Yahoo Tuning Groups Ultimate Backup tuning Re: Questions (was: Paper?)

Dear Jacques,

thanks for your detailed reply.

On 30.04.2010, at 19:21, Jacques Dudon wrote:
> First there is no information avalaible about recurrent series, I
> have to write it ! [...] So to continue to ask questions through the Tuning List might be a
> good way to push me to communicate more ! :)

I see. The problem is that I don't quite know what to ask or where to start asking :)

> Septimal harmony : [...] In order to understand what you want, tell me first if what you want
> is : simply 7 with 3, or 7 with 3 and 5 ?

Both options would be nice to have. I assume I could have the three prime-limits 3, 5, and 7, with certain variants of meantone, discussed below.

> For 7 with 3, Slendro matrix has lots of transpositions, not only
> chords but full slendro (pentatonic) scales, and all in JI. I can try
> to scan you at least the matrix page of it, with the scales it contains.

Yes, I saw slendro_matrix.scl and therefore I asked you about your 1/1 paper on Seven-Limit Slendro Mutations. It would be very kind if you could scan at least this page you mentioned.

If you have some further scale in your collection in this department that I missed please point me to them. I have great problems to comprehend large ratios, even more so if they actually temper more simple ratios :) So, I may have missed related scales.

> For 7 with 3 AND 5, some of my meantones would do better than others,
> but as you said, would need extensions to more notes.
> You are right about 1/4 comma meantone, it would be quite OK for
> approximations of factor 7. Since my "Mezzo" version uses
> transpositions by 5, it should be easy to extend it the same way to
> 19 or even 24 tones to fit in the Ethno mappings.

Great! My problem is that without a detailed explanation of your tuning approach I am unable to extend your scales myself :(

> I would think a more septimal because perfectly eq-b -optimised
> solution would be the "golden_h7eb.scl" (EU7), that integrates the
> 5th with the 7th harmonic :
> The Tara algorithm x^4 = x + 7/2 itself invites you to multiply the
> series directly by 7, in order to play on both the -c and the eq-b
> (and I just found a third synchronous-beating property for it). The
> memory I have from it was something incredibly vibrant in the
> septimal chords concerned by the eq-b !
> It has more character than mezzo, and would be my choice for septimal
> harmonies. Plus it's a super fractal-approximation of the Golden
> meantone meta-temperament.

I would love to follow your recommendation to use an extended version of golden_h7eb.scl. But again, unfortunately I am so far unable to extend it myself.

You mentioned in your discussion that your tunings are linear temperaments. So, if you tell me the generators for these scales than I could create a 24 tone version myself. I would show that to you first for approval :)

> By definition all recurrent series are based on transposition, they are only that ! but
> they will transpose very different things, and not necessarily
> "harmonic" chords... except for some like Mohajira, Ishku, Buzurg,
> Chandrak, Amlak etc.

If your recurrent series scales are linear temperaments then I fully understand this comment and it makes sense :)

> coherent_shrutis.scl is based on a cycle of fifths, only with very
> light retunings - so I do not think it would make transpositions
> difficult, as long as you accept 361/360, 513/512 and 1216/1215 as
> more or less "schismic" variations.The cycle of fifths goes from A
> (5/3) to D(9/8), so if you transpose the whole scale by 81/80, or
> 96/95, you will have a full quasi 5-limit 24 shrutis system in which
> 19 is quite transparent but very useful.

I think I understand. So, would it be OK for these demos to extend coherent_shrutis.scl by 12 tones where each tone transposes an existing tone in coherent_shrutis.scl by a syntonic comma?

Thank you!

Best wishes,
Torsten

--
Torsten Anders
Interdisciplinary Centre for Computer Music Research
University of Plymouth
Office: +44-1752-586219
Private: +44-1752-558917
http://strasheela.sourceforge.net
http://www.torsten-anders.de

🔗genewardsmith <genewardsmith@...>

🔗jacques.dudon <fotosonix@...>

Hi Torsten,

--- In tuning@yahoogroups.com, Torsten Anders <torsten.anders@...> wrote:

> Dear Jacques,
>
> thanks for your detailed reply.
>
> On 30.04.2010, at 19:21, Jacques Dudon wrote:
> > First there is no information avalaible about recurrent series, I
> > have to write it ! [...]

I meant about "my" recurrent series of course...

