Igs

Igs,

you said <it's not true that you've improved on 12-tET--you've simply reinstated Meantone and given it your own personal well-tempered twist>

I compared my Blue Temperament to 1/4 comma meantone (1/4 CMT) and I was surprised to find that, mostly, they were very similar indeed, except for two notes in BT that are more than 44 cents away from those in 1/4 CMT.

The fact that my Blue Temperament is somewhat similar to 1/4 CMT is an interesting coincidence but I did *not* give 1/4 CMT my "own personal well-tempered twist".

The two exceptions below are, I think, significant (121.6c and 816.9c in Blue Temperament (BT)).

Here's how the two tunings compare (use fixed width font)...

1/4 CMT BT difference
  0:    0.0c  0.0c
  1:   76.0c 121.6c 45.6c !!!
  2:  193.2c 200.7c 7.5c
  3:  310.3c 313.5c 3.2c
  4:  386.3c 388.4c 2.1c
  5:  503.4c 501.2c 2.2c
  6:  579.5c 580.4c 0.9c
  7:  696.6c 702.0c 5.4c
  8:  772.6c 816.9c 44.3c !!!
  9:  889.7c 889.4c 0.3c
 10: 1006.8c 1012.5c 5.7c
 11: 1082.9c 1085.1c 2.2c
 12: 1200.0c 1200.0c    0.0c  

The next thing I'm going to do is identify how many good harmony intervals (an octave or less wide) occur in both tunings over 12 keys and this should decide which of the two is better. This will take a while.

Thanks Chris for the scales.

John.

On Thu, Feb 10, 2011 at 1:41 PM, john777music <jfos777@...> wrote:
>
> Here's how the two tunings compare (use fixed width font)...
>
> 1/4 CMT BT difference
>   0:    0.0c  0.0c
>   1:   76.0c 121.6c 45.6c !!!
>   2:  193.2c 200.7c 7.5c
>   3:  310.3c 313.5c 3.2c
>   4:  386.3c 388.4c 2.1c
>   5:  503.4c 501.2c 2.2c
>   6:  579.5c 580.4c 0.9c
>   7:  696.6c 702.0c 5.4c
>   8:  772.6c 816.9c 44.3c !!!
>   9:  889.7c 889.4c 0.3c
>  10: 1006.8c 1012.5c 5.7c
>  11: 1082.9c 1085.1c 2.2c
>  12: 1200.0c 1200.0c    0.0c

Hi John,

You are comparing from a 12-note meantone chain going from Eb to G#.
If you instead compare to a 12-note meantone chain going from Db to
F#, you end up with the following scale:

0.000
117.110
193.156
310.266
386.312
503.422
579.468
696.578
813.688
889.734
1006.844
1082.890
1200.000

The 117.11000 cent interval is ~4.5 cents flat of Blue Temperament's,
and the 813.688 cent interval is ~3.3 cents flat. The rest are the
same as what you listed.

-Mike

Mike,

I don't know what you mean by a chain going from Eb to G# or Db to F#. Can you explain? I thought chains were either circular or went on forever and covered all keys.

I was going by the scale Chris posted for me. Is the scale you posted just a mode of the scale I used and if so which note of the scale I used is the starting point of the scale you posted. Looking at it again it seems it can't be a mode of the scale I was given. If the scale you posted is not a mode then what is it and what is it called and how is it made?

I'm surprised at how similar the scale you posted is to Blue Temperament but like I said, this is coincidental and not by design.

I'm going to test Blue Temperament and the scale you posted for number of good dyads. It will be interesting to see which has more.

John.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Thu, Feb 10, 2011 at 1:41 PM, john777music <jfos777@...> wrote:
> >
> > Here's how the two tunings compare (use fixed width font)...
> >
> > 1/4 CMT BT difference
> >   0:    0.0c  0.0c
> >   1:   76.0c 121.6c 45.6c !!!
> >   2:  193.2c 200.7c 7.5c
> >   3:  310.3c 313.5c 3.2c
> >   4:  386.3c 388.4c 2.1c
> >   5:  503.4c 501.2c 2.2c
> >   6:  579.5c 580.4c 0.9c
> >   7:  696.6c 702.0c 5.4c
> >   8:  772.6c 816.9c 44.3c !!!
> >   9:  889.7c 889.4c 0.3c
> >  10: 1006.8c 1012.5c 5.7c
> >  11: 1082.9c 1085.1c 2.2c
> >  12: 1200.0c 1200.0c    0.0c
>
> Hi John,
>
> You are comparing from a 12-note meantone chain going from Eb to G#.
> If you instead compare to a 12-note meantone chain going from Db to
> F#, you end up with the following scale:
>
> 0.000
> 117.110
> 193.156
> 310.266
> 386.312
> 503.422
> 579.468
> 696.578
> 813.688
> 889.734
> 1006.844
> 1082.890
> 1200.000
>
> The 117.11000 cent interval is ~4.5 cents flat of Blue Temperament's,
> and the 813.688 cent interval is ~3.3 cents flat. The rest are the
> same as what you listed.
>
> -Mike
>

