Yahoo Tuning Groups Ultimate Backup tuning Definitions of JI (was: Digest Number 3539

Message: 9 Date: Mon, 06 Jun 2005 15:55:12 -0000
From: "Dave Keenan" <d.keenan@bigpond.net.au>
Subject: Re: Subject: Definitions of JI (was: Digest Number 3539)

>> >
Well yes, I can see that is a logical conclusion from taking "just" to
mean "rational".

So if someone says to you that some interval is Just, this would tell
you absolutely nothing about it?

I would say that while JI might be not able to tell us much about a single interval,
It tells us how a scales or choice of pitches are organized and structured,.
This method produces scales of pitch structure that are for th most part
unequal in scale size ( but i repeat my self) Meta slendro BTW has an interval of 400 cents, although the context makes it have nothing to do with E.

The same interval could exist in a myiad of ETs also for all practical purposes. it is the context in which it exist that is important

For example, if asked to play a just
major third, you'd be just as likely to play a 12-equal major third,
or to ask which of the infinite number of possible rational major
thirds they would like you to aim for?

What ratio do you consider 75 cents to be?

-- Dave Keenan

Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

🔗klaus schmirler <KSchmir@online.de>

Kraig Grady wrote:

> I would say that while JI might be not able to tell us much about a
> single interval, It tells us how a scales or choice of pitches are
> organized and structured,. This method produces scales of pitch
> structure that are for th most part unequal in scale size ( but i
> repeat my self)

Ah! There is that other view that would not call a pythagorean scale just, no matter the number of decimals. It's the first time I notice this view in this discussion, and I think it is entirely valid.

For me justness implies intervals arrived at by harmonic divisions, and as long as the actual intervals reflect the sizes of the just intervals, the intention of justness must be acknowledged. 12-et is badly tuned just up to a point, because (or as long as) it doesn't divide the octave into two 600 cent tritones nor the fifth into equal thirds of 350 or 351 cents. But as with any other meantone, the justness stops at the seconds.

The harmonic divisions, however, bring up another problem which concerns scales and chords equally, I guess.

All is fine as long as all divisions are arranged otonally. All divisions together reinforce the 1/1, and well tuned justness becomes audible. Is this still the case when the ordering is utonal? (Is there a 1/1, or does it have to be distinguished form a lcm/lcm?) What happens when a chord is mixed? Do the unconnected otonal regions take over, so you get polytonality?

klaus

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, klaus schmirler <KSchmir@o...> wrote:

> Ah! There is that other view that would not call a pythagorean scale
> just, no matter the number of decimals. It's the first time I notice
> this view in this discussion, and I think it is entirely valid.

Who supports this view, and with what justification? Is this from a
Dave Keenan type of argument, that connections by just 3-limit
intervals are not enough to call something just?

> For me justness implies intervals arrived at by harmonic divisions,
> and as long as the actual intervals reflect the sizes of the just
> intervals, the intention of justness must be acknowledged.

And a harmonic division is what--a low integer ratio?

🔗klaus schmirler <KSchmir@online.de>

Gene Ward Smith wrote:

> --- In tuning@yahoogroups.com, klaus schmirler <KSchmir@o...> wrote:
> > >>Ah! There is that other view that would not call a pythagorean scale >>just, no matter the number of decimals. It's the first time I notice >>this view in this discussion, and I think it is entirely valid.
> > > Who supports this view, and with what justification? Is this from a
> Dave Keenan type of argument, that connections by just 3-limit
> intervals are not enough to call something just?

Kraig Grady in the part you snipped:

> JI might be not able to tell us much about a
>> single interval, It tells us how a scales or choice of pitches are
>> organized and structured,. This method produces scales of pitch
>> structure that are for th most part unequal in scale size

> > >>For me justness implies intervals arrived at by harmonic divisions, >>and as long as the actual intervals reflect the sizes of the just >>intervals, the intention of justness must be acknowledged. > > > And a harmonic division is what--a low integer ratio?

If it's the division of an integer ratio, it's generally a higher integer ratio.

Besides, it's a possibly improper verbal shortcut on my part for "division at the harmonic mean". But first of all, it's a mistake, and "harmonic" should read "arithmetic". Must be that frequency vs. string length thing. Sorry!

