a new VRWT

Inspired by a Victorian Rational Well Temperament created
a few years ago by George Secor and Aaron Johnson (which
aimed to make its bad 3rds be 24/19), I have come up with
the following...

!
{2 3 17 19} well temperament.
12
!
17/16
9/8
68/57
24/19
4/3
17/12
3/2
51/32
32/19
57/32
36/19
2/1
!
! 17/12-----17/16-----51/32
! .'/ \ / \ /|\
! 68/57/ \ / \ / | \
! | / \ / \ /57/32\
! |/ \ / \ /.' '.\
! 4/3-------1/1-------3/2-------9/8
! '. .' '. .' '. .'
! 32/19-----24/19-----36/19

It's a periodicity block, and I think it's the best one in
this space. It's got a little more key contrast than
Kalle's scale. Neither scale provides a great fit for the
classical pattern of key consonance, as George points out.
I would invite George to help choose the best key given the
pattern it has. I'd invite him to do the same for Kalle's
scale, which I'll copy below for easy reference:

!
Kalle Aho's 12-tone rational well temperament.
12
!
164/155 !..164
174/155 !..174
184/155 !..184
39/31 !..195
207/155 !..207
219/155 !..219
232/155 !..232
246/155 !..246
52/31 !..260
276/155 !..276
292/155 !..292
2/1 !..310
!
! Lowest mode of the harmonic series which has no harmonic
! waste when interpreted as a 5-limit well temperament.

Thanks!

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Inspired by a Victorian Rational Well Temperament created
> a few years ago by George Secor and Aaron Johnson (which
> aimed to make its bad 3rds be 24/19), I have come up with
> the following...
>
> !
> {2 3 17 19} well temperament.
> 12
> !
> 17/16
> 9/8
> 68/57
> 24/19
> 4/3
> 17/12
> 3/2
> 51/32
> 32/19
> 57/32
> 36/19
> 2/1
> !
> ! 17/12-----17/16-----51/32
> ! .'/ \ / \ /|\
> ! 68/57/ \ / \ / | \
> ! | / \ / \ /57/32\
> ! |/ \ / \ /.' '.\
> ! 4/3-------1/1-------3/2-------9/8
> ! '. .' '. .' '. .'
> ! 32/19-----24/19-----36/19
>
> It's a periodicity block, and I think it's the best one in
> this space. It's got a little more key contrast than
> Kalle's scale. Neither scale provides a great fit for the
> classical pattern of key consonance, as George points out.
> I would invite George to help choose the best key given the
> pattern it has.

The best major triads are on Db, Ab, Gb, and Eb (in that order), so
transposing this down 1/2-step would make it a better well-
temperament (in the traditional sense).

> I'd invite him to do the same for Kalle's
> scale, which I'll copy below for easy reference:
>
> !
> Kalle Aho's 12-tone rational well temperament.
> 12
> !
> 164/155 !..164
> 174/155 !..174
> 184/155 !..184
> 39/31 !..195
> 207/155 !..207
> 219/155 !..219
> 232/155 !..232
> 246/155 !..246
> 52/31 !..260
> 276/155 !..276
> 292/155 !..292
> 2/1 !..310
> !
> ! Lowest mode of the harmonic series which has no harmonic
> ! waste when interpreted as a 5-limit well temperament.

Kalle and I already discussed this. See:
/tuning/topicId_75816.html#75877
That "weekend" I spent working on the kitchen floor took much longer
than I expected, and I didn't remember to get back to this.

Thanks for bringing this to my attention. I'll have to listen to it
(and your new VRWT as well). But that won't be till next week, since
I'll be on vacation over the next several days.

--George

Thanks George!!

-Carl

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
>
> Thanks for bringing this to my attention. I'll have to listen to it
> (and your new VRWT as well). But that won't be till next week, since
> I'll be on vacation over the next several days.
>
> --George

Okay, Carl and Kalle, I'm back, I've listened to your WT's, and I'm
ready to offer a few comments.

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> >
> > Inspired by a Victorian Rational Well Temperament created
> > a few years ago by George Secor and Aaron Johnson (which
> > aimed to make its bad 3rds be 24/19), I have come up with
> > the following...
> >
> > !
> > {2 3 17 19} well temperament.
> > 12
> > !
> > 17/16
> > 9/8
> > 68/57
> > 24/19
> > 4/3
> > 17/12
> > 3/2
> > 51/32
> > 32/19
> > 57/32
> > 36/19
> > 2/1
> > !
> > ! 17/12-----17/16-----51/32
> > ! .'/ \ / \ /|\
> > ! 68/57/ \ / \ / | \
> > ! | / \ / \ /57/32\
> > ! |/ \ / \ /.' '.\
> > ! 4/3-------1/1-------3/2-------9/8
> > ! '. .' '. .' '. .'
> > ! 32/19-----24/19-----36/19
> >
> > It's a periodicity block, and I think it's the best one in
> > this space. It's got a little more key contrast than
> > Kalle's scale. Neither scale provides a great fit for the
> > classical pattern of key consonance, as George points out.
> > I would invite George to help choose the best key given the
> > pattern it has.
>
> The best major triads are on Db, Ab, Gb, and Eb (in that order), so
> transposing this down 1/2-step would make it a better well-
> temperament (in the traditional sense).