Hi John,

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:
> I don't know what you mean by a chain going from Eb to G# or Db to F#. Can you
> explain? I thought chains were either circular or went on forever and covered all keys.

Try this version of Meantone, from 31-EDO; this is incidentally what we get when we approximate your "Blue JI" in 31-EDO:

0.00000000000000
116.129032258064517
193.548387096774195
309.677419354838712
387.09677419354839
503.225806451612907
580.645161290322585
696.774193548387102
812.903225806451619
890.322580645161297
1006.451612903225814
1083.870967741935492
1200.000000000000000

This scale in nominals is C-Db-D-Eb-E-F-F#-G-Ab-A-Bb-B-C, in other words it is the chain of fifths starting on Db going up to F#, but rearranged as a mode of C (assuming C is taken to be 0.00000 cents). This 31-EDO version is VERY VERY VERY CLOSE to 1/4-comma, and is probably better to compare anyway. It's worth noting that this version favors the flats more than the naturals or sharps, you can't go above the key of G or you'll run into wolves (for instance, in the key of A, you have A B Db D E F# Ab A, where Db is approximately a 7/6 instead of a 6/5, and Ab is something horrendous like 48/25 or something, and F# to Db is a wolf fifth). Remember that in Meantone, enharmonics are not equivalent.

> I'm surprised at how similar the scale you posted is to Blue Temperament but like I said, > this is coincidental and not by design.

Of course. You reinvented the wheel, I'm afraid. Doing a bit of homework might have saved you a lot of trouble. You might also want to look into the work of Charles Lucy, who arrived also at a scale very similar to both your own Blue Temperament as well as 1/4-comma Meantone, by his own circuitous route. His site is here:

http://www.lucytune.com/

-Igs

Hi Mike,

I compared the scale you gave me (which you tell me is a form of 1/4 comma Meantone, I would like to know more about it and how it was made) with my Blue Temperament.

My Blue Temperament has 97 good intervals an octave or less wide over 12 keys and the scale you gave me has only 90 good intervals, seven less than mine.

This is a little sweet victory for me.

I'll give the details of what intervals I deem to be good, the maximum deviation from pure in cents (6.776 cents) and how often each good interval occurs in both scales tomorrow.

Gene, I'll get to your scale tomorrow as well.

John.

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:
>
> Mike,
>
> I don't know what you mean by a chain going from Eb to G# or Db to F#. Can you explain? I thought chains were either circular or went on forever and covered all keys.
>
> I was going by the scale Chris posted for me. Is the scale you posted just a mode of the scale I used and if so which note of the scale I used is the starting point of the scale you posted. Looking at it again it seems it can't be a mode of the scale I was given. If the scale you posted is not a mode then what is it and what is it called and how is it made?
>
> I'm surprised at how similar the scale you posted is to Blue Temperament but like I said, this is coincidental and not by design.
>
> I'm going to test Blue Temperament and the scale you posted for number of good dyads. It will be interesting to see which has more.
>
> John.
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> >
> > On Thu, Feb 10, 2011 at 1:41 PM, john777music <jfos777@> wrote:
> > >
> > > Here's how the two tunings compare (use fixed width font)...
> > >
> > > 1/4 CMT BT difference
> > >   0:    0.0c  0.0c
> > >   1:   76.0c 121.6c 45.6c !!!
> > >   2:  193.2c 200.7c 7.5c
> > >   3:  310.3c 313.5c 3.2c
> > >   4:  386.3c 388.4c 2.1c
> > >   5:  503.4c 501.2c 2.2c
> > >   6:  579.5c 580.4c 0.9c
> > >   7:  696.6c 702.0c 5.4c
> > >   8:  772.6c 816.9c 44.3c !!!
> > >   9:  889.7c 889.4c 0.3c
> > >  10: 1006.8c 1012.5c 5.7c
> > >  11: 1082.9c 1085.1c 2.2c
> > >  12: 1200.0c 1200.0c    0.0c
> >
> > Hi John,
> >
> > You are comparing from a 12-note meantone chain going from Eb to G#.
> > If you instead compare to a 12-note meantone chain going from Db to
> > F#, you end up with the following scale:
> >
> > 0.000
> > 117.110
> > 193.156
> > 310.266
> > 386.312
> > 503.422
> > 579.468
> > 696.578
> > 813.688
> > 889.734
> > 1006.844
> > 1082.890
> > 1200.000
> >
> > The 117.11000 cent interval is ~4.5 cents flat of Blue Temperament's,
> > and the 813.688 cent interval is ~3.3 cents flat. The rest are the
> > same as what you listed.
> >
> > -Mike
> >
>

