Some scale analysis

If one looks at the 5-limit relationships of the Phillips scale, it
can be associated with the scale obtained by adding three notes to the
16-note Euler genus 3^4 5^4, where we take the chain of fifths at the bottom of the chain of thirds, and add two fifths at one end and a fourth at the other. Tempering this b 225/224-planar produces a scale related to Phillips I've called "quasi-phillips", which has some advantages.

Here it is:

! quasi-phillips.scl
Quasi-Phillips 19 note scale
19
!
83.67670365
152.1394612
199.6830428
268.1458003
384.1521395
399.3660856
467.8288431
500.1584786
583.8351823
652.2979398
699.8415214
768.3042789
851.9809826
899.5245642
967.9873217
1083.993661
1099.207607
1152.456418
2/1

If we set a margin of accuracy of 2.5 cents, then phillips has
4 major tetrads, 3 minor tetrads, 7 major triads and 7 minor triads.
Quasi-phillips has 4 major tetrads, 5 minor tetrads, 10 major triads and 9 minor triads. Raising the error bar to 5 gives phillips
two additional major terrads, an additional minor tetrad, and two aditional major triads.

Scales we might compare to are the 20 and 24 note Euler genuses for
3^5 5^4 and 3^6 5^4, and the 24 note schismic "2MOS".

! euler20.scl
3^5 5^4 Euler genus tempered by 225/224-planar
20
!
83.67670365
152.1394612
199.6830428
268.1458003
351.8225040
384.1521395
399.3660856
467.8288431
583.8351823
652.2979398
699.8415214
768.3042789
783.5182250
851.9809826
899.5245642
967.9873217
1083.993661
1152.456418
1167.670365
2/1

This has 6 major and 6 minor tetrads and 12 major and 12 minor tetrads within 2.5 cents.

! euler24.scl
3^6 5^4 Euler genus tempered by 225/224-planar
24
!
83.67670365
152.1394612
199.6830428
268.1458003
283.3597464
351.8225040
384.1521395
399.3660856
467.8288431
583.8351823
652.2979398
667.5118859
699.8415214
768.3042789
783.5182250
851.9809826
899.5245642
967.9873217
1051.664025
1083.993661
1099.207607
1152.456418
1167.670365
2/1

This has 8 major and 8 minor tetrads and 15 major and 15 minor tetrads within 2.5 cents.

! schis24.scl
!
! [0, 2, 7, 9, 14, 16, 23, 25, 30, 32, 39, 41,
! 46, 48, 53, 55, 62, 64, 69, 71, 78, 80, 85, 87]
24 tone schismic temperament in its 94-et incarnation
24
!
25.53191489
89.36170213
114.8936170
178.7234043
204.2553191
293.6170213
319.1489362
382.9787234
408.5106383
497.8723404
523.4042553
587.2340426
612.7659574
676.5957447
702.1276596
791.4893617
817.0212766
880.8510638
906.3829787
995.7446809
1021.276596
1085.106383
1110.638298
2/1

This has 9 major and 9 minor tetrads within 5 cents, and 15 major and 15 minor triads within 5 cents (actually, within 3.6 cents with this tuning.) It also has a long chain of 23 consectutive fifths.

--- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:

> This has 6 major and 6 minor tetrads and 12 major and 12 minor tetrads within 2.5 cents.

12 major and 12 minor *triads*.

> This has 8 major and 8 minor tetrads and 15 major and 15 minor tetrads within 2.5 cents.

15 major and 15 minor triads.

In a message dated 10/25/02 1:35:38 PM Central Daylight Time,
genewardsmith@juno.com writes:

> Tempering this b 225/224-planar produces a scale related to Phillips I've
> called "quasi-phillips", which has some advantages.
>
I fail to see any advantages of this scale over my Phillips JT scale. The
tuning in my scale is generally closer to the theoretical values .

Pauline

--- In tuning@y..., prophecyspirit@a... wrote:
> In a message dated 10/25/02 1:35:38 PM Central Daylight Time,
> genewardsmith@j... writes:
>
>
> > Tempering this b 225/224-planar produces a scale related to Phillips I've
> > called "quasi-phillips", which has some advantages.
> >
> I fail to see any advantages of this scale over my Phillips JT scale. The
> tuning in my scale is generally closer to the theoretical values .

