Why is 8192/6561 more consonant than 5/4 between rast-segah

In the context of investigating why perde segah is supposed to be
lower than 5/4, I have come up with an ascending diatonic Rast scale
that sounds truer than any I have previously attempted. This might
very well be the ultimate theoretical scale model for the correct
Rast. It is curious that the scale is 11-limit in every way, including
a stretched octave, but just as sweet and consonant as a 5-limit Major
scale.

Here it is:

0: 1/1 0 cents C unison, perfect prime
1: 432/385 199 cents D
2: 96/77 382 cents Ed
3: 385/288 503 cents F
4: 3/2 702 cents G perfect fifth
5: 648/385 901 cents A
6: 144/77 1084 cents Bd
7: 385/192 1205 cents C undecimal kleisma +1
octave

Yes, even the octave is 11-limit, stretched by an undecimal kleisma.
Due to the undecimal kleisma being referred to as "Keenan's Kleisma",
I decided to call this scale, "Keenanish Rast".

385/192 appears to be a resonant octave with a pleasing beat. Could
this be a clue as to why human ears seek stretched octaves?

Still, it baffles me why the chord 382 320 | 154:192:231 and the
chord 320 382 | 64:77:96 of this scale sounds so savoury. What is
so special about these beating chords, I wonder...

One thing I noticed in the scale is that, the numerator of 432/385
ordinarily replacable by 9/8, of 648/385 ordinarily replacable by
27/16, of 96/77 ordinarily replacable by 5/4, and 144/77 ordinarily
replacable by 15/8 are all 3-limit. In other words, the counterpart
ratios' numerators comprise odd number primes also.

In retrospect, the counterpart ratios of 385/288 and 385/192, which
are 4/3 and 2/1, include even number primes (that is to say, number
2). Take note, that 385 does not contain the prime 2 at all.

Quod erat demonstrandum: Where there is ordinarily, in the
factorization of either the numerator or the denominator, prime number
2, expect never to find number 2 in Keenanish Rast scale. Where there
is ordinarily, again in the factorization of either the numerator or
the denominator, odd prime numbers such as 3, 5, 7... expect to find
only 2 and 3 in Keenanish Rast scale.

Strange indeed! Some uniform warping of tone-space is occuring here,
so much so that a 5-limit harmonic Major jumps hyperspace toward an
equally valid 11-limit harmonic Major.

Next on the agenda is to introduce Yarman36b and its natural extension
Yarman48, which, in my opinion, are so far the best temperaments for
representing the above-given Keenanish Rast, and consequently, Turkish
Maqam music practice as a whole.

Cordially,
Oz.

> From: Ozan Yarman <ozanyarman@...>
> Date: October 7, 2009 12:23:22 AM GMT+03:00
> To: tuning@yahoogroups.com
> Subject: Re: [tuning] Why is 8192/6561 more consonant that 5/4
> between rast-segah
>
> Dear Aaron,
>
> Melodic step-sizes in a maqam scale are, most of the time, not
> abiding MOS. There may be three or more different step sizes.
> Consider this Hüzzam:
>
> Ed 120 F 204 G 150 Ab 228 Bd 120 C 294 D# 84 Ed
>
> See, there are 6 different step sizes, with Ed at 56/45.
>
> Replacing 15:14 between Ed-F and Bd-C with 16:15 will ruin the rast-
> segah relationship. Maybe the ear is inclining towards 56/45 instead
> of 5/4? But I fear that is too low. The proper perde segah should be
> at 96/77:
>
> Ed 116 F 204 G 150 Ab 228 Bd 116 C 294 D# 88 Ed
>
> That's an even better Hüzzam, again with 6 different step sizes,
> none of which is a multiple of another.
>
> While there are fourth and fifth relationships between given tones,
> this scale is not constructed via a chain of fifths.
>
> Now I wonder, why 96/77 instead of 5/4 (if we are to think in terms
> of JI)? Or, why the beating of the major third?
>
> Aaron, this is not restricted to my taste, it is a phenomenon of
> Maqam music according to my observations. It may not be a scientific > one, for I have not tested the rast-segah relatinship in a
> laboratory condition with test subjects and equipment, but it is
> rather obvious to one who listens to, say, "Ey gul-i bag eda" by
> Dede Efendi in Hüzzam:
>
> http://www.vbox7.com/play:feba93b8?r=google
>
> Can you also observe the low perde segah?
>
> Oz.
>
> ✩ ✩ ✩
> www.ozanyarman.com
>
> On Oct 5, 2009, at 11:31 PM, Aaron Johnson wrote:
>
>>
>>
>> Oz,
>>
>> My ad-hoc theory is that you are probably noticing the subtle
>> effects of having all the melodic step sizes be similar, and/or >> multiples of each other, due to the chain-of-fifths? Or, you prefer
>> to notice a slight beating in the third.
>>
>> I'm really curious how scientifically you have tested your
>> perception of this....do you have a 100% success rate? Perhaps you
>> can set up a kind of blind listening test?
>>
>> All best,
>> Aaron.
>>
>> On Mon, Oct 5, 2009 at 2:44 PM, Ozan Yarman <ozanyarman@...
>> > wrote:
>>
>>
>> Aaron, thank you for sharing your thoughts. But why does the ear of
>> a maqam musician incline toward 8192/6561 instead of 5/4? I have
>> many times tried to accompany an actual performance with either 53-
>> tET and 41-tET, and observed the latter to agree with practice more.
>>
>> Oz.
>>
>> ✩ ✩ ✩
>> www.ozanyarman.com
>>
>> On Oct 2, 2009, at 9:04 PM, Aaron Johnson wrote:
>>
>>>
>>>
>>>
>>>
>>> On Fri, Oct 2, 2009 at 12:06 PM, Ozan Yarman <ozanyarman@...
>>> > wrote:
>>>
>>>
>>> Aaron,
>>>
>>> I also suspect PI has little to do with the actual phenomenon of
>>> why a 381-384 cent major third sounds just as sweet as a 386 cent
>>> major third. Note, that Rauf Yekta pointed out the rast-segah
>>> interval to be 384 cents, not 386 cents, claiming that the ear is
>>> sensitive to the minute difference of 2 cents.
>>>
>>>
>>> That's a schismatic (Pythagorean) diminished 4th, so that makes
>>> sense.
>>>
>>>
>>>
>>> I don't know why, but I also experience 5/4 as an inappropriate
>>> segah. It's just too high. There must be some logical psycho-
>>> acoustical reason behind it. Through trial and error, I located
>>> the desired interval to be somewhere about 382 cents.
>>>
>>>
>>> Any thoughts?
>>>
>>>
>>> My thoughts? You are a very sensitive musician with keen hearing,
>>> brought up in a cultural/musical tradition you understand well,
>>> and your instincts favor a Pythagorean diminshed 4th, rather than
>>> a 5/4!
>>>
>>> I suspect that this is also is variable in certain lab conditions
>>> like tempo, etc. IOW, there might be 'lab' conditions possible
>>> where you'd also confuse the two 3rds.
>>>
>>> Best,
>>> Aaron.
>>>
>>>
>>> Cordially,
>>> Oz.
>>>
>>> ✩ ✩ ✩
>>> www.ozanyarman.com
>>>
>>> On Oct 2, 2009, at 5:42 PM, Aaron Johnson wrote:
>>>
>>>>
>>>>
>>>> Hey, what was the $50 challenge? I forget....
>>>>
>>>> I do think that so-called LucyTuning© sounds good for certain
>>>> things, BTW. But I have to agree that _why_ it sounds good has
>>>> pretty much nothing to do with PI. In other words, people don't
>>>> start floating out of their seats, arms outstretched in mystical
>>>> oneness with creation, surrounded by incandescent blue orbs of
>>>> cosmic energy, when the major third is 2^(1/PI), as opposed to
>>>> say 28 steps of 88-equal.....in 99.99% of cases, i.e. most music
>>>> that doesn't have sustained chords 5 minutes long where you could
>>>> measure things with an oscilloscope and software, *as it
>>>> happens*, I highly doubt anyone could tell the difference.
>>>>
>>>> AKJ
>>>>
>>>> On Fri, Oct 2, 2009 at 7:50 AM, Charles Lucy <lucy@...>>>>> wrote:
>>>>
>>>>
>>>> it is convenient that it works in a meantone pattern. That makes
>>>> it easy to notate, conceptualise the harmony, and to use for
>>>> educated musicians.
>>>>
>>>> As for the snake oil (marketing?):
>>>>
>>>> Snake oil seems to refer to an ancient Chinese remedy; after all
>>>> the Chinese do represent the largest market for just about
>>>> everything nowadays;-)
>>>>
>>>> Numerology?
>>>>
>>>> Can we count on Carl to be able to explain what that is?
>>>>
>>>> BTW Carl I'm still waiting for my $50 payout from a previous
>>>> challenge which you reneged on a couple of years ago.
>>>>
>>>> Have fun with it!
>>>>
>>>> C.L.
>>>>
>>>> On 2 Oct 2009, at 03:16, Carl Lumma wrote:
>>>>
>>>>> --- In tuning@yahoogroups.com, jrinkel@... wrote:
>>>>> >
>>>>> > In addition to the many suggestions so far, you may want
>>>>> > to check out
>>>>> >
>>>>> > www.lucytune.com
>>>>> >
>>>>> > It talks about a microtonal scale based on the value of pi.
>>>>>
>>>>> The uninitiated should be aware that lucytuning is simply
>>>>> meantone (in the neighborhood of 1/3-comma meantone), mixed
>>>>> with liberal quantities of numerology and snake oil.
>>>>>
>>>>> -Carl
>>>>>
>>>>>
>>>>
>>>> Charles Lucy
>>>> lucy@...
>>>>
>>>> - Promoting global harmony through LucyTuning -
>>>>
>>>> for information on LucyTuning go to:
>>>> http://www.lucytune.com
>>>>
>>>> For LucyTuned Lullabies go to:
>>>> http://www.lullabies.co.uk
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> --
>>>>
>>>>
>>>> Aaron Krister Johnson
>>>> http://www.akjmusic.com
>>>> http://www.untwelve.org
>>>>
>>>>
>>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>> --
>>>
>>> Aaron Krister Johnson
>>> http://www.akjmusic.com
>>> http://www.untwelve.org
>>>
>>>
>>>
>>
>>
>>
>>
>>
>>
>> --
>>
>> Aaron Krister Johnson
>> http://www.akjmusic.com
>> http://www.untwelve.org
>>
>>
>>
>>
>

