An alternative to Miracle

The following is, I think, important to know for anyone using 72-et.

We can define a sequence of Miracle scales, Mir(n), by octave reducing a
chain of n secors, leading to a scale with n+1 notes to the octave.
Similarly, let us define a sequence of quasi-Miracle scales, Qm(n), as a
triple chain of secors:
one chain of n secors, and then from both a fourth and a minor sixth
above the base of the chain, two other chains of n-1 secors, leading to
scales with 3n+1 notes to the octave. This has the effect of introducing
two triads bridging the secors in the original chain, so that we now have
a chain of connected triads.

Mir(9) and Qm(3) both have ten notes to an octave, and here is a
side-by-side:

Mir(9) [0, 7, 14, 21, 28, 35, 42, 49, 56, 63]

5-limit: 7 intervals 0 triads
7-limit: 22 intervals 10 triads
9-limit: 22 intervals 10 triads
11-limit: 30 intervals 32 triads

Qm(3) [0, 7, 14, 21, 30, 37, 44, 49, 56, 63]

5-limit: 15 intervals 6 triads
7-limit: 27 intervals 22 triads
9-limit: 31 intervals 36 triads
11-limit: 35 intervals 52 triads

The harmonic advantages of using Qm(3) instead of Mir(9) are clear from
this table, but I would maintain Qm(3) is melodically superior also,
though of course YMMV. Qm(3) however gets away from the bland smoothness
of Mir(9) to some extent, raising three of the degrees by a quomma,
whatever your feelings about that smoothness may be.

--- In tuning@y..., Gene W Smith <genewardsmith@j...> wrote:
> The following is, I think, important to know for anyone using 72-et.
>
> We can define a sequence of Miracle scales, Mir(n), by octave
reducing a
> chain of n secors, leading to a scale with n+1 notes to the octave.
> Similarly, let us define a sequence of quasi-Miracle scales, Qm(n),
as a
> triple chain of secors:
> one chain of n secors, and then from both a fourth and a minor sixth
> above the base of the chain, two other chains of n-1 secors, leading
to
> scales with 3n+1 notes to the octave. This has the effect of
introducing
> two triads bridging the secors in the original chain, so that we now
have
> a chain of connected triads.
>
> Mir(9) and Qm(3) both have ten notes to an octave, and here is a
> side-by-side:
>
> Mir(9) [0, 7, 14, 21, 28, 35, 42, 49, 56, 63]
>
> 5-limit: 7 intervals 0 triads
> 7-limit: 22 intervals 10 triads
> 9-limit: 22 intervals 10 triads
> 11-limit: 30 intervals 32 triads
>
> Qm(3) [0, 7, 14, 21, 30, 37, 44, 49, 56, 63]
>
> 5-limit: 15 intervals 6 triads
> 7-limit: 27 intervals 22 triads
> 9-limit: 31 intervals 36 triads
> 11-limit: 35 intervals 52 triads
>
> The harmonic advantages of using Qm(3) instead of Mir(9) are clear
from
> this table, but I would maintain Qm(3) is melodically superior also,
> though of course YMMV. Qm(3) however gets away from the bland
smoothness
> of Mir(9) to some extent, raising three of the degrees by a quomma,
> whatever your feelings about that smoothness may be.

I quite agree, both on harmonic and melodic properties. I don't think
anyone thinks the 10 note Miracle MOS is much use for anything except
a basis for miracle-specific notation. And it has been clear from the
early days of miracle that once you go smaller than blackjack (21
notes) there are these tunings with short segments of secor chains
spaced apart by fifths or major thirds that have many more harmonies
per note than contiguous miracle chains.

I see them as coming from that excellent planar temperament that
predated miracle, where the 224:225 septimal kleisma vanishes in
7-limit, or where both 224:225 and 384:385 vanish in 11-limit. I
wasn't previously aware of this melodically good 10-limit one (Qm(3)),
but apparently Carl Lumma was. I was however aware of 12 and 19 note
supersets of it.

Qm(3) can also be seen as a 72-ET detempering of Paul Erlich's
Asymmetrical decatonic in 22-ET.

--- In tuning@y..., Gene W Smith <genewardsmith@j...> wrote:

> The following is, I think, important to know for anyone using 72-et.

