Topic: 159 and trikleismic temperament

3 scales

Thread (56 messages)

From: Gene Ward Smith (2007-02-16)
Subject: 159 and trikleismic temperament

I've been thinking of late about temperaments tempering out 385/384, 
one of which is Ozan's 159-et. While good approximations to rational 
intervals does not seem to be the focus of Ozan's thinking, from the 
point of view of low complexity for rational numbers the 72&87 
temperament, known as trikleismic or tritikleismic, is an obvious 
choice. It was mentioned briefly when Ozan brought up the subject of 
159, and it is one of the better 11 limit temperaments out there. It 
also does 13 accurately (using eg the 159 tuning, which is an excellent 
and recommended trikleismic tuning) but that's more complex. 

Trikleismic consists of three chains of hanson/kleismic temperament, 
separated by 400 cent periods. That is, we have generators which are 
minor thirds a cent+ sharp, and a period of 1/3 of an octave. This 
gives the 5-limit harmony on each chain as in hanson, with the 7 and 11 
limits are in easy reach on the other chains: 4 periods up, which is an 
octave and a period, and two minor thirds down gives the 7/4, and  
three minor thirds up and a period down gives the 11/8. The 13/8 is 
considerably more complex, requiring 14 minor thirds to get to. 
However, if we take three chains of 19, for a scale of 57 notes, which 
is one of the most obvious possibilities, we will find a good number. 
While 57 is obviously a lot of notes, it's also 22 fewer notes than 
Ozan's 79.


! trikleismic57.scl
Trikleismic[57] in 159-et tuning
57
!
15.094340
52.830189
67.924528
83.018868
98.113208
135.849057
150.943396
166.037736
181.132075
218.867925
233.962264
249.056604
264.150943
301.886792
316.981132
332.075472
347.169811
384.905660
400.000000
415.094340
452.830189
467.924528
483.018868
498.113208
535.849057
550.943396
566.037736
581.132075
618.867925
633.962264
649.056604
664.150943
701.886792
716.981132
732.075472
747.169811
784.905660
800.000000
815.094340
852.830189
867.924528
883.018868
898.113208
935.849057
950.943396
966.037736
981.132075
1018.867925
1033.962264
1049.056604
1064.150943
1101.886792
1116.981132
1132.075472
1147.169811
1184.905660
1200.000000

From: Ozan Yarman (2007-02-16)
Subject: Re: [tuning] 159 and trikleismic temperament

Goodness gracious, it is like a trainwreck... I can't even get a decent D,
A, B or Bb by a chain of perfect fifths, the notation for even the simplest
maqams is a nightmare, certain transpositions of tetrachords & major and
minor triads are intolerable, the only apparent benefit compared to 79/80
MOS 159-tET is the dispersed 14:13, which happens to be found in equal
number in both tunings.

I'll have you know, that my focus is as much a good approximation of complex
simple integer ratios as a suitable cyclic notation (and by that, I mean
circulating with minimal damage to whatever interval is chosen) and the
ability to modulate & transpose scales without wincing.

Clearly, this trikleismic temperament fails in the last two criteria for a
master tuning worthy of expressing maqamat at every key.

As to the nagging question, "do we need so many keys at all?", my answer,
from the point of view that the option for polyphony is a requisite, is
"absolutely yes".

Oz.


----- Original Message -----
From: "Gene Ward Smith" <genewardsmith@coolgoose.com>
To: <tuning@yahoogroups.com>
Sent: 16 \ufffdubat 2007 Cuma 10:42
Subject: [tuning] 159 and trikleismic temperament


> I've been thinking of late about temperaments tempering out 385/384,
> one of which is Ozan's 159-et. While good approximations to rational
> intervals does not seem to be the focus of Ozan's thinking, from the
> point of view of low complexity for rational numbers the 72&87
> temperament, known as trikleismic or tritikleismic, is an obvious
> choice. It was mentioned briefly when Ozan brought up the subject of
> 159, and it is one of the better 11 limit temperaments out there. It
> also does 13 accurately (using eg the 159 tuning, which is an excellent
> and recommended trikleismic tuning) but that's more complex.
>
> Trikleismic consists of three chains of hanson/kleismic temperament,
> separated by 400 cent periods. That is, we have generators which are
> minor thirds a cent+ sharp, and a period of 1/3 of an octave. This
> gives the 5-limit harmony on each chain as in hanson, with the 7 and 11
> limits are in easy reach on the other chains: 4 periods up, which is an
> octave and a period, and two minor thirds down gives the 7/4, and
> three minor thirds up and a period down gives the 11/8. The 13/8 is
> considerably more complex, requiring 14 minor thirds to get to.
> However, if we take three chains of 19, for a scale of 57 notes, which
> is one of the most obvious possibilities, we will find a good number.
> While 57 is obviously a lot of notes, it's also 22 fewer notes than
> Ozan's 79.
>
>
> ! trikleismic57.scl
> Trikleismic[57] in 159-et tuning
> 57
> !
> 15.094340
> 52.830189
> 67.924528
> 83.018868
> 98.113208
> 135.849057
> 150.943396
> 166.037736
> 181.132075
> 218.867925
> 233.962264
> 249.056604
> 264.150943
> 301.886792
> 316.981132
> 332.075472
> 347.169811
> 384.905660
> 400.000000
> 415.094340
> 452.830189
> 467.924528
> 483.018868
> 498.113208
> 535.849057
> 550.943396
> 566.037736
> 581.132075
> 618.867925
> 633.962264
> 649.056604
> 664.150943
> 701.886792
> 716.981132
> 732.075472
> 747.169811
> 784.905660
> 800.000000
> 815.094340
> 852.830189
> 867.924528
> 883.018868
> 898.113208
> 935.849057
> 950.943396
> 966.037736
> 981.132075
> 1018.867925
> 1033.962264
> 1049.056604
> 1064.150943
> 1101.886792
> 1116.981132
> 1132.075472
> 1147.169811
> 1184.905660
> 1200.000000
>
>

From: Gene Ward Smith (2007-02-16)
Subject: Re: 159 and trikleismic temperament

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> Goodness gracious, it is like a trainwreck... I can't even get a 
decent D,
> A, B or Bb by a chain of perfect fifths, the notation for even the 
simplest
> maqams is a nightmare, certain transpositions of tetrachords & 
major and
> minor triads are intolerable, the only apparent benefit compared to 
79/80
> MOS 159-tET is the dispersed 14:13, which happens to be found in 
equal
> number in both tunings.

Obviously, a 57 note scale and a 79 note scale are not directly 
comparable, so I suggest we compare 79 notes of each. However, be it 
noted I was not proposing Trikleismic[57] as a maqam scale, but just 
as a scale.

The truncated wedgie for ozan temperament, the 2deg159 temperament, is
<<33 54 -64 43 9 ... ||. From this we may deduce, by subtraction, 
that there are 79-33 = 46 of the best fifths, and get similarly:

3/2s: 46
5/4s: 25
7/4s: 15
9/8s: 13
11/8s: 36
13/8s: 70

Major triads: 25
Otonal tetrads: 0
Otonal pentads, etc.: 0

The truncated wedgie for trikleismic temperament is
<<18 15 -6 9 42 ... ||. From this we may similarly derive:

3/2s: 61
5/4s: 64
7/4s: 73
9/8s: 43
11/8s: 70
13/8s: 37

Major triads: 61
Otonal tetrads: 55
Otonal pentads: 37
Otonal hexads (11-limit): 37
Otonal septads (13-limit): 31
Otonal octads (15-limit): 31

The greater efficiency of trikleismic is obvious.

> I'll have you know, that my focus is as much a good approximation 
of complex
> simple integer ratios as a suitable cyclic notation (and by that, I 
mean
> circulating with minimal damage to whatever interval is chosen) and 
the
> ability to modulate & transpose scales without wincing.
> 
> Clearly, this trikleismic temperament fails in the last two 
criteria for a
> master tuning worthy of expressing maqamat at every key.

I don't know what you mean by "circulating with minimal damage" 
exactly, but I would guess it has to do with the intervals in a given 
interval class, and that will depend, not just on the temperament, 
but ezactly which scale of the temperament you use. As for the 
ability to modulate a scale, that depends on the complexity of the 
scale in that temperament, if you mean by "modulate" to transpose 
key. What scales do you want to modulate with without wincing?

The 79 notes of trikleismic I was analyzing do not constitute a MOS, 
and if you wanted to have one (which would be more regular in 
behavior) then past 57 you'd need to go all the way up to 102 notes. 
If you used 66 or 90 notes, you'd at least get a 2MOS, however.

From: Gene Ward Smith (2007-02-16)
Subject: Re: 159 and trikleismic temperament

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

> The truncated wedgie for ozan temperament, the 2deg159 temperament, is
> <<33 54 -64 43 9 ... ||. 

Or maybe <<33 54 95 43 9 ...||, but that doesn't help us here, despite 
a lower graham complexity.

From: Ozan Yarman (2007-02-16)
Subject: Re: [tuning] Re: 159 and trikleismic temperament

SNIP


> > Goodness gracious, it is like a trainwreck... I can't even get a
> decent D,
> > A, B or Bb by a chain of perfect fifths, the notation for even the
> simplest
> > maqams is a nightmare, certain transpositions of tetrachords &
> major and
> > minor triads are intolerable, the only apparent benefit compared to
> 79/80
> > MOS 159-tET is the dispersed 14:13, which happens to be found in
> equal
> > number in both tunings.
>
> Obviously, a 57 note scale and a 79 note scale are not directly
> comparable, so I suggest we compare 79 notes of each. However, be it
> noted I was not proposing Trikleismic[57] as a maqam scale, but just
> as a scale.
>



I do not believe it will make a very suitable one. Maqams require at least
near pure fourths for the construction of agreeable tetrachords, plus
tolerable fifths for correct modulations.