> > Septimal harmony : [...] In order to understand what you want, tell me first if what you want
> > is : simply 7 with 3, or 7 with 3 and 5 ?
>
> Both options would be nice to have. I assume I could have the three prime-limits 3, 5, and 7, with certain variants of meantone, discussed below.
>
> > For 7 with 3, Slendro matrix has lots of transpositions, not only
> > chords but full slendro (pentatonic) scales, and all in JI. I can try
> > to scan you at least the matrix page of it, with the scales it contains.
>
> Yes, I saw slendro_matrix.scl and therefore I asked you about your 1/1 paper on Seven-Limit Slendro Mutations. It would be very kind if you could scan at least this page you mentioned.

I look for it tomorrow

> If you have some further scale in your collection in this department that I missed please point me to them. I have great problems to comprehend large ratios, even more so if they actually temper more simple ratios :) So, I may have missed related scales.

If you try them and hear how they sound to you, comprehension will come by itself -

> > For 7 with 3 AND 5, some of my meantones would do better than others,
> > but as you said, would rather need extensions to more notes.
> > You are right about 1/4 comma meantone, it would be quite OK for
> > approximations of factor 7. Since my "Mezzo" version uses
> > transpositions by 5, it should be easy to extend it the same way to
> > 19 or even 24 tones to fit in the Ethno mappings.
>
> Great! My problem is that without a detailed explanation of your tuning approach I am unable to extend your scales myself :(

I was just talking of multiplications by 5 here, I am sure you can do that ! but the scala files, in whatever mapping you want, this you should do by yourself. It is really easy actually, I never knew Scala myself before I wrote these tunings...

> > I would think a more septimal, because perfectly eq-b-optimised
> > solution would be the "golden_h7eb.scl" (EU7), that integrates the
> > 5th and the 7th harmonic :
> > The Tara algorithm x^4 = x + 7/2 itself invites you to multiply the
> > series directly by 7, in order to play on both the -c and the eq-b
> > (and I just found a third synchronous-beating property for it).
> > Memory I have from it was something incredibly vibrant in the
> > septimal chords concerned by the eq-b !

> I would love to follow your recommendation to use an extended version of golden_h7eb.scl. But again, unfortunately I am so far unable to extend it myself.

It is again a simple multiplication of the series by 7/2 (or of the ratios by 7/4 or 7/8 depending on the notes) I am suggesting here, in order to play on two parralel series of 12 notes, rather than extending the original series. This suggestion is justified by the algorithm itself : x^4 - 7/2 = x means that 8B - 7G = D/2,in other words the first-order difference tone of the tritone B - 7G/8 in such a meantone is tuned in unisson with D, 4 octaves lower. That's why I suggest to integrate here 7G/8 = 999/668 * 7/8 = 6993/5344 , and so on with all the 12 notes of the scale. Write a new scala file with "24" instead of "12" and tell me how it feels...

> You mentioned in your discussion that your tunings are linear temperaments.
I have not said that. I am interested to find if some of those fractal tunings can be considered as variations of known linear temperaments. Some of them are very close and can be, apparently.
Golden_h7eb is obviously a fractal clone of the Golden meantone.

> So, if you tell me the generators for these scales than I could create a 24 tone version myself. I would show that to you first for approval :)
Well, I already published here a list of those fractals, and others are in the Scala files so it's something you can try, except that they would not be *exactly* admitted as "extensions" of the original tunings anymore ! The extensions allowed were concerning the series and the creation of new series.

> > By definition all recurrent series are based on transposition, they are only that ! but
> > they will transpose very different things, and not necessarily
> > "harmonic" chords... except for some like Mohajira, Ishku, Buzurg,
> > Chandrak, Amlak etc.
>
> If your recurrent series scales are linear temperaments then I fully understand this comment and it makes sense :)

They are not the same thing, but I think you understood well.

> > coherent_shrutis.scl is based on a cycle of fifths, only with very
> > light retunings - so I do not think it would make transpositions
> > difficult, as long as you accept 361/360, 513/512 and 1216/1215 as
> > more or less "schismic" variations.The cycle of fifths goes from A
> > (5/3) to D(9/8), so if you transpose the whole scale by 81/80, or
> > 96/95, you will have a full quasi 5-limit 24 shrutis system in
> > which 19 is quite transparent but very useful.
>
> I think I understand. So, would it be OK for these demos to extend > coherent_shrutis.scl by 12 tones where each tone transposes an existing tone in coherent_shrutis.scl by a syntonic comma ?

Perfectly OK and Hindustani correct ! Syntonic comma would be the best choice for the 22 shrutis, 96/95 the best for a more "kind of schismic temperament" I would think.

> Thank you!
>
> Best wishes,
> Torsten

Your're welcome !
- - - - - -
Jacques

🔗Torsten Anders <torsten.anders@...>

Dear Jacques,

On 01.05.2010, at 23:33, jacques.dudon wrote:
--- In tuning@yahoogroups.com, Torsten Anders <torsten.anders@...> wrote:
> > Yes, I saw slendro_matrix.scl and therefore I asked you about your 1/1 paper on Seven-Limit Slendro Mutations. It would be very kind if you could scan at least this page you mentioned.