So you divide the octave by adding 1/1 and 2/1 and divide by 2: 3/2. New notes are added by dividing the resulting intervals in the same manner, which is inherently different from the method of linking generators. One difference (a big one for people away from their pocket calculators) is that dividing intervals calls for integer ratios as they are deemed typical for just intonation. Generator chains on the other hand are commonly not very choosy about the size and rationality of their generator. Is this justification enough to draw a sharp divide?

klaus

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, klaus schmirler <KSchmir@o...> wrote:
> Gene Ward Smith wrote:
>
> > --- In tuning@yahoogroups.com, klaus schmirler <KSchmir@o...> wrote:
> >
> >
> >>Ah! There is that other view that would not call a pythagorean scale
> >>just, no matter the number of decimals. It's the first time I notice
> >>this view in this discussion, and I think it is entirely valid.
> >
> >
> > Who supports this view, and with what justification? Is this from a
> > Dave Keenan type of argument, that connections by just 3-limit
> > intervals are not enough to call something just?
>
> Kraig Grady in the part you snipped:
>
> > JI might be not able to tell us much about a
> >> single interval, It tells us how a scales or choice of pitches are
> >> organized and structured,. This method produces scales of pitch
> >> structure that are for th most part unequal in scale size

Pythagorean has generators 2 and 3, and hence produces unequal step sizes.

> So you divide the octave by adding 1/1 and 2/1 and divide by 2: 3/2.
> New notes are added by dividing the resulting intervals in the same
> manner, which is inherently different from the method of linking
> generators. One difference (a big one for people away from their
> pocket calculators) is that dividing intervals calls for integer
> ratios as they are deemed typical for just intonation. Generator
> chains on the other hand are commonly not very choosy about the size
> and rationality of their generator. Is this justification enough to
> draw a sharp divide?

Not really--you can take arithmetic means with real numbers, not just
rational numbers.

🔗klaus schmirler <KSchmir@online.de>

Gene Ward Smith wrote:
> --- In tuning@yahoogroups.com, klaus schmirler <KSchmir@o...> wrote:
> >>Gene Ward Smith wrote:
>>
>>
>>>--- In tuning@yahoogroups.com, klaus schmirler <KSchmir@o...> wrote:
>>>
>>>
>>>
>>>>Ah! There is that other view that would not call a pythagorean scale >>>>just, no matter the number of decimals. It's the first time I notice >>>>this view in this discussion, and I think it is entirely valid.
>>>
>>>
>>>Who supports this view, and with what justification? Is this from a
>>>Dave Keenan type of argument, that connections by just 3-limit
>>>intervals are not enough to call something just?
>>
>>Kraig Grady in the part you snipped:
>>
>>
>>>JI might be not able to tell us much about a
>>>
>>>>single interval, It tells us how a scales or choice of pitches are
>>>>organized and structured,. This method produces scales of pitch
>>>>structure that are for th most part unequal in scale size
> > > Pythagorean has generators 2 and 3, and hence produces unequal step sizes.

It's certainly contentiuos what "for the most part unequal" means exactly, but when three out of five intervals in a pythagorean pentatonic are major tones, the likelihood that it meets Kraig's criteria of "just" is less than slim.

Wasn't there a convention once to silently assume the octave as a generator, or even to call that kind of "generator" "period"?

> > >>So you divide the octave by adding 1/1 and 2/1 and divide by 2: 3/2. >>New notes are added by dividing the resulting intervals in the same >>manner, which is inherently different from the method of linking >>generators. One difference (a big one for people away from their >>pocket calculators) is that dividing intervals calls for integer >>ratios as they are deemed typical for just intonation. Generator >>chains on the other hand are commonly not very choosy about the size >>and rationality of their generator. Is this justification enough to >>draw a sharp divide?
> > > Not really--you can take arithmetic means with real numbers, not just
> rational numbers. I could wiggle and squirm and we'd end up where the rest of the parent discussion got stuck: with arguments about tuning precision. (Actually, I had hoped that mentioning the absence of pocket calculators would have spared me your objection. Do I have to say or do you know that I'm not talking about cents either and that the arithmetic mean of a logarithmic measure corresponds to a geometric mean in frequencies?)