Ozan, as I expected from listening to your tuning, it is almost perfectly an equal/proportional beating affair.

Try this very slight tweak:

0: 1/1 0.000 unison, perfect prime
1: 198.970 cents 198.970
2: 381.869 cents 381.869
3: 502.202 cents 502.202
4: 3/2 701.955 perfect fifth
5: 900.926 cents 900.926
6: 1085.307 cents 1085.307
7: 1205.540 cents 1205.540

Check the various beating especially around the 4th and 8th partial.

As I have been insisting for the last few years, neither primes nor "approximations" are the answer. I heard a "red thread" in your tuning, a coherence and family resemblance amongst the intervals, in your tuning, and there it is. The harmonic spectrum here is a physical reference- the timbre of the tuning has consistently beating stripes of color, so to speak.

--- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
>
> In the context of investigating why perde segah is supposed to be
> lower than 5/4, I have come up with an ascending diatonic Rast scale
> that sounds truer than any I have previously attempted. This might
> very well be the ultimate theoretical scale model for the correct
> Rast. It is curious that the scale is 11-limit in every way, including
> a stretched octave, but just as sweet and consonant as a 5-limit Major
> scale.
>
> Here it is:
>
> 0: 1/1 0 cents C unison, perfect prime
> 1: 432/385 199 cents D
> 2: 96/77 382 cents Ed
> 3: 385/288 503 cents F
> 4: 3/2 702 cents G perfect fifth
> 5: 648/385 901 cents A
> 6: 144/77 1084 cents Bd
> 7: 385/192 1205 cents C undecimal kleisma +1
> octave
>
> Yes, even the octave is 11-limit, stretched by an undecimal kleisma.
> Due to the undecimal kleisma being referred to as "Keenan's Kleisma",
> I decided to call this scale, "Keenanish Rast".
>
> 385/192 appears to be a resonant octave with a pleasing beat. Could
> this be a clue as to why human ears seek stretched octaves?
>
> Still, it baffles me why the chord 382 320 | 154:192:231 and the
> chord 320 382 | 64:77:96 of this scale sounds so savoury. What is
> so special about these beating chords, I wonder...
>
> One thing I noticed in the scale is that, the numerator of 432/385
> ordinarily replacable by 9/8, of 648/385 ordinarily replacable by
> 27/16, of 96/77 ordinarily replacable by 5/4, and 144/77 ordinarily
> replacable by 15/8 are all 3-limit. In other words, the counterpart
> ratios' numerators comprise odd number primes also.
>
> In retrospect, the counterpart ratios of 385/288 and 385/192, which
> are 4/3 and 2/1, include even number primes (that is to say, number
> 2). Take note, that 385 does not contain the prime 2 at all.
>
> Quod erat demonstrandum: Where there is ordinarily, in the
> factorization of either the numerator or the denominator, prime number
> 2, expect never to find number 2 in Keenanish Rast scale. Where there
> is ordinarily, again in the factorization of either the numerator or
> the denominator, odd prime numbers such as 3, 5, 7... expect to find
> only 2 and 3 in Keenanish Rast scale.
>
> Strange indeed! Some uniform warping of tone-space is occuring here,
> so much so that a 5-limit harmonic Major jumps hyperspace toward an
> equally valid 11-limit harmonic Major.
>
> Next on the agenda is to introduce Yarman36b and its natural extension
> Yarman48, which, in my opinion, are so far the best temperaments for
> representing the above-given Keenanish Rast, and consequently, Turkish
> Maqam music practice as a whole.
>
> Cordially,
> Oz.
>
>
>
>
> > From: Ozan Yarman <ozanyarman@...>
> > Date: October 7, 2009 12:23:22 AM GMT+03:00
> > To: tuning@yahoogroups.com
> > Subject: Re: [tuning] Why is 8192/6561 more consonant that 5/4
> > between rast-segah
> >
> > Dear Aaron,
> >
> > Melodic step-sizes in a maqam scale are, most of the time, not
> > abiding MOS. There may be three or more different step sizes.
> > Consider this Hüzzam:
> >
> > Ed 120 F 204 G 150 Ab 228 Bd 120 C 294 D# 84 Ed
> >
> > See, there are 6 different step sizes, with Ed at 56/45.
> >
> > Replacing 15:14 between Ed-F and Bd-C with 16:15 will ruin the rast-
> > segah relationship. Maybe the ear is inclining towards 56/45 instead
> > of 5/4? But I fear that is too low. The proper perde segah should be
> > at 96/77:
> >
> > Ed 116 F 204 G 150 Ab 228 Bd 116 C 294 D# 88 Ed
> >
> > That's an even better Hüzzam, again with 6 different step sizes,
> > none of which is a multiple of another.
> >
> > While there are fourth and fifth relationships between given tones,
> > this scale is not constructed via a chain of fifths.
> >
> > Now I wonder, why 96/77 instead of 5/4 (if we are to think in terms
> > of JI)? Or, why the beating of the major third?
> >
> > Aaron, this is not restricted to my taste, it is a phenomenon of
> > Maqam music according to my observations. It may not be a scientific
> > one, for I have not tested the rast-segah relatinship in a
> > laboratory condition with test subjects and equipment, but it is
> > rather obvious to one who listens to, say, "Ey gul-i bag eda" by
> > Dede Efendi in Hüzzam:
> >
> > http://www.vbox7.com/play:feba93b8?r=google
> >
> > Can you also observe the low perde segah?
> >
> > Oz.
> >
> > ✩ ✩ ✩
> > www.ozanyarman.com
> >
> > On Oct 5, 2009, at 11:31 PM, Aaron Johnson wrote:
> >
> >>
> >>
> >> Oz,
> >>
> >> My ad-hoc theory is that you are probably noticing the subtle
> >> effects of having all the melodic step sizes be similar, and/or
> >> multiples of each other, due to the chain-of-fifths? Or, you prefer
> >> to notice a slight beating in the third.
> >>
> >> I'm really curious how scientifically you have tested your
> >> perception of this....do you have a 100% success rate? Perhaps you