> I look for it tomorrow

Thanks you!

> > If you have some further scale in your collection in this
> department that I missed please point me to them. I have great
> problems to comprehend large ratios, even more so if they actually
> temper more simple ratios :) So, I may have missed related scales.

> If you try them and hear how they sound to you, comprehension will come by itself -

:) Actually, if I want to use Strasheela then I need to fully understand the structure of a scale, i.e. the intervallic relations it allows. Unfortuantely, I do not have the time to go through > 150 scales and do that by ear (I afraid that even for a single irregular temperament it could take me quite some time to understand all its possible intervals by ear only).

> > > For 7 with 3 AND 5, some of my meantones would do better than others,
> > > but as you said, would rather need extensions to more notes.
> > > You are right about 1/4 comma meantone, it would be quite OK for
> > > approximations of factor 7. Since my "Mezzo" version uses
> > > transpositions by 5, it should be easy to extend it the same way to
> > > 19 or even 24 tones to fit in the Ethno mappings.
> >
> > Great! My problem is that without a detailed explanation of your tuning approach I am unable to extend your scales myself :(

> I was just talking of multiplications by 5 here, I am sure you can
> do that ! but the scala files, in whatever mapping you want, this
> you should do by yourself. It is really easy actually, I never knew
> Scala myself before I wrote these tunings...

Oops, now I get confused. We were talking to have the limits 3, 5, and 7 in meantone temperaments by having a long meantone spiral of fifths that then allows for septimal intervals like augmented second and augmented sixth.

It would certainly be easy for me to multiply all ratios in your Scala file by 5 (including wrapping it into a single octave :). However, I don't understand how this would extend the spiral of meantone fifths to add further septimal intervals.

Is 5 one of your "generators" to create this particular meantone variant? What are the other "generators" (it can't be the octave only)?

> > > I would think a more septimal, because perfectly eq-b-optimised
> > > solution would be the "golden_h7eb.scl" (EU7), that integrates the
> > > 5th and the 7th harmonic :
> > > The Tara algorithm x^4 = x + 7/2 itself invites you to multiply the
> > > series directly by 7, in order to play on both the -c and the eq-b
> > > (and I just found a third synchronous-beating property for it).
> > > Memory I have from it was something incredibly vibrant in the
> > > septimal chords concerned by the eq-b !

> > I would love to follow your recommendation to use an extended
> version of golden_h7eb.scl. But again, unfortunately I am so far
> unable to extend it myself.

> It is again a simple multiplication of the series by 7/2 (or of the
> ratios by 7/4 or 7/8 depending on the notes) I am suggesting here,
> in order to play on two parralel series of 12 notes, rather than
> extending the original series.

Of course, multiplying by octave (variants of) 7 would add septimal intervals. Still, I don't understand why the result should be a meantone temperament.

Nevertheless, I will look into this as an option, it sounds interesting :)

> This suggestion is justified by the algorithm itself : x^4 - 7/2 = x
> means that 8B - 7G = D/2,in other words the first-order difference
> tone of the tritone B - 7G/8 in such a meantone is tuned in unisson
> with D, 4 octaves lower. That's why I suggest to integrate here 7G/8
> = 999/668 * 7/8 = 6993/5344 , and so on with all the 12 notes of the
> scale. Write a new scala file with "24" instead of "12" and tell me
> how it feels...

Thank you for your explanation, but I don't follow, sorry.

First of all, I would like to understand the purpose of your formulas like x^4 - 7/2 = x. Are you using these to recursively approximating your x? I just tried it, and with most values it very quickly results in infinity, so I obviously don't understand what you are doing with this formula :)

Secondly, I don't understand the relation between your formula and your conclusion that 8B - 7G = D/2, but perhaps this is secondary.

More important, I not sure I understand your notation 8B or 7G. Are you talking about the difference between 8th partial of B and the 7th partial of G and the octave below D, where B, G and D are in the same octave?

> > You mentioned in your discussion that your tunings are linear temperaments.
> I have not said that. I am interested to find if some of those
> fractal tunings can be considered as variations of known linear
> temperaments. Some of them are very close and can be, apparently.
> Golden_h7eb is obviously a fractal clone of the Golden meantone.

OK. I don't understand the actual difference though.

> > So, if you tell me the generators for these scales than I could create a 24 tone version myself. I would show that to you first for approval :)
> Well, I already published here a list of those fractals, and others
> are in the Scala files so it's something you can try, except that
> they would not be *exactly* admitted as "extensions" of the original
> tunings anymore ! The extensions allowed were concerning the series
> and the creation of new series.