> The truncated wedgie for ozan temperament, the 2deg159 temperament, is
> <<33 54 -64 43 9 ... ||. From this we may deduce, by subtraction,
> that there are 79-33 = 46 of the best fifths, and get similarly:
>
> 3/2s: 46
> 5/4s: 25
> 7/4s: 15
> 9/8s: 13
> 11/8s: 36
> 13/8s: 70
>
> Major triads: 25
> Otonal tetrads: 0
> Otonal pentads, etc.: 0
>
> The truncated wedgie for trikleismic temperament is
> <<18 15 -6 9 42 ... ||. From this we may similarly derive:
>
> 3/2s: 61
> 5/4s: 64
> 7/4s: 73
> 9/8s: 43
> 11/8s: 70
> 13/8s: 37
>
> Major triads: 61
> Otonal tetrads: 55
> Otonal pentads: 37
> Otonal hexads (11-limit): 37
> Otonal septads (13-limit): 31
> Otonal octads (15-limit): 31
>
> The greater efficiency of trikleismic is obvious.
>


Efficiency? What kind of a fancy word is that in regards to musical meaning?
And how did you deduce these numbers anyway? There are exactly 80
practicable harmonic major thirds, as much minor sevenths, unidecimal
semi-augmented fourths, and tridecimal neutral sixths, 113 perfect fifths
and as many whole tones in 80 MOS 159-tET, all of which sound quite
acceptable, if not pleasing, to the ear.

Besides, 79/80 MOS 159-tET comprises just as many, if not more, functional
otonal chords up to at least the 13 limit compared to your trikleismic.

What is the point of temperament if not to provide the means for consistent
modulation and transposition at distant keys?



> > I'll have you know, that my focus is as much a good approximation
> of complex
> > simple integer ratios as a suitable cyclic notation (and by that, I
> mean
> > circulating with minimal damage to whatever interval is chosen) and
> the
> > ability to modulate & transpose scales without wincing.
> >
> > Clearly, this trikleismic temperament fails in the last two
> criteria for a
> > master tuning worthy of expressing maqamat at every key.
>
> I don't know what you mean by "circulating with minimal damage"
> exactly, but I would guess it has to do with the intervals in a given
> interval class, and that will depend, not just on the temperament,
> but ezactly which scale of the temperament you use.



Minimal damage, as in, the preserving of the size - thus, the character - of
an interval as best as can be done without precluding functional harmony...
something your trikleismic fails on even the simplest keys due to a lack of
modulational integrity.



As for the
> ability to modulate a scale, that depends on the complexity of the
> scale in that temperament, if you mean by "modulate" to transpose
> key. What scales do you want to modulate with without wincing?
>



All the possible scales of all maqamat at any Ahenk, the lot of which can be
made to correspond to certain degrees of 79/80 MOS 159-tET, hence the idea
of a 79-tone qanun for any setting and circumstance.



> The 79 notes of trikleismic I was analyzing do not constitute a MOS,
> and if you wanted to have one (which would be more regular in
> behavior) then past 57 you'd need to go all the way up to 102 notes.
> If you used 66 or 90 notes, you'd at least get a 2MOS, however.
>
>


What is a 2MOS? it it the TOS of Margo Schulter? Let's see this 66 and 90.

Oz.

From: Ozan Yarman (2007-02-16)
Subject: Re: [tuning] Re: 159 and trikleismic temperament

What's a wedgie?


----- Original Message -----
From: "Gene Ward Smith" <genewardsmith@coolgoose.com>
To: <tuning@yahoogroups.com>
Sent: 16 \ufffdubat 2007 Cuma 22:15
Subject: [tuning] Re: 159 and trikleismic temperament


> --- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
> wrote:
>
> > The truncated wedgie for ozan temperament, the 2deg159 temperament, is
> > <<33 54 -64 43 9 ... ||.
>
> Or maybe <<33 54 95 43 9 ...||, but that doesn't help us here, despite
> a lower graham complexity.
>
>

From: monz (2007-02-16)
Subject: Re: 159 and trikleismic temperament

Hi Ozan,


--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> What's a wedgie?


http://tonalsoft.com/enc/w/wedgie.aspx


-monz
http://tonalsoft.com
Tonescape microtonal music software

From: Ozan Yarman (2007-02-16)
Subject: Re: [tuning] Re: 159 and trikleismic temperament

Don't you have anything in layman's terms?

Oz.

----- Original Message -----
From: "monz" <monz@tonalsoft.com>
To: <tuning@yahoogroups.com>
Sent: 16 \ufffdubat 2007 Cuma 22:54
Subject: [tuning] Re: 159 and trikleismic temperament


> Hi Ozan,
>
>
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> >
> > What's a wedgie?
>
>
> http://tonalsoft.com/enc/w/wedgie.aspx
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>
>
>

From: Gene Ward Smith (2007-02-16)
Subject: Re: 159 and trikleismic temperament

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

> I do not believe it will make a very suitable one. Maqams require 
at least
> near pure fourths for the construction of agreeable tetrachords, 
plus
> tolerable fifths for correct modulations.

Trikelismic[57] has 39 near pure fourths and the same number of near-
pure fifths with roots in a given octave. This is not much less than 
your scale with 46 each, and using 22 fewer notes. So this complaintm 
is not justifiable.

> Efficiency? What kind of a fancy word is that in regards to musical 
meaning?

Meantone, for example, is very efficient, since it packs a lot of 
fifths and thirds into a small space. That sort of efficiency is 
important if approximating JI intervals is important.

> And how did you deduce these numbers anyway? There are exactly 80
> practicable harmonic major thirds, as much minor sevenths, 
unidecimal
> semi-augmented fourths, and tridecimal neutral sixths, 113 perfect 
fifths
> and as many whole tones in 80 MOS 159-tET, all of which sound quite
> acceptable, if not pleasing, to the ear.

I got the numbers by counting all and only the best fifths, thirds, 
etc. But any reasonably even scale of about the same size gives 
somewhat similar results in terms of second-best. 

> Besides, 79/80 MOS 159-tET comprises just as many, if not more, 
functional
> otonal chords up to at least the 13 limit compared to your 
trikleismic.

If you would define what "functional" means, we could talk. And 
trikelismic is a temperament, not a scale. Suppose we use the 
following scale:

! trikelismic102.scl
Trikleimsic[102] in 159-et tuning
102
!
15.094340
30.188679
37.735849
45.283019
52.830189
67.924528
83.018868
98.113208
113.207547
120.754717
128.301887
135.849057
150.943396
166.037736
181.132075
188.679245
196.226415
203.773585
218.867925
233.962264
249.056604
264.150943
271.698113
279.245283
286.792453
301.886792
316.981132
332.075472
347.169811
354.716981
362.264151
369.811321
384.905660
400.000000
415.094340
430.188679
437.735849
445.283019
452.830189
467.924528
483.018868
498.113208
513.207547
520.754717
528.301887
535.849057
550.943396
566.037736
581.132075
588.679245
596.226415
603.773585
618.867925
633.962264
649.056604
664.150943
671.698113
679.245283
686.792453
701.886792
716.981132
732.075472
747.169811
754.716981
762.264151
769.811321
784.905660
800.000000
815.094340
830.188679
837.735849
845.283019
852.830189
867.924528
883.018868
898.113208
913.207547
920.754717
928.301887
935.849057
950.943396
966.037736
981.132075
988.679245
996.226415
1003.773585
1018.867925
1033.962264
1049.056604
1064.150943
1071.698113
1079.245283
1086.792453
1101.886792
1116.981132
1132.075472
1147.169811
1154.716981
1162.264151
1169.811321
1184.905660
1200.000000

By any reasonable standard I this has more functional otonal chords 
that your scale. Of course, it also has more notes and is less 
regular, but you didn't mention those conditions.

Near-JI fifths: 74
694-cent meantone fifths: 48
709-cent sharp fifths: 45
395-cent major thirds: 87
392-cent major thirds: 45
400-cent major thirds: 10
370-cent major thirds: 72

And so forth.

> What is the point of temperament if not to provide the means for 
consistent
> modulation and transposition at distant keys?

Another major issue are the useful approximatios. But of course, the 
best way to get your consistent modulation and transposition working 
for distant keys is an equal temperament.

> Minimal damage, as in, the preserving of the size - thus, the 
character - of
> an interval as best as can be done without precluding functional 
harmony...
> something your trikleismic fails on even the simplest keys due to a 
lack of
> modulational integrity.

Trikelismic once again is a temperament, not a scale, so this claim 
appears to make no sense. But what do you mean by "modulational 
integrity"?

> > What scales do you want to modulate with without wincing?
 
> All the possible scales of all maqamat at any Ahenk...

I'll repeat the request which has been repeatedly made that you 
specify this precisely, in terms of cents or ratios.

> What is a 2MOS?

Twice as many notes as a MOS, making it somewhat more regular.

> it it the TOS of Margo Schulter? Let's see this 66 and 90.

Let's see what you think of 102 first.

From: Gene Ward Smith (2007-02-16)
Subject: Re: 159 and trikleismic temperament

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> What's a wedgie?

I wouldn't worry about it. The truncated or OE wedgie for your 
temperament is here:

> > > The truncated wedgie for ozan temperament, the 2deg159 
temperament, is
> > > <<33 54 -64 43 9 ... ||.

This tells you it takes 33 generators to get to the fifth, 54 to the 
major third, -64 (go in the other direction) for the 7/4, 43 for the 
11/8 and 9 for the 13/8. If the period was n to the octave, you'd 
count by multiplying through by that, but here we don't have to think 
about that. Since a major triad is 0-33-54, there are 79-54 = 25 in 
your scale (using the best tuning.) There are no otonal tetrads in 
the best tuning, since 79-|33-(-64)| is negative.

> > Or maybe <<33 54 95 43 9 ...||, but that doesn't help us here, 
despite
> > a lower graham complexity.

This would be a different wedgie, with 95 steps to get to 7/4, but 
it's the same in 159-et.

From: Ozan Yarman (2007-02-16)
Subject: Re: [tuning] Re: 159 and trikleismic temperament

SNIP

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> > I do not believe it will make a very suitable one. Maqams require
> at least
> > near pure fourths for the construction of agreeable tetrachords,
> plus
> > tolerable fifths for correct modulations.
>
> Trikelismic[57] has 39 near pure fourths and the same number of near-
> pure fifths with roots in a given octave. This is not much less than
> your scale with 46 each, and using 22 fewer notes. So this complaintm
> is not justifiable.
>



I mean, an unbroken chain of them.



> > Efficiency? What kind of a fancy word is that in regards to musical
> meaning?
>
> Meantone, for example, is very efficient, since it packs a lot of
> fifths and thirds into a small space. That sort of efficiency is
> important if approximating JI intervals is important.
>



Define "small space".