Before I put my question I described a criterion for differentiating just from unjust independently of accuracy. It's nothing new and has a name here, but I don't remember it: odd limit something. It would call 12-et "just" up to the 7 limit and "just" and consistent up to the 5 limit, just because it can represent the "unequal" (Kraig Grady), "harmonic" (in terms of string lengths), "arithmetic" (in terms of frequencies) division by mapping them to different sizes. Whether the accuracy is good enough for a given purpose would then be a different question that every individual case and user may answer differently.

klaus

🔗Kraig Grady <kraiggrady@anaphoria.com>

________________________________________________________________________
________________________________________________________________________

Message: 3 Date: Tue, 07 Jun 2005 15:19:13 +0200
From: klaus schmirler <KSchmir@online.de>
Subject: Over and Under [was: Definitions of JI]

Stuctures like the eikosany have no implied 1/1 , or one can place the 1/1 anywhere to generate the whole structure with the same degree of simmplicity and complexity.
It takes the same material as th diamond and arranges it is a complementry way which in some of ervs papers, he shows how they overlap.
I built quite a few instruments in this tuning and was in the proces of rebuilding many of them with added tones ( back problems for the last 4 months has prevented that). That it is possible to have just intonation without an implied tonic , greatly appealed to me at the time as a sort of atonality ( i actiually prefer the Rudolph Reti term - pantonality)constructed and held in place by a series of consonances.

I do consider 3 limit just, a special type of just though maybe prejust.

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

🔗klaus schmirler <KSchmir@online.de>

Kraig Grady wrote:

> Stuctures like the eikosany have no implied 1/1 , or one can place
> the 1/1 anywhere to generate the whole structure with the same
> degree of simmplicity and complexity. It takes the same material as
> th diamond and arranges it is a complementry way which in some of
> ervs papers, he shows how they overlap.

I never understood that. does it have to do with that one diagonal of 1/1s in the diamond?

I built quite a few
> instruments in this tuning and was in the proces of rebuilding many
> of them with added tones ( back problems for the last 4 months has
> prevented that). That it is possible to have just intonation
> without an implied tonic , greatly appealed to me at the time as a
> sort of atonality ( i actiually prefer the Rudolph Reti term -
> pantonality)constructed and held in place by a series of
> consonances.
> > I do consider 3 limit just, a special type of just though maybe
> prejust.

I guess in this case just or non-just is indeed a matter of accuracy.

I would have made a strict division between generator chains and interval division. They are often mixed (as in common practice 5 limit with its chains of many fifths but only up to two thirds), but only an interval constellation arrived at by division would be pure JI (divide the octave: 3/2 and 4/3; divide each of these: 5/4 and 6/5 and 7/6 and 8/7; divide the 5/4: 9/8 and 10/9). I think that no interval is repeated. Now compare it to a chain of fifths. There are only two different intervals; the first division is identical, the second division can be mapped to different interval sizes; but the 10/9 is mapped to another 9/8, so the "imitation of justness" fails at this point. For the 5/4 and 7/6, you'd acknowledge the attempt and call it just as well, but add badly tuned or well enough or something of that sort.

I thought this was uniqueness, but it isn't.

klaus

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, klaus schmirler <KSchmir@o...> wrote:

> > Pythagorean has generators 2 and 3, and hence produces unequal
step sizes.

> It's certainly contentiuos what "for the most part unequal" means
> exactly, but when three out of five intervals in a pythagorean
> pentatonic are major tones, the likelihood that it meets Kraig's
> criteria of "just" is less than slim.

Apparently it does, however. You are getting at a property here, but
the property has nothing to do with JI really; it is the number of
generators, or "rank". If one generator is a period, and you are
constructing a periodic scale, the number of different step sizes in
your scale is at least equal to the rank of the group you are using to
define the notes. Of course, if the group you are using to define the
notes is all the real numbers, that tells you nothing, but very often
you will be using a regular tuning system defined by a free abelian
group of some finite rank, meaning some finite number of independent
generators.

Hence, Pythagorean scales must have at least two step sizes. The same
is true of linear temperaments such as meantone or miracle. Five limit
JI scales must have at least three step sizes. The same is true of
planar temperaments such as marvel or breed. Consequently this
property, which is really the rank in a disguised form, cannot be used
to define Just Intonation.

> Wasn't there a convention once to silently assume the octave as a
> generator, or even to call that kind of "generator" "period"?