> >> can set up a kind of blind listening test?
> >>
> >> All best,
> >> Aaron.
> >>
> >> On Mon, Oct 5, 2009 at 2:44 PM, Ozan Yarman <ozanyarman@...
> >> > wrote:
> >>
> >>
> >> Aaron, thank you for sharing your thoughts. But why does the ear of
> >> a maqam musician incline toward 8192/6561 instead of 5/4? I have
> >> many times tried to accompany an actual performance with either 53-
> >> tET and 41-tET, and observed the latter to agree with practice more.
> >>
> >> Oz.
> >>
> >> ✩ ✩ ✩
> >> www.ozanyarman.com
> >>
> >> On Oct 2, 2009, at 9:04 PM, Aaron Johnson wrote:
> >>
> >>>
> >>>
> >>>
> >>>
> >>> On Fri, Oct 2, 2009 at 12:06 PM, Ozan Yarman <ozanyarman@...
> >>> > wrote:
> >>>
> >>>
> >>> Aaron,
> >>>
> >>> I also suspect PI has little to do with the actual phenomenon of
> >>> why a 381-384 cent major third sounds just as sweet as a 386 cent
> >>> major third. Note, that Rauf Yekta pointed out the rast-segah
> >>> interval to be 384 cents, not 386 cents, claiming that the ear is
> >>> sensitive to the minute difference of 2 cents.
> >>>
> >>>
> >>> That's a schismatic (Pythagorean) diminished 4th, so that makes
> >>> sense.
> >>>
> >>>
> >>>
> >>> I don't know why, but I also experience 5/4 as an inappropriate
> >>> segah. It's just too high. There must be some logical psycho-
> >>> acoustical reason behind it. Through trial and error, I located
> >>> the desired interval to be somewhere about 382 cents.
> >>>
> >>>
> >>> Any thoughts?
> >>>
> >>>
> >>> My thoughts? You are a very sensitive musician with keen hearing,
> >>> brought up in a cultural/musical tradition you understand well,
> >>> and your instincts favor a Pythagorean diminshed 4th, rather than
> >>> a 5/4!
> >>>
> >>> I suspect that this is also is variable in certain lab conditions
> >>> like tempo, etc. IOW, there might be 'lab' conditions possible
> >>> where you'd also confuse the two 3rds.
> >>>
> >>> Best,
> >>> Aaron.
> >>>
> >>>
> >>> Cordially,
> >>> Oz.
> >>>
> >>> ✩ ✩ ✩
> >>> www.ozanyarman.com
> >>>
> >>> On Oct 2, 2009, at 5:42 PM, Aaron Johnson wrote:
> >>>
> >>>>
> >>>>
> >>>> Hey, what was the $50 challenge? I forget....
> >>>>
> >>>> I do think that so-called LucyTuning© sounds good for certain
> >>>> things, BTW. But I have to agree that _why_ it sounds good has
> >>>> pretty much nothing to do with PI. In other words, people don't
> >>>> start floating out of their seats, arms outstretched in mystical
> >>>> oneness with creation, surrounded by incandescent blue orbs of
> >>>> cosmic energy, when the major third is 2^(1/PI), as opposed to
> >>>> say 28 steps of 88-equal.....in 99.99% of cases, i.e. most music
> >>>> that doesn't have sustained chords 5 minutes long where you could
> >>>> measure things with an oscilloscope and software, *as it
> >>>> happens*, I highly doubt anyone could tell the difference.
> >>>>
> >>>> AKJ
> >>>>
> >>>> On Fri, Oct 2, 2009 at 7:50 AM, Charles Lucy <lucy@...>
> >>>> wrote:
> >>>>
> >>>>
> >>>> it is convenient that it works in a meantone pattern. That makes
> >>>> it easy to notate, conceptualise the harmony, and to use for
> >>>> educated musicians.
> >>>>
> >>>> As for the snake oil (marketing?):
> >>>>
> >>>> Snake oil seems to refer to an ancient Chinese remedy; after all
> >>>> the Chinese do represent the largest market for just about
> >>>> everything nowadays;-)
> >>>>
> >>>> Numerology?
> >>>>
> >>>> Can we count on Carl to be able to explain what that is?
> >>>>
> >>>> BTW Carl I'm still waiting for my $50 payout from a previous
> >>>> challenge which you reneged on a couple of years ago.
> >>>>
> >>>> Have fun with it!
> >>>>
> >>>> C.L.
> >>>>
> >>>> On 2 Oct 2009, at 03:16, Carl Lumma wrote:
> >>>>
> >>>>> --- In tuning@yahoogroups.com, jrinkel@ wrote:
> >>>>> >
> >>>>> > In addition to the many suggestions so far, you may want
> >>>>> > to check out
> >>>>> >
> >>>>> > www.lucytune.com
> >>>>> >
> >>>>> > It talks about a microtonal scale based on the value of pi.
> >>>>>
> >>>>> The uninitiated should be aware that lucytuning is simply
> >>>>> meantone (in the neighborhood of 1/3-comma meantone), mixed
> >>>>> with liberal quantities of numerology and snake oil.
> >>>>>
> >>>>> -Carl
> >>>>>
> >>>>>
> >>>>
> >>>> Charles Lucy
> >>>> lucy@...
> >>>>
> >>>> - Promoting global harmony through LucyTuning -
> >>>>
> >>>> for information on LucyTuning go to:
> >>>> http://www.lucytune.com
> >>>>
> >>>> For LucyTuned Lullabies go to:
> >>>> http://www.lullabies.co.uk
> >>>>
> >>>>
> >>>>
> >>>>
> >>>>
> >>>>
> >>>>
> >>>>
> >>>> --
> >>>>
> >>>>
> >>>> Aaron Krister Johnson
> >>>> http://www.akjmusic.com
> >>>> http://www.untwelve.org
> >>>>
> >>>>
> >>>>
> >>>
> >>>
> >>>
> >>>
> >>>
> >>>
> >>> --
> >>>
> >>> Aaron Krister Johnson
> >>> http://www.akjmusic.com
> >>> http://www.untwelve.org
> >>>
> >>>
> >>>
> >>
> >>
> >>
> >>
> >>
> >>
> >> --
> >>
> >> Aaron Krister Johnson
> >> http://www.akjmusic.com
> >> http://www.untwelve.org
> >>
> >>
> >>
> >>
> >
>

Here is what I found in the first few bars of the Turkish piece which
Ozan put a link to:

Notes found in first few seconds of Turkish piece:

Max pitch is A5

A5 +3 to _ 5 cents
G5 + 12 cents
F5 + 8 cents
E5 +17 cents
E5 + 3 cents
D5 + 22 cents
C#5 -9 cents
C5 +19 cents
B4 -27 cents
A#4 +23 cents
A4 +18 cents
G4 + 7 cents
F#4 +26 cents
F4 + 1 cent
E4 + 37 cents

All notes from D4 and below are 12 edo with less than 10 cents.

I don't know what the instrument is which is playing the "melody" but
it seems to have all its notes in the range E4 to A5 and to be
"improved" on a player-controlled variable-pitch instrument.

All other instruments seem to be regular common or garden 12edo.