> Qm(3) [0, 7, 14, 21, 30, 37, 44, 49, 56, 63]
>
> 5-limit: 15 intervals 6 triads
> 7-limit: 27 intervals 22 triads
> 9-limit: 31 intervals 36 triads
> 11-limit: 35 intervals 52 triads

i believe dave keenan posted this scale last year -- this was either
in connection with tetrachordality, or as a partial detempering of my
decatonic scale (http://www-math.cudenver.edu/~jstarret/22ALL.pdf).

the scale *is* a subset of blackjack (how many transpositions of it
can you find in blackjack)? hence, i'd call it a *feature* of
miracle, rather than an *alternative* to miracle (just as the melodic
minor scale and other altered diatonic scales are a feature of
meantone, rather than being alternatives to meantone).

**********************************************************************

gene: just having caught up on the tuning-math list, it appears that
this scale is also a subset of the fokker/lumma(/keenan) scale, which
is a *specific* 12-tone scale in the planar temperament where 225:224
is tempered out. here's one possible tuning:

The 12-note Fokker-Lumma-Keenan scale:

0 115.58 200.05 268.80 384.39 499.97 584.44 700.03 815.61 884.36
968.83 1084.41 (1200)

fokker proposed a just intonation version in the 40s (?):

5/3-------5/4------15/8------45/32
/|\ /|\`. ,'/ \`. ,'/
/ | \ / | \15/14/---\45/28/
/ 7/6-------7/4 \ | / \ | /
/,' `.\ /,' `.\|/ \|/
4/3-------1/1-------3/2-------9/8

(click "message index" and then "expand messages" to view this
properly on the website)

dave keenan independently came up with the meantone (that is, linear
temperament, both 225:224 and 81:80 tempered out) version of this
scale:

Db Ab . . F C G D A E B F# . . D# A#.

there's a lot of discussion about it near the beginning of the
archives on this site, and/or near the end of the previous (mills)
archives . . . i may have been the first person to suggest 72-equal
for the non-meantone version of it . . .

here are some posts that mention fokker/lumma(/keenan):

/tuning/topicId_10730.html#10730

/tuning/topicId_4885.html#4885

/tuning/topicId_3539.html#3570

/tuning/topicId_1012.html#1273

/tuning/topicId_1213.html#1213

/tuning/topicId_1188.html#1188

/tuning/topicId_53.html#53

--- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> I
> wasn't previously aware of this melodically good 10-limit one (Qm
(3)),
[...]
> Qm(3) can also be seen as a 72-ET detempering of Paul Erlich's
> Asymmetrical decatonic in 22-ET.

dave, i'm pretty sure you posted this very scale some time last year -
- whether you were aware of it or not!

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:
> --- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > I
> > wasn't previously aware of this melodically good 10-limit one (Qm
> (3)),
> [...]
> > Qm(3) can also be seen as a 72-ET detempering of Paul Erlich's
> > Asymmetrical decatonic in 22-ET.
>
> dave, i'm pretty sure you posted this very scale some time last year
-
> - whether you were aware of it or not!

Oh yeah. So I did. Must be Alzheimer's. Thanks Paul. See
/tuning/topicId_27221.html#27221

I actually give both of what Gene calls Qm(3) 10 note and Qm(2) 7
note.

Thanks Gene, for reminding us about these and pointing out 11-limit
harmonic resources that I missed. Maybe these scales are the best to
be considered the miracle "white note" scales.

--- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:
> > --- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> >
> > > I
> > > wasn't previously aware of this melodically good 10-limit one
(Qm
> > (3)),
> > [...]
> > > Qm(3) can also be seen as a 72-ET detempering of Paul Erlich's
> > > Asymmetrical decatonic in 22-ET.
> >
> > dave, i'm pretty sure you posted this very scale some time last
year
> -
> > - whether you were aware of it or not!
>
> Oh yeah. So I did. Must be Alzheimer's.

naah . . . i'm probably the freak, with an insane capacity for
recalling precise numerical constructions such as obscure scales in
72-tET, while having only the faintest idea of what's going on in the
world directly around me . . .