No; the octave is not a generator unless so stated, but if the octave
is a period, and there are two generators, then the other generator is
called "the generator". If you say the period is an octave, or 1/n
octave, that defines one of the generators. However, you don't need to
refer to octaves at all, even if the tuning system has them, and to
call something a "period" assumes you are constructing scales, which
you don't necessarily need to be doing either.

> (Actually, I had hoped that mentioning the absence of pocket
> calculators would have spared me your objection. Do I have to say or
> do you know that I'm not talking about cents either and that the
> arithmetic mean of a logarithmic measure corresponds to a geometric
> mean in frequencies?)

Absent a pocket calculator it is still not too difficult to take a
geometric mean; the basic tools for doing that is the arithmetic mean
and iteration.

> Before I put my question I described a criterion for differentiating
> just from unjust independently of accuracy. It's nothing new and has a
> name here, but I don't remember it: odd limit something.

"Odd limit" defines a method of deciding which intervals are going to
count as consonances for the nonce. If q is an odd number and all
ratios n/m with n,m <= q are considered to be consonant, you are in
the q odd limit.

It would call
> 12-et "just" up to the 7 limit and "just" and consistent up to the 5
> limit, just because it can represent the "unequal" (Kraig Grady),
> "harmonic" (in terms of string lengths), "arithmetic" (in terms of
> frequencies) division by mapping them to different sizes.

This doesn't make much sense to me. When and why did string lengths
enter the discussion? Anyway, this just shows the weakness of trying
to define JI in terms of rank; it won't work.

Whether the
> accuracy is good enough for a given purpose would then be a different
> question that every individual case and user may answer differently.

Not very helpful if you want to know what someone means by calling
something "JI". If you knew they meant "within 1/4 cent of accuracy in
the relevant limit" it might be more specific than you'd like, but it
would at least be specific.

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> Apparently it does, however. You are getting at a property here, but
> the property has nothing to do with JI really; it is the number of
> generators, or "rank". If one generator is a period, and you are
> constructing a periodic scale, the number of different step sizes in
> your scale is at least equal to the rank of the group you are using to
> define the notes.

I should say, rather, to *construct* the notes. You could define notes
in 5-limit JI and only construct them in 3-limit JI, and then you
could have two step sizes. Or you could leave off 3s and only use 5s,
or only use combinations of 2 and 5/3. But if the notes of your scale
generate a rank 3 group, then you cannot just have two step sizes in
it. You must have at least 3, but can have many more.

🔗Ozan Yarman <ozanyarman@superonline.com>

Gene, can a diatonical scale out of 43tET or 55tET be considered just?
----- Original Message -----
From: Gene Ward Smith
To: tuning@yahoogroups.com
Sent: 08 Haziran 2005 Çarşamba 21:46
Subject: [tuning] Re: Definitions of JI [was: Over and Under [was: Definitions of JI]}

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

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Ozan Yarman <ozanyarman@superonline.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> What about a good approximation to just?

People call 53 or 72 good approximations to just, or near-just,
etc, but I don't think anyone says that about 43 or 55, and if they do
they really shouldn't. Nearly always they are used, or suggested for
use, for meantone tunings. Neither is a standout so far as tuning
accuracy goes, and 55 in particular is inconsistent in the 7-limit.
You get septimal meantone tempering out 81/80 and 126/125, but a
different temperament tempering out 81/80 and 686/675, which uses a
different mapping to primes; what I call a "val". It has two
contending 7-limit vals, in other words. If 43 works you might stick
with it. Of course I was suggesting 103 precisely because it supports
both meantone and miracle using different mappins, and the ability to
do that isn't a disaster.

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, klaus schmirler <KSchmir@o...> wrote:
>
> > Ah! There is that other view that would not call a pythagorean scale
> > just, no matter the number of decimals. It's the first time I notice
> > this view in this discussion, and I think it is entirely valid.
>
> Who supports this view, and with what justification? Is this from a
> Dave Keenan type of argument, that connections by just 3-limit
> intervals are not enough to call something just?

I note that even by my "more than linear connectivity" criterion
Pythagorean could still be called Just since it has Just 4:9's as well
as 2:3's.

I could easily go either way on whether Pythagorean is JI, and I don't
think it's very important. Historically it was definitely not
considered JI, and I don't see much point in considering it so now.
But I think this is only about whether you should use scare-quotes or
not when you write that Pythagorean is "3-limit JI". Normally that is
said only to illustrate the meaning of "limit".

