The C-Fb-G major triad: Pythag-Just tuning

I had to come back to the list; I came up with something just now. Again,
it might have already been done before, but here it goes.

I took the 53-tone scale, which is usually based on the Pythagorean system
(recurring 3/2 intervals, i.e. 3-limit). I made it into 5-limit just, but
instead of defining the 5/4 interval as the notes C to E, I calculated the
relationship of the interval 5/4 to the 3/2-based structure instead of the
octave, the 2/1 interval.

In the 53-tone scale, if note #0 is C, the 3/2 interval gives you note #31,
which then is defined as G. And 4/3 is note #22, defined as F. Going
around the circle of fifths a bit, you come up with E (interval 81/64) being
assigned to note #18. But the interval 5/4 is calculated to note #17 (and
very accurately too). That would be, in strict Pythagorean terms, F-flat.
Or, in an alternative notation, E-quarter-flat.

I defined the intervals like this: D is the central note (not C). I go back
and forward four fifths from there, giving me a Pythagorean sequence from
B-flat to F-sharp. I then define G-flat as 5/4 and A-sharp as 8/5. I then
fill in the notes by fifths using the 3/2 interval, three towards D and four
away from D. Then I use 25/16 for C-double flat and 32/25 for E-double
sharp, and so.

So this is what I got:

0 D 1/1
1 Cx 81/80
2 Bx# 128/125

3 Fbb 25/24
4 Eb 135/128
5 D# 16/15
6 Cx# 27/25

7 Gbbb 1125/1024
8 Fb 10/9
9 E 9/8
10 Dx 512/225
11 Cxx 144/125

12 Gbb 75/64
13 F 32/27
14 E# 6/5
15 Dx# 4096/3375

16 Abbb 100/81
17 Gb 5/4
18 F# 81/64
19 Ex 32/25

20 Bbbbb 125/96
21 Abb 675/512
22 G 4/3
23 Fx 27/20
24 Ex# 512/375

25 Bbbb 25/18
26 Ab 45/32
27 G# 64/45
28 Fx# 36/25

29 Cbbb 375/256
30 Bbb 40/27
31 A 3/2
32 Gx 1024/675
33 Fxx 192/125

34 Cbb 25/16
35 Bb 128/81
36 A# 8/5
37 Gx# 81/50

38 Dbbb 3375/2048
39 Cb 5/3
40 B 27/16
41 Ax 128/75

42 Ebbbb 125/72
43 Dbb 225/128
44 C 16/9
45 B# 9/5
46 Ax# 2048/1125

47 Ebbb 50/27
48 Db 15/8
49 C# 256/135
50 Bx 48/25

51 Fbbb 125/64
52 Ebb 160/81
53 D' 2/1

A semi-radical idea, at least looking at 5-limit.

Danny

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--- In tuning@y..., "Danny Wier" <dawier@y...> wrote:

> A semi-radical idea, at least looking at 5-limit.

This is a 5-limit detempering of the 53-et which looks to be based on
the commas 15625/15552 (kleisma) and 32805/32768 (schisma), so it
would be worth comparing this to a Fokker block based on these two if
you wanted to persue it. I would ditch the name "Pythagorean", since
it isn't at all Pythagorean.

Danny Wier:
> > A semi-radical idea, at least looking at 5-limit.

Gene:
> This is a 5-limit detempering of the 53-et which looks to be based on
> the commas 15625/15552 (kleisma) and 32805/32768 (schisma), so it
> would be worth comparing this to a Fokker block based on these two if
> you wanted to persue it. I would ditch the name "Pythagorean", since
> it isn't at all Pythagorean.

Where does the kleisma come in? It looks like standard schismic to me. A
radical idea indeed when it was new -- back in the 13th Century.

Graham

--- In tuning@y..., "Danny Wier" <dawier@y...> wrote:

> A semi-radical idea, at least looking at 5-limit.
>
> Danny

This idea actually seems to have dominated keyboard tuning in the
1420-1480 era. Keyboards were tuned in a Pythagorean chain from Gb to
B, and the major triads on D, A, and E, since they were actually D Gb
A, A Db E, and E Ab B, were nearly just.