> Thanks Paul. See
> /tuning/topicId_27221.html#27221
>
> I actually give both of what Gene calls Qm(3) 10 note and Qm(2) 7
> note.

whoa . . . that i did not catch!

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:
> --- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> > Thanks Paul. See
> > /tuning/topicId_27221.html#27221
> >
> > I actually give both of what Gene calls Qm(3) 10 note and Qm(2) 7
> > note.
>
> whoa . . . that i did not catch!

Qm(2) turns out to be

! xenakis_schrom.scl
!
Xenakis's Byzantine Liturgical mode, 7 + 16 + 7 parts

which is practically indistinguishable from

! al-farabi_chrom2.scl
!
Al-Farabi's Chromatic permuted
7
!
16/15 56/45 4/3 3/2 8/5 28/15 2/1

Using Gene's terminology, in 72-EDO, Qm(6) is of course the same as
Mir(18), a 19 note scale that still fits in blackjack. The secor
chains run into each other.

But beyond 10 notes, I suspect it turns out to be better not to
continue growing the chains, but to start more secor-chains a
fifth/minor-sixth from the others.

Define
Mir(n) = QM(0,n)
Qm(n) = QM(2,n)

Where the first parameter is one less than the number of secor chains,
and the second is the number of secors in the longest chain.

Of course this can be turned sideways and looked at instead as a
number of chains of fourths (or fifths) spaced a secor apart. In this
case the first parameter can be read as the number of fifths in the
longest chains and the second parameter as the number of chains of
fifths.

Just what is the optimum way to choose these two parameters? This is
something that needs to be explored.

We know how to calculate the cardinalities of the most even scales in
_linear_ temperaments while standing on our heads, but I haven't a
clue how we do that for planar temperaments except by trial and error.
This relates to an earlier discussion on "Trihills property".

Similarly for the number of harmonies. This is easy for linear, but
other than counting them, I don't have a handle on this for the
various subsets of planar temperaments, except it seems they should be
as close to "circular" as possible. But how to scale the axes?

Another data point for this investigation is the 19-tone Byzantine
tetrachord superset scale that I derived from Rami Vitale's 23 note
rational one. It might be called QM(4,3). The middle chain has 3 notes
while the other 4 chains have 4.

I suspect QM(1,2) and QM(1,3) will be harmonically and melodically
good 6 and 8 note scales/chords. QM(0,30) ?= QM(4,5) may be another
optimum.

Unfortunately I don't have time to investigate this, so I hope you do
Gene.

P.S. It seems like we should call this the Byzantine planar
temperament.

P.P.S. This feels like coming full-circle. My first post to the list
many years ago involved the 12 note scale in 31-EDO, written on a
chain of fifths, as

Db Ab . . F C G D A E B F# . . D# A#

The possibility has existed all along, that it could also be written
on a chain of secors, as

A . . . . D# E F . . . A# B C Db . . . F# G Ab . . .
. D

and is then seen to be in this "Byzantine" planar temperament.

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:

> the scale *is* a subset of blackjack (how many transpositions of it
> can you find in blackjack)? hence, i'd call it a *feature* of
> miracle, rather than an *alternative* to miracle (just as the melodic
> minor scale and other altered diatonic scales are a feature of
> meantone, rather than being alternatives to meantone).

I wasn't thinking of the scale as an alternative to miracle, but of the construction--replacing a chain of secors with a chain of "segars", consisting of four notes and two triads. If you do a side-by-side, these seem to become similar at higher values, but the
Qm construction starts out with an advantage. Here is Mir(3n) compared to Qm(n) for n from 1 to 6, in the 7-limit:

Intervals: 5, 15, 27, 41, 57, 75 vs 2, 8, 22, 37, 73

Triads: 1, 10, 22, 34, 48, 70 vs 0, 0, 10, 22, 46, 70

Is there a reason to prefer the miracle rather than the qm alternative? It seems like a good alternative, at any rate.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:

> Intervals: 5, 15, 27, 41, 57, 75 vs 2, 8, 22, 37, 73

Should be 5, 15, 27, 41, 57, 73; as Dave points out, Qm(6) and Mir(18)
are identical.

--- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> P.S. It seems like we should call this the Byzantine planar
> temperament.