> > And how did you deduce these numbers anyway? There are exactly 80
> > practicable harmonic major thirds, as much minor sevenths,
> unidecimal
> > semi-augmented fourths, and tridecimal neutral sixths, 113 perfect
> fifths
> > and as many whole tones in 80 MOS 159-tET, all of which sound quite
> > acceptable, if not pleasing, to the ear.
>
> I got the numbers by counting all and only the best fifths, thirds,
> etc. But any reasonably even scale of about the same size gives
> somewhat similar results in terms of second-best.
>



Let us not forget, that I can alternate between two types of harmonic major
thirds, which increases their number by +55.



> > Besides, 79/80 MOS 159-tET comprises just as many, if not more,
> functional
> > otonal chords up to at least the 13 limit compared to your
> trikleismic.
>
> If you would define what "functional" means, we could talk. And
> trikelismic is a temperament, not a scale.



QUOTE
Obviously, a 57 note scale and a 79 note scale are not directly
comparable, so I suggest we compare 79 notes of each. However, be it
noted I was not proposing Trikleismic[57] as a maqam scale, but just
as a scale.
UNQUOTE



Suppose we use the
> following scale:
>
> ! trikelismic102.scl
> Trikleimsic[102] in 159-et tuning
> 102
> !
> 15.094340
> 30.188679
> 37.735849
> 45.283019
> 52.830189
> 67.924528
> 83.018868
> 98.113208
> 113.207547
> 120.754717
> 128.301887
> 135.849057
> 150.943396
> 166.037736
> 181.132075
> 188.679245
> 196.226415
> 203.773585
> 218.867925
> 233.962264
> 249.056604
> 264.150943
> 271.698113
> 279.245283
> 286.792453
> 301.886792
> 316.981132
> 332.075472
> 347.169811
> 354.716981
> 362.264151
> 369.811321
> 384.905660
> 400.000000
> 415.094340
> 430.188679
> 437.735849
> 445.283019
> 452.830189
> 467.924528
> 483.018868
> 498.113208
> 513.207547
> 520.754717
> 528.301887
> 535.849057
> 550.943396
> 566.037736
> 581.132075
> 588.679245
> 596.226415
> 603.773585
> 618.867925
> 633.962264
> 649.056604
> 664.150943
> 671.698113
> 679.245283
> 686.792453
> 701.886792
> 716.981132
> 732.075472
> 747.169811
> 754.716981
> 762.264151
> 769.811321
> 784.905660
> 800.000000
> 815.094340
> 830.188679
> 837.735849
> 845.283019
> 852.830189
> 867.924528
> 883.018868
> 898.113208
> 913.207547
> 920.754717
> 928.301887
> 935.849057
> 950.943396
> 966.037736
> 981.132075
> 988.679245
> 996.226415
> 1003.773585
> 1018.867925
> 1033.962264
> 1049.056604
> 1064.150943
> 1071.698113
> 1079.245283
> 1086.792453
> 1101.886792
> 1116.981132
> 1132.075472
> 1147.169811
> 1154.716981
> 1162.264151
> 1169.811321
> 1184.905660
> 1200.000000
>



Ouch! I see no way of implementing this on a qanun. The notation is
completely irregular and there are many unnecessary tones. Don't you have
anything simpler?


> By any reasonable standard I this has more functional otonal chords
> that your scale. Of course, it also has more notes and is less
> regular, but you didn't mention those conditions.
>


I did, several times before.


> Near-JI fifths: 74
> 694-cent meantone fifths: 48
> 709-cent sharp fifths: 45
> 395-cent major thirds: 87
> 392-cent major thirds: 45
> 400-cent major thirds: 10
> 370-cent major thirds: 72
>
> And so forth.
>


Numbers, numbers...


> > What is the point of temperament if not to provide the means for
> consistent
> > modulation and transposition at distant keys?
>
> Another major issue are the useful approximatios. But of course, the
> best way to get your consistent modulation and transposition working
> for distant keys is an equal temperament.
>



This is so obvious that it is unnecessary to specify. Unavoidable is the
fact, that no two digit equal tuning will suffice in the case of Maqam Music
(not even 53 or 72), and there is no way to implement any division exceeding
80 or so on the qanun.

Besides, modulation should be distinct from direct transposition as a thumb
of rule. This is achieved in well-temperaments via non-linear mapping.


> > Minimal damage, as in, the preserving of the size - thus, the
> character - of
> > an interval as best as can be done without precluding functional
> harmony...
> > something your trikleismic fails on even the simplest keys due to a
> lack of
> > modulational integrity.
>
> Trikelismic once again is a temperament, not a scale, so this claim
> appears to make no sense. But what do you mean by "modulational
> integrity"?
>



What doesn't make sense? I can't have a Rast scale on natural tones without
a breach in the chain of generators. That's what I mean by modulational
integrity.


> > > What scales do you want to modulate with without wincing?
>
> > All the possible scales of all maqamat at any Ahenk...
>
> I'll repeat the request which has been repeatedly made that you
> specify this precisely, in terms of cents or ratios.
>



What? All of them here, now? Impossible. I have already divulged the ones in
my knowledge throughout the past year. Simply put, one needs comma steps in
a voluminous, but not altogether unwiedly temperament, yielding satisfactory
approximations of such epimoric ratios as 10:9, 11:10, 12:11, 13:12, 14:13
(the infamous mujannab zone) and sufficiently precise thirds (harmonic and
pythagorean), fifths (wide, pure, narrow) and whole tones.

Priority number 1: map principal Rast scale to white keys.
Priority number 2: allow consistent alteration from Rast to Mahur/Suzidilara
at every key
Priority number 3: reconcile the traditional perde-system with notation
(sharps a comma below flats as in AEU)
Priority number 4: achieve a 12-tone closed-cycle subset.



> > What is a 2MOS?
>
> Twice as many notes as a MOS, making it somewhat more regular.
>
> > it it the TOS of Margo Schulter? Let's see this 66 and 90.
>
> Let's see what you think of 102 first.
>



I find it too unwieldy. Let me see the others now.

Oz.

From: Ozan Yarman (2007-02-16)
Subject: Re: [tuning] Re: 159 and trikleismic temperament

----- Original Message -----
From: "Gene Ward Smith" <genewardsmith@coolgoose.com>
To: <tuning@yahoogroups.com>
Sent: 17 \ufffdubat 2007 Cumartesi 0:54
Subject: [tuning] Re: 159 and trikleismic temperament


> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> >
> > What's a wedgie?
>
> I wouldn't worry about it. The truncated or OE wedgie for your
> temperament is here:
>
> > > > The truncated wedgie for ozan temperament, the 2deg159
> temperament, is
> > > > <<33 54 -64 43 9 ... ||.
>
> This tells you it takes 33 generators to get to the fifth, 54 to the
> major third, -64 (go in the other direction) for the 7/4, 43 for the
> 11/8 and 9 for the 13/8. If the period was n to the octave, you'd
> count by multiplying through by that, but here we don't have to think
> about that. Since a major triad is 0-33-54, there are 79-54 = 25 in
> your scale (using the best tuning.) There are no otonal tetrads in
> the best tuning, since 79-|33-(-64)| is negative.
>


There are, on the contrary, 80 (degs 0-26-47 of 80MOS159tET) + 54 (degs
0-25-46 of 79MOS159tET) possible major triads, and a parallel number of
otonal tetrads that are very much agreeable to the ear.



> > > Or maybe <<33 54 95 43 9 ...||, but that doesn't help us here,
> despite
> > > a lower graham complexity.
>
> This would be a different wedgie, with 95 steps to get to 7/4, but
> it's the same in 159-et.
>
>
>


Wedgie or not, the sounds of the 79-tone qanun decry this kind of
numerology.

Oz.

From: Carl Lumma (2007-02-17)
Subject: Re: 159 and trikleismic temperament

> By any reasonable standard I this has more functional otonal chords 
> that your scale. Of course, it also has more notes and is less 
> regular, but you didn't mention those conditions.

He didn't mention number of otonal chords, either.

-Carl

From: Gene Ward Smith (2007-02-17)
Subject: Re: 159 and trikleismic temperament

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

> QUOTE
> Obviously, a 57 note scale and a 79 note scale are not directly
> comparable, so I suggest we compare 79 notes of each. However, be it
> noted I was not proposing Trikleismic[57] as a maqam scale, but just
> as a scale.
> UNQUOTE

Which says I was *not* proposing it as a maqam scale.

> Ouch! I see no way of implementing this on a qanun. The notation is
> completely irregular and there are many unnecessary tones. Don't 
you have
> anything simpler?

Something other than trikleismic is going to be better if you need 
more regularity.

> What doesn't make sense? I can't have a Rast scale on natural tones 
without
> a breach in the chain of generators. That's what I mean by 
modulational
> integrity.

Rast is supposed to be, more or less, a diatonic scale?

> What? All of them here, now? Impossible. I have already divulged 
the ones in
> my knowledge throughout the past year. Simply put, one needs comma 
steps in
> a voluminous, but not altogether unwiedly temperament, yielding 
satisfactory
> approximations of such epimoric ratios as 10:9, 11:10, 12:11, 
13:12, 14:13
> (the infamous mujannab zone) and sufficiently precise thirds 
(harmonic and
> pythagorean), fifths (wide, pure, narrow) and whole tones.

If you want a lot of all three versions of the same interval, then 
you need to have the complexity of a single step of 159 to be low, 
which means a completely different kind of scale, one with a lot of 
single steps of 159, and then some big jumps.


> Priority number 1: map principal Rast scale to white keys.
> Priority number 2: allow consistent alteration from Rast to 
Mahur/Suzidilara
> at every key
> Priority number 3: reconcile the traditional perde-system with 
notation
> (sharps a comma below flats as in AEU)
> Priority number 4: achieve a 12-tone closed-cycle subset.

If you could explain what all that means people could discuss things 
more intelligently.

> I find it too unwieldy. Let me see the others now.