It has within the limits of 225/224-planar temperament (within 2.5 cents, say) one less major tetrad, two fewer minor tetrads, one less major triad, and three less minor triads. It is also in a little better tune in general, fitting everything in an error bar of 2.2 cents. Those seem to me to be advantages. If you mean your scale is generally closer to JI, it isn't.

--- In tuning@y..., "Gene Ward Smith" <genewardsmith@j...> wrote:
> --- In tuning@y..., prophecyspirit@a... wrote:
> > In a message dated 10/25/02 1:35:38 PM Central Daylight Time,
> > genewardsmith@j... writes:
> >
> >
> > > Tempering this b 225/224-planar produces a scale related to
Phillips I've
> > > called "quasi-phillips", which has some advantages.
> > >
> > I fail to see any advantages of this scale over my Phillips JT
scale. The
> > tuning in my scale is generally closer to the theoretical values .
>
> It has within the limits of 225/224-planar temperament (within 2.5
>cents, say) one less major tetrad, two fewer minor tetrads, one less
>major triad, and three less minor triads. It is also in a little
>better tune in general, fitting everything in an error bar of 2.2
>cents. Those seem to me to be advantages. If you mean your scale is
>generally closer to JI, it isn't.

so pauline's scale isn't in 225/224-planar after all? where does it
depart? perhaps pauline is only shooting for a subset of the chords
you're finding, and is including some other chords as well, including
something involving a ratio of 17, if i recall correctly . . .

In a message dated 10/25/02 5:01:28 PM Central Daylight Time,
genewardsmith@juno.com writes:

> If you mean your scale is generally closer to JI, it isn't.
>
>
My Phillips JT scale has intervals approx only 2 cents or less off the
theoretical values in relation to each other. When you take into account the
5th are - 2 cents short, the other intervals are right on target with each
other with no noticeable beats while playing. And I've played hundreds of
pieces with my tuning, and know every chord I provide notes for is needed to
accurately play music composed for tempered scales, whether MT, WM or 12-ET.

And my scale allows playing in septimal minor. What other scale does?

Pauline

In a message dated 10/25/02 6:08:14 PM Central Daylight Time,
wallyesterpaulrus@yahoo.com writes:

> , including
> something involving a ratio of 17, if i recall correctly . . .
>
I've used 30/17 worth 983 cnets as a large minor 7th for E and B. As more
tone generators are needed for the 7/4 values worth 155 cents for them than I
have right now. But I plan to eventually use the regular values for these
7ths.

Pauline

--- In tuning@y..., prophecyspirit@a... wrote:

> And my scale allows playing in septimal minor. What other scale
>does?

gene's tetrads were septimal ones. i reckon there's at least as
much "septimal minor" in his variation as in your original scale.

>>It has within the limits of 225/224-planar temperament (within
>>2.5 cents, say) one less major tetrad, two fewer minor tetrads,
>>one less major triad, and three less minor triads. It is also in
>>a little better tune in general, fitting everything in an error
>>bar of 2.2 cents. Those seem to me to be advantages. If you mean
>>your scale is generally closer to JI, it isn't.
>
>so pauline's scale isn't in 225/224-planar after all? where does
>it depart? perhaps pauline is only shooting for a subset of the
>chords you're finding, and is including some other chords as well,
>including something involving a ratio of 17, if i recall
>correctly . . .

I think Gene switched "it"s on us mid-stream, there. First, he's
talking about Pauline's scale, then his. No?

Also, did anybody see that Gene wrote 171-tet, while Pauline wrote
17-tet?

I'm with you, Pauline, on the issue of comma shifts.

-Carl

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:

> so pauline's scale isn't in 225/224-planar after all? where does it
> depart? perhaps pauline is only shooting for a subset of the chords
> you're finding, and is including some other chords as well, including
> something involving a ratio of 17, if i recall correctly . . .

Pauline's scale has 7-limit intervals tempered by .3587, .6863, .8259, 1.1291, 1.4878, 1.955, 2.174, 2.314, 2.641, 4.129, and 4.488 cents--which ones are a part of the system?