Thanks Igs,

read my last post where Blue Temperament seems to be more versatile than the 1/4 comma Meantone Mike gave me.

I'll test your 31 EDO version tomorrow and Gene's 197 version. If mine wins then I haven't just reinvented the wheel, I've improved on it.

John.

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> Hi John,
>
> --- In tuning@yahoogroups.com, "john777music" <jfos777@> wrote:
> > I don't know what you mean by a chain going from Eb to G# or Db to F#. Can you
> > explain? I thought chains were either circular or went on forever and covered all keys.
>
> Try this version of Meantone, from 31-EDO; this is incidentally what we get when we approximate your "Blue JI" in 31-EDO:
>
> 0.00000000000000
> 116.129032258064517
> 193.548387096774195
> 309.677419354838712
> 387.09677419354839
> 503.225806451612907
> 580.645161290322585
> 696.774193548387102
> 812.903225806451619
> 890.322580645161297
> 1006.451612903225814
> 1083.870967741935492
> 1200.000000000000000
>
> This scale in nominals is C-Db-D-Eb-E-F-F#-G-Ab-A-Bb-B-C, in other words it is the chain of fifths starting on Db going up to F#, but rearranged as a mode of C (assuming C is taken to be 0.00000 cents). This 31-EDO version is VERY VERY VERY CLOSE to 1/4-comma, and is probably better to compare anyway. It's worth noting that this version favors the flats more than the naturals or sharps, you can't go above the key of G or you'll run into wolves (for instance, in the key of A, you have A B Db D E F# Ab A, where Db is approximately a 7/6 instead of a 6/5, and Ab is something horrendous like 48/25 or something, and F# to Db is a wolf fifth). Remember that in Meantone, enharmonics are not equivalent.
>
> > I'm surprised at how similar the scale you posted is to Blue Temperament but like I said, > this is coincidental and not by design.
>
> Of course. You reinvented the wheel, I'm afraid. Doing a bit of homework might have saved you a lot of trouble. You might also want to look into the work of Charles Lucy, who arrived also at a scale very similar to both your own Blue Temperament as well as 1/4-comma Meantone, by his own circuitous route. His site is here:
>
> http://www.lucytune.com/
>
> -Igs
>

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:

> I'll test your 31 EDO version tomorrow and Gene's 197 version. If mine wins then I
> haven't just reinvented the wheel, I've improved on it.

Like I said in my initial post, your temperament is *not* a Meantone but a well-temperament that is loosely based on Meantone (even if you didn't design it with that in mind). The way well-temperaments work is by tempering different intervals in the scale by different amounts, so that more keys (and thus, more chords) become usable but some pure harmonies are retained. I know diddly-squat about well-temperaments in general, except that they historically preceded equal temperament and there were probably hundreds (if not thousands) of them. They aren't hard to come up with, really. George Secor and Margo Schulter are the resident experts on WT's and could tell you an awful lot more about them than I can. Hopefully one of them will chime in, or Carl perhaps. But for a start, see here:

http://en.wikipedia.org/wiki/Werckmeister_temperament

Werckmeister was around in the late 1600's, just FYI.

-Igs

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, "john777music" <jfos777@> wrote:
>
> > I'll test your 31 EDO version tomorrow and Gene's 197 version. If mine wins then I
> > haven't just reinvented the wheel, I've improved on it.
>
> Like I said in my initial post, your temperament is *not* a Meantone but a well-temperament that is loosely based on Meantone (even if you didn't design it with that in mind).