They are going to be even more irregular. What about the following as 
a maqam tuning; it's not proper, and that might be important to you 
from what you've been saying, so I'm curious to know what you think:

! tertia78.scl
Tertiaseptal[78] in 140-et tuning
78
!
8.571429
34.285714
42.857143
68.571429
77.142857
85.714286
111.428571
120.000000
145.714286
154.285714
162.857143
188.571429
197.142857
222.857143
231.428571
240.000000
265.714286
274.285714
300.000000
308.571429
317.142857
342.857143
351.428571
377.142857
385.714286
394.285714
420.000000
428.571429
454.285714
462.857143
471.428571
497.142857
505.714286
531.428571
540.000000
548.571429
574.285714
582.857143
591.428571
617.142857
625.714286
651.428571
660.000000
668.571429
694.285714
702.857143
728.571429
737.142857
745.714286
771.428571
780.000000
805.714286
814.285714
822.857143
848.571429
857.142857
882.857143
891.428571
900.000000
925.714286
934.285714
960.000000
968.571429
977.142857
1002.857143
1011.428571
1037.142857
1045.714286
1054.285714
1080.000000
1088.571429
1114.285714
1122.857143
1131.428571
1157.142857
1165.714286
1191.428571
1200.000000

From: Ozan Yarman (2007-02-17)
Subject: Re: [tuning] Re: 159 and trikleismic temperament

Give me a few, and I'll tell you their number.

Oz.

----- Original Message -----
From: "Carl Lumma" <clumma@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 17 \ufffdubat 2007 Cumartesi 2:45
Subject: [tuning] Re: 159 and trikleismic temperament


> > By any reasonable standard I this has more functional otonal chords
> > that your scale. Of course, it also has more notes and is less
> > regular, but you didn't mention those conditions.
>
> He didn't mention number of otonal chords, either.
>
> -Carl
>
>
>

From: Ozan Yarman (2007-02-17)
Subject: Re: [tuning] Re: 159 and trikleismic temperament

SNIP

>
> > QUOTE
> > Obviously, a 57 note scale and a 79 note scale are not directly
> > comparable, so I suggest we compare 79 notes of each. However, be it
> > noted I was not proposing Trikleismic[57] as a maqam scale, but just
> > as a scale.
> > UNQUOTE
>
> Which says I was *not* proposing it as a maqam scale.
>


But as a scale nonetheless.


> > Ouch! I see no way of implementing this on a qanun. The notation is
> > completely irregular and there are many unnecessary tones. Don't
> you have
> > anything simpler?
>
> Something other than trikleismic is going to be better if you need
> more regularity.
>


Surely.


> > What doesn't make sense? I can't have a Rast scale on natural tones
> without
> > a breach in the chain of generators. That's what I mean by
> modulational
> > integrity.
>
> Rast is supposed to be, more or less, a diatonic scale?
>



Good gracious, we've been over this a dozen times already. Rast is an
harmonic major scale on natural notes:

Ra = Ut  (1)
Du = Re  (9/8)
Se = Mi   (5/4 to 27/22)
Cha=Fa   (4/3)
Nev=Sol  (3/2)
Hu = La   (27/16, and occasionally 5/3)
Ve = Si   (15/8 to 81/44)
ra  = ut   (2/1)



> > What? All of them here, now? Impossible. I have already divulged
> the ones in
> > my knowledge throughout the past year. Simply put, one needs comma
> steps in
> > a voluminous, but not altogether unwiedly temperament, yielding
> satisfactory
> > approximations of such epimoric ratios as 10:9, 11:10, 12:11,
> 13:12, 14:13
> > (the infamous mujannab zone) and sufficiently precise thirds
> (harmonic and
> > pythagorean), fifths (wide, pure, narrow) and whole tones.
>
> If you want a lot of all three versions of the same interval, then
> you need to have the complexity of a single step of 159 to be low,
> which means a completely different kind of scale, one with a lot of
> single steps of 159, and then some big jumps.
>



I'll settle for 79/80 MOS 159-tET for the time being, if you don't mind.


>
> > Priority number 1: map principal Rast scale to white keys.
> > Priority number 2: allow consistent alteration from Rast to
> Mahur/Suzidilara
> > at every key
> > Priority number 3: reconcile the traditional perde-system with
> notation
> > (sharps a comma below flats as in AEU)
> > Priority number 4: achieve a 12-tone closed-cycle subset.
>
> If you could explain what all that means people could discuss things
> more intelligently.
>


OK, one more time:

1. The principal diatonic scale of maqam Rast, which I gave above, MUST be
mapped to natural keys without breaking the chain of fifths. 79 MOS 159-tET
maps at least the ascending scale to these notes without any accidentals,
and requires only a comma-down modifier at the 3rd, 6th, and 7th degrees on
descent.

2. Transition to the principal scale of Mahur, which is the Pythagorean
version of Rast, should be made possible, again without breaking the chain
of fifths and at every key.

3. The traditional 17-tone system of (clustering) perdes, the first octave
of which I had given a week ago, MUST be notated appropriately on the staff
the way I have shown through 79 MOS 159-tET. Do I need to repeat myself?

4. For chromatic passages like in Western common-practice, one needs a
12-tone cyclic subset from the master tuning, which is achieved in 79 MOS
159-tET.


As a bonus, one has the option to compose in microtonal polyphony.


> > I find it too unwieldy. Let me see the others now.
>
> They are going to be even more irregular. What about the following as
> a maqam tuning; it's not proper, and that might be important to you
> from what you've been saying, so I'm curious to know what you think:
>
> ! tertia78.scl
> Tertiaseptal[78] in 140-et tuning
> 78
> !
> 8.571429
> 34.285714
> 42.857143
> 68.571429
> 77.142857
> 85.714286
> 111.428571
> 120.000000
> 145.714286
> 154.285714
> 162.857143
> 188.571429
> 197.142857
> 222.857143
> 231.428571
> 240.000000
> 265.714286
> 274.285714
> 300.000000
> 308.571429
> 317.142857
> 342.857143
> 351.428571
> 377.142857
> 385.714286
> 394.285714
> 420.000000
> 428.571429
> 454.285714
> 462.857143
> 471.428571
> 497.142857
> 505.714286
> 531.428571
> 540.000000
> 548.571429
> 574.285714
> 582.857143
> 591.428571
> 617.142857
> 625.714286
> 651.428571
> 660.000000
> 668.571429
> 694.285714
> 702.857143
> 728.571429
> 737.142857
> 745.714286
> 771.428571
> 780.000000
> 805.714286
> 814.285714
> 822.857143
> 848.571429
> 857.142857
> 882.857143
> 891.428571
> 900.000000
> 925.714286
> 934.285714
> 960.000000
> 968.571429
> 977.142857
> 1002.857143
> 1011.428571
> 1037.142857
> 1045.714286
> 1054.285714
> 1080.000000
> 1088.571429
> 1114.285714
> 1122.857143
> 1131.428571
> 1157.142857
> 1165.714286
> 1191.428571
> 1200.000000
>
>


While I appreciate the preservation of interval sizes via an 8 cent lowering
of pitch almost each time I modulate a major or minor triad by a fifth up, I
regret the poor approximation of certain 11 & 13-limit intervals, absence of
Pythagorean diatonic major scales at every degree, and consistency in
notating pitches.

Oz.

From: Gene Ward Smith (2007-02-17)
Subject: Re: 159 and trikleismic temperament

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

> > Rast is supposed to be, more or less, a diatonic scale?

> Good gracious, we've been over this a dozen times already.

Yes, but your answers keep changing.

From: Carl Lumma (2007-02-17)
Subject: Re: 159 and trikleismic temperament

I meant, you didn't say having lots of otonal chords
was important to you (that I saw).  Is it?

-Carl

> Give me a few, and I'll tell you their number.
> 
> Oz.
> 
> ----- Original Message -----
> From: "Carl Lumma" <clumma@...>
> To: <tuning@yahoogroups.com>
> Sent: 17 Þubat 2007 Cumartesi 2:45
> Subject: [tuning] Re: 159 and trikleismic temperament
> 
> > > By any reasonable standard I this has more functional
> > > otonal chords that your scale. Of course, it also has
> > > more notes and is less regular, but you didn't mention
> > > those conditions.
> >
> > He didn't mention number of otonal chords, either.
> >
> > -Carl

From: Carl Lumma (2007-02-17)
Subject: Re: 159 and trikleismic temperament

Ozan, (and Gene),

I feel like you have a really good start at a dialog with
Gene for perhaps the first time.  I hope you will both have
patience and try to work out something together.  I think
it could be very fruitful.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
> I meant, you didn't say having lots of otonal chords
> was important to you (that I saw).  Is it?
> 
> -Carl
> 
> > Give me a few, and I'll tell you their number.
> > 
> > Oz.
> > 
> > ----- Original Message -----
> > From: "Carl Lumma" <clumma@>
> > To: <tuning@yahoogroups.com>
> > Sent: 17 Þubat 2007 Cumartesi 2:45
> > Subject: [tuning] Re: 159 and trikleismic temperament
> > 
> > > > By any reasonable standard I this has more functional
> > > > otonal chords that your scale. Of course, it also has
> > > > more notes and is less regular, but you didn't mention
> > > > those conditions.
> > >
> > > He didn't mention number of otonal chords, either.
> > >
> > > -Carl

From: Carl Lumma (2007-02-17)
Subject: Re: 159 and trikleismic temperament

> OK, one more time:
> 
> 1. The principal diatonic scale of maqam Rast, which I gave
> above, MUST be mapped to natural keys without breaking the
> chain of fifths. 79 MOS 159-tET maps at least the ascending
> scale to these notes without any accidentals, and requires
> only a comma-down modifier at the 3rd, 6th, and 7th degrees
> on descent.
> 
> 2. Transition to the principal scale of Mahur, which is the
> Pythagorean version of Rast, should be made possible, again
> without breaking the chain of fifths and at every key.

Every key of what -- the Rast scale?

> 3. The traditional 17-tone system of (clustering) perdes,
> the first octave of which I had given a week ago, MUST be
> notated appropriately on the staff the way I have shown
> through 79 MOS 159-tET. Do I need to repeat myself?

It would be good if you could define what "notated
appropriately" means.

Also, what I have from you on the perdes is:

Here are the 17 traditional perdes:

0: RAST
1: Shuri
2: Zengule cluster
3: DUGAH
4: Kurdi/Nihavend cluster
5: SEGAH cluster
6: Buselik
7: CHARGAH
8: Hijaz
9: Uzzal/Saba cluster
10: NEVA
11: Bayati
12: Hisar cluster
13: HUSEYNI
14: Ajem cluster
15: EVDJ cluster
16: Mahur
17: GERDANIYE

But these are just names.  This dosen't tell us what
these 17 things are.