Your file golden_h7eb.scl contains following line below. So, 696.143 cent is the generator for the meantone temperament of which your scale golden_h7eb.scl is an non-regular variation?

! x^4 = x + 7/2 Tara recurrent sequence, x = 1.494972766 or 696.143 c.

> > > By definition all recurrent series are based on transposition, they are only that ! but
> > > they will transpose very different things, and not necessarily
> > > "harmonic" chords... except for some like Mohajira, Ishku, Buzurg,
> > > Chandrak, Amlak etc.
> >
> > If your recurrent series scales are linear temperaments then I fully understand this comment and it makes sense :)

> They are not the same thing, but I think you understood well.

Hm, I am not quite sure. With a linear temperament, all pitches are created by repeated transpositions of a single interval (plus "octavations"). How many distinct generators (intervals required to create all scale pitches) are involved in your recurrent scales? Are there several small variations of a single interval?

> > > coherent_shrutis.scl is based on a cycle of fifths, only with very
> > > light retunings - so I do not think it would make transpositions
> > > difficult, as long as you accept 361/360, 513/512 and 1216/1215 as
> > > more or less "schismic" variations.The cycle of fifths goes from A
> > > (5/3) to D(9/8), so if you transpose the whole scale by 81/80, or
> > > 96/95, you will have a full quasi 5-limit 24 shrutis system in
> > > which 19 is quite transparent but very useful.
> >
> > I think I understand. So, would it be OK for these demos to extend > coherent_shrutis.scl by 12 tones where each tone transposes an existing tone in coherent_shrutis.scl by a syntonic comma ?

> Perfectly OK and Hindustani correct ! Syntonic comma would be the best choice for the 22 shrutis, 96/95 the best for a more "kind of schismic temperament" I would think.

OK.

Thank you!

Best wishes,
Torsten

🔗Jacques Dudon <fotosonix@...>

Hi Torsten,
Did you succeed to print "Seven-Limit Slendro Mutations" (sent in Tuning files) ?

You wrote :

> Oops, now I get confused. We were talking to have the limits 3, 5, > and 7 in meantone temperaments by having a long meantone spiral of > fifths that then allows for septimal intervals like augmented > second and augmented sixth.
> It would certainly be easy for me to multiply all ratios in your > Scala file by 5 (including wrapping it into a single octave :). > However, I don't understand how this would extend the spiral of > meantone fifths to add further septimal intervals.

10 meantone fifths (of around let's say 696 or 697 c.) approximate the 7h harmonic, 3 octaves higher.
(and augmented sixths would be approximated by -9 meantone fifths).
In 1/4 syntonic comma meantone, four fifths give 5/1, eight give 25/1 and two fiths more arrive to 55,9017/1, almost 56 = 8 * 7.
In a series of 11 fifths (a 12 tones scale), you have only 2 of those "7/4 approximations".
Each extension by a fifth will add one more : 24 notes = 14 "7/4". Is that the question ?

> Is 5 one of your "generators" to create this particular meantone > variant? What are the other "generators" (it can't be the octave > only)?

Yes, you're right ! in this particular kind, 5 can be considered as a auxilliary generator and the others are the meantone fifth and the octave.
But you can also make abstraction, in practice, of the subtelty of the rational intervals and consider the fifth 5^(1/4) to be the average generator.

> Of course, multiplying by octave (variants of) 7 would add septimal > intervals. Still, I don't understand why the result should be a > meantone temperament.
> Nevertheless, I will look into this as an option, it sounds > interesting :)
I did not say that doing this would be identical to a extension of the starting meantone (it would actually result in a similar effect), I suggested it for differential coherence reasons, if x = the ratio of the fifth, x^4 - 7/2 = x being the algorithm of the recurrent sequence where for example if I am in G (= 1/1), which means that D = x/2 and x^4 = 4B, then x^4 - 7/2 = x is translated 8B - 7G = D/2.

> First of all, I would like to understand the purpose of your > formulas like x^4 - 7/2 = x. Are you using these to recursively > approximating your x?
No, it expresses that x (the fifth) is identical to the difference tone of x^4 (= 5/1) less 7/2 (= octave of 7/4 ). But I don't know I can make it simpler. What is it you don't understand ?

> I just tried it, and with most values it very quickly results in > infinity
???
> , so I obviously don't understand what you are doing with this > formula :)
> Secondly, I don't understand the relation between your formula and > your conclusion that 8B - 7G = D/2, but perhaps this is secondary.
It was one example with 1/1 in G.