-- Dave Keenan

🔗Ozan Yarman <ozanyarman@superonline.com>

----- Original Message -----
From: Gene Ward Smith
To: tuning@yahoogroups.com
Sent: 09 Haziran 2005 Perşembe 1:10
Subject: [tuning] Re: Definitions of JI [was: Over and Under [was: Definitions of JI]}

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> What about a good approximation to just?

Then how about this well-temperament with a period of 1202.333 cents?

|
0: 0.000 cents 0.000 0 0 commas C
39: 696.955 cents -5.000 -153 G
11: 701.955 cents -5.000 -153 D
50: 696.955 cents -10.000 -307 A
22: 701.955 cents -9.999 -307 E
61: 696.955 cents -14.999 -460 B
33: 701.955 cents -14.999 -460 F#
5: 701.955 cents -14.998 -460 C#
44: 696.955 cents -19.998 -614 G#
16: 701.955 cents -19.998 -614 D#
55: 696.955 cents -24.998 -767 A#
27: 701.955 cents -24.998 -767 E#
66: 696.955 cents -29.998 -921 B#
38: 701.955 cents -29.997 -921 G(
10: 701.955 cents -29.997 -921 D(
49: 696.955 cents -34.997 -1074 A(
21: 701.955 cents -34.997 -1074 E(
60: 696.955 cents -39.997 -1228 B(
32: 701.955 cents -39.996 -1228 GbL
4: 701.955 cents -39.996 -1227 DbL
43: 696.955 cents -44.996 -1381 AbL
15: 701.955 cents -44.995 -1381 EbL
54: 696.955 cents -49.995 -1534 BbL
26: 701.955 cents -49.995 -1534 FL
65: 696.955 cents -54.995 -1688 CL
37: 701.955 cents -54.995 -1688 GL
9: 701.955 cents -54.994 -1688 DL
48: 696.955 cents -59.994 -1841 AL
20: 701.955 cents -59.994 -1841 EL
59: 696.955 cents -64.994 -1995 BL
31: 701.955 cents -64.994 -1995 F7)
3: 701.955 cents -64.993 -1995 C7)
42: 696.955 cents -69.993 -2148 G7)
14: 701.955 cents -69.993 -2148 D7)
53: 696.955 cents -74.993 -2302 A7)
25: 701.955 cents -74.992 -2302 FL(
64: 696.955 cents -79.992 -2455 CL(
36: 701.955 cents -79.992 -2455 GL(
8: 701.955 cents -79.992 -2455 DL(
47: 696.955 cents -84.992 -2608 AL(
19: 701.955 cents -84.991 -2608 EL(
58: 696.955 cents -89.991 -2762 BL(
30: 701.955 cents -89.991 -2762 F7
2: 701.955 cents -89.991 -2762 C7
41: 696.955 cents -94.991 -2915 G7
13: 701.955 cents -94.990 -2915 D7
52: 696.955 cents -99.990 -3069 A7
24: 701.955 cents -99.990 -3069 E7
63: 696.955 cents -104.990 -3222 B7
35: 701.955 cents -104.989 -3222 F#7
7: 701.955 cents -104.989 -3222 C#7
46: 696.955 cents -109.989 -3376 G#7
18: 701.955 cents -109.989 -3376 D#7
57: 696.955 cents -114.989 -3529 A#7
29: 701.955 cents -114.988 -3529 F)
1: 701.955 cents -114.988 -3529 C)
40: 696.955 cents -119.988 -3682 G)
12: 701.955 cents -119.987 -3682 D)
51: 696.955 cents -124.987 -3836 A)
23: 701.955 cents -124.987 -3836 Fb
62: 696.955 cents -129.987 -3989 Cb
34: 701.955 cents -129.987 -3989 Gb
6: 701.955 cents -129.986 -3989 Db
45: 696.955 cents -134.986 -4143 Ab
17: 701.955 cents -134.986 -4143 Eb
56: 696.955 cents -139.986 -4296 Bb
28: 701.955 cents -139.986 -4296 F
67: 701.955 cents -139.985 -4296 C

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> Then how about this well-temperament with a period of 1202.333 cents?

I don't think anyone is going to agree to callin g octaves sharp by
two cents "just". You seem to have migrated back to 67, and to have
some fifths pure and some flat by five cents. That would certainly
work for meantone music, but I take it that it also does for maqams?
The simplicity of an equal temperament would seem to be better, but
perhaps this has something to offer also. But it seems to me that if
you object to 103 on the grounds of excessive complexity, then this is
even more of a problem.