Are you proposing this as a name for the 7-limit planar temperament with 225/224 as a comma?

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > P.S. It seems like we should call this the Byzantine planar
> > temperament.
>
> Are you proposing this as a name for the 7-limit planar temperament
with 225/224 as a comma?

As the comma that vanishes, yes. As the unison vector. But more
generally as a name for the practically indistinguishable 11-limit
planar temperament where both 224:225 and 384:385 vanish.

--- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> > Are you proposing this as a name for the 7-limit planar temperament
> with 225/224 as a comma?
>
> As the comma that vanishes, yes. As the unison vector. But more
> generally as a name for the practically indistinguishable 11-limit
> planar temperament where both 224:225 and 384:385 vanish.

In this temperament, in either version, the Qm(n) series keeps rolling along, and does not merge with Miracle. Whether the improvement in the already excellent tuning is worth the detempering is one of those fun questions, but it certainly would provide an "alternative to Miracle" in a sense that Qm(n) in its linear temperament incarnation doesn't do. With wedge products I have the math tools to investigate it, and may report on it sometime.

--- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

/tuning/topicId_37039.html#37116

> We know how to calculate the cardinalities of the most even scales
in _linear_ temperaments while standing on our heads, but I haven't a
> clue how we do that for planar temperaments except by trial and
error.
> This relates to an earlier discussion on "Trihills property".
>
> Similarly for the number of harmonies. This is easy for linear, but
> other than counting them, I don't have a handle on this for the
> various subsets of planar temperaments, except it seems they should
be as close to "circular" as possible. But how to scale the axes?
>

***This is terribly interesting, but I'm not getting it. I looked up
the difference between "linear" and "planar" temperaments, and have
just a "glimmer..." but could somebody please back up with a few
examples so I can understand this better, if you have time??

Thanks so much!

J. Pehrson

--- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> P.P.S. This feels like coming full-circle. My first post to the
list
> many years ago involved the 12 note scale in 31-EDO, written on a
> chain of fifths, as
>
> Db Ab . . F C G D A E B F# . . D# A#
>
> The possibility has existed all along, that it could also be
written
> on a chain of secors, as
>
> A . . . . D# E F . . . A# B C Db . . . F# G
Ab . . .
> . D
>
> and is then seen to be in this "Byzantine" planar temperament.

dave, did you see this from yesterday:

/tuning/topicId_37039.html#37102

?

you'll see that i quote your very same 12-out-of-31 scale, and tie it
in with the fokker/lumma scale, which i said uses the planar
temperament where 225:224 is tempered out. this all came up because
carl lumma noticed (on tuning-math) that qm(3) is a subset of the
fokker/lumma scale.

in your last few messages, were you trying to say that you'd like to
call this planar temperament where 225:224 is tempered
out, "byzantine"?

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:
>
> > the scale *is* a subset of blackjack (how many transpositions of
it
> > can you find in blackjack)?

no one answered this.

> > hence, i'd call it a *feature* of
> > miracle, rather than an *alternative* to miracle (just as the
melodic
> > minor scale and other altered diatonic scales are a feature of
> > meantone, rather than being alternatives to meantone).
>
> I wasn't thinking of the scale as an alternative to miracle, but
>of the construction

gotcha.

>--replacing a chain of secors with a chain
>of "segars",

where does that term come from?

> consisting of four notes and two triads.

can you explicate this construction again?

> If you do a side-by-side, these seem to become similar at higher
values, but the
> Qm construction starts out with an advantage. Here is Mir(3n)
compared to Qm(n) for n from 1 to 6, in the 7-limit:
>
> Intervals: 5, 15, 27, 41, 57, 75 vs 2, 8, 22, 37, 73
>
> Triads: 1, 10, 22, 34, 48, 70 vs 0, 0, 10, 22, 46, 70
>
> Is there a reason to prefer the miracle rather than the qm
>alternative? It seems like a good alternative, at any rate.

sho'nuff!