I have put a link to the image that I am seeing with melodyne DNA beta
here:

http://www.lucytune.com/turkish/turkishmelodyne.jpg

On 7 Oct 2009, at 05:31, Ozan Yarman wrote:

>
> In the context of investigating why perde segah is supposed to be
> lower than 5/4, I have come up with an ascending diatonic Rast scale
> that sounds truer than any I have previously attempted. This might
> very well be the ultimate theoretical scale model for the correct
> Rast. It is curious that the scale is 11-limit in every way,
> including a stretched octave, but just as sweet and consonant as a 5-
> limit Major scale.
>
> Here it is:
>
> 0: 1/1 0 cents C unison, perfect prime
> 1: 432/385 199 cents D
> 2: 96/77 382 cents Ed
> 3: 385/288 503 cents F
> 4: 3/2 702 cents G perfect fifth
> 5: 648/385 901 cents A
> 6: 144/77 1084 cents Bd
> 7: 385/192 1205 cents C undecimal kleisma +1
> octave
>
> Yes, even the octave is 11-limit, stretched by an undecimal kleisma.
> Due to the undecimal kleisma being referred to as "Keenan's
> Kleisma", I decided to call this scale, "Keenanish Rast".
>
> 385/192 appears to be a resonant octave with a pleasing beat. Could
> this be a clue as to why human ears seek stretched octaves?
>
> Still, it baffles me why the chord 382 320 | 154:192:231 and the > chord 320 382 | 64:77:96 of this scale sounds so savoury. What
> is so special about these beating chords, I wonder...
>
> One thing I noticed in the scale is that, the numerator of 432/385
> ordinarily replacable by 9/8, of 648/385 ordinarily replacable by
> 27/16, of 96/77 ordinarily replacable by 5/4, and 144/77 ordinarily
> replacable by 15/8 are all 3-limit. In other words, the counterpart
> ratios' numerators comprise odd number primes also.
>
> In retrospect, the counterpart ratios of 385/288 and 385/192, which
> are 4/3 and 2/1, include even number primes (that is to say, number
> 2). Take note, that 385 does not contain the prime 2 at all.
>
> Quod erat demonstrandum: Where there is ordinarily, in the
> factorization of either the numerator or the denominator, prime
> number 2, expect never to find number 2 in Keenanish Rast scale.
> Where there is ordinarily, again in the factorization of either the
> numerator or the denominator, odd prime numbers such as 3, 5, 7...
> expect to find only 2 and 3 in Keenanish Rast scale.
>
> Strange indeed! Some uniform warping of tone-space is occuring here,
> so much so that a 5-limit harmonic Major jumps hyperspace toward an
> equally valid 11-limit harmonic Major.
>
> Next on the agenda is to introduce Yarman36b and its natural
> extension Yarman48, which, in my opinion, are so far the best
> temperaments for representing the above-given Keenanish Rast, and
> consequently, Turkish Maqam music practice as a whole.
>
> Cordially,
> Oz.
>
>
>
>
>> From: Ozan Yarman <ozanyarman@...>
>> Date: October 7, 2009 12:23:22 AM GMT+03:00
>> To: tuning@yahoogroups.com
>> Subject: Re: [tuning] Why is 8192/6561 more consonant that 5/4
>> between rast-segah
>>
>> Dear Aaron,
>>
>> Melodic step-sizes in a maqam scale are, most of the time, not
>> abiding MOS. There may be three or more different step sizes.
>> Consider this Hüzzam:
>>
>> Ed 120 F 204 G 150 Ab 228 Bd 120 C 294 D# 84 Ed
>>
>> See, there are 6 different step sizes, with Ed at 56/45.
>>
>> Replacing 15:14 between Ed-F and Bd-C with 16:15 will ruin the rast-
>> segah relationship. Maybe the ear is inclining towards 56/45
>> instead of 5/4? But I fear that is too low. The proper perde segah
>> should be at 96/77:
>>
>> Ed 116 F 204 G 150 Ab 228 Bd 116 C 294 D# 88 Ed
>>
>> That's an even better Hüzzam, again with 6 different step sizes,
>> none of which is a multiple of another.
>>
>> While there are fourth and fifth relationships between given tones,
>> this scale is not constructed via a chain of fifths.
>>
>> Now I wonder, why 96/77 instead of 5/4 (if we are to think in terms
>> of JI)? Or, why the beating of the major third?
>>
>> Aaron, this is not restricted to my taste, it is a phenomenon of
>> Maqam music according to my observations. It may not be a
>> scientific one, for I have not tested the rast-segah relatinship in
>> a laboratory condition with test subjects and equipment, but it is
>> rather obvious to one who listens to, say, "Ey gul-i bag eda" by
>> Dede Efendi in Hüzzam:
>>
>> http://www.vbox7.com/play:feba93b8?r=google
>>
>> Can you also observe the low perde segah?
>>
>> Oz.
>>
>> ✩ ✩ ✩
>> www.ozanyarman.com
>>
>> On Oct 5, 2009, at 11:31 PM, Aaron Johnson wrote:
>>
>>>
>>>
>>> Oz,
>>>
>>> My ad-hoc theory is that you are probably noticing the subtle
>>> effects of having all the melodic step sizes be similar, and/or
>>> multiples of each other, due to the chain-of-fifths? Or, you
>>> prefer to notice a slight beating in the third.
>>>
>>> I'm really curious how scientifically you have tested your
>>> perception of this....do you have a 100% success rate? Perhaps you
>>> can set up a kind of blind listening test?
>>>
>>> All best,
>>> Aaron.
>>>
>>> On Mon, Oct 5, 2009 at 2:44 PM, Ozan Yarman <ozanyarman@...
>>> > wrote:
>>>
>>>
>>> Aaron, thank you for sharing your thoughts. But why does the ear
>>> of a maqam musician incline toward 8192/6561 instead of 5/4? I
>>> have many times tried to accompany an actual performance with
>>> either 53-tET and 41-tET, and observed the latter to agree with
>>> practice more.
>>>
>>> Oz.
>>>
>>> ✩ ✩ ✩
>>> www.ozanyarman.com
>>>
>>> On Oct 2, 2009, at 9:04 PM, Aaron Johnson wrote:
>>>
>>>>
>>>>
>>>>
>>>>
>>>> On Fri, Oct 2, 2009 at 12:06 PM, Ozan Yarman <ozanyarman@...
>>>> > wrote:
>>>>
>>>>
>>>> Aaron,
>>>>
>>>> I also suspect PI has little to do with the actual phenomenon of
>>>> why a 381-384 cent major third sounds just as sweet as a 386 cent
>>>> major third. Note, that Rauf Yekta pointed out the rast-segah
>>>> interval to be 384 cents, not 386 cents, claiming that the ear is
>>>> sensitive to the minute difference of 2 cents.
>>>>
>>>>
>>>> That's a schismatic (Pythagorean) diminished 4th, so that makes
>>>> sense.
>>>>
>>>>
>>>>
>>>> I don't know why, but I also experience 5/4 as an inappropriate
>>>> segah. It's just too high. There must be some logical psycho-
>>>> acoustical reason behind it. Through trial and error, I located
>>>> the desired interval to be somewhere about 382 cents.
>>>>
>>>>
>>>> Any thoughts?
>>>>
>>>>
>>>> My thoughts? You are a very sensitive musician with keen hearing,
>>>> brought up in a cultural/musical tradition you understand well,
>>>> and your instincts favor a Pythagorean diminshed 4th, rather than
>>>> a 5/4!
>>>>
>>>> I suspect that this is also is variable in certain lab conditions
>>>> like tempo, etc. IOW, there might be 'lab' conditions possible
>>>> where you'd also confuse the two 3rds.
>>>>
>>>> Best,
>>>> Aaron.
>>>>
>>>>
>>>> Cordially,
>>>> Oz.
>>>>
>>>> ✩ ✩ ✩
>>>> www.ozanyarman.com
>>>>
>>>> On Oct 2, 2009, at 5:42 PM, Aaron Johnson wrote:
>>>>
>>>>>
>>>>>
>>>>> Hey, what was the $50 challenge? I forget....
>>>>>
>>>>> I do think that so-called LucyTuning© sounds good for certain
>>>>> things, BTW. But I have to agree that _why_ it sounds good has
>>>>> pretty much nothing to do with PI. In other words, people don't
>>>>> start floating out of their seats, arms outstretched in mystical
>>>>> oneness with creation, surrounded by incandescent blue orbs of
>>>>> cosmic energy, when the major third is 2^(1/PI), as opposed to
>>>>> say 28 steps of 88-equal.....in 99.99% of cases, i.e. most music
>>>>> that doesn't have sustained chords 5 minutes long where you
>>>>> could measure things with an oscilloscope and software, *as it
>>>>> happens*, I highly doubt anyone could tell the difference.
>>>>>
>>>>> AKJ
>>>>>
>>>>> On Fri, Oct 2, 2009 at 7:50 AM, Charles Lucy
>>>>> <lucy@...> wrote:
>>>>>
>>>>>
>>>>> it is convenient that it works in a meantone pattern. That makes
>>>>> it easy to notate, conceptualise the harmony, and to use for
>>>>> educated musicians.
>>>>>
>>>>> As for the snake oil (marketing?):
>>>>>
>>>>> Snake oil seems to refer to an ancient Chinese remedy; after
>>>>> all the Chinese do represent the largest market for just about
>>>>> everything nowadays;-)
>>>>>
>>>>> Numerology?
>>>>>
>>>>> Can we count on Carl to be able to explain what that is?
>>>>>
>>>>> BTW Carl I'm still waiting for my $50 payout from a previous
>>>>> challenge which you reneged on a couple of years ago.
>>>>>
>>>>> Have fun with it!
>>>>>
>>>>> C.L.
>>>>>
>>>>> On 2 Oct 2009, at 03:16, Carl Lumma wrote:
>>>>>
>>>>>> --- In tuning@yahoogroups.com, jrinkel@... wrote:
>>>>>> >
>>>>>> > In addition to the many suggestions so far, you may want
>>>>>> > to check out
>>>>>> >
>>>>>> > www.lucytune.com
>>>>>> >
>>>>>> > It talks about a microtonal scale based on the value of pi.
>>>>>>
>>>>>> The uninitiated should be aware that lucytuning is simply
>>>>>> meantone (in the neighborhood of 1/3-comma meantone), mixed
>>>>>> with liberal quantities of numerology and snake oil.
>>>>>>
>>>>>> -Carl
>>>>>>
>>>>>>
>>>>>
>>>>> Charles Lucy
>>>>> lucy@...
>>>>>
>>>>> - Promoting global harmony through LucyTuning -
>>>>>
>>>>> for information on LucyTuning go to:
>>>>> http://www.lucytune.com
>>>>>
>>>>> For LucyTuned Lullabies go to:
>>>>> http://www.lullabies.co.uk
>>>>>
>>>>>
>>>>>
>>>>>
>>>>>
>>>>>
>>>>>
>>>>>
>>>>> --
>>>>>
>>>>>
>>>>> Aaron Krister Johnson
>>>>> http://www.akjmusic.com
>>>>> http://www.untwelve.org
>>>>>
>>>>>
>>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> --
>>>>
>>>> Aaron Krister Johnson
>>>> http://www.akjmusic.com
>>>> http://www.untwelve.org
>>>>
>>>>
>>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>> --
>>>
>>> Aaron Krister Johnson
>>> http://www.akjmusic.com
>>> http://www.untwelve.org
>>>
>>>
>>>
>>
>
>
>