If you take 270-et, you have a meantone fifth between 43 and 55,
though closer to 43. You have another near-just fifth which you can
use when needed. And as a truly terrific bonus, you have a 13-limit
interpretation of everything--270 does the 13-limit quite nicely.
While it's a heck of a lot more complex than 43, it seems to me that
using it to define maqams would be much preferable than using a scale
like the one below.

🔗Ozan Yarman <ozanyarman@superonline.com>

Ok, then stretch the octave as 2/1. Then you have two tempered fifths, the smaller 695.597 cents, the larger 700.597 cents. That gives you an awfully good well temperament of great resolution through 67 tones, which can easily be considered equal:

0: 1/1 C unison, perfect prime
1: 17.836 cents C)
2: 35.672 cents C7
3: 53.507 cents C7) DbL(
4: 71.343 cents C#( DbL
5: 89.179 cents C# Db(
6: 107.015 cents C#) Db
7: 124.851 cents C#7 Db)
8: 142.687 cents C#7) DL(
9: 160.522 cents DL
10: 178.358 cents D(
11: 196.194 cents D
12: 214.030 cents D)
13: 231.866 cents D7
14: 249.701 cents D7) EbL(
15: 267.537 cents D#( EbL
16: 285.373 cents D# Eb(
17: 303.209 cents D#) Eb
18: 321.045 cents D#7 Eb)
19: 338.880 cents D#7) EL(
20: 356.716 cents EL
21: 374.552 cents E(
22: 392.388 cents E
23: 410.224 cents E) Fb
24: 428.060 cents E7 Fb)
25: 445.895 cents E7) FL(
26: 463.731 cents E#( FL
27: 481.567 cents E# F(
28: 499.403 cents F
29: 517.239 cents F)
30: 535.074 cents F7
31: 552.910 cents F7) GbL(
32: 570.746 cents F#( GbL
33: 588.582 cents F# Gb(
34: 606.418 cents F#) Gb
35: 624.254 cents F#7 Gb)
36: 642.089 cents F#7) GL(
37: 659.925 cents GL
38: 677.761 cents G(
39: 695.597 cents G
40: 713.433 cents G)
41: 731.268 cents G7
42: 749.104 cents G7) AbL(
43: 766.940 cents G#( AbL
44: 784.776 cents G# Ab(
45: 802.612 cents G#) Ab
46: 820.447 cents G#7 Ab)
47: 838.283 cents G#7) AL(
48: 856.119 cents AL
49: 873.955 cents A(
50: 891.791 cents A
51: 909.627 cents A)
52: 927.462 cents A7
53: 945.298 cents A7) BbL(
54: 963.134 cents A#( BbL
55: 980.970 cents A# Bb(
56: 998.806 cents A#) Bb
57: 1016.641 cents A#7 Bb)
58: 1034.477 cents A#7) BL(
59: 1052.313 cents BL
60: 1070.149 cents B(
61: 1087.985 cents B
62: 1105.821 cents B) Cb
63: 1123.656 cents B7 Cb)
64: 1141.492 cents B7) CL(
65: 1159.328 cents B#( CL
66: 1177.164 cents B# C(
67: 1200.000 cents C

And remember, this is supposed to be the sound system for my personal Qanun. I am mighty pleased with it as it is, because indeed, both meantone and maqam music notation requires such "tempered fifths". The issue is, I will eventually require to merge this with pythagorean tuning when certain maqams demand so.

However, I just cannot have reversed-order sharps/flats while the diatonic major of the white keys remains 3-limit pythagorean, and I hate to consider limma-alterations for it. Perhaps a quick retuning to 65tET will fix the problem?

270tET? Not in your life. I will already have a hard time convincing people to abondon the Arel-Ezgi notation (which doesn't suit 24-pythagorean or 53tET anyway) and adopt 55tET.

Cordially,
Ozan
----- Original Message -----
From: Gene Ward Smith
To: tuning@yahoogroups.com
Sent: 09 Haziran 2005 Perşembe 7:59
Subject: [tuning] Re: Definitions of JI [was: Over and Under [was: Definitions of JI]}

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> Then how about this well-temperament with a period of 1202.333 cents?