 3|, <0 -4 3 -2|]. There's been a question all along as to
>whether it should be called "negri", and I suppose it should be; it's
>pretty closely tied to 2/19 as a generator either way.

It should; these other maps should be sent back to the cleaners.

-Carl

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

> Tertiathirds is a name for the 7-limit temperament with TM comma base
> {49/48, 225/224}, wedgie <<4 -3 2 -14 -8 13|| and mapping
> [<1 2 2 3|, <0 -4 3 -2|]. There's been a question all along as to
> whether it should be called "negri", and I suppose it should be; it's
> pretty closely tied to 2/19 as a generator either way.

I assumed that tertiathirds and negri were alternative names for the
same thing, in my own 7-limit temperament spreadsheet from long ago.
One being a systematic name, the other a common name, like sodium
chloride and salt.

>> Tertiathirds is a name for the 7-limit temperament with TM comma base
>> {49/48, 225/224}, wedgie <<4 -3 2 -14 -8 13|| and mapping
>> [<1 2 2 3|, <0 -4 3 -2|]. There's been a question all along as to
>> whether it should be called "negri", and I suppose it should be; it's
>> pretty closely tied to 2/19 as a generator either way.
>
>I assumed that tertiathirds and negri were alternative names for the
>same thing, in my own 7-limit temperament spreadsheet from long ago.
>One being a systematic name, the other a common name, like sodium
>chloride and salt.

Works for me.

-Carl