>4. For chromatic passages like in Western common-practice,
> one needs a 12-tone cyclic subset from the master tuning,
> which is achieved in 79 MOS 159-tET.

Why is this important?  Must a maqam tuning be expected
to reproduce Western common-practice also?
 
-Carl

From: Gene Ward Smith (2007-02-17)
Subject: Re: 159 and trikleismic temperament

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
>
> Ozan, (and Gene),
> 
> I feel like you have a really good start at a dialog with
> Gene for perhaps the first time.  I hope you will both have
> patience and try to work out something together.  I think
> it could be very fruitful.

Figuring out if propriety sufficies to ensure "modulational integrity" 
would be a good start.

From: Carl Lumma (2007-02-17)
Subject: Re: 159 and trikleismic temperament

> > Ozan, (and Gene),
> > 
> > I feel like you have a really good start at a dialog with
> > Gene for perhaps the first time.  I hope you will both have
> > patience and try to work out something together.  I think
> > it could be very fruitful.
> 
> Figuring out if propriety sufficies to ensure "modulational
> integrity" would be a good start.

My approach would be, rather than guessing huge master scales,
trying to figure out what basic scales are needed, and what the
modulation requirements are, and then build the larger scale
from there.  Unfortunately my attempts at this have mostly
ended in frustration.

-Carl

From: Gene Ward Smith (2007-02-18)
Subject: Re: 159 and trikleismic temperament

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:

> My approach would be, rather than guessing huge master scales,
> trying to figure out what basic scales are needed, and what the
> modulation requirements are, and then build the larger scale
> from there.  Unfortunately my attempts at this have mostly
> ended in frustration.

It's a fine idea, but you need to get the list of required scales, with 
indication of how much leeway you have.

From: Gene Ward Smith (2007-02-18)
Subject: Re: 159 and trikleismic temperament

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

> It's a fine idea, but you need to get the list of required scales, 
with 
> indication of how much leeway you have.

Here's some of what we know about Ozan's scale:

(1) It is a MOS

(2) It has a near-JI fifth

(3) It is strictly proper

(4) The interval class of the JI fifth contains a meantone fifth also

These conditions may be necessary, but they are not sufficient, as 
all of the above is also true of Ennealimmal[72], which Ozan 
dismissed as hopeless. One of my problems is that I don't know *why*.
What, specifically, does Ozan's scale have that Ennealimmal[72] ain't 
got? I know there are questions about rational approximations lurking 
out there, but he didn't say anything about that. And I don't know 
what, specifically, is required. I do get that having the 13-limit be 
distinct would be good, but how accurately?

Anyway, I really would like a specific answer to the question about 
Ennealimmal[72]. What's the problem with it?

From: Cameron Bobro (2007-02-18)
Subject: Re: 159 and trikleismic temperament

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...> 
wrote:
 
>  Ennealimmal[72]

Could you post this one again, sorry, I lost track of which one was 
which. 

-Cameron Bobro

From: Cameron Bobro (2007-02-18)
Subject: Re: 159 and trikleismic temperament

This must be it, in case anyone else got a little swamped by the
deluge of tunings- and these threads are unravelling.

! ennea72.scl
Ennealimmal[72] in 612-et tuning (strictly proper)
72
!
13.725490
35.294118
49.019608
62.745098
84.313725
98.039216
119.607843
133.333333
147.058824
168.627451
182.352941
196.078431
217.647059
231.372549
252.941176
266.666667
280.392157
301.960784
315.686275
329.411765
350.980392
364.705882
386.274510
400.000000
413.725490
435.294118
449.019608
462.745098
484.313725
498.039216
519.607843
533.333333
547.058824
568.627451
582.352941
596.078431
617.647059
631.372549
652.941176
666.666667
680.392157
701.960784
715.686275
729.411765
750.980392
764.705882
786.274510
800.000000
813.725490
835.294118
849.019608
862.745098
884.313725
898.039216
919.607843
933.333333
947.058824
968.627451
982.352941
996.078431
1017.647059
1031.372549
1052.941176
1066.666667
1080.392157
1101.960784
1115.686275
1129.411765
1150.980392
1164.705882
1186.274510
1200.000000

From: Cameron Bobro (2007-02-18)
Subject: Re: 159 and trikleismic temperament

Also known as Pipedum 72a by de Coup, within a fraction of a cent by 
all measures, especially on the 64th degree.

375/4374, 2401/2400 and 15625/15552, Manuel Op de Coul, 2002
|
  0:          1/1               0.000 unison, perfect prime
  1:        126/125            13.795 small septimal comma
  2:         49/48             35.697 slendro diesis, eptimal 1/6-
  3:         36/35             48.770 septimal diesis, 1/4-tone
  4:        648/625            62.565 major diesis
  5:         21/20             84.467 minor semitone
  6:       1323/1250           98.262
  7:         15/14            119.443 major diatonic semitone
  8:         27/25            133.238 large limma, BP small semitone
  9:         49/45            147.428 BP minor semitone
 10:        625/567           168.609 BP great semitone
 11:         10/9             182.404 minor whole tone
 12:         28/25            196.198 middle second
 13:        245/216           218.101
 14:          8/7             231.174 septimal whole tone
 15:        125/108           253.076 semi-augmented whole tone
 16:          7/6             266.871 septimal minor third
 17:        147/125           280.666
 18:         25/21            301.847 BP second, quasi-tempered 
 19:          6/5             315.641 minor third
 20:        756/625           329.436
 21:         49/40            351.338 larger approximation to 
 22:        216/175           364.412
 23:          5/4             386.314 major third
 24:         63/50            400.108 quasi-equal major third
 25:       3969/3125          413.903
 26:          9/7             435.084 septimal major third, BP third
 27:         35/27            449.275 9/4-tone, septimal semi
  28:         98/75            463.069
 29:        250/189           484.250
 30:          4/3             498.045 perfect fourth
 31:        875/648           519.947
 32:         49/36            533.742 Arabic lute acute fourth
 33:         48/35            546.815 septimal semi-augmented fourth
 34:         25/18            568.717 classic augmented fourth
 35:          7/5             582.512 septimal or Huygens' tritone, 
 36:        882/625           596.307
 37:         10/7             617.488 Euler's tritone
 38:         36/25            631.283 classic diminished fifth
 39:         35/24            653.185 septimal semi-diminished fifth
 40:        147/100           666.979
 41:       1296/875           680.053
 42:          3/2             701.955 perfect fifth
 43:        189/125           715.750
 44:      46656/30625         728.823
 45:        125/81            751.121
 46:         14/9             764.916 septimal minor sixth
 47:       6125/3888          786.818
 48:        100/63            799.892 quasi-equal minor sixth
 49:          8/5             813.686 minor sixth
 50:        175/108           835.588
 51:         49/30            849.383 larger approximation to 
 52:        288/175           862.457
 53:          5/3             884.359 major sixth, BP sixth
 54:         42/25            898.153 quasi-tempered major sixth
 55:        245/144           920.056
 56:         12/7             933.129 septimal major sixth
 57:        216/125           946.924 semi-augmented sixth
 58:          7/4             968.826 harmonic seventh
 59:        441/250           982.621
 60:       7776/4375          995.694
 61:          9/5            1017.596 just minor seventh, BP seventh
 62:         49/27           1031.787
 63:       3125/1701         1052.968
 64:         50/27           1066.762 grave major seventh
 65:         28/15           1080.557 grave major seventh
 66:       1225/648          1102.459
 67:         40/21           1115.533 acute major seventh
 68:         48/25           1129.328 classic diminished octave
 69:         35/18           1151.230 septimal semi-diminished octave
 70:         49/25           1165.024 BP eighth
 71:        125/63           1186.205
 72:          2/1            1200.000 octave

Judging by photos of the qanun and seeing how the guys flip the 
tuning levers on the fly, I'd hazard a guess that the actual 
position of an interval in a tuning, not just the presence or 
absence of the interval, makes the difference between playable and 
theoretical.

From: ciarán maher (2007-02-18)
Subject: new tenney pieces: spectral variations nos. 1 to 3

hi

thanks to those who helped me with the scord issues for the new  
tenney player piano pieces.

they're done now, and we'll be premiering them on the st. conlon  
disklaiver from gaudeamus at dnk amstermdam on 19th march. it'd be  
great if anyone in the area could make it.

i have some tests you can hear at:

http://www.rhizomecowboy.com/spectral_variations/

some notes get dropped but you can get the idea.

we'll also be playing clarence barlow's extended version of the  
oiriginal Spectral CANON for CONLON Nancoarrow, and the 'phi'  
realisation of For Ann (rising) i made in 1998.

thanks again to those who helped.

i've pasted a little blurb below.

ciarán


JAMES TENNEY: WORLD PREMIERES at DNK Amsterdam
Spectral Variations Nos. 1 to 3 (2006)
–––––––––––––––––––––––– 
–––––––––––––––––––––––– 
––––––––––––––––––

	Φ For Ann (rising). Electronic (1998) James Tenney.
	Spectral CANON for CONLON Nancarrow (Extended). Player Piano (1991)  
James Tenney.
	Spectral Variations Nos. 1 to 3. Player Piano (2006) James Tenney.
	Accidental for Jim. Player Piano (2006) Ciarán Maher.


DNK Amsterdam in association with Ciarán Maher and Steim will present  
the world premieres of three new pieces for player piano by the late  
James Tenney (1934 - 2006).

James Tenney had a long association with the music of Conlon  
Nancarrow and played a crucial role in bringing him to the attention  
of the world. He wrote the very substantial liner notes for the  
original release of Nancarrow's player piano studies and would go on  
to transcribe some of that work for orchestra in Five Studies for  
Player Piano (Conlon Nancarrow), 2000.

In 1972 Tenney began work on Spectral CANON for CONLON Nancarrow  
which is a stunningly intricate process exposition of the rhythmicon  
idea for retuned player piano. The piece was realised in 1974 after  
Nancarrow himself had punched the roll.