> More important, I not sure I understand your notation 8B or 7G. Are > you talking about the difference between 8th partial of B and the > 7th partial of G and the octave below D, where B, G and D are in > the same octave?
Yes, that's it !

> Your file golden_h7eb.scl contains following line below. So, > 696.143 cent is the generator for the meantone temperament of which > your scale golden_h7eb.scl is an non-regular variation ?

Yes.

> > > If your recurrent series scales are linear temperaments then I > fully understand this comment and it makes sense :)
>
> > They are not the same thing, but I think you understood well.
>
> Hm, I am not quite sure. With a linear temperament, all pitches are > created by repeated transpositions of a single interval (plus > "octavations").

OK, let's say it is some sort of a linear temperament if I apply the fractal solution, x = 1.494972766 or 696.143 c.
In the recurrent sequence form applied to rational numbers, it is a recurrent sequence whose successive ratios converges in one direction or the other towards the same generator x.

> How many distinct generators (intervals required to create all > scale pitches) are involved in your recurrent scales?

Octave and the generator (x), that may slightly vary, in the rational form, along the sequence.
- - - - - - - - -
Jacques

🔗genewardsmith <genewardsmith@...>

🔗jacques.dudon <fotosonix@...>

🔗Torsten Anders <torsten.anders@...>

Dear Jacques,

Thank you very much for your reply, and sorry for my late reaction.

On 04.05.2010, at 22:52, Jacques Dudon wrote:
> Did you succeed to print "Seven-Limit Slendro Mutations" (sent in > Tuning files) ?

I found it :)

> Torsten wrote :
>> Oops, now I get confused. We were talking to have the limits 3, 5, >> and 7 in meantone temperaments by having a long meantone chain/>> spiral of fifths that then allows for septimal intervals like >> augmented second and augmented sixth.
>> It would certainly be easy for me to multiply all ratios in your >> Scala file by 5 (including wrapping it into a single octave :). >> However, I don't understand how this would extend the spiral of >> meantone fifths to add further septimal intervals.
>
>
> 10 meantone fifths (of around let's say 696 or 697 c.) approximate > the 7h harmonic, 3 octaves higher.
> (and augmented sixths would be approximated by -9 meantone fifths).
> In 1/4 syntonic comma meantone, four fifths give 5/1, eight give > 25/1 and two fiths more arrive to 55,9017/1, almost 56 = 8 * 7.
> In a series of 11 fifths (a 12 tones scale), you have only 2 of > those "7/4 approximations".
> Each extension by a fifth will add one more : 24 notes = 14 "7/4". > Is that the question ?

Exactly. As you say, for septimal harmony it is desirable to have a rather long chain of fifths (clearly more than 11), instead of complementing this chain of meantone fifths by another generator (factor 5). As I understand it, multiplying each note of your 1/4-comma meantone variant by fifth would result in many tones which are already part of the tuning anyway (or are very close to them), but the chain of fifths is only extended by 4 fifths instead of the possible 12 additional fifths.

Would it be possible to extend the chain of fifths in your 1/4-comma meantone variant into a chain of 23 fifths?

> > > > I would think a more septimal, because perfectly eq-b-optimised
> > > > solution would be the "golden_h7eb.scl" (EU7), that integrates > the
> > > > 5th and the 7th harmonic :
> > > > The Tara algorithm x^4 = x + 7/2 itself invites you to > multiply the
> > > > series directly by 7, in order to play on both the -c and the > eq-b
> > > > (and I just found a third synchronous-beating property for it).
> > > > Memory I have from it was something incredibly vibrant in the
> > > > septimal chords concerned by the eq-b !
>
>> Of course, multiplying by octave (variants of) 7 would add septimal >> intervals. Still, I don't understand why the result should be a >> meantone temperament.
>> Nevertheless, I will look into this as an option, it sounds >> interesting :)
> I did not say that doing this would be identical to a extension of > the starting meantone (it would actually result in a similar > effect), I suggested it for differential coherence reasons, if x = > the ratio of the fifth, x^4 - 7/2 = x being the algorithm of the > recurrent sequence where for example if I am in G (= 1/1), which > means that D = x/2 and x^4 = 4B, then x^4 - 7/2 = x is > translated 8B - 7G = D/2.
>
>> First of all, I would like to understand the purpose of your >> formulas like x^4 - 7/2 = x. Are you using these to recursively >> approximating your x?
> No, it expresses that x (the fifth) is identical to the difference > tone of x^4 (= 5/1) less 7/2 (= octave of 7/4 ). But I don't know I > can make it simpler. What is it you don't understand ?

Ah, I see: 5/1-7/2=3/2. Are you transposing intervals expressed as ratios by subtraction?