Arrgh! It is NOT a well-temperament. It has a wolf sharp by 50/49, which is 34.976 cents. It has two other fifths flat by 81/80, which is 21.806 cents. Two more fifths are actually reasonable, being flat by 225/224, which is 7.712 cents. Smooth out the two 225/224 fifths, and you get the Marveldene, not a well temperament. Smooth out all four flat fifths and you get Meantone[12], not a well temperament. Smooth out all the fifths, and you get 12et, not a well temperament. There's not a well temperament in sight, anywhere.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
> Arrgh! It is NOT a well-temperament. It has a wolf sharp by 50/49, which is 34.976 cents. It has two other fifths flat by 81/80, which is 21.806 cents. Two more fifths are actually reasonable, being flat by 225/224, which is 7.712 cents. Smooth out the two 225/224 fifths, and you get the Marveldene, not a well temperament. Smooth out all four flat fifths and you get Meantone[12], not a well temperament. Smooth out all the fifths, and you get 12et, not a well temperament. There's not a well temperament in sight, anywhere.
>

I don't understand how this isn't a well-temperament of some variety. Note we're talking not about his Blue JI scale, but the Blue Temperament scale (in case there is any confusion on that). I was under the impression that any tuning where the fifths are irregularly tempered so as to produce more consonant intervals than would be possible in a regular temperament, is a well-temperament. Is there some more precise definition of which I am unaware?

-Igs

Gene,

by my reckoning 9 of my 12 Fifths over 12 keys are within 6.776 cents acccuracy (256/255) of 3/2 in Blue Temperament. Here's Blue Temperament again if you want to check...

0.0c, 121.6c, 200.7c, 313.5c, 388.4c, 501.2c, 580.4c,
702.0c, 816.9c, 889.4c, 1012.5c, 1085.1c, 1200c.

John.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@> wrote:
> >
> > --- In tuning@yahoogroups.com, "john777music" <jfos777@> wrote:
> >
> > > I'll test your 31 EDO version tomorrow and Gene's 197 version. If mine wins then I
> > > haven't just reinvented the wheel, I've improved on it.
> >
> > Like I said in my initial post, your temperament is *not* a Meantone but a well-temperament that is loosely based on Meantone (even if you didn't design it with that in mind).
>
> Arrgh! It is NOT a well-temperament. It has a wolf sharp by 50/49, which is 34.976 cents. It has two other fifths flat by 81/80, which is 21.806 cents. Two more fifths are actually reasonable, being flat by 225/224, which is 7.712 cents. Smooth out the two 225/224 fifths, and you get the Marveldene, not a well temperament. Smooth out all four flat fifths and you get Meantone[12], not a well temperament. Smooth out all the fifths, and you get 12et, not a well temperament. There's not a well temperament in sight, anywhere.
>

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:
>
> Gene,
>
> by my reckoning 9 of my 12 Fifths over 12 keys are within 6.776 cents acccuracy (256/255) of 3/2 in Blue Temperament. Here's Blue Temperament again if you want to check...
>
> 0.0c, 121.6c, 200.7c, 313.5c, 388.4c, 501.2c, 580.4c,
> 702.0c, 816.9c, 889.4c, 1012.5c, 1085.1c, 1200c.

I was talking about BlueJI, not Blue Temperament. When did the conversatuion switch tracks?

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

> I don't understand how this isn't a well-temperament of some variety. Note we're talking not about his Blue JI scale, but the Blue Temperament scale (in case there is any confusion on that).

There's plenty of confusion about that. Bad choice of names!

I was under the impression that any tuning where the fifths are irregularly tempered so as to produce more consonant intervals than would be possible in a regular temperament, is a well-temperament. Is there some more precise definition of which I am unaware?

Well-temperaments are just circulating temperaments. If you go around a circle of fifths, they had better be reasonable fifths.

Glad to bring some needed clarification. I transcribe below updated descriptions from my publications on the topic.

"Well" Temperaments a grammatically-erroneous translation from German: the correct term is "Good" Temperaments. And they are NOT the same as Circular either. (Both errors were introduced by Jorgensen in 1977 and have been commented by scholars for decades).

CIRCULAR temperaments are unequal 12-note temperaments that allow all the tonalities and intervals to be playable. Some are better than in Equal Temperament, others are instead quite dissonant but not reaching the "wolves" of meantone. These are enharmonic systems where every sharp functions as flat as well, though some accidentals function better as flats, others as sharps. Circular temperaments are grouped in two families: FRENCH and GOOD temperaments.