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:

> ***This is terribly interesting, but I'm not getting it. I looked
up
> the difference between "linear" and "planar" temperaments, and have
> just a "glimmer..." but could somebody please back up with a few
> examples so I can understand this better, if you have time??
>
> Thanks so much!
>
> J. Pehrson

a linear temperament is, essentially, generated by a single interval -
- examples: meantone, miracle. there is an additional interval that
acts as a generator, but is usually thought of as the "interval of
repetition" or "period" instead, and this is usually the octave, but
occasionally turns out to be a half-octave, third-octave, etc., or
a "tritave", etc. all the notes in the temperament can be expressed
in a unique way as an integer number of generators, plus an integer
number of "periods", from an arbitrary starting point.

a planar temperament has _two independent generators_ in addition to
the interval of repetition. 5-limit ji is an example of this -- your
two generators are the just perfect fifth and the just major third,
for example, and the two are not derivable from one another. any note
in 5-limit ji can be expressed in a unique way as an integer number
of perfect fifths, plus an integer number of major thirds, plus an
integer number of octaves, from an arbitrary starting point (usually
denoted 1/1). now, 5-limit ji is not really a temperament (it's a
planar *tuning*), so you normally have to go beyond the 5-limit to
find examples of planar temperament. they haven't come up too much.

the example we're talking about here (225:224-vanishing planar
temperament) is not too hard to understand. basically, the perfect
fifth and major third remain pretty close to just, and still can't be
expressed in terms of one another. but one or both of them is
slightly adjusted so that two perfect fifths plus two major thirds
(225:128) moves closer to, or all the way to, 7:4 (224:128). thus you
have a 7-limit tuning, but instead of the three independent
generators (plus the octave) that you need to generate 7-limit ji,
you only need two independent generators. this is still more than the
one generator that suffices for a linear temperament such as meantone
or miracle -- hence this is a planar temperament, rather than a
linear temperament.

still foggy?

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:

/tuning/topicId_37039.html#37133

> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
>
> > ***This is terribly interesting, but I'm not getting it. I
looked
> up
> > the difference between "linear" and "planar" temperaments, and
have
> > just a "glimmer..." but could somebody please back up with a few
> > examples so I can understand this better, if you have time??
> >
> > Thanks so much!
> >
> > J. Pehrson
>
> a linear temperament is, essentially, generated by a single
interval -
> - examples: meantone, miracle. there is an additional interval that
> acts as a generator, but is usually thought of as the "interval of
> repetition" or "period" instead, and this is usually the octave,
but
> occasionally turns out to be a half-octave, third-octave, etc., or
> a "tritave", etc. all the notes in the temperament can be expressed
> in a unique way as an integer number of generators, plus an integer
> number of "periods", from an arbitrary starting point.
>
> a planar temperament has _two independent generators_ in addition
to
> the interval of repetition. 5-limit ji is an example of this --
your
> two generators are the just perfect fifth and the just major third,
> for example, and the two are not derivable from one another. any
note
> in 5-limit ji can be expressed in a unique way as an integer number
> of perfect fifths, plus an integer number of major thirds, plus an
> integer number of octaves, from an arbitrary starting point
(usually
> denoted 1/1). now, 5-limit ji is not really a temperament (it's a
> planar *tuning*), so you normally have to go beyond the 5-limit to
> find examples of planar temperament. they haven't come up too much.
>
> the example we're talking about here (225:224-vanishing planar
> temperament) is not too hard to understand. basically, the perfect
> fifth and major third remain pretty close to just, and still can't
be
> expressed in terms of one another. but one or both of them is
> slightly adjusted so that two perfect fifths plus two major thirds
> (225:128) moves closer to, or all the way to, 7:4 (224:128). thus
you
> have a 7-limit tuning, but instead of the three independent
> generators (plus the octave) that you need to generate 7-limit ji,
> you only need two independent generators. this is still more than
the
> one generator that suffices for a linear temperament such as
meantone
> or miracle -- hence this is a planar temperament, rather than a
> linear temperament.
>
> still foggy?

****Thanks so much, Paul. That's a great explanation, and perfectly
clear! I understood about the two different *dimensions* of 5-limit
JI from your colorful paper, _Forms of Tonality_ and how that
differed from meantone, but I never thought about what would happen
if you were to "adjust" the *two* different axes simultaneously!

Thanks!