Later, in 1991, Clarence Barlow generated the version now called  
Spectral CANON for CONLON Nancarrow (Extended), in which the strict  
process of the canon, which Tenney had cut short on a 24 voice unison  
in the original, is allowed to work itself through to reveal further  
the extraordinary richness of this material. Barlow will be present  
at the current concert and has kindly allowed this piece to be part  
of the programme.

Tenney wanted to explore the material further and in 1998 sent his  
friend and student Ciarán Maher the maths for a new variation. Maher  
generated a rough demo which the pair reviewed together but a refined  
version was never produced.

In 2006 Tenney visited Ireland for a residency at Trinity College  
Dublin, and Maher had the chance to resurrect the project. Working  
with Tenney, he wrote programs in Flash which generated note  
durations and voice start times in list and graphic form. During this  
process, two further variations of the form occurred to Tenney, and  
together they worked out and tested the requisite formulas.

The result is the three new pieces Spectral Variations Nos. 1 to 3  
which are premiered in the present concert.

In addition we will present Accidental for Jim, a further  
manipulation of the material by Maher in which the rhythmicon  
relationships for the pitch set are inverted ( i.e. harmonic ratios  
24:23 have the rhythmic relationships 1:2 etc. ).

We will also present a subtle refinement of Tenney's 1969  
electroacoustic classic For Ann (rising). The refinement which  
approximates a golden section measurement of the minor 6th which  
separates the rising glissandi is described by Larry Polansky in  
Soundings 13  as follows:

I have heard Tenney consider a possible modification of this piece  
which would, I think, be an interesting exploration. He suggests that  
each glissando be related by the ratio of successive Fibonacci terms  
[…] or about 1.618033988749894 (etc.), a minor sixth. This interval  
[…] would result in the property of all first order difference tones  
of any given glissando pair being present in some lower glissando.  
[…] and the piece might be conceivably be smoother, or more "perfect".

Maher made the 'phi' version with the above properties at Dartington  
in 1998 where Tenney had the opportunity finally to hear it. This is  
Maher's first presentation of the piece since the Dartington concert.  
More recently Marc Sabat (www.plainsound.org) has made realisations  
of further interesting and beautiful refinements of this process.

Unfortunately James Tenney died in August 2006 before he could hear  
the new player piano pieces and this concert is dedicated to his  
memory and his contagious enthusiasm.

cm belfast, jan 2007

DNK: 			www.dnk-amsterdam.com
Ciarán Maher: 		www.rhizomecowboy.com
Steim:			www.steim.org/steim/

From: Carl Lumma (2007-02-18)
Subject: Re: 159 and trikleismic temperament

> >  Ennealimmal[72]
> 
> Could you post this one again, sorry, I lost track of which one was 
> which. 
> 
> -Cameron Bobro

People, a groups search for "Ennealimmal[72]" turns it right up.

http://groups.yahoo.com/group/tuning/message/69822

-Carl

From: Carl Lumma (2007-02-18)
Subject: Re: new tenney pieces: spectral variations nos. 1 to 3

--- In tuning@yahoogroups.com, ciarán maher <ciaran@...> wrote:
>
> hi
> 
> thanks to those who helped me with the scord issues for the new  
> tenney player piano pieces.
> 
> they're done now, and we'll be premiering them on the st. conlon  
> disklaiver from gaudeamus at dnk amstermdam on 19th march. it'd be  
> great if anyone in the area could make it.

In Amsterdam?  Man, I wish.

-Carl

From: Cameron Bobro (2007-02-18)
Subject: Re: 159 and trikleismic temperament

"Unicode 8", hehe.

As you can see, I found it, and the same tuning in rational form by de 
Coup.

From: Carl Lumma (2007-02-18)
Subject: Re: 159 and trikleismic temperament

> As you can see, I found it,
> and the same tuning in rational form by de Coup.

Yes, good work!  But are you sure that isn't de Coul?

-Carl

From: Gene Ward Smith (2007-02-18)
Subject: Re: 159 and trikleismic temperament

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> 
wrote:
>
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@> 
> wrote:
>  
> >  Ennealimmal[72]
> 
> Could you post this one again, sorry, I lost track of which one was 
> which. 

Ennealimmal is the 72&99 7-limit super-temperament, with a period of 
1/9 of an octave. It's really essentially a 7-limit temperament, but 
if you use eg <<18 27 18 45 18 1 ...|| you have a 13-limit mapping 
extending it: [<9 15 22 26 33 34|, <0 -2 -3 -2 -5 -2|].

! ennea72.scl
Ennealimmal[72] in 612-et tuning (strictly proper)
72
!
13.725490
35.294118
49.019608
62.745098
84.313725
98.039216
119.607843
133.333333
147.058824
168.627451
182.352941
196.078431
217.647059
231.372549
252.941176
266.666667
280.392157
301.960784
315.686275
329.411765
350.980392
364.705882
386.274510
400.000000
413.725490
435.294118
449.019608
462.745098
484.313725
498.039216
519.607843
533.333333
547.058824
568.627451
582.352941
596.078431
617.647059
631.372549
652.941176
666.666667
680.392157
701.960784
715.686275
729.411765
750.980392
764.705882
786.274510
800.000000
813.725490
835.294118
849.019608
862.745098
884.313725
898.039216
919.607843
933.333333
947.058824
968.627451
982.352941
996.078431
1017.647059
1031.372549
1052.941176
1066.666667
1080.392157
1101.960784
1115.686275
1129.411765
1150.980392
1164.705882
1186.274510
1200.000000

From: Gene Ward Smith (2007-02-18)
Subject: Re: 159 and trikleismic temperament

--- In tuning@yahoogroups.com, "Cameron Bobro" <misterbobro@...> wrote:
>
> Also known as Pipedum 72a by de Coup, within a fraction of a cent by 
> all measures, especially on the 64th degree.

4375/4374 and 2401/2400 are so small you can use JI tuning for 
ennealimmal, but note also that you don't actually now divide the 
octave into 9 equal parts with this tuning.

From: Carl Lumma (2007-02-19)
Subject: Re: new tenney pieces: spectral variations nos. 1 to 3

--- In tuning@yahoogroups.com, ciarán maher <ciaran@...> wrote:
> hi
> 
> thanks to those who helped me with the scord issues for the new  
> tenney player piano pieces.
> 
> they're done now, and we'll be premiering them on the st. conlon  
> disklaiver from gaudeamus at dnk amstermdam on 19th march. it'd be  
> great if anyone in the area could make it.
> 
> i have some tests you can hear at:
> 
> http://www.rhizomecowboy.com/spectral_variations/
> 
> some notes get dropped but you can get the idea.

Stunning!  I wasn't expecting to like this, but it turns the
piano into a synthesizer in a way I've never experienced
before.  And a very good one at that.

> we'll also be playing clarence barlow's extended version of the  
> oiriginal Spectral CANON for CONLON Nancoarrow, and the 'phi'  
> realisation of For Ann (rising) i made in 1998.

Wish I could be there.

-Carl

From: Gene Ward Smith (2007-02-19)
Subject: Re: new tenney pieces: spectral variations nos. 1 to 3

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:

> > http://www.rhizomecowboy.com/spectral_variations/
> > 
> > some notes get dropped but you can get the idea.
> 
> Stunning!  I wasn't expecting to like this, but it turns the
> piano into a synthesizer in a way I've never experienced
> before.  And a very good one at that.

Are there urls for downloadable files to be found anywhere?

From: Carl Lumma (2007-02-19)
Subject: Re: new tenney pieces: spectral variations nos. 1 to 3

> > Stunning!  I wasn't expecting to like this, but it turns the
> > piano into a synthesizer in a way I've never experienced
> > before.  And a very good one at that.
> 
> Are there urls for downloadable files to be found anywhere?

I tried to decompile the flash to get them, but for some
reason gave up ("some reason" being my kid needing attention).
I assume they aren't meant to be had.

-Carl

From: monz (2007-02-19)
Subject: Re: 159 and trikleismic temperament

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@> wrote:
> >
> > What's a wedgie?
> 
> I wouldn't worry about it. The truncated or OE wedgie for
> your temperament is here:
> 
> > > > The truncated wedgie for ozan temperament, 
> > > > the 2deg159 temperament, is <<33 54 -64 43 9 ... ||.
> 
> This tells you it takes 33 generators to get to the fifth,
> 54 to the major third, -64 (go in the other direction) for
> the 7/4, 43 for the 11/8 and 9 for the 13/8. 


I didn't know that the wedgie gave away that info!

Do they always work like this? I checked the meantone 
wedgie <<1, 4, 10, 4, 13, 12|| and can see that 3 maps
to 1 generator, 5 to 4 generators, and 7 to 10 generators
... but then what does the next "4" represent? And the
"13" which comes next does represent the 13-identity,
but then what is the "12" after that?


-monz
http://tonalsoft.com
Tonescape microtonal music software

From: Gene Ward Smith (2007-02-19)
Subject: Re: new tenney pieces: spectral variations nos. 1 to 3

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:

> I tried to decompile the flash to get them, but for some
> reason gave up ("some reason" being my kid needing attention).
> I assume they aren't meant to be had.

The flash didn't work for me. If people don't want to share their 
music, and easier way to do it is not put it up.

From: Gene Ward Smith (2007-02-19)
Subject: Re: 159 and trikleismic temperament

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:

> I didn't know that the wedgie gave away that info!
> 
> Do they always work like this? I checked the meantone 
> wedgie <<1, 4, 10, 4, 13, 12|| and can see that 3 maps
> to 1 generator, 5 to 4 generators, and 7 to 10 generators
> ... but then what does the next "4" represent? And the
> "13" which comes next does represent the 13-identity,
> but then what is the "12" after that?

The 13 doesn't represent the 13-identity. The second 4 means that if 
the period is a 3, then there are 4 generators (3/2 being a 
generator) to the 5; the 13 then means there are 13 generators to the 
7 (27*7 = 189 is 13 fifths.) Finally, the period can't be 5, but can 
be taken to be 5^(1/4), in which case the generator can be taken as 
1/2, or 5^(1/2)/2 if you reduce to [1,5^(1/4)]. Now 12 is the number 
of generators (3) to get to 7, times the number of periods in the 
period-prime of 5 (4.) Since we usually are interested in octaves or 
reciprocal-integer fractions of octaves as periods, it's the first n-
1 numbers, when the number of odd primes is n, which are most 
interesting in terms of reading off the meaning of the wedgie (which 
has more meanings to it than I've said, I'm afraid.)