Charles Lucy
lucy@...m

- Promoting global harmony through LucyTuning -

for information on LucyTuning go to:
http://www.lucytune.com

For LucyTuned Lullabies go to:
http://www.lullabies.co.uk

Dear Charles,

F# is the tonic of the piece, that is to say, it is perde segah in
maqam Hüzzam. There is something wrong with the Melodyne DNAmeasurements. Segah cannot be 26 cents above the equal tempered F# in
reference to A4=440 Hz.

Please scrutinize this video I made:

http://www.ozanyarman.com/misc/Ey_guli_bagi_eda_Murat_Sakaryali_Soloexplorer_analysis_of_perde_segah.mov

You can see that perde segah is executed correctly at around -30 to
-20 cents.

Oz.

✩ ✩ ✩
www.ozanyarman.com

On Oct 7, 2009, at 11:05 PM, Charles Lucy wrote:

>
>
> Here is what I found in the first few bars of the Turkish piece
> which Ozan put a link to:
>
> Notes found in first few seconds of Turkish piece:
>
> Max pitch is A5
>
> A5 +3 to _ 5 cents
> G5 + 12 cents
> F5 + 8 cents
> E5 +17 cents
> E5 + 3 cents
> D5 + 22 cents
> C#5 -9 cents
> C5 +19 cents
> B4 -27 cents
> A#4 +23 cents
> A4 +18 cents
> G4 + 7 cents
> F#4 +26 cents
> F4 + 1 cent
> E4 + 37 cents
>
>
> All notes from D4 and below are 12 edo with less than 10 cents.
>
> I don't know what the instrument is which is playing the "melody"
> but it seems to have all its notes in the range E4 to A5 and to be
> "improved" on a player-controlled variable-pitch instrument.
>
> All other instruments seem to be regular common or garden 12edo.
>
> I have put a link to the image that I am seeing with melodyne DNA
> beta here:
>
> http://www.lucytune.com/turkish/turkishmelodyne.jpg
>
>
> On 7 Oct 2009, at 05:31, Ozan Yarman wrote:
>
>>
>> In the context of investigating why perde segah is supposed to be
>> lower than 5/4, I have come up with an ascending diatonic Rast
>> scale that sounds truer than any I have previously attempted. This
>> might very well be the ultimate theoretical scale model for the
>> correct Rast. It is curious that the scale is 11-limit in every
>> way, including a stretched octave, but just as sweet and consonant
>> as a 5-limit Major scale.
>>
>> Here it is:
>>
>> 0: 1/1 0 cents C unison, perfect prime
>> 1: 432/385 199 cents D
>> 2: 96/77 382 cents Ed
>> 3: 385/288 503 cents F
>> 4: 3/2 702 cents G perfect fifth
>> 5: 648/385 901 cents A
>> 6: 144/77 1084 cents Bd
>> 7: 385/192 1205 cents C undecimal kleisma
>> +1 octave
>>
>> Yes, even the octave is 11-limit, stretched by an undecimal
>> kleisma. Due to the undecimal kleisma being referred to as
>> "Keenan's Kleisma", I decided to call this scale, "Keenanish Rast".
>>
>> 385/192 appears to be a resonant octave with a pleasing beat. Could
>> this be a clue as to why human ears seek stretched octaves?
>>
>> Still, it baffles me why the chord 382 320 | 154:192:231 and
>> the chord 320 382 | 64:77:96 of this scale sounds so savoury. >> What is so special about these beating chords, I wonder...
>>
>> One thing I noticed in the scale is that, the numerator of 432/385
>> ordinarily replacable by 9/8, of 648/385 ordinarily replacable by
>> 27/16, of 96/77 ordinarily replacable by 5/4, and 144/77 ordinarily
>> replacable by 15/8 are all 3-limit. In other words, the counterpart
>> ratios' numerators comprise odd number primes also.
>>
>> In retrospect, the counterpart ratios of 385/288 and 385/192, which
>> are 4/3 and 2/1, include even number primes (that is to say, number
>> 2). Take note, that 385 does not contain the prime 2 at all.
>>
>> Quod erat demonstrandum: Where there is ordinarily, in the
>> factorization of either the numerator or the denominator, prime
>> number 2, expect never to find number 2 in Keenanish Rast scale.
>> Where there is ordinarily, again in the factorization of either the>> numerator or the denominator, odd prime numbers such as 3, 5, 7...
>> expect to find only 2 and 3 in Keenanish Rast scale.
>>
>> Strange indeed! Some uniform warping of tone-space is occuring
>> here, so much so that a 5-limit harmonic Major jumps hyperspace
>> toward an equally valid 11-limit harmonic Major.
>>
>> Next on the agenda is to introduce Yarman36b and its natural
>> extension Yarman48, which, in my opinion, are so far the best
>> temperaments for representing the above-given Keenanish Rast, and
>> consequently, Turkish Maqam music practice as a whole.
>>
>> Cordially,
>> Oz.
>>
>>
>>
>>
>>> From: Ozan Yarman <ozanyarman@...>
>>> Date: October 7, 2009 12:23:22 AM GMT+03:00
>>> To: tuning@...m
>>> Subject: Re: [tuning] Why is 8192/6561 more consonant that 5/4
>>> between rast-segah
>>>
>>> Dear Aaron,
>>>
>>> Melodic step-sizes in a maqam scale are, most of the time, not
>>> abiding MOS. There may be three or more different step sizes.
>>> Consider this Hüzzam:
>>>
>>> Ed 120 F 204 G 150 Ab 228 Bd 120 C 294 D# 84 Ed
>>>
>>> See, there are 6 different step sizes, with Ed at 56/45.
>>>
>>> Replacing 15:14 between Ed-F and Bd-C with 16:15 will ruin the
>>> rast-segah relationship. Maybe the ear is inclining towards 56/45
>>> instead of 5/4? But I fear that is too low. The proper perde segah
>>> should be at 96/77:
>>>
>>> Ed 116 F 204 G 150 Ab 228 Bd 116 C 294 D# 88 Ed
>>>
>>> That's an even better Hüzzam, again with 6 different step sizes,
>>> none of which is a multiple of another.
>>>
>>> While there are fourth and fifth relationships between given >>> tones, this scale is not constructed via a chain of fifths.
>>>
>>> Now I wonder, why 96/77 instead of 5/4 (if we are to think in
>>> terms of JI)? Or, why the beating of the major third?
>>>
>>> Aaron, this is not restricted to my taste, it is a phenomenon of
>>> Maqam music according to my observations. It may not be a
>>> scientific one, for I have not tested the rast-segah relatinship
>>> in a laboratory condition with test subjects and equipment, but it
>>> is rather obvious to one who listens to, say, "Ey gul-i bag eda"
>>> by Dede Efendi in Hüzzam:
>>>
>>> http://www.vbox7.com/play:feba93b8?r=google
>>>
>>> Can you also observe the low perde segah?