FRENCH temperaments have meantone (almost always 1/4 S.c.) diatonic fifths, the others modified to close the Circle: this implies that a few fifths have to be wider than pure. These are extremely unequal circular temperaments. (well-known examples: Rameau and d'Alembert)

GOOD temperaments show diatonic fifths that may vary in size (unlike meantone) and arranged so that there is no need for wide fifths to close the Circle: chromatic fifths are usually pure. These temperaments range from decidedly unequal to mildly so, almost equal. (well-known examples: Neidhardt's temperaments).

Kind Regards

Claudio

http://temper.braybaroque.ie/

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
> I was under the impression that any tuning where the fifths are irregularly tempered so as to produce more consonant intervals than would be possible in a regular temperament, is a well-temperament. Is there some more precise definition of which I am unaware?
> Well-temperaments are just circulating temperaments. If you go around a circle of fifths, they had better be reasonable fifths.
>

--- In tuning@yahoogroups.com, "Claudio Di Veroli" <dvc@...> wrote:
>
> Glad to bring some needed clarification. I transcribe below updated descriptions from my publications on the topic.
>
> "Well" Temperaments a grammatically-erroneous translation from
> German: the correct term is "Good" Temperaments.

To play the quibbler: I wouldn't say "Well temperament" is an erroneous translation from german. I have hardly ever heard the term "Wohltemperament". There is the term "wohltemperiert", whose correct translation is indeed "well-tempered". Erroneous in the strict grammatical sense is the conversion of the adjective into a noun - and this is in german as erroneous as in english.
--
Hans Straub

Thank you so much Claudio, and Hans, for these important precisions.
(though I don't know now if I should say "Good temperament", or perhaps "Well-tempered tuning" ??)
Just two complementary questions :
Can we say that "temperaments ordinaires" = french temperaments ?
and that "temperaments extraordinaires" (more rarely used but that I hear sometimes) = well temperered tunings ?
thanks !
- - - -
Jacques

--- In tuning@yahoogroups.com, "Claudio Di Veroli" <dvc@...> wrote:
>
> Glad to bring some needed clarification. I transcribe below updated descriptions from my publications on the topic.
>
> "Well" Temperaments a grammatically-erroneous translation from German: the correct term is "Good" Temperaments. And they are NOT the same as Circular either. (Both errors were introduced by Jorgensen in 1977 and have been commented by scholars for decades).
>
> CIRCULAR temperaments are unequal 12-note temperaments that allow all the tonalities and intervals to be playable. Some are better than in Equal Temperament, others are instead quite dissonant but not reaching the "wolves" of meantone. These are enharmonic systems where every sharp functions as flat as well, though some accidentals function better as flats, others as sharps. Circular temperaments are grouped in two families: FRENCH and GOOD temperaments.
>
> FRENCH temperaments have meantone (almost always 1/4 S.c.) diatonic fifths, the others modified to close the Circle: this implies that a few fifths have to be wider than pure. These are extremely unequal circular temperaments. (well-known examples: Rameau and d'Alembert)
>
> GOOD temperaments show diatonic fifths that may vary in size (unlike meantone) and arranged so that there is no need for wide fifths to close the Circle: chromatic fifths are usually pure. These temperaments range from decidedly unequal to mildly so, almost equal. (well-known examples: Neidhardt's temperaments).
>
> Kind Regards
>
> Claudio
>
> http://temper.braybaroque.ie/

On Thu, Feb 10, 2011 at 2:34 PM, john777music <jfos777@...> wrote:
>
> Mike,
>
> I don't know what you mean by a chain going from Eb to G# or Db to F#. Can you explain? I thought chains were either circular or went on forever and covered all keys.

Quarter-comma meantone does go on forever. The generating interval is
a slightly flat 3/2. The original scale posted was 12 notes ranging
from 3 fifths down to 9 fifths up. I sent you one ranging from 5
fifths down to 7 fifths up again.

> I was going by the scale Chris posted for me. Is the scale you posted just a mode of the scale I used and if so which note of the scale I used is the starting point of the scale you posted. Looking at it again it seems it can't be a mode of the scale I was given. If the scale you posted is not a mode then what is it and what is it called and how is it made?

The scale I posted is in fact a mode of the scale Chris posted; I
believe that the "C" in my scale is the "D" in Chris's scale.