--- In tuning@y..., "wallyesterpaulrus" <wallyesterpaulrus@y...> wrote:
> --- In tuning@y..., prophecyspirit@a... wrote:
>
> > And my scale allows playing in septimal minor. What other scale
> >does?
>
> gene's tetrads were septimal ones. i reckon there's at least as
> much "septimal minor" in his variation as in your original scale.

Five vs three tetrads and nine vs six triads if you confine yourself to errors under 2.5 cents. It would be nice to have a list of which chords are essential to this business in Pauline's understanding.

In a message dated 10/25/02 10:51:28 PM Central Daylight Time,
clumma@yahoo.com writes:

> I'm with you, Pauline, on the issue of comma shifts.
>
> -Carl
>
While the difference in cents betwee 386 (384) and 408 (400) looks large in
print, the difference in pitch is very small in midrange and below.

Pauline

In a message dated 10/26/02 5:26:30 AM Central Daylight Time,
genewardsmith@juno.com writes:

> Pauline's scale has 7-limit intervals tempered by 4.129, and 4.488 cents--

None of my inervals are tempered that much. Becaue the tempered 5ths reduce
the apparent tempering. For example, 9/7 worth 435 cents is 432 cents in my
JT scale. and just as smooth as the other. Some intervals can be tempered a
great deal and sill be smooth.

Pauiine

In a message dated 10/26/02 5:33:39 AM Central Daylight Time,
genewardsmith@juno.com writes:

> Five vs three tetrads and nine vs six triads if you confine yourself to
> errors under 2.5 cents. It would be nice to have a list of which chords are
> essential to this business in Pauline's understanding.
>
My C scale has 7c, 7d, 7 e, 7f, 7g, and 7a minor triads. How nany are needed
depends on how a composition is written.

Pauline

--- In tuning@y..., prophecyspirit@a... wrote:
> In a message dated 10/26/02 5:26:30 AM Central Daylight Time,
> genewardsmith@j... writes:
>
>
> > Pauline's scale has 7-limit intervals tempered by 4.129, and 4.488 cents--
>
> None of my inervals are tempered that much.

They are there if you want to use them, at any rate.

--- In tuning@y..., prophecyspirit@a... wrote:
> In a message dated 10/26/02 5:33:39 AM Central Daylight Time,
> genewardsmith@j... writes:
>
>
> > Five vs three tetrads and nine vs six triads if you confine yourself to
> > errors under 2.5 cents. It would be nice to have a list of which chords are
> > essential to this business in Pauline's understanding.
> >
> My C scale has 7c, 7d, 7 e, 7f, 7g, and 7a minor triads.

That's six, all right. What are the major triads? Major and minor tetrads?

In a message dated 10/26/02 7:27:23 PM Central Daylight Time,
genewardsmith@juno.com writes:

> My C scale has 7c, 7d, 7e, 7f, 7g, and 7a minor triads.
>
> What are the major triads? Major and minor tetrads?

Major = C, D, E, F#, G, A, B

minor = d, d#, e, f #, g#, a, a#, b . The F major and a tertian tminor triad
are in the F scale.

Pauline

--- In tuning@y..., prophecyspirit@a... wrote:
> In a message dated 10/26/02 7:27:23 PM Central Daylight Time,
> genewardsmith@j... writes:
>
>
> > My C scale has 7c, 7d, 7e, 7f, 7g, and 7a minor triads.
> >
> > What are the major triads? Major and minor tetrads?
>
> Major = C, D, E, F#, G, A, B
>
> minor = d, d#, e, f #, g#, a, a#, b . The F major and a tertian tminor triad
> are in the F scale.

I'm having trouble sorting this out. Could you give chains of fifths for your C scale for major and minor triads, and *also* for major and minor tetrads--meaning things intended to play a version of
1--5/4--3/2--7/4 (major tetrad) and 1--6/5--3/2--12/7 (minor tetrad.)
You've given a chain for the minor triads, and I presume the next chain is for major triads. What about tetrads? To what does

minor = d, d#, e, f #, g#, a, a#, b

refer?