Let's consider pajara. This has a wedgie <<2 -4 -4 -11 -12 2||, so 
the truncated or octave-equivalent part is <2 -4 -4 ...||. The gcd of 
2,-4, and -4 is 2, so the period is half an octave. The generator 
part of the mapping to primes is then <2 -4 -4|/2 = <1 -2 -2|, so 
there is one genrerator to get to 3 (ie 3 or 3/2 can be taken as the 
generator), but since the period is 1/2 octave, to get complexity 
figures you need to double things--which means to use the numbers 
from the wedgie.

So, a fifth has a complexity of 2, a major third 4, and a 7/4 4. In 
Pajara[10], the symmetrical decatonic scale, there are therefore 
10-2=8 fifths, 10-4=6 major thirds, and 10-4=6 7/4s. The otonal 
tetrad has a complexity of 2-(-4)=6, so there are 10-6=4 otonal 
tetrads.

From: ciarán maher (2007-02-19)
Subject: Re: [tuning] Re: new tenney pieces: spectral variations nos. 1 to 3

well they're just tests, and of course the music is jim's not mine so  
i have to respect publishing rights etc.

but i'm sure it's cool to disseminate here. the files the flash  
player loads are here:

http://www.rhizomecowboy.com/spectral_variations/specVariation1.mp3
http://www.rhizomecowboy.com/spectral_variations/specVariation2.mp3
http://www.rhizomecowboy.com/spectral_variations/specVariation3.mp3

ciarán


On 19 Feabh 2007, at 02:22, Gene Ward Smith wrote:

> Are there urls for downloadable files to be found anywhere?

ciarán maher
rhizomecowboy.com

From: ciarán maher (2007-02-19)
Subject: Re: [tuning] Re: new tenney pieces: spectral variations nos. 1 to 3

thanks man.

jim was a genius.

it's really so sad that he died before hearing the pieces. but great  
that they're going to get heard.
ciarán.


On 19 Feabh 2007, at 00:53, Carl Lumma wrote:

> Stunning! I wasn't expecting to like this, but it turns the
> piano into a synthesizer in a way I've never experienced
> before. And a very good one at that.
>
> > we'll also be playing clarence barlow's extended version of the
> > oiriginal Spectral CANON for CONLON Nancoarrow, and the 'phi'
> > realisation of For Ann (rising) i made in 1998.
>
> Wish I could be there.

ciarán maher
rhizomecowboy.com

From: ciarán maher (2007-02-19)
Subject: Re: [tuning] Re: new tenney pieces: spectral variations nos. 1 to 3

nice

On 19 Feabh 2007, at 08:00, Gene Ward Smith wrote:

> The flash didn't work for me. If people don't want to share their
> music, and easier way to do it is not put it up.

ciarán maher
rhizomecowboy.com

From: ozanyarman@ozanyarman.com (2007-02-19)
Subject: Re: [tuning] Re: 159 and trikleismic temperament

Great!

----- Original Message -----
From: "Carl Lumma" <clumma@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 17 \ufffdubat 2007 Cumartesi 20:56
Subject: [tuning] Re: 159 and trikleismic temperament


Ozan, (and Gene),

I feel like you have a really good start at a dialog with
Gene for perhaps the first time.  I hope you will both have
patience and try to work out something together.  I think
it could be very fruitful.

-Carl

From: ozanyarman@ozanyarman.com (2007-02-19)
Subject: Re: [tuning] Re: 159 and trikleismic temperament

----- Original Message -----
From: "Carl Lumma" <clumma@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 17 \ufffdubat 2007 Cumartesi 21:01
Subject: [tuning] Re: 159 and trikleismic temperament


> > OK, one more time:
> >
> > 1. The principal diatonic scale of maqam Rast, which I gave
> > above, MUST be mapped to natural keys without breaking the
> > chain of fifths. 79 MOS 159-tET maps at least the ascending
> > scale to these notes without any accidentals, and requires
> > only a comma-down modifier at the 3rd, 6th, and 7th degrees
> > on descent.
> >
> > 2. Transition to the principal scale of Mahur, which is the
> > Pythagorean version of Rast, should be made possible, again
> > without breaking the chain of fifths and at every key.
>
> Every key of what -- the Rast scale?
>


Every key of the master tuning proposed of course.


> > 3. The traditional 17-tone system of (clustering) perdes,
> > the first octave of which I had given a week ago, MUST be
> > notated appropriately on the staff the way I have shown
> > through 79 MOS 159-tET. Do I need to repeat myself?
>
> It would be good if you could define what "notated
> appropriately" means.
>


Well, it meas that sharps and flats must be aligned properly with perdes.
That is to say, sharpened notes must be lower in pitch by about a comma's
worth compared to flattened notes.



> Also, what I have from you on the perdes is:
>
> Here are the 17 traditional perdes:
>
> 0: RAST
> 1: Shuri
> 2: Zengule cluster
> 3: DUGAH
> 4: Kurdi/Nihavend cluster
> 5: SEGAH cluster
> 6: Buselik
> 7: CHARGAH
> 8: Hijaz
> 9: Uzzal/Saba cluster
> 10: NEVA
> 11: Bayati
> 12: Hisar cluster
> 13: HUSEYNI
> 14: Ajem cluster
> 15: EVDJ cluster
> 16: Mahur
> 17: GERDANIYE
>
> But these are just names.  This dosen't tell us what
> these 17 things are.
>


79 MOS 159-tET does say.


> >4. For chromatic passages like in Western common-practice,
> > one needs a 12-tone cyclic subset from the master tuning,
> > which is achieved in 79 MOS 159-tET.
>
> Why is this important?  Must a maqam tuning be expected
> to reproduce Western common-practice also?
>



Nowadays, yes.


> -Carl
>
>

Oz.

From: ozanyarman@ozanyarman.com (2007-02-19)
Subject: Re: [tuning] Re: 159 and trikleismic temperament

----- Original Message -----
From: "Carl Lumma" <clumma@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 17 \ufffdubat 2007 Cumartesi 23:57
Subject: [tuning] Re: 159 and trikleismic temperament


> > > Ozan, (and Gene),
> > >
> > > I feel like you have a really good start at a dialog with
> > > Gene for perhaps the first time.  I hope you will both have
> > > patience and try to work out something together.  I think
> > > it could be very fruitful.
> >
> > Figuring out if propriety sufficies to ensure "modulational
> > integrity" would be a good start.
>
> My approach would be, rather than guessing huge master scales,
> trying to figure out what basic scales are needed, and what the
> modulation requirements are, and then build the larger scale
> from there.  Unfortunately my attempts at this have mostly
> ended in frustration.
>


How come?


> -Carl
>
>
>

Oz.

From: ozanyarman@ozanyarman.com (2007-02-19)
Subject: Re: [tuning] Re: 159 and trikleismic temperament

If otonal chords suit my endeavours in maqam polyphony, why not?


----- Original Message -----
From: "Carl Lumma" <clumma@yahoo.com>
To: <tuning@yahoogroups.com>
Sent: 17 \ufffdubat 2007 Cumartesi 20:52
Subject: [tuning] Re: 159 and trikleismic temperament


I meant, you didn't say having lots of otonal chords
was important to you (that I saw).  Is it?

-Carl

> Give me a few, and I'll tell you their number.
>
> Oz.
>

From: ozanyarman@ozanyarman.com (2007-02-19)
Subject: Re: [tuning] Re: 159 and trikleismic temperament

----- Original Message -----
From: "Gene Ward Smith" <genewardsmith@coolgoose.com>
To: <tuning@yahoogroups.com>
Sent: 18 \ufffdubat 2007 Pazar 8:19
Subject: [tuning] Re: 159 and trikleismic temperament


> --- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
> wrote:
>
> > It's a fine idea, but you need to get the list of required scales,
> with
> > indication of how much leeway you have.
>
> Here's some of what we know about Ozan's scale:
>
> (1) It is a MOS
>
> (2) It has a near-JI fifth
>
> (3) It is strictly proper
>
> (4) The interval class of the JI fifth contains a meantone fifth also
>



Also, a superpythagorean fifth, with which 80 MOS 159-tET closes.



> These conditions may be necessary, but they are not sufficient, as
> all of the above is also true of Ennealimmal[72], which Ozan
> dismissed as hopeless. One of my problems is that I don't know *why*.
> What, specifically, does Ozan's scale have that Ennealimmal[72] ain't
> got?



For one thing, the ascending principal Rast scale is not correctly mapped to
natural keys. Instead, one gets a very bad Mahur. Usable superpythagorean
fifths are absent, sharps and flats converge on the same notes, etc...
etc...



I know there are questions about rational approximations lurking
> out there, but he didn't say anything about that.



Come on... I have made my case several times already, expounding the
cardinal intervals I aimed for.



And I don't know
> what, specifically, is required. I do get that having the 13-limit be
> distinct would be good, but how accurately?
>



Just making 13:12 distinct from 12:11 should be enough for starters.



> Anyway, I really would like a specific answer to the question about
> Ennealimmal[72]. What's the problem with it?
>
>
>


I told you already.

Oz.

From: monz (2007-02-19)
Subject: Re: 159 and trikleismic temperament

Hi Gene,


--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
> 
> > I didn't know that the wedgie gave away that info!
>
> <snip>
> 
> The 13 doesn't represent the 13-identity. The second 4 means
> that if the period is a 3, then there are 4 generators (3/2
> being a generator) to the 5; 

You already lost me here. Is there some reason why
meantone would ever have a period of 3? Can you tabulate
step by step how the 4 generators map the 5 when the
period is 3?


> the 13 then means there are 13 generators to the 7
> (27*7 = 189 is 13 fifths.) 

Since i don't understand the simpler mapping to 5,
i'm afraid this doesn't make sense either. Where did
the 27 come from?


> Finally, the period can't be 5, but can be taken to be
> 5^(1/4), in which case the generator can be taken as 1/2,
> or 5^(1/2)/2 if you reduce to [1,5^(1/4)]. Now 12 is the
> number of generators (3) to get to 7, times the number of
> periods in the period-prime of 5 (4.)

Again, i think i need to work out the math here step by step
to see it happen.