>> How many distinct generators (intervals required to create all >> scale pitches) are involved in your recurrent scales?
>
> Octave and the generator (x), that may slightly vary, in the > rational form, along the sequence.

OK, I get roughly the idea :)

BTW: did you experimentally test with Ethno2 sounds whether you can actually perceive the difference between your scales that are close to a linear temperament and their corresponding linear temperament? It appears to me that (at least some of) the Ethno2 sounds are not necessarily tuned extremely accurately. Which sounds can you recommend for their accurate tuning (and clearly perceivable tuning in long sustained notes)?

Thank you!

Best wishes,
Torsten

🔗Jacques Dudon <fotosonix@...>

> Torsten wrote :

> (Jacques) :
> > 10 meantone fifths (of around let's say 696 or 697 c.) approximate
> > the 7h harmonic, 3 octaves higher.
> > (and augmented sixths would be approximated by -9 meantone fifths).
> > In 1/4 syntonic comma meantone, four fifths give 5/1, eight give
> > 25/1 and two fiths more arrive to 55,9017/1, almost 56 = 8 * 7.
> > In a series of 11 fifths (a 12 tones scale), you have only 2 of
> > those "7/4 approximations".
> > Each extension by a fifth will add one more : 24 notes = 14 "7/4".
> > Is that the question ?
>
> Exactly. As you say, for septimal harmony it is desirable to have a
> rather long chain of fifths (clearly more than 11), instead of
> complementing this chain of meantone fifths by another generator
> (factor 5). As I understand it, multiplying each note of your 1/4-
> comma meantone variant by fifth would result in many tones which are
> already part of the tuning anyway (or are very close to them), but the
> chain of fifths is only extended by 4 fifths instead of the possible
> 12 additional fifths.
> Would it be possible to extend the chain of fifths in your 1/4-comma
> meantone variant into a chain of 23 fifths?

(= you meant multiply by "five", I guess) ;
What I suggested was to extend the same structure (3 Mezzo fifths> multiplication by 5 ad lib)
Since the tuning does it already 3 times, you just need to transpose the whole scale by 125/128 (5^3) to extend it to 24 tones.
I think it's rather simple myself, and keeps the scale entirely rational, but you prefer other solutions, like using the fractal generators go ahead.

> > > > > I would think a more septimal, because perfectly eq-b-> optimised
> > > > > solution would be the "golden_h7eb.scl" (EU7), that integrates
> > the
> > > > > 5th and the 7th harmonic :
> > > > > The Tara algorithm x^4 = x + 7/2 itself invites you to
> > multiply the
> > > > > series directly by 7, in order to play on both the -c and the
> > eq-b
> > > > > (and I just found a third synchronous-beating property for > it).
> > > > > Memory I have from it was something incredibly vibrant in the
> > > > > septimal chords concerned by the eq-b !
> >
> >> Of course, multiplying by octave (variants of) 7 would add septimal
> >> intervals. Still, I don't understand why the result should be a
> >> meantone temperament.
> >> Nevertheless, I will look into this as an option, it sounds
> >> interesting :)
> > I did not say that doing this would be identical to a extension of
> > the starting meantone (it would actually result in a similar
> > effect), I suggested it for differential coherence reasons, if x =
> > the ratio of the fifth, x^4 - 7/2 = x being the algorithm of the
> > recurrent sequence where for example if I am in G (= 1/1), which
> > means that D = x/2 and x^4 = 4B, then x^4 - 7/2 = x is
> > translated 8B - 7G = D/2.
> >
> >> First of all, I would like to understand the purpose of your
> >> formulas like x^4 - 7/2 = x. Are you using these to recursively
> >> approximating your x?
> > No, it expresses that x (the fifth) is identical to the difference
> > tone of x^4 (~ 5/1) less 7/2 (= octave of 7/4 ). But I don't know I
> > can make it simpler. What is it you don't understand ?
>
> Ah, I see: 5/1 - 7/2 = 3/2. Are you transposing intervals expressed as
> ratios by subtraction ?

Sorry, I can't understand the question.

> >> How many distinct generators (intervals required to create all
> >> scale pitches) are involved in your recurrent scales?
> >
> > Octave and the generator (x), that may slightly vary, in the
> > rational form, along the sequence.
>
> OK, I get roughly the idea :)
>
> BTW: did you experimentally test with Ethno2 sounds whether you can
> actually perceive the difference between your scales that are close to
> a linear temperament and their corresponding linear temperament? It
> appears to me that (at least some of) the Ethno2 sounds are not
> necessarily tuned extremely accurately. Which sounds can you recommend
> for their accurate tuning (and clearly perceivable tuning in long
> sustained notes)?