Joseph

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > > Are you proposing this as a name for the 7-limit planar
temperament
> > with 225/224 as a comma?
> >
> > As the comma that vanishes, yes. As the unison vector. But more
> > generally as a name for the practically indistinguishable 11-limit
> > planar temperament where both 224:225 and 384:385 vanish.
>
> In this temperament, in either version, the Qm(n) series keeps
rolling along, and does not merge with Miracle.
>

Sure, in general. But they are all very well represented in 72-ET,
where they do meet up. I still refer to the QMs as being in the planar
temperament when they are in 72-ET, because their boundaries are best
understood as two dimensional; as bounds on the number of fifths and
the number of secors.

> Whether the
improvement in the already excellent tuning is worth the detempering
is one of those fun questions, but it certainly would provide an
"alternative to Miracle" in a sense that Qm(n) in its linear
temperament incarnation doesn't do. With wedge products I have the
math tools to investigate it, and may report on it sometime.
>

True, but that wasn't what I meant. It probably isn't worth the
trouble of detempering them outside of 72-ET, although I made a
particular recommendation of how best to do so, some years ago.

What I'd like to know is how the melodic evenness and number of
harmonies relates to the size and shape of the 2D boundary, in the way
that we know how it works for linear temperaments with their 1D
boundary.

--- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> Sure, in general. But they are all very well represented in 72-ET,
> where they do meet up. I still refer to the QMs as being in the
planar
> temperament when they are in 72-ET, because their boundaries are
best
> understood as two dimensional; as bounds on the number of fifths
and
> the number of secors.

aren't these scales perfect examples of 'hyper-MOS' scales (as
defined in the discussion of the Hypothesis) -- periodicity blocks
(possibly of the Fokker variety) where more than one unison vector
remains un-tempered-out? also, didn't gene post a considerable bunch
of such scales a while back -- under rubrics such as "meantone-
miracle"?

> What I'd like to know is how the melodic evenness and number of
> harmonies relates to the size and shape of the 2D boundary, in the
way
> that we know how it works for linear temperaments with their 1D
> boundary.

it's time to delve into this hyper-mos stuff in earnest. thanks,
carl, for inspiring this terminology (though i recall mats oljare
proposing something equivalent, 2nd-order MOS or some such) . . .

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:

/tuning/topicId_37039.html#37132

> --- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > --- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:
> >
> > > the scale *is* a subset of blackjack (how many transpositions
of
> it
> > > can you find in blackjack)?
>
> no one answered this.
>
> > > hence, i'd call it a *feature* of
> > > miracle, rather than an *alternative* to miracle (just as the
> melodic
> > > minor scale and other altered diatonic scales are a feature of
> > > meantone, rather than being alternatives to meantone).
> >
> > I wasn't thinking of the scale as an alternative to miracle, but
> >of the construction
>
> gotcha.
>
> >--replacing a chain of secors with a chain
> >of "segars",
>
> where does that term come from?
>

***My presumption was that the "segar" is the generating interval for
the "Groucho" temperament, a very bad temperament on a scale
of "badness..."

JP

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:
> dave, did you see this from yesterday:
>
> /tuning/topicId_37039.html#37102
>
> ?
>
> you'll see that i quote your very same 12-out-of-31 scale, and tie
it
> in with the fokker/lumma scale, which i said uses the planar
> temperament where 225:224 is tempered out. this all came up because
> carl lumma noticed (on tuning-math) that qm(3) is a subset of the
> fokker/lumma scale.

Yes, I saw it. That's what made me want to show that scale on a chain
of secors. Sorry I didn't reference your post, or Carl's. Thanks for
finding all those old posts.

> in your last few messages, were you trying to say that you'd like to
> call this planar temperament where 225:224 is tempered
> out, "byzantine"?

Yes. Where "Byzantine" has the meaning: Pertaining to the style of
music of the Eastern Church. The term should also cover the 11-limit
planar temperament where 224:225 and 384:385 are tempered out.

When I claim that a certain subset of 72-ET belongs to this planar
temperament, I particularly mean that when the notes of 72-ET are
arranged in a square 2D lattice with fifths in the horizontal
direction and secors in the vertical, then the scale in question is an
almost rectangular (typically almost square) block. Such rectangles
seem to contain an inordinate number of Byzantine tetrachords.