>>>
>>> Oz.
>>>
>>> ✩ ✩ ✩
>>> www.ozanyarman.com
>>>
>>> On Oct 5, 2009, at 11:31 PM, Aaron Johnson wrote:
>>>
>>>>
>>>>
>>>> Oz,
>>>>
>>>> My ad-hoc theory is that you are probably noticing the subtle
>>>> effects of having all the melodic step sizes be similar, and/or
>>>> multiples of each other, due to the chain-of-fifths? Or, you
>>>> prefer to notice a slight beating in the third.
>>>>
>>>> I'm really curious how scientifically you have tested your
>>>> perception of this....do you have a 100% success rate? Perhaps
>>>> you can set up a kind of blind listening test?
>>>>
>>>> All best,
>>>> Aaron.
>>>>
>>>> On Mon, Oct 5, 2009 at 2:44 PM, Ozan Yarman <ozanyarman@...
>>>> > wrote:
>>>>
>>>>
>>>> Aaron, thank you for sharing your thoughts. But why does the ear
>>>> of a maqam musician incline toward 8192/6561 instead of 5/4? I
>>>> have many times tried to accompany an actual performance with
>>>> either 53-tET and 41-tET, and observed the latter to agree with
>>>> practice more.
>>>>
>>>> Oz.
>>>>
>>>> ✩ ✩ ✩
>>>> www.ozanyarman.com
>>>>
>>>> On Oct 2, 2009, at 9:04 PM, Aaron Johnson wrote:
>>>>
>>>>>
>>>>>
>>>>>
>>>>>
>>>>> On Fri, Oct 2, 2009 at 12:06 PM, Ozan Yarman <ozanyarman@...
>>>>> > wrote:
>>>>>
>>>>>
>>>>> Aaron,
>>>>>
>>>>> I also suspect PI has little to do with the actual phenomenon of
>>>>> why a 381-384 cent major third sounds just as sweet as a 386
>>>>> cent major third. Note, that Rauf Yekta pointed out the rast-
>>>>> segah interval to be 384 cents, not 386 cents, claiming that the
>>>>> ear is sensitive to the minute difference of 2 cents.
>>>>>
>>>>>
>>>>> That's a schismatic (Pythagorean) diminished 4th, so that makes
>>>>> sense.
>>>>>
>>>>>
>>>>>
>>>>> I don't know why, but I also experience 5/4 as an inappropriate
>>>>> segah. It's just too high. There must be some logical psycho-
>>>>> acoustical reason behind it. Through trial and error, I located
>>>>> the desired interval to be somewhere about 382 cents.
>>>>>
>>>>>
>>>>> Any thoughts?
>>>>>
>>>>>
>>>>> My thoughts? You are a very sensitive musician with keen
>>>>> hearing, brought up in a cultural/musical tradition you
>>>>> understand well, and your instincts favor a Pythagorean >>>>> diminshed 4th, rather than a 5/4!
>>>>>
>>>>> I suspect that this is also is variable in certain lab
>>>>> conditions like tempo, etc. IOW, there might be 'lab' conditions
>>>>> possible where you'd also confuse the two 3rds.
>>>>>
>>>>> Best,
>>>>> Aaron.
>>>>>
>>>>>
>>>>> Cordially,
>>>>> Oz.
>>>>>
>>>>> ✩ ✩ ✩
>>>>> www.ozanyarman.com
>>>>>
>>>>> On Oct 2, 2009, at 5:42 PM, Aaron Johnson wrote:
>>>>>
>>>>>>
>>>>>>
>>>>>> Hey, what was the $50 challenge? I forget....
>>>>>>
>>>>>> I do think that so-called LucyTuning© sounds good for certain
>>>>>> things, BTW. But I have to agree that _why_ it sounds good has
>>>>>> pretty much nothing to do with PI. In other words, people don't
>>>>>> start floating out of their seats, arms outstretched in
>>>>>> mystical oneness with creation, surrounded by incandescent blue
>>>>>> orbs of cosmic energy, when the major third is 2^(1/PI), as
>>>>>> opposed to say 28 steps of 88-equal.....in 99.99% of cases,
>>>>>> i.e. most music that doesn't have sustained chords 5 minutes
>>>>>> long where you could measure things with an oscilloscope and
>>>>>> software, *as it happens*, I highly doubt anyone could tell the
>>>>>> difference.
>>>>>>
>>>>>> AKJ
>>>>>>
>>>>>> On Fri, Oct 2, 2009 at 7:50 AM, Charles Lucy
>>>>>> <lucy@...> wrote:
>>>>>>
>>>>>>
>>>>>> it is convenient that it works in a meantone pattern. That
>>>>>> makes it easy to notate, conceptualise the harmony, and to use
>>>>>> for educated musicians.
>>>>>>
>>>>>> As for the snake oil (marketing?):
>>>>>>
>>>>>> Snake oil seems to refer to an ancient Chinese remedy; after
>>>>>> all the Chinese do represent the largest market for just about
>>>>>> everything nowadays;-)
>>>>>>
>>>>>> Numerology?
>>>>>>
>>>>>> Can we count on Carl to be able to explain what that is?
>>>>>>
>>>>>> BTW Carl I'm still waiting for my $50 payout from a previous
>>>>>> challenge which you reneged on a couple of years ago.
>>>>>>
>>>>>> Have fun with it!
>>>>>>
>>>>>> C.L.
>>>>>>
>>>>>> On 2 Oct 2009, at 03:16, Carl Lumma wrote:
>>>>>>
>>>>>>> --- In tuning@yahoogroups.com, jrinkel@... wrote:
>>>>>>> >
>>>>>>> > In addition to the many suggestions so far, you may want
>>>>>>> > to check out
>>>>>>> >
>>>>>>> > www.lucytune.com
>>>>>>> >
>>>>>>> > It talks about a microtonal scale based on the value of pi.
>>>>>>>
>>>>>>> The uninitiated should be aware that lucytuning is simply
>>>>>>> meantone (in the neighborhood of 1/3-comma meantone), mixed
>>>>>>> with liberal quantities of numerology and snake oil.
>>>>>>>
>>>>>>> -Carl
>>>>>>>
>>>>>>>
>>>>>>
>>>>>> Charles Lucy
>>>>>> lucy@...
>>>>>>
>>>>>> - Promoting global harmony through LucyTuning -
>>>>>>
>>>>>> for information on LucyTuning go to:
>>>>>> http://www.lucytune.com
>>>>>>
>>>>>> For LucyTuned Lullabies go to:
>>>>>> http://www.lullabies.co.uk
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>> --
>>>>>>
>>>>>>
>>>>>> Aaron Krister Johnson
>>>>>> http://www.akjmusic.com
>>>>>> http://www.untwelve.org
>>>>>>
>>>>>>
>>>>>>
>>>>>
>>>>>
>>>>>
>>>>>
>>>>>
>>>>>
>>>>> --
>>>>>
>>>>> Aaron Krister Johnson
>>>>> http://www.akjmusic.com
>>>>> http://www.untwelve.org
>>>>>
>>>>>
>>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> --
>>>>
>>>> Aaron Krister Johnson
>>>> http://www.akjmusic.com
>>>> http://www.untwelve.org
>>>>
>>>>
>>>>
>>>
>>
>>
>
> Charles Lucy
> lucy@lucytune.com
>
> - Promoting global harmony through LucyTuning -
>
> for information on LucyTuning go to:
> http://www.lucytune.com
>
> For LucyTuned Lullabies go to:
> http://www.lullabies.co.uk
>
>
>
>
>
>

Dear Cameron,

I scrutinized the tweak you suggested. The beating of 5/4 to 2/1 at
the zeroth step yields -2 (between 5:4 and 2:1), -4.5 (between 8:5 and
5:4) and 9 (between 8:5 and 2:1) relative beat rates.

