Special temperaments

Three little questions for temperaments specialists.
Among tuning systems I came up to, many share this feature :
They are basically developped like linear temperaments, but integrate one transposition of the whole sequence by a specific interval (other than octave or equal divisions of octave and other than a interval produced by the first generator of course). The result is two parralel chains of a linear temperament, and I think Petr has also used this feature.
They can be considered as a subsets of planar temperaments, but I wanted to know if a special nomenclature existed (I call them myself "Ribbon temperaments"), and if some known emblematic temperaments show this specific 3D feature.

The other question concerning some of my investigations (that can also be found in Ethno2) is about temperaments that would use a certain generator, and after a certain number of reiterations would shift to another generator. I call them "Hybrid" temperaments.
In more particular cases they would come back to the starting point, and I call them "Double face temperaments". I have been using that on 12 tones keyboard mappings, where for example the white keys from F to B would use a meantone generator, that shifts on black keys, from B or F# back to Bb or F or whatever to a superpyth. (Multi-system EA7, NA6, SEA9). A WT like Comptine (1/4 pyth. comma meantone sequence from C to E, completed by 8 pure fifths) could also be considered a special case of those, where one of the generators is a perfect fifth.

Then I have a third case between the two precedents, where 2 different generators are used in succession according to some specific patterns, without necessarily making a circle (SEA3) - I have been calling those "Entrelacs" (= Interlace, Intertwine, Tracery ?).
Simple zig-zag alternance of the two generators would be of course equivalent to a "ribbon temperament" (where one of the generators is defined differently).
- - - - - - -
Jacques

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
>
> Three little questions for temperaments specialists.
> Among tuning systems I came up to, many share this feature :
> They are basically developped like linear temperaments, but integrate
> one transposition of the whole sequence by a specific interval (other
> than octave or equal divisions of octave and other than a interval
> produced by the first generator of course). The result is two
> parralel chains of a linear temperament, and I think Petr has also
> used this feature.
> They can be considered as a subsets of planar temperaments, but I
> wanted to know if a special nomenclature existed (I call them myself
> "Ribbon temperaments"), and if some known emblematic temperaments
> show this specific 3D feature.

This I would call a specific type of scale in a planar temperament. Mostly when people are talking about a ribbon type of scale they mean a scale for a rank two temperament like compton or mystery, and it seems to me the two types of ribbons probably have a lot in common in practice, if not in theory. Could you give some examples of generators for ribbons?

> The other question concerning some of my investigations (that can
> also be found in Ethno2) is about temperaments that would use a
> certain generator, and after a certain number of reiterations would
> shift to another generator. I call them "Hybrid" temperaments.
> In more particular cases they would come back to the starting point,
> and I call them "Double face temperaments".

So there are three generators--octaves, and the two generators generating scale steps? That would again be a specific type of scale in a rank three temperament.

> Then I have a third case between the two precedents, where 2
> different generators are used in succession according to some
> specific patterns, without necessarily making a circle (SEA3) - I
> have been calling those "Entrelacs" (= Interlace, Intertwine,
> Tracery ?).

That would describe a lot of rank three temperament scales, such a pairwise well-formed scales.

> Posted by: "Jacques Dudon" fotosonix@...   jacques.dudon
>
> Wed Jul 28, 2010 2:07 am (PDT)
> > >
> Three little questions for temperaments specialists.
> Among tuning systems I came up to, many share this feature :
> They are basically developped like linear temperaments, but integrate
> one transposition of the whole sequence by a specific interval (other
> than octave or equal divisions of octave and other than a interval
> produced by the first generator of course). The result is two
> parralel chains of a linear temperament, and I think Petr has also
> used this feature.

Hello, Jacques!

As it happens, what you are describing is the structure of some of
my favorite temperaments, including one that I'm planning to use
for lots of my "Margo Schulter's Ethno Extras" as you have titled
them, a title I accept with due humility, especially as lots of them
will be "Margo's Variations on Jacques Dudon's Ethno and Related
Tunings."

> They can be considered as a subsets of planar temperaments, but I
> wanted to know if a special nomenclature existed (I call them myself
> "Ribbon temperaments"), and if some known emblematic temperaments
> show this specific 3D feature.

While I've been very actively using these for about eight years, and just
in the last month and a bit more have devised a new one, I must admit
that I haven't come up with a good descriptive name for the genre, and
"Ribbon temperaments" sounds fine to me.

People sometimes speak of "Bicycle chain" temperaments, but this tends to
imply that each "chain" forms a circulating structure, often of an EDO
type (e.g. 36-EDO as three chains of 12-EDO spaced at 33-1/3 cents apart.

My first application of this principle, as I recall, was actually with
Pythagorean tuning: two 12-note chains (Eb-G#, evidently a typical
14th-century European tuning) at a 64:63 apart (27.26 cents), which
in effect meant an extended fifth at G#-D# connecting the two 12-note
chains, a fifth wide by about 3.804 cents, comparable to 17-EDO and
slightly wider than the basic generator in Aulos/Soria.