No, I did not find the time to do any music myself. All the sounds I heard where tuned together, but of course some of them show fluctuations during their play, like in any sampler.
You have to make a selection.
- - - - - - -
Jacques

🔗Torsten Anders <torsten.anders@...>

Dear Jacques,

thanks for your reply.

On 09.05.2010, at 19:30, Jacques Dudon wrote:
> > Torsten wrote :
>
>> (Jacques) :
>> > 10 meantone fifths (of around let's say 696 or 697 c.) approximate
>> > the 7h harmonic, 3 octaves higher. [...]
>> > In a series of 11 fifths (a 12 tones scale), you have only 2 of
>> > those "7/4 approximations".
>> > Each extension by a fifth will add one more : 24 notes = 14 "7/4".
>> > Is that the question ?
>>
>> Exactly. As you say, for septimal harmony it is desirable to have a
>> rather long chain of fifths (clearly more than 11), instead of
>> complementing this chain of meantone fifths by another generator
>> (factor 5). [...]
>
>> Would it be possible to extend the chain of fifths in your 1/4-comma
>> meantone variant into a chain of 23 fifths?
>
> (= you meant multiply by "five", I guess) ;
> What I suggested was to extend the same structure (3 Mezzo fifths> > multiplication by 5 ad lib)
> Since the tuning does it already 3 times, you just need to transpose > the whole scale by 125/128 (5^3) to extend it to 24 tones.
> I think it's rather simple myself, and keeps the scale entirely > rational, but you prefer other solutions, like using the fractal > generators go ahead.

I think I fully understood what you were suggesting, namely to extend this particular meantone variant by multiplying its pitches with 5. Instead, I was asking whether would alternatively be possible to extend its chain of fifths to reach more 7-limit intervals.

Never mind. I take your reaction as a no :) So, I assume for 7-limit harmony I will instead use your "golden_h7eb.scl", and extend it as you suggest by multiplying all its pitches with (octave variants of) 7.

BTW: in order to be able to use Strasheela for these demos I just added support for arbitrary (octave-repeating) regular temperaments of any rank. So, I can then use your meantone generator of "golden_h7eb.scl" and the factor 7 as another generator to create a scale that is internally used by the software for the composition process (e.g., to know which pitch sets for which chords etc.).

Best wishes,
Torsten

🔗genewardsmith <genewardsmith@...>

🔗Ozan Yarman <ozanyarman@...>

The ultimate trick is to find a modified meantone which is
extraordinary for featuring simple proportional beat rates in all
major and minor triads. I have never seen anyone come closer to this
ideal than George with his 5/25 syntonic comma tempered 12-tone tuning.

Oz.

✩ ✩ ✩
www.ozanyarman.com

On May 11, 2010, at 5:15 AM, genewardsmith wrote:

>
>
> --- In tuning@yahoogroups.com, Torsten Anders <torsten.anders@...>
> wrote:
>
>
>> Never mind. I take your reaction as a no :)
>
> I think in fact the answer is yes. Here is an extended chain of
> fifths giving brats of 4, cooked up via a recurrence derived from
> the polynomial for brats of 4, namely x^4 + 2x - 8.
>
> ! mean24rat.scl
> Meantone[24] in a rational tuning with brats of 4
> 24
> !
> 3287219/3225600
> 1121/1050
> 449272979/412876800
> 3527/3150
> 459763/403200
> 1882/1575
> 62841811/51609600
> 64307/50400
> 2863099537/2202009600
> 421/315
> 585917/430080
> 257/180
> 133478939/91750400
> 2356/1575
> 245879/161280
> 2516/1575
> 168032819/103219200
> 527/315
> 171923/100800
> 2816/1575
> 23501171/12902400
> 8017/4200
> 458892629/235929600
> 2
>
>
>
>
> ------------------------------------
>
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🔗Mike Battaglia <battaglia01@...>

🔗Ozan Yarman <ozanyarman@...>

🔗Torsten Anders <torsten.anders@...>

🔗jacques.dudon <fotosonix@...>

As a Skisni sequence (I suppose that's what Gene did, if of a brat 4), it is allowed by the rules competition. What I don't understand is what that has to see with what you asked before, namely using a meantone for both 5 and 7 harmonics, if I understood well - that's not the purpose of Skisni and there are better sequences for that in the ethno collection, among which Mezzo and for other reasons, Tara (Golden h7) and others.
What's the problem with the rational solutions I suggested you with Mezzo and Tara ?