--- In tuning@y..., graham@m... wrote:
>
> Where does the kleisma come in? It looks like standard schismic to
me. A
> radical idea indeed when it was new -- back in the 13th Century.

Was that when the medieval Arabic tuning system of 17-out-of-
Pythagorean, which is preserved in classical Turkish music theory,
was new?

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_30893.html#30900

> --- In tuning@y..., "Danny Wier" <dawier@y...> wrote:
>
> > A semi-radical idea, at least looking at 5-limit.
> >
> > Danny
>
> This idea actually seems to have dominated keyboard tuning in the
> 1420-1480 era. Keyboards were tuned in a Pythagorean chain from Gb
to B, and the major triads on D, A, and E, since they were actually D
Gb A, A Db E, and E Ab B, were nearly just.

This is the tuning that Margo Schulter was discussing in detail as
advanced by Mark Lindley, correct??

/tuning/topicId_28704.html#28704

JP

--- In tuning@y..., jpehrson@r... wrote:

> This is the tuning that Margo Schulter was discussing in detail as
> advanced by Mark Lindley, correct??

Yup!

Me:
> > Where does the kleisma come in? It looks like standard schismic to
> me. A
> > radical idea indeed when it was new -- back in the 13th Century.

Paul:
> Was that when the medieval Arabic tuning system of 17-out-of-
> Pythagorean, which is preserved in classical Turkish music theory,
> was new?

I assume so. It's when al-Urmawi specified the 17 note Pythagorean, and
defined Rast using it to look suspiciously like an equal tetrachord JI
diatonic. The idea could have been around longer for all I'd know.

Graham

--- In tuning@y..., graham@m... wrote:

> Where does the kleisma come in?

If you look at ratios of scale steps, you get a lot of
(3125/3072)/(81/80) = 15625/15552 as well as
(81/80)/(2048/2025) - 32805/32768.

It looks like standard schismic to me. A
> radical idea indeed when it was new -- back in the 13th Century.

It looked as if he was going to do schismic, but he didn't. Instead,
he did something which isn't a temperament at all, but a 53-note
5-limit JI scale. By way of comparison, here is a Fokker block based
on the kleisma and the schisma:

[1, 81/80, 128/125, 25/24, 135/128, 16/15, 27/25, 800/729, 10/9, 9/8,
256/225, 144/125, 75/64, 32/27, 6/5, 243/200, 100/81, 5/4, 81/64,
32/25, 125/96, 320/243, 4/3, 27/20, 512/375, 25/18, 45/32, 64/45,
36/25, 375/256, 40/27, 3/2, 243/160, 192/125, 25/16, 128/81, 8/5,
81/50, 400/243, 5/3, 27/16, 128/75, 125/72, 225/128, 16/9, 9/5,
729/400, 50/27, 15/8, 256/135, 48/25, 125/64, 160/81]

Danny might want to look at this and see what he thinks of it.

Hey I'm reading your posts; I'll try to get together a competent response to
some of them later on. I just have a question and a comment:

First, what is the name of the comma 3^53:2^84
(19,383,245,667,680,019,896,796,723:19,342,813,113,834,066,795,298,816 --
which comes out 1.002090:1 or 3.6150 cents)?

Second, the question of whether Pythagorean tuning is really Pythagorean.
Since it's there in nature, it can't be invented like the light bulb. It's
probably been known to the Egyptians five thousand years ago or the
Sumerians around that time, and very likely the Australians thousands of
years before that. As far as the naming is concerned, it's not so much a
matter of discovery as it is promotion. We just happen to be culturally
closer to the classical Greeks, so that's where we get our names. The
Arabic numerals, after all, aren't really Arabic; they're Indian.

Like I had just "rediscovered" a medieval tuning system and gave it my own
name. I'm glad somebody thought of it before; it's been "out there", like
the Pythagorean fifths, since time began and it needed to be found sometime.

~DaW~

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--- In tuning@y..., "Danny Wier" <dawier@y...> wrote:

> First, what is the name of the comma 3^53:2^84
>
(19,383,245,667,680,019,896,796,723:19,342,813,113,834,066,795,298,816
--
> which comes out 1.002090:1 or 3.6150 cents)?

It's the last entry on Manuel's list--Mercator's comma. Mercator took
time off from cartography and gave us this, it seems, though the
Chinese may have beaten him to it.