That was not, of course, a "temperament," and nor was Sesquisexta of
earlier 2001, two 12-note chains of 3:2 fifths at a pure 7:6 apart
(sesquisexta being the Latin term for a ratio of 7:6).

However, I would say that Peppermint is an emblematic example. The
2:1 octave is the period. The "Noble fifth" generator of 704.096 cents
was suggested by Keenan Pepper in September 2000: this is an extended
fifth which produces a temperament where the apotome or chromatic
semitone (128.669 cents) and limma or diatonic semitone (79.522 cents)
have the ratio of Phi between their logarithmic sizes.

In 2002, from Pepper's regular temperament, I developed the 24-note
Peppermint tuning, with two 12-note chains at 58.680 cents apart,
the different between the regular Pepper major second at 208.191 cents
and a pure 7:6 (266.071 cents). This arrangements gives a very
accurate division of 11:12:13:14, or as applied to a 4:3, something
like 33:36:39:42:44.

One way of viewing the ribbon structure (in Tuning list terminology,
planar or rank 3) is to note that each 12-note chain produces near-just
ratios of 11:13:14 (the 14:11 major third, 13:11 minor third, and 14:13
apotome), while the two chains together add 12:11, 13:12 -- and, of
course, the septimal intervals including the pure 7:6 that, less the
regular Pepper major second or tone, defines the third generator.

An interesting feature is that the "artificial diesis" or generator
defining the space between the two chains of this ribbon temperament,
58.680 cents, serves to represent both 28:27 (the septimal thirdtone
or semitone of Archytas, e.g. 9/8-7/6) and 33:32 (e.g. 12/11-9/8).

This temperament is available in the Scala scale archive as schulter_pepr.scl.

Here I should mention that George Secor's HTT-29, his "High Tolerance
Temperament" in 29 notes, makes use of a similar principle, and thus
certainly influenced me in applying the "ribbon" feature to Keenan
Pepper's regular temperament (earlier noted on Erv Wilson's Scale
Tree and as Golden Horagram 26). However, HTT-29 has other refinements
that make it more complex in structure than a simple "ribbon
temperament" of the kind we are discussing.

> The other question concerning some of my investigations (that can
> also be found in Ethno2) is about temperaments that would use a
> certain generator, and after a certain number of reiterations would
> shift to another generator. I call them "Hybrid" temperaments.
> In more particular cases they would come back to the starting point,
> and I call them "Double face temperaments". I have been using that on
> 12 tones keyboard mappings, where for example the white keys from F
> to B would use a meantone generator, that shifts on black keys, from
> B or F# back to Bb or F or whatever to a superpyth. (Multi-system
> EA7, NA6, SEA9). A WT like Comptine (1/4 pyth. comma meantone
> sequence from C to E, completed by 8 pure fifths) could also be
> considered a special case of those, where one of the generators is a
> perfect fifth.

This sounds like the kind of technique used in my Zest-24, or "Zarlino
Encompassing Spectrum Temperament" with two 12-note circles, each
with eight 2/7-comma meantone fifths (695.810 cents) at F-C#, and
four extended fifths of 708.379 cents at C#/Db-F. The two circles
are placed at the distance of 50.276 cents, the diesis of Zarlino's
regular 2/7-comma meantone.

Since this doesn't seem present in the most recent version of the
Scala archive I downloaded the other day, I'll post it here:

! zest24.scl
!
Zarlino Encompassing Spectrum Temperament (two circles at ~50.28c apart)
24
!
50.27584
25/24
120.94826
191.62069
241.89653
287.43104
337.70688
383.24139
433.51722
504.18965
554.46549
574.86208
625.13792
695.81035
746.08619
779.05173
829.32757
887.43104
937.70688
995.81035
1046.08619
1079.05173
48/25
2/1

There are lots of hemifourths, which I used in _La Pacifica_.

Gene Ward Smith has also designed a number of these "Hybrid"
temperaments, with Cauldron as one name that occurs to me.

> Then I have a third case between the two precedents, where 2
> different generators are used in succession according to some
> specific patterns, without necessarily making a circle (SEA3) - I
> have been calling those "Entrelacs" (= Interlace, Intertwine,
> Tracery ?).

While this equivalence is a new concept to me, I much like the term "Entrelacs" or "Interlace."

> Simple zig-zag alternance of the two generators would be of course
> equivalent to a "ribbon temperament" (where one of the generators is
> defined differently).
> - - - - - - -

From looking at your septimal_3.scl in Ethno2 (7 Indonesia), I think I may
understand this idea. And Entralacs or Interlace does seem very descriptive!
Let's see if some of my Ethno Extras fit this pattern.

With many thanks,

Margo