> Since we usually are interested in octaves or
> reciprocal-integer fractions of octaves as periods,
> it's the first n-1 numbers, when the number of
> odd primes is n, which are most interesting in terms
> of reading off the meaning of the wedgie (which has
> more meanings to it than I've said, I'm afraid.)
> 
> Let's consider pajara. This has a wedgie <<2 -4 -4 -11 -12 2||,
> so the truncated or octave-equivalent part is <2 -4 -4 ...||. 
>
> <snip>


Now *this* i understand, and it looks to me like
"truncated wedgie" should have an Encyclopedia entry
of its own. What do you think?

And please do feel free to expansively relate those
"more meanings" to us, so that i can include them on
the "wedgie" page.


-monz
http://tonalsoft.com
Tonescape microtonal music software

From: monz (2007-02-19)
Subject: wedgie info (was: 159 and trikleismic temperament)

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:

> You already lost me here. Is there some reason why
> meantone would ever have a period of 3? Can you tabulate
> step by step how the 4 generators map the 5 when the
> period is 3?


I've changed the subject line and suggest that we
migrate this over to tuning-math.


-monz
http://tonalsoft.com
Tonescape microtonal music software

From: Gene Ward Smith (2007-02-19)
Subject: Re: 159 and trikleismic temperament

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

> > The 13 doesn't represent the 13-identity. The second 4 means
> > that if the period is a 3, then there are 4 generators (3/2
> > being a generator) to the 5; 
> 
> You already lost me here. Is there some reason why
> meantone would ever have a period of 3?

Why not? I think you'd be well-advised to use a tuning where the 3 
was flattened a bit--for instance TOP tuning, which is the same no 
matter what generators you choose. This says to tune the "3" to 
1899.2629 cents. But if using a 3 as period and having no octaves 
makes sense, as some people seem to think, then using it as a period 
and having octaves does too. I've experimented a bit with this stuff, 
and it works fine, in my opinion.

> Can you tabulate
> step by step how the 4 generators map the 5 when the
> period is 3?

The same way they do when the period is a 2. (3/2)^4/5 = 81/80, so if 
81/80 is tempered out, four generators of size a tempered 3/2 makes 
the tempered 5.

> > the 13 then means there are 13 generators to the 7
> > (27*7 = 189 is 13 fifths.) 
> 
> Since i don't understand the simpler mapping to 5,
> i'm afraid this doesn't make sense either. Where did
> the 27 come from?

27=3^3, and if 3 is the period, you can adjust by powers of 3. Hence 
getting to 189 in terms of generators is the same as getting to 7, 
just as when the period is 2, getting to 5 via four fifths is the 
same as getting to 5/4.

However, this period business may be confusing the issue. The first 
number of <<1 4 10 4 13 12|| tells us that 2 and 3 together work as a 
pair of generators; this is because it's the 2 and 3 slot. You can 
call the 2 the period, and 3 the generator, or do it the other way 
around. The second number, 4, is in the 2 and 5 slot, and tells us 
that using 2 and 5 as generators gives only 1/4 of the total 
intervals. The 5 slots are the second (2 and 5) the fourth (3 and 5) 
and the sixth (7 and 5.) These are 4, 4, 12, with gcd 4, which tells 
us that we need to take 5^(1/4) as a generator, not 5. So, 2 and 5^
(1/4) will work, or 3 and 5^(1/4), or 7 and 5^(1/4). Yhe other primes,
2, 3, and 7, have a gcd of 1 and hence may be used as generators.

> > Now *this* i understand, and it looks to me like
> "truncated wedgie" should have an Encyclopedia entry
> of its own. What do you think?

Or else "OE part", which we've been calling it. Yes, it wouldn't hurt 
I guess.

From: ozanyarman@ozanyarman.com (2007-02-19)
Subject: Re: [tuning] Re: 159 and trikleismic temperament

What's propriety?

----- Original Message -----
From: "Gene Ward Smith" <genewardsmith@coolgoose.com>
To: <tuning@yahoogroups.com>
Sent: 17 \ufffdubat 2007 Cumartesi 23:47
Subject: [tuning] Re: 159 and trikleismic temperament


> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@...> wrote:
> >
> > Ozan, (and Gene),
> >
> > I feel like you have a really good start at a dialog with
> > Gene for perhaps the first time.  I hope you will both have
> > patience and try to work out something together.  I think
> > it could be very fruitful.
>
> Figuring out if propriety sufficies to ensure "modulational integrity"
> would be a good start.
>
>

From: ozanyarman@ozanyarman.com (2007-02-19)
Subject: Re: 159 and trikleismic temperament

And how do you presume they are changing, Gene? I've said nothing so far
that suggests Rast is anything but a maqam based on the harmonic major
scale.

Oz.


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

> > Rast is supposed to be, more or less, a diatonic scale?

> Good gracious, we've been over this a dozen times already.

Yes, but your answers keep changing.

From: Carl Lumma (2007-02-19)
Subject: Re: 159 and trikleismic temperament

> > > 1. The principal diatonic scale of maqam Rast, which I gave
> > > above, MUST be mapped to natural keys without breaking the
> > > chain of fifths. 79 MOS 159-tET maps at least the ascending
> > > scale to these notes without any accidentals, and requires
> > > only a comma-down modifier at the 3rd, 6th, and 7th degrees
> > > on descent.
> > >
> > > 2. Transition to the principal scale of Mahur, which is the
> > > Pythagorean version of Rast, should be made possible, again
> > > without breaking the chain of fifths and at every key.
> >
> > Every key of what -- the Rast scale?
>
> Every key of the master tuning proposed of course.

Wow, that's pretty strong.  Usually only ETs have this
kind of ability.

> > > 3. The traditional 17-tone system of (clustering) perdes,
> > > the first octave of which I had given a week ago, MUST be
> > > notated appropriately on the staff the way I have shown
> > > through 79 MOS 159-tET. Do I need to repeat myself?
> >
> > It would be good if you could define what "notated
> > appropriately" means.
> 
> Well, it meas that sharps and flats must be aligned properly
> with perdes.
> That is to say, sharpened notes must be lower in pitch by about
> a comma's worth compared to flattened notes.

By sharps and flats, I assume you're referring to approximate
apotomes and limma.  D# = D + 7f and Eb = D - 5f.  You want
7f-4o < 3o-5f, which simplifies to 12f < 7o.  This means you
want a fifth flatter than 700 cents.

Does a "comma's worth" have an exact definition?  If so, it's
straightforward to pinpoint the size of your flat fifth.

I guess I'm still struggling to understand the other
requirements.  You need Rast and Mahur on every degree of
the master tuning... then indeed you need an ET with
both a meantone and a pure fifth.  I don't see how a MOS
can do it.

> > >4. For chromatic passages like in Western common-practice,
> > > one needs a 12-tone cyclic subset from the master tuning,
> > > which is achieved in 79 MOS 159-tET.
> >
> > Why is this important?  Must a maqam tuning be expected
> > to reproduce Western common-practice also?
> 
> Nowadays, yes.

From my point of view, though I'm willing to subscribe to
something like 'maqam music today has taken Western bits
and made them uniquely its own', I think it would also be
interesting to take a snapshot of maqam music from around
the time of the earliest recordings and try to come up with
a tuning for that.

-Carl

From: Ozan Yarman (2007-02-20)
Subject: Re: [tuning] Re: 159 and trikleismic temperament

SNIP

> > > >
> > > > 2. Transition to the principal scale of Mahur, which is the
> > > > Pythagorean version of Rast, should be made possible, again
> > > > without breaking the chain of fifths and at every key.
> > >
> > > Every key of what -- the Rast scale?
> >
> > Every key of the master tuning proposed of course.
>
> Wow, that's pretty strong.  Usually only ETs have this
> kind of ability.
>


79/80 MOS 159-tET as well.


SNIP

> >
> > Well, it meas that sharps and flats must be aligned properly
> > with perdes.
> > That is to say, sharpened notes must be lower in pitch by about
> > a comma's worth compared to flattened notes.
>
> By sharps and flats, I assume you're referring to approximate
> apotomes and limma.  D# = D + 7f and Eb = D - 5f.  You want
> 7f-4o < 3o-5f, which simplifies to 12f < 7o.  This means you
> want a fifth flatter than 700 cents.
>


On the average, yes. But I also want the super-pythagorean cycle of 80 MOS
159-tET when the need arises, which effectively pushes sharps over the
flats, and flats under the sharps.


> Does a "comma's worth" have an exact definition?  If so, it's
> straightforward to pinpoint the size of your flat fifth.
>



The proponents of the comma will tell you that their model is the
Pythagorean, otherwise approximated by the Holdrian. That gives 53-equal of
course, which makes no distinction between 11:10 and 12:11.



> I guess I'm still struggling to understand the other
> requirements.  You need Rast and Mahur on every degree of
> the master tuning... then indeed you need an ET with
> both a meantone and a pure fifth.  I don't see how a MOS
> can do it.
>


79/80 MOS 159-tET can suffice in quasi-equal transpositions at every key.
I've shown you how it can be done.


> > > >4. For chromatic passages like in Western common-practice,
> > > > one needs a 12-tone cyclic subset from the master tuning,
> > > > which is achieved in 79 MOS 159-tET.
> > >
> > > Why is this important?  Must a maqam tuning be expected
> > > to reproduce Western common-practice also?
> >
> > Nowadays, yes.
>
> >From my point of view, though I'm willing to subscribe to
> something like 'maqam music today has taken Western bits
> and made them uniquely its own', I think it would also be
> interesting to take a snapshot of maqam music from around
> the time of the earliest recordings and try to come up with
> a tuning for that.
>



Those recordings tell us that the cardinal intervals to be aimed for are the
7, 11 and 13 limit tones encapsulated in my tuning.



> -Carl
>
>
>



Oz.

From: Carl Lumma (2007-02-20)
Subject: Re: new tenney pieces: spectral variations nos. 1 to 3

> but i'm sure it's cool to disseminate here. the files the flash  
> player loads are here:
> 
> http://www.rhizomecowboy.com/spectral_variations/specVariation1.mp3
> http://www.rhizomecowboy.com/spectral_variations/specVariation2.mp3
> http://www.rhizomecowboy.com/spectral_variations/specVariation3.mp3
> 
> ciaran

Thanks!  -Carl