The beating of 4/3 to 2/1 again at the zeroth step yields 1.5, 0.333
and 0.5 relative beat rates between 4:3 and 2:1, 3:2 and 4:3, and 3:2
and 2:1.

Now for my tweak. Keeping the philosophy behind the Keenanish Rast,
but replacing Keenan's kleisma (385/384) with an even better interval,
that is, Werckmeister's schisma (441/440), I arrived again at an 11-
limit Rast with near perfect beat rates:

0: 1/1 C unison, perfect prime
1: 55/49 D quasi-equal major second
2: 550/441 Ed
3: 147/110 F
4: 3/2 G perfect fifth
5: 165/98 A
6: 275/147 Bd
7: 441/220 C Werckmeister's undecimal
septenarian schisma +1 octave

In cents:

200.0
382.0
502.0
702.0
902.0
1984.0
1204.0

The absolute difference between this "Werckmeisterian Rast" and
"Keenanish Rast" is no more than a cent at any degree. The octave is
slightly reduced and is now in proper alignment.

Observe the beat rates between 5/4 and 2/1 at the first and fifth
degrees of the scale:

5/4 : 2/1 = -2.5
8:5 : 5/4 = -4
8:5 : 2/1 = 9.9

Observe the beat rates between 4/3 and 2/1 at the first, second,
fifth, sixth, seventh and eighth degrees:

4/3 : 2/1 = 2
3/2 : 4/3 = 0
3/2 : 2/1 = 0

At the third degree:

4/3 : 2/1 = 11
3/2 : 4/3 = -0.818181
3/2 : 2/1 = -9

More beat rates can be examined in SCALA.

Werckmeisterian Rast scale can be factorized as follows:

1 / 1
5 x 11 / 7^2
2 x 5^2 x 11 / 3^2 x 7^2
3 x 7^2 / 2 x 5 x 11
3 / 2
3 x 5 x 11 / 2 x 7^2
5^2 x 11 / 3 x 7^2
3^2 x 7^2 / 2^2 x 5 x 11

Compare with 5-limit Harmonic major's ratios and their factorizations:

1/1............1/1
9/8.........3^2 / 2^3
5/4...........5 / 2^2
4/3.........2^2 / 3
3/2...........3 / 2
27/16.......3^3 / 2^4
15/8......3 x 5 / 2^3
2/1...........2 / 1

Note that, where there was prime 3 in the numerators, there are nowprimes 3, 5, 11 in Werckmeisterian Rast. Where there was prime 2 in
the numerators and denominators, there are now 3 and 7. Where there
was prime 5 in the numerators or prime 3 in the denominators, there
are now 2, 5, 11.

Again the tone-space has been warped in a uniform fashion:

2 (both) => 3,7
3 (num.) => 3, 5, 11
3 (den.) => 2, 5, 11
5 (num.) => 2, 5, 11

I'm guessing that 7 will be warped to 2, 3 and 11. Conversely, 11
should be warped to 2, 3, 5.

Next: Yarman48, a bike-chain of 4 stretched octave 12-tone equal
temperaments.

Oz.

✩ ✩ ✩
www.ozanyarman.com

On Oct 7, 2009, at 11:13 AM, cameron wrote:

> Ozan, as I expected from listening to your tuning, it is almost
> perfectly an equal/proportional beating affair.
>
> Try this very slight tweak:
>
> 0: 1/1 0.000 unison, perfect prime
> 1: 198.970 cents 198.970
> 2: 381.869 cents 381.869
> 3: 502.202 cents 502.202
> 4: 3/2 701.955 perfect fifth
> 5: 900.926 cents 900.926
> 6: 1085.307 cents 1085.307
> 7: 1205.540 cents 1205.540
>
> Check the various beating especially around the 4th and 8th partial.
>
> As I have been insisting for the last few years, neither primes nor
> "approximations" are the answer. I heard a "red thread" in your
> tuning, a coherence and family resemblance amongst the intervals, in
> your tuning, and there it is. The harmonic spectrum here is a
> physical reference- the timbre of the tuning has consistently
> beating stripes of color, so to speak.
>
>
>
> --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
>>
>> In the context of investigating why perde segah is supposed to be
>> lower than 5/4, I have come up with an ascending diatonic Rast scale
>> that sounds truer than any I have previously attempted. This might
>> very well be the ultimate theoretical scale model for the correct
>> Rast. It is curious that the scale is 11-limit in every way,
>> including
>> a stretched octave, but just as sweet and consonant as a 5-limit
>> Major
>> scale.
>>
>> Here it is:
>>
>> 0: 1/1 0 cents C unison, perfect prime
>> 1: 432/385 199 cents D
>> 2: 96/77 382 cents Ed
>> 3: 385/288 503 cents F
>> 4: 3/2 702 cents G perfect fifth
>> 5: 648/385 901 cents A
>> 6: 144/77 1084 cents Bd
>> 7: 385/192 1205 cents C undecimal kleisma +1
>> octave
>>
>> Yes, even the octave is 11-limit, stretched by an undecimal kleisma.
>> Due to the undecimal kleisma being referred to as "Keenan's Kleisma",
>> I decided to call this scale, "Keenanish Rast".
>>
>> 385/192 appears to be a resonant octave with a pleasing beat. Could
>> this be a clue as to why human ears seek stretched octaves?
>>
>> Still, it baffles me why the chord 382 320 | 154:192:231 and the
>> chord 320 382 | 64:77:96 of this scale sounds so savoury. What is
>> so special about these beating chords, I wonder...
>>
>> One thing I noticed in the scale is that, the numerator of 432/385
>> ordinarily replacable by 9/8, of 648/385 ordinarily replacable by
>> 27/16, of 96/77 ordinarily replacable by 5/4, and 144/77 ordinarily
>> replacable by 15/8 are all 3-limit. In other words, the counterpart
>> ratios' numerators comprise odd number primes also.
>>
>> In retrospect, the counterpart ratios of 385/288 and 385/192, which
>> are 4/3 and 2/1, include even number primes (that is to say, number
>> 2). Take note, that 385 does not contain the prime 2 at all.
>>
>> Quod erat demonstrandum: Where there is ordinarily, in the
>> factorization of either the numerator or the denominator, prime
>> number
>> 2, expect never to find number 2 in Keenanish Rast scale. Where there
>> is ordinarily, again in the factorization of either the numerator or
>> the denominator, odd prime numbers such as 3, 5, 7... expect to find
>> only 2 and 3 in Keenanish Rast scale.
>>
>> Strange indeed! Some uniform warping of tone-space is occuring here,
>> so much so that a 5-limit harmonic Major jumps hyperspace toward an
>> equally valid 11-limit harmonic Major.
>>
>> Next on the agenda is to introduce Yarman36b and its natural
>> extension
>> Yarman48, which, in my opinion, are so far the best temperaments for
>> representing the above-given Keenanish Rast, and consequently,
>> Turkish
>> Maqam music practice as a whole.
>>
>> Cordially,
>> Oz.
>>
>>
>>
>>
>>> From: Ozan Yarman <ozanyarman@...>
>>> Date: October 7, 2009 12:23:22 AM GMT+03:00
>>> To: tuning@yahoogroups.com
>>> Subject: Re: [tuning] Why is 8192/6561 more consonant that 5/4
>>> between rast-segah
>>>
>>> Dear Aaron,
>>>
>>> Melodic step-sizes in a maqam scale are, most of the time, not
>>> abiding MOS. There may be three or more different step sizes.
>>> Consider this Hüzzam:
>>>
>>> Ed 120 F 204 G 150 Ab 228 Bd 120 C 294 D# 84 Ed
>>>
>>> See, there are 6 different step sizes, with Ed at 56/45.
>>>
>>> Replacing 15:14 between Ed-F and Bd-C with 16:15 will ruin the rast-
>>> segah relationship. Maybe the ear is inclining towards 56/45 instead
>>> of 5/4? But I fear that is too low. The proper perde segah should be
>>> at 96/77:
>>>
>>> Ed 116 F 204 G 150 Ab 228 Bd 116 C 294 D# 88 Ed
>>>
>>> That's an even better Hüzzam, again with 6 different step sizes,
>>> none of which is a multiple of another.
>>>
>>> While there are fourth and fifth relationships between given tones,
>>> this scale is not constructed via a chain of fifths.
>>>
>>> Now I wonder, why 96/77 instead of 5/4 (if we are to think in terms
>>> of JI)? Or, why the beating of the major third?
>>>
>>> Aaron, this is not restricted to my taste, it is a phenomenon of
>>> Maqam music according to my observations. It may not be a scientific
>>> one, for I have not tested the rast-segah relatinship in a
>>> laboratory condition with test subjects and equipment, but it is
>>> rather obvious to one who listens to, say, "Ey gul-i bag eda" by
>>> Dede Efendi in Hüzzam:
>>>
>>> http://www.vbox7.com/play:feba93b8?r=google
>>>
>>> Can you also observe the low perde segah?
>>>
>>> Oz.
>>>
>>> ✩ ✩ ✩
>>> www.ozanyarman.com
>>>
>>> On Oct 5, 2009, at 11:31 PM, Aaron Johnson wrote:
>>>
>>>>
>>>>
>>>> Oz,
>>>>
>>>> My ad-hoc theory is that you are probably noticing the subtle
>>>> effects of having all the melodic step sizes be similar, and/or
>>>> multiples of each other, due to the chain-of-fifths? Or, you prefer
>>>> to notice a slight beating in the third.
>>>>
>>>> I'm really curious how scientifically you have tested your
>>>> perception of this....do you have a 100% success rate? Perhaps you
>>>> can set up a kind of blind listening test?
>>>>
>>>> All best,
>>>> Aaron.
>>>>
>>>> On Mon, Oct 5, 2009 at 2:44 PM, Ozan Yarman <ozanyarman@...
>>>>> wrote:
>>>>
>>>>
>>>> Aaron, thank you for sharing your thoughts. But why does the ear of
>>>> a maqam musician incline toward 8192/6561 instead of 5/4? I have
>>>> many times tried to accompany an actual performance with either 53-
>>>> tET and 41-tET, and observed the latter to agree with practice
>>>> more.
>>>>
>>>> Oz.
>>>>
>>>> ✩ ✩ ✩
>>>> www.ozanyarman.com
>>>>
>>>> On Oct 2, 2009, at 9:04 PM, Aaron Johnson wrote:
>>>>
>>>>>
>>>>>
>>>>>
>>>>>
>>>>> On Fri, Oct 2, 2009 at 12:06 PM, Ozan Yarman <ozanyarman@...
>>>>>> wrote:
>>>>>
>>>>>
>>>>> Aaron,
>>>>>
>>>>> I also suspect PI has little to do with the actual phenomenon of
>>>>> why a 381-384 cent major third sounds just as sweet as a 386 cent
>>>>> major third. Note, that Rauf Yekta pointed out the rast-segah
>>>>> interval to be 384 cents, not 386 cents, claiming that the ear is
>>>>> sensitive to the minute difference of 2 cents.
>>>>>
>>>>>
>>>>> That's a schismatic (Pythagorean) diminished 4th, so that makes
>>>>> sense.
>>>>>
>>>>>
>>>>>
>>>>> I don't know why, but I also experience 5/4 as an inappropriate
>>>>> segah. It's just too high. There must be some logical psycho-
>>>>> acoustical reason behind it. Through trial and error, I located
>>>>> the desired interval to be somewhere about 382 cents.
>>>>>
>>>>>
>>>>> Any thoughts?
>>>>>
>>>>>
>>>>> My thoughts? You are a very sensitive musician with keen hearing,
>>>>> brought up in a cultural/musical tradition you understand well,
>>>>> and your instincts favor a Pythagorean diminshed 4th, rather than
>>>>> a 5/4!
>>>>>
>>>>> I suspect that this is also is variable in certain lab conditions
>>>>> like tempo, etc. IOW, there might be 'lab' conditions possible
>>>>> where you'd also confuse the two 3rds.
>>>>>
>>>>> Best,
>>>>> Aaron.
>>>>>
>>>>>
>>>>> Cordially,
>>>>> Oz.
>>>>>
>>>>> ✩ ✩ ✩
>>>>> www.ozanyarman.com
>>>>>
>>>>> On Oct 2, 2009, at 5:42 PM, Aaron Johnson wrote:
>>>>>
>>>>>>
>>>>>>
>>>>>> Hey, what was the $50 challenge? I forget....
>>>>>>
>>>>>> I do think that so-called LucyTuning© sounds good for certain
>>>>>> things, BTW. But I have to agree that _why_ it sounds good has
>>>>>> pretty much nothing to do with PI. In other words, people don't
>>>>>> start floating out of their seats, arms outstretched in mystical
>>>>>> oneness with creation, surrounded by incandescent blue orbs of
>>>>>> cosmic energy, when the major third is 2^(1/PI), as opposed to
>>>>>> say 28 steps of 88-equal.....in 99.99% of cases, i.e. most music
>>>>>> that doesn't have sustained chords 5 minutes long where you could
>>>>>> measure things with an oscilloscope and software, *as it
>>>>>> happens*, I highly doubt anyone could tell the difference.
>>>>>>
>>>>>> AKJ
>>>>>>
>>>>>> On Fri, Oct 2, 2009 at 7:50 AM, Charles Lucy <lucy@...>
>>>>>> wrote:
>>>>>>
>>>>>>
>>>>>> it is convenient that it works in a meantone pattern. That makes
>>>>>> it easy to notate, conceptualise the harmony, and to use for
>>>>>> educated musicians.
>>>>>>
>>>>>> As for the snake oil (marketing?):
>>>>>>
>>>>>> Snake oil seems to refer to an ancient Chinese remedy; after all
>>>>>> the Chinese do represent the largest market for just about
>>>>>> everything nowadays;-)
>>>>>>
>>>>>> Numerology?
>>>>>>
>>>>>> Can we count on Carl to be able to explain what that is?
>>>>>>
>>>>>> BTW Carl I'm still waiting for my $50 payout from a previous
>>>>>> challenge which you reneged on a couple of years ago.
>>>>>>
>>>>>> Have fun with it!
>>>>>>
>>>>>> C.L.
>>>>>>
>>>>>> On 2 Oct 2009, at 03:16, Carl Lumma wrote:
>>>>>>
>>>>>>> --- In tuning@yahoogroups.com, jrinkel@ wrote:
>>>>>>>>
>>>>>>>> In addition to the many suggestions so far, you may want
>>>>>>>> to check out
>>>>>>>>
>>>>>>>> www.lucytune.com
>>>>>>>>
>>>>>>>> It talks about a microtonal scale based on the value of pi.
>>>>>>>
>>>>>>> The uninitiated should be aware that lucytuning is simply
>>>>>>> meantone (in the neighborhood of 1/3-comma meantone), mixed
>>>>>>> with liberal quantities of numerology and snake oil.
>>>>>>>
>>>>>>> -Carl
>>>>>>>
>>>>>>>
>>>>>>
>>>>>> Charles Lucy
>>>>>> lucy@...
>>>>>>
>>>>>> - Promoting global harmony through LucyTuning -
>>>>>>
>>>>>> for information on LucyTuning go to:
>>>>>> http://www.lucytune.com
>>>>>>
>>>>>> For LucyTuned Lullabies go to:
>>>>>> http://www.lullabies.co.uk
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>> --
>>>>>>
>>>>>>
>>>>>> Aaron Krister Johnson
>>>>>> http://www.akjmusic.com
>>>>>> http://www.untwelve.org
>>>>>>
>>>>>>
>>>>>>
>>>>>
>>>>>
>>>>>
>>>>>
>>>>>
>>>>>
>>>>> --
>>>>>
>>>>> Aaron Krister Johnson
>>>>> http://www.akjmusic.com
>>>>> http://www.untwelve.org
>>>>>
>>>>>
>>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> --
>>>>
>>>> Aaron Krister Johnson
>>>> http://www.akjmusic.com
>>>> http://www.untwelve.org
>>>>
>>>>
>>>>
>>>>
>>>
>>
>
>
>
>
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