So it's the particular choice of the fifth (or fourth) and the secor
(or major seventh) as the generators that I think of as Byzantine.

This brings up the question of whether a radically different choice of
generators should be considered a different planar temperament, even
if they have the same unison vectors. Or it it merely that every
planar temperament can be shown to have an optimum choice of
generators (as far as scale boundaries go) and fifths and secors is it
for this one?

Rami Vitale's first post on the subject of Byzantine supersets is
/tuning/topicId_27464.html#27464

In it he claims (and later shows) that five Byzantine heptatonics
(which he expresses in 7-limit JI) are all contained in a 23 note
7-limit tuning that he has constructed.

When the septimal kleisma (224:225) is tempered out (which Rami
objected to), this becomes a 19 note scale that fits into a 5 * 5
square on the abovementioned fifths * secors lattice.

Maybe some tetrachords considered essential to Byzantine liturgy are
not contained in a sufficiently small rectangle on this lattice, in
which case it might not be deserving of the name. Does anyone know?

I guess they are all in the Scala archive, but this gives no sense of
which are more important. Are you reading, John Chalmers?

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:
> --- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>
> > Sure, in general. But they are all very well represented in 72-ET,
> > where they do meet up. I still refer to the QMs as being in the
> planar
> > temperament when they are in 72-ET, because their boundaries are
> best
> > understood as two dimensional; as bounds on the number of fifths
> and
> > the number of secors.
>
> aren't these scales perfect examples of 'hyper-MOS' scales (as
> defined in the discussion of the Hypothesis) -- periodicity blocks
> (possibly of the Fokker variety) where more than one unison vector
> remains un-tempered-out?

Almost certainly.

> also, didn't gene post a considerable bunch
> of such scales a while back -- under rubrics such as "meantone-
> miracle"?

Quite possibly.

> > What I'd like to know is how the melodic evenness and number of
> > harmonies relates to the size and shape of the 2D boundary, in the
> way
> > that we know how it works for linear temperaments with their 1D
> > boundary.
>
> it's time to delve into this hyper-mos stuff in earnest. thanks,
> carl, for inspiring this terminology (though i recall mats oljare
> proposing something equivalent, 2nd-order MOS or some such) . . .

Hoorah. I get the feeling we should move it to tuning-math.

I wonder how big a rectangle Partch's Genesis scale needs on the
fifths*secors grid? 9*5? No time to check it out myself just now.

--- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> When I claim that a certain subset of 72-ET belongs to this planar
> temperament, I particularly mean that when the notes of 72-ET are
> arranged in a square 2D lattice with fifths in the horizontal
> direction and secors in the vertical, then the scale in question is
an
> almost rectangular (typically almost square) block.

yes, gene was producing quite a few such square 2D lattice scales
quite some time ago. i searched for "miracle-meantone" and "meantone-
miracle" but couldn't find them. gene?

--- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> This brings up the question of whether a radically different choice of
> generators should be considered a different planar temperament, even
> if they have the same unison vectors. Or it it merely that every
> planar temperament can be shown to have an optimum choice of
> generators (as far as scale boundaries go) and fifths and secors is it
> for this one?

I've found that using the Hermite reduction of the generator matrix works very well for suggesting how to find scales--it generalizes the idea of "bearings" I posted on a while back. In essence, the idea is to stick as closely as possible to 2, 3 and 5 for the generators, and then plot the consonant intervals one is interested in (a single matrix multiplication finds the coordinates expeditiously.) I think I'll go into this on the math list.

--- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

/tuning/topicId_37039.html#37138

> This brings up the question of whether a radically different choice
of generators should be considered a different planar temperament,
even if they have the same unison vectors. Or it it merely that every
> planar temperament can be shown to have an optimum choice of
> generators (as far as scale boundaries go) and fifths and secors is
it for this one?
>

***Does this mean that two different "planar" temperaments are
actually eliminating the same unison vectors?? That means that they
are both, essentially, the *same* temperament, yes/no??

If so, does it really make any difference in the *means* to the
results, or should only the *final* result be considered?

Am I understanding this at all?? It's pretty interesting, the
glimmer of it that I'm getting, anyway.

Joseph