> Second, the question of whether Pythagorean tuning is really
Pythagorean.
> Since it's there in nature, it can't be invented like the light
bulb.

All I meant by that is that "Pythagorean" usually means 3-limit, and
this was 5-limit. The 5 is there in nature also.

> [1, 81/80, 128/125, 25/24, 135/128, 16/15, 27/25, 800/729, 10/9, 9/8,
> 256/225, 144/125, 75/64, 32/27, 6/5, 243/200, 100/81, 5/4, 81/64,
> 32/25, 125/96, 320/243, 4/3, 27/20, 512/375, 25/18, 45/32, 64/45,
> 36/25, 375/256, 40/27, 3/2, 243/160, 192/125, 25/16, 128/81, 8/5,
> 81/50, 400/243, 5/3, 27/16, 128/75, 125/72, 225/128, 16/9, 9/5,
> 729/400, 50/27, 15/8, 256/135, 48/25, 125/64, 160/81]
>
> Danny might want to look at this and see what he thinks of it.

It's an improvement on my scale, since it shrunk the numerators and
denominators of some of the intervals:

1,125/1,024 > 800/729
4,096/3,375 > 243/200
675/512 > 320/243
1,024/675 > 243/160
3,375/2,048 > 400/243
2,048/1,125 > 729/400

I also made a mistake with one interval which had a numerator of 512 and a
denominator of less than half that; the numerator should be 256.

How do you make Fokker periodicities with Scala, by the way?

~DaW~

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From: <genewardsmith@juno.com>

> --- In tuning@y..., "Danny Wier" <dawier@y...> wrote:
>
> > First, what is the name of the comma 3^53:2^84
> >
> (19,383,245,667,680,019,896,796,723:19,342,813,113,834,066,795,298,816
> --
> > which comes out 1.002090:1 or 3.6150 cents)?
>
> It's the last entry on Manuel's list--Mercator's comma. Mercator took
> time off from cartography and gave us this, it seems, though the
> Chinese may have beaten him to it.

Uh oh, I found a better one...

3^665:2^1054

(this comes out to 1.930343e+317:1.930258e+317 = 1.000043655063 ~ 0.075575
cents!)

~DaW~

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Danny!
This might save you some time
http://www.anaphoria.com/viggo.PDF
wilson and Hanson have covered this area up to 232565518
you might want to start there!

Danny Wier wrote:

> From: <genewardsmith@juno.com>
>
> > --- In tuning@y..., "Danny Wier" <dawier@y...> wrote:
> >
> > > First, what is the name of the comma 3^53:2^84
> > >
> > (19,383,245,667,680,019,896,796,723:19,342,813,113,834,066,795,298,816
> > --
> > > which comes out 1.002090:1 or 3.6150 cents)?
> >
> > It's the last entry on Manuel's list--Mercator's comma. Mercator took
> > time off from cartography and gave us this, it seems, though the
> > Chinese may have beaten him to it.
>
> Uh oh, I found a better one...
>
> 3^665:2^1054
>
> (this comes out to 1.930343e+317:1.930258e+317 = 1.000043655063 ~ 0.075575
> cents!)
>
> ~DaW~
>
>

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

Gene wrote:
>It's the last entry on Manuel's list--Mercator's comma. Mercator took
>time off from cartography and gave us this, it seems, though the
>Chinese may have beaten him to it.

This is Nicolaus Mercator and the map maker is Gerard Mercator.
There's some reason to confound them since Nicolaus also worked on
Gerard's map projection. For a link go to
http://www.xs4all.nl/~huygensf/english then click on "Persons" and
the link to Mercator.

Manuel

>How do you make Fokker periodicities with Scala, by the way?

In case I misinterpreted your question, use the PIPEDUM command,
or go to File->New->Periodicity block.

Manuel

--- In tuning@y..., "Danny Wier" <dawier@y...> wrote:

> Uh oh, I found a better one...
>
> 3^665:2^1054
>
> (this comes out to 1.930343e+317:1.930258e+317 = 1.000043655063 ~
0.075575
> cents!)

Yup . . . it's been mentioned quite a bit that the 666th note in a
chain of fifths is awfully close to the first.