Carl, I transposed your WT 1/2-step down, thus:

! LummaVRWT.scl
!
Carl Lumma's (2,3,17,19) VRWT (transposed 1/2-step downward)
12
!
18/17
64/57
384/323
64/51
4/3
24/17
3/2
512/323
57/34
576/323
32/17
2/1

This is indeed the best key for a WT. However, the beating of the
narrow fifth between D and A (tempered 8 cents) tends to stick out
like the proverbial sore thumb. Also, the brats on G and D major are
not simple, so the small numbers in the ratios didn't really buy you
very much. (Sorry!)

> > I'd invite him to do the same for Kalle's
> > scale, which I'll copy below for easy reference:
> >
> > !
> > Kalle Aho's 12-tone rational well temperament.
> > 12
> > !
> > 164/155 !..164
> > 174/155 !..174
> > 184/155 !..184
> > 39/31 !..195
> > 207/155 !..207
> > 219/155 !..219
> > 232/155 !..232
> > 246/155 !..246
> > 52/31 !..260
> > 276/155 !..276
> > 292/155 !..292
> > 2/1 !..310
> > !
> > ! Lowest mode of the harmonic series which has no harmonic
> > ! waste when interpreted as a 5-limit well temperament.
>
> Kalle and I already discussed this. See:
> /tuning/topicId_75816.html#75877

Kalle, I tried your WT without transposing the ratios, since I was
not sure what would be the best key. Bb was clearly the best major
triad (suggesting transposition upward a whole step), and most of the
remaining ones sound similar to 12-ET in the amount of tempering.
The simple brats do contribute to a more cohesive sound than with 12-
ET (though I admit it's very subtle).

You wrote earlier that you were wanting the "beating patterns of all
possible chords [to] be in a synchronous relationship to the tempo."
Only the fifths beat slowly enough for this to be of any use. Half
of them are just (and therefore don't beat). The remaining ones do
indeed beat at either 1, 2, or 4 times the rate of the slowest
beating one(s), so you've evidently achieved what you were after.
(As we noted before, it doesn't make a very good WT in the
traditional sense.)

As long as I'm on this subject, I happened to be comparing these with
some of the WT's I found recently and was startled to see something
completely unintentional crop up in this one (which I've labeled 24d):

! SecorWTPB-24d.scl
!
George Secor's 24-triad proportional-beating well-temperament (24d)
12
!
315/298
167/149
177/149
187/149
199/149
210/149
223/149
236/149
250/149
531/298
280/149
2/1

For the record, this was first described in Andreas' correction to an
oversight in one of my WT attempts:
/tuning/topicId_75816.html#76060
The primary objective was to find a well-constructed rational WT with
low numbers in which none of the fifths is tempered by >4.7 cents
(5/23-comma).

I happened to notice that the beat rate of the fifths on F, C, G, D,
A, and E is exactly the same (and, because the brats are all simple,
the beat rates of the major 3rds in their major triads are all exact
multiples)!

--George

> Okay, Carl and Kalle, I'm back, I've listened to your WT's,
> and I'm ready to offer a few comments.

Hey George, thanks for getting back.

I've been auditioning the WTs with pianoteq. It's a truly
incredible synth (physically modeled piano). It also happens
to load Scala files (I haven't tried non-12-tone files yet),
has a standalone mode for when you just don't want to mess
with a VST host, and even plays and renders MIDI files.
Perfect!

> However, the beating of the narrow fifth between
> D and A (tempered 8 cents) tends to stick out like the
> proverbial sore thumb.

Yes, that is a problem. Kalle's has a sore thumb fifth
too, but it's 695 cents vs. 694 cents here, and that extra
cent seems to makes a big difference!

> Also, the brats on G and D major are not simple, so the
> small numbers in the ratios didn't really buy you very much.
> (Sorry!)

The point wasn't simple brats, but rather lots of simple
ratios. In fact I'm tempted to claim it has the simplest
ratios possible for any WT with no 5-limit harmonic waste,
no major third worse than 24/19, and no fifth worse than
1/3-comma meantone. Play the tritones up the scale. I
can really hear the 17th harmonic lock there. Also, the
19/12 minor 6ths.

> > > !
> > > Kalle Aho's 12-tone rational well temperament.
> > > 12
> > > !
> > > 164/155 !..164
> > > 174/155 !..174
> > > 184/155 !..184
> > > 39/31 !..195
> > > 207/155 !..207
> > > 219/155 !..219
> > > 232/155 !..232
> > > 246/155 !..246
> > > 52/31 !..260
> > > 276/155 !..276
> > > 292/155 !..292
> > > 2/1 !..310
> > > !
> > > ! Lowest mode of the harmonic series which has no harmonic
> > > ! waste when interpreted as a 5-limit well temperament.
//
> Kalle, I tried your WT without transposing the ratios, since I was
> not sure what would be the best key.

Just FYI, the key above was one I chose, not the one Kalle
originally gave.

> Bb was clearly the best major triad (suggesting transposition
> upward a whole step), and most of the remaining ones sound
> similar to 12-ET in the amount of tempering.

It's remarkably close to 12-ET actually. Closer than most
other WTs in terms of melodic deviation from equal!

-Carl