Someone also suggested the Lucy tuning, it would not be better for that but just in case, there is a eq-b version of LT if you want with OCE2 (Lucie...): x^4 = 15 - 3x^3
x= 1.4944195778683 or 695.5023126 c.
- - - - -
Jacques

--- In tuning@yahoogroups.com, Torsten Anders <torsten.anders@...> wrote:
>
> Dear Gene,
>
> Thank you very much for this! Jacques, is the scale below OK for you :)
>
> Best,
> Torsten
>
> ________________________________________
> From: tuning@yahoogroups.com [tuning@yahoogroups.com] On Behalf Of genewardsmith [genewardsmith@...]
> Sent: 11 May 2010 03:15
> To: tuning@yahoogroups.com
> Subject: [tuning] Re: Questions (was: Paper?)
>
> --- In tuning@yahoogroups.com<mailto:tuning%40yahoogroups.com>, Torsten Anders <torsten.anders@> wrote:
>
> > Never mind. I take your reaction as a no :)
>
> I think in fact the answer is yes. Here is an extended chain of fifths giving brats of 4, cooked up via a recurrence derived from the polynomial for brats of 4, namely x^4 + 2x - 8.
>
> ! mean24rat.scl
> Meantone[24] in a rational tuning with brats of 4
> 24
> !
> 3287219/3225600
> 1121/1050
> 449272979/412876800
> 3527/3150
> 459763/403200
> 1882/1575
> 62841811/51609600
> 64307/50400
> 2863099537/2202009600
> 421/315
> 585917/430080
> 257/180
> 133478939/91750400
> 2356/1575
> 245879/161280
> 2516/1575
> 168032819/103219200
> 527/315
> 171923/100800
> 2816/1575
> 23501171/12902400
> 8017/4200
> 458892629/235929600
> 2
>

🔗Torsten Anders <torsten.anders@...>

Dear Jacques,

> What I don't understand is what that has to see with what you asked before, namely using a meantone for both 5 and 7 harmonics, if I understood well - that's not the purpose of Skisni

thanks for your kind reply and this clarification. To be honest, all these poetic names for meantone variants do not mean much to me :)

I failed to find "Mezzo" in your list of tunings, but because you were suggesting it just after confirming that 1/4-comma meantone would be a good candidate for 7-limit harmony, I just assumed that Mezzo is a close variant of 1/4-comma meantone. I also assumed that Gene's proposal was an extension of a variant of 1/4-comma meantone to 24 tones, because that was what originally I asked for (this might have been lost in this long email exchange trying to clarify that I actually meant a long sequence of fifths :)

> What's the problem with the rational solutions I suggested you with Mezzo and Tara ?

As I just mentioned last night, I am about to address an extension of golden_h7eb.scl (extension with factor 7) directly in Strasheela :)

Best,
Torsten

________________________________________
From: tuning@yahoogroups.com [tuning@yahoogroups.com] On Behalf Of jacques.dudon [fotosonix@...]
Sent: 11 May 2010 15:21
To: tuning@yahoogroups.com
Subject: [tuning] Re: Questions (was: Paper?)

--- In tuning@yahoogroups.com<mailto:tuning%40yahoogroups.com>, Torsten Anders <torsten.anders@...> wrote:
>
> Dear Gene,
>
> Thank you very much for this! Jacques, is the scale below OK for you :)
>
> Best,
> Torsten
>
> ________________________________________
> From: tuning@yahoogroups.com<mailto:tuning%40yahoogroups.com> [tuning@yahoogroups.com<mailto:tuning%40yahoogroups.com>] On Behalf Of genewardsmith [genewardsmith@...]
> Sent: 11 May 2010 03:15
> To: tuning@yahoogroups.com<mailto:tuning%40yahoogroups.com>
> Subject: [tuning] Re: Questions (was: Paper?)
>
> --- In tuning@yahoogroups.com<mailto:tuning%40yahoogroups.com><mailto:tuning%40yahoogroups.com>, Torsten Anders <torsten.anders@> wrote:
>
> > Never mind. I take your reaction as a no :)
>
> I think in fact the answer is yes. Here is an extended chain of fifths giving brats of 4, cooked up via a recurrence derived from the polynomial for brats of 4, namely x^4 + 2x - 8.
>
> ! mean24rat.scl
> Meantone[24] in a rational tuning with brats of 4
> 24
> !
> 3287219/3225600
> 1121/1050
> 449272979/412876800
> 3527/3150
> 459763/403200
> 1882/1575
> 62841811/51609600
> 64307/50400
> 2863099537/2202009600
> 421/315
> 585917/430080
> 257/180
> 133478939/91750400
> 2356/1575
> 245879/161280
> 2516/1575
> 168032819/103219200
> 527/315
> 171923/100800
> 2816/1575
> 23501171/12902400
> 8017/4200
> 458892629/235929600
> 2
>