The 31-et 11-limit heptads

These are obviously more in the nature of scales than chords, but whatever you want to call them they are very interesting.

[0, 5, 9, 13, 18, 23, 27]

[1, 9/8, 11/9, 4/3, 3/2, 5/3, 11/6]
[9/8, 1, 11/10, 6/5, 4/3, 3/2, 18/11]
[11/9, 11/10, 1, 11/10, 11/9, 11/8, 3/2]
[4/3, 6/5, 11/10, 1, 9/8, 5/4, 11/8]
[3/2, 4/3, 11/9, 9/8, 1, 9/8, 11/9]
[5/3, 3/2, 11/8, 5/4, 9/8, 1, 11/10]
[11/6, 18/11, 3/2, 11/8, 11/9, 11/10, 1]

[0, 5, 9, 14, 18, 23, 27]

[1, 9/8, 11/9, 11/8, 3/2, 5/3, 11/6]
[9/8, 1, 11/10, 11/9, 4/3, 3/2, 18/11]
[11/9, 11/10, 1, 9/8, 11/9, 11/8, 3/2]
[11/8, 11/9, 9/8, 1, 11/10, 11/9, 4/3]
[3/2, 4/3, 11/9, 11/10, 1, 9/8, 11/9]
[5/3, 3/2, 11/8, 11/9, 9/8, 1, 11/10]
[11/6, 18/11, 3/2, 4/3, 11/9, 11/10, 1]

In-Reply-To: <a22e1o+vqal@eGroups.com>
genewardsmith wrote:

> These are obviously more in the nature of scales than chords, but
> whatever you want to call them they are very interesting.
>
> [0, 5, 9, 13, 18, 23, 27]

5 4 4 5 5 4 4

> [0, 5, 9, 14, 18, 23, 27]

5 4 5 4 5 4 4

See <http://x31eq.com/7plus3.htm#7note>.

Graham

--- In tuning@y..., graham@m... wrote:
> In-Reply-To: <a22e1o+vqal@e...>
> genewardsmith wrote:
>
> > These are obviously more in the nature of scales than chords, but
> > whatever you want to call them they are very interesting.
> >
> > [0, 5, 9, 13, 18, 23, 27]
>
> 5 4 4 5 5 4 4

This is the Arabic Diatonic.

>
> > [0, 5, 9, 14, 18, 23, 27]
>
> 5 4 5 4 5 4 4

This is a mode of Mohajira.

So these are 11-limit SSSs in 31-tET? Fascinating. Of course, the
fact that certain 'consonant' ratios are conflated in 31-tET (that
is, 31-tET is not unique in the 11-limit) weakens this somewhat.

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

> So these are 11-limit SSSs in 31-tET? Fascinating.

I think so; I've been playing with it and it works nicely.

Of course, the
> fact that certain 'consonant' ratios are conflated in 31-tET (that
> is, 31-tET is not unique in the 11-limit) weakens this somewhat.

It doesn't weaken it so far as the general effect goes, and I think the versions which conflate something may be the most interesting.

We have:

81/80 & 100/99, so that 9/8~10/9~11/10:
7, 12, 19, 26, 45

81/80 & 121/100, so that 9/8~10/9 and 11/10~12/11:
7, 24, 31, 38, 55

100/99 & 121/100, so that 10/9~11/10~12/11:
7, 8, 15, 22, 29, 37, 51, 59, 66

We can even conflate 28/25 and 10/9, by for example 126/125 & 121/120:
15, 16, 31, 46, 70, 77, 101

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> > So these are 11-limit SSSs in 31-tET? Fascinating.
>
> I think so; I've been playing with it and it works nicely.
>
> Of course, the
> > fact that certain 'consonant' ratios are conflated in 31-tET
(that
> > is, 31-tET is not unique in the 11-limit) weakens this somewhat.
>
> It doesn't weaken it so far as the general effect goes,

It depends on the example -- if there's enough otonality to go
around, as it were, then you're right; however, for example, a dyad
like 600 cents alone, though a 'consonance' in tunings where 50:49
vanishes, is nothing special without a (probably otonal) context in
which its ratio-intepretation is clarified.

If we look at the ets which have a version of the Arabic Diatonic which is 11-limit consonant relative to the tuning of the et (which also means they also have a "consonant" Mohajira) we obtain
10, 13, 17, 24, 31, 38, and 55. The only condition I put on "version of" is that the large step is larger than the small step. Arabic music has been associated with both 17 and 24, but the 11-limit consonance is better in 31 and 55, and I wonder what the result of retuning any availble examples (and where might these be found?) would be.