> I'm surprised at how similar the scale you posted is to Blue Temperament but like I said, this is coincidental and not by design.
>
> I'm going to test Blue Temperament and the scale you posted for number of good dyads. It will be interesting to see which has more.

-Mike

> To play the quibbler: I wouldn't say "Well temperament" is an erroneous translation from german.

I see. You want explanations. No problem. Let us go to the origin of the term: it was coined by Werckmeister who, in the same work, wrote that the keyboard shouldhave "eine Gute Temperatur" (in English: a Good Temperament). He went on referring to a keyboard tuned this way as "Wohltemperierte" (in English: Well-Tempered).

Accordingly, the proper way in English is to say that I tuned a keyboard in a Good Temperament, and that as a result I got a Well-Tempered keyboard.

(While I should instead refrain from saying that I tuned a keyboard in a Well Temperament and that as a result I got a Good-Tempered keyboard).

Kind regards,

Claudio

PS: Note, I recall the above having been discussed and agreed quite a few times (will try to find some references) including German and English native speakers.

Nothing to thank Jaques. Please note however that BOTH groups of circular temperaments are in wide use among early music players and ensembles (those specialising mainly in the German and Italian repertoire obviously use the related Good temperaments, while ensembles specialising in the French and English repertoire use the French temperaments.)

Claudio

> Thank you so much Claudio, and Hans, for these important precisions.
> (though I don't know now if I should say "Good temperament", or perhaps "Well-tempered tuning" ??)
> Just two complementary questions :
> Can we say that "temperaments ordinaires" = french temperaments ?
> and that "temperaments extraordinaires" (more rarely used but that I hear sometimes) = well temperered tunings ?
> thanks !
> - - - -
> Jacques
>
>
> --- In tuning@yahoogroups.com, "Claudio Di Veroli" <dvc@> wrote:
> >
> > Glad to bring some needed clarification. I transcribe below updated descriptions from my publications on the topic.
> >
> > "Well" Temperaments a grammatically-erroneous translation from German: the correct term is "Good" Temperaments. And they are NOT the same as Circular either. (Both errors were introduced by Jorgensen in 1977 and have been commented by scholars for decades).
> >
> > CIRCULAR temperaments are unequal 12-note temperaments that allow all the tonalities and intervals to be playable. Some are better than in Equal Temperament, others are instead quite dissonant but not reaching the "wolves" of meantone. These are enharmonic systems where every sharp functions as flat as well, though some accidentals function better as flats, others as sharps. Circular temperaments are grouped in two families: FRENCH and GOOD temperaments.
> >
> > FRENCH temperaments have meantone (almost always 1/4 S.c.) diatonic fifths, the others modified to close the Circle: this implies that a few fifths have to be wider than pure. These are extremely unequal circular temperaments. (well-known examples: Rameau and d'Alembert)
> >
> > GOOD temperaments show diatonic fifths that may vary in size (unlike meantone) and arranged so that there is no need for wide fifths to close the Circle: chromatic fifths are usually pure. These temperaments range from decidedly unequal to mildly so, almost equal. (well-known examples: Neidhardt's temperaments).
> >
> > Kind Regards
> >
> > Claudio
> >
> > http://temper.braybaroque.ie/
>

Thanks Mike.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Thu, Feb 10, 2011 at 2:34 PM, john777music <jfos777@...> wrote:
> >
> > Mike,
> >
> > I don't know what you mean by a chain going from Eb to G# or Db to F#. Can you explain? I thought chains were either circular or went on forever and covered all keys.
>
> Quarter-comma meantone does go on forever. The generating interval is
> a slightly flat 3/2. The original scale posted was 12 notes ranging
> from 3 fifths down to 9 fifths up. I sent you one ranging from 5
> fifths down to 7 fifths up again.
>
> > I was going by the scale Chris posted for me. Is the scale you posted just a mode of the scale I used and if so which note of the scale I used is the starting point of the scale you posted. Looking at it again it seems it can't be a mode of the scale I was given. If the scale you posted is not a mode then what is it and what is it called and how is it made?
>
> The scale I posted is in fact a mode of the scale Chris posted; I
> believe that the "C" in my scale is the "D" in Chris's scale.
>
> > I'm surprised at how similar the scale you posted is to Blue Temperament but like I said, this is coincidental and not by design.
> >
> > I'm going to test Blue Temperament and the scale you posted for number of good dyads. It will be interesting to see which has more.
>
> -Mike
>