I've tried combing the net for Arabic, Turkish etc. music, but I only find pop sort of stuff which doesn't interest me. It would be nice to know the best resources in this area.

Here's what these scales look like in the various ets:

10

[0, 2, 3, 4, 6, 8, 9]
[0, 2, 3, 5, 6, 8, 9]

13

[0, 3, 4, 5, 8, 11, 12]
[0, 3, 4, 7, 8, 11, 12]

17

[0, 3, 5, 7, 10, 13, 15]
[0, 3, 5, 8, 10, 13, 15]

24

[0, 4, 7, 10, 14, 18, 21]
[0, 4, 7, 11, 14, 18, 21]

31

[0, 5, 9, 13, 18, 23, 27]
[0, 5, 9, 14, 18, 23, 27]

38

[0, 6, 11, 16, 22, 28, 33]
[0, 6, 11, 17, 22, 28, 33]

55

[0, 9, 16, 23, 32, 41, 48]
[0, 9, 16, 25, 32, 41, 48]

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> If we look at the ets which have a version of the Arabic Diatonic
which is 11-limit consonant relative to the tuning of the et (which
also means they also have a "consonant" Mohajira) we obtain
> 10, 13, 17, 24, 31, 38, and 55. The only condition I put
on "version of" is that the large step is larger than the small step.
Arabic music has been associated with both 17 and 24, but the 11-
limit consonance is better in 31 and 55, and I wonder what the result
of retuning any availble examples (and where might these be found?)
would be.
>
> I've tried combing the net for Arabic, Turkish etc. music, but I
only find pop sort of stuff which doesn't interest me.

Well, if you're looking for anything that explores many of the
available dyads, you'll be disappointed. Arabic music is melodic, not
harmonic.

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

> Well, if you're looking for anything that explores many of the
> available dyads, you'll be disappointed. Arabic music is melodic, not
> harmonic.

I was thinking of suggesting that the planar temperament of 81/80 and
121/120 be called "Arabic", but perhaps that would not be a good name? By "not harmonic" do you perhaps mean that there is a tendency to treat everything as a consonance, so to speak? I find it a little hard to believe that the appearance of the Arabic diatonic and Mohajira out of an investigation which was purely one of harmony is a complete accident. How do you explain it?

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> > Well, if you're looking for anything that explores many of the
> > available dyads, you'll be disappointed. Arabic music is melodic,
not
> > harmonic.
>
> I was thinking of suggesting that the planar temperament of 81/80
and
> 121/120 be called "Arabic",

Shouldn't the two-step-size nature of the scales be associated with a
linear, not planar, temperament?

>but perhaps that would not be a good >name? By "not harmonic" do you
perhaps mean that there is a tendency >to treat everything as a
consonance, so to speak?

Not really -- it's just that there is no harmony harmonizing the
melody, aside from, in certain styles, a tonic drone. No simultaneous
use of most of the possible pairs of tones.

>I find it a little hard to believe that the appearance of the Arabic
>diatonic and Mohajira out of an investigation which was purely one
>of harmony is a complete accident.

I don't find it that hard to believe -- the only accident is that the
11:9 nearly bisects the 3:2. And I don't think many 'conflated'
intervals, or even non-conflated, just intervals beyond the 7- or 9-
limit, are sufficiently consonant to 'explain' anything outside of a
clear otonal-chord context. In fact they often occur at 'peaks' of
dyadic dissonance.

Gene!
First i thought you might enjoy this document given to me by Erv when i was first working in 31 in the 70's
http://www.anaphoria.com/harmin31.gif
it was within this tuning that i first started experimenting with the Eikosany. After meeting with Ben Johnson at Webster College in St. Louis I came back and tuned it up just.
I agree with Erv that 31 is one the the best tunings for beginners in that there is enough familiar material from which to explore from. I will try to put up more 31 tone material if there is any interest

It appears that cultures that chose to develop in the direction of melody seem to develop more refinement in their intonations, hence Just intonation is often the result. Those based on harmony realize that developing along the line of temperments and/or dual or multiple meanings of it tones allows more musical possibilities.

genewardsmith wrote:

> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> > Well, if you're looking for anything that explores many of the
> > available dyads, you'll be disappointed. Arabic music is melodic, not
> > harmonic.
>
> I was thinking of suggesting that the planar temperament of 81/80 and
> 121/120 be called "Arabic", but perhaps that would not be a good name? By "not harmonic" do you perhaps mean that there is a tendency to treat everything as a consonance, so to speak? I find it a little hard to believe that the appearance of the Arabic diatonic and Mohajira out of an investigation which was purely one of harmony is a complete accident. How do you explain it?
>
>

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

--- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> Shouldn't the two-step-size nature of the scales be associated with a
> linear, not planar, temperament?

Would you buy the temperament of 81/80, 121/120 and 6144/6125 as
"Arabic"? It is compatible with 7, 24, 31, 38 and 55.

And I don't think many 'conflated'
> intervals, or even non-conflated, just intervals beyond the 7- or 9-
> limit, are sufficiently consonant to 'explain' anything outside of a
> clear otonal-chord context.

I've been playing with the 31-et version of Arabic diatonic and it seems pretty good, concordance-wise.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
> >
> > Shouldn't the two-step-size nature of the scales be associated
with a
> > linear, not planar, temperament?
>
> Would you buy the temperament of 81/80, 121/120 and 6144/6125 as
> "Arabic"? It is compatible with 7, 24, 31, 38 and 55.

Maybe get rid of the 5-axis altogether -- would it then be just 81/80
and 121/120, and compatible with 17 as well? What's the generator,
the neutral third?

So you liked it concordance-wise. I thought you thought 11:9s were
discordant. And how about those 16:11s?

I wrote,

> Maybe get rid of the 5-axis altogether -- would it then be just
81/80
> and 121/120,

Obviously not, as these are ratios involving 5!

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

> Maybe get rid of the 5-axis altogether -- would it then be just 81/80
> and 121/120, and compatible with 17 as well? What's the generator,
> the neutral third?

I like all the nice 5-harmony also. As I explain over on tuning-math,
81/80, 121/120 and 176/175 define a temperament with a neutral third generator which I propose to call the "Arabic".

> So you liked it concordance-wise. I thought you thought 11:9s were
> discordant. And how about those 16:11s?

What I said was that they seemed a tad more dissonant that 12/7s, but actually I tend to treat 11/9, 12/7 and 11/8 all as more or less consonant. How about you?

In-Reply-To: <a25tfq+mdgg@eGroups.com>
Gene wrote:

> Would you buy the temperament of 81/80, 121/120 and 6144/6125 as
> "Arabic"? It is compatible with 7, 24, 31, 38 and 55.

Add 2, 10 and 17 to that list, and you have the start of the septimally
double-positive family of ETs. So you could call this "the septimally
double-positive temperament".

I'm generally against non-Arabs defining anything as "Arabic". What
happens when you want to describe real Arabic music? At least, you should
make sure "Arabic" is always placed in scare-quotes.

Ignoring the 7-approximation, I've been calling these "meantone-like
neutral third scales". That's because they combine the syntonic comma
81:80 with the 11-limit neutral third comma 243:242. This doesn't cover
the approximation to 7. But then it doesn't figure in the neutral-third
diatonics (the mohajira- and rast-like 7 note scales).

So, my preferences for terminology:

"Neutral third scales" for neutral third MOS scales, or the 7-note scales
that are subsets of the 10-note neutral third MOS and contain the same
steps as the 7-note neutral third MOS but in a different order.

"Septimally double-positive scales" as a rough synonym for "neutral third
scales" but without the implication that a neutral third be treated as an
approximation to 11:9.

"Meantone-like neutral third scales/temperament" for scales with a neutral
third of 11:9 that also temper out 81:80.

"Septimally double-positive temperament" for the temperament common to 24,
31, 38 and 55-equal, including the approximation to 7.

"Neutral third diatonics" for the 7-note neutral third scales, implying
the meantone-like approximation.

"Mohajira-like" and "rast-like" for the 7-note neutral third MOS and the
other 7-note neutral third scale respectively, regardless of the tonic.

"7++" pronounced "seven plus plus" as a synonym for "septimally
double-positive".

"Arabic scales" for scales actually being used in Arabic music.

> I've been playing with the 31-et version of Arabic diatonic and it
> seems pretty good, concordance-wise.

The 24-equal version may be better, I've never really decided. The fifths
are better, and help reign in everything else. The neutral thirds are
also good. The major and minor thirds are no worse than people are used
to. The 11:8 is better, and the neutral second becomes a clear 12:11
instead of a wishy-washy compromise between 12:11 and 11:10. As I said
above, you don't need 7.

Graham

In-Reply-To: <memo.18765@cix.compulink.co.uk>
Gene:
> > Would you buy the temperament of 81/80, 121/120 and 6144/6125 as
> > "Arabic"? It is compatible with 7, 24, 31, 38 and 55.

Me:
> Add 2, 10 and 17 to that list, and you have the start of the septimally
> double-positive family of ETs. So you could call this "the septimally
> double-positive temperament".

Actually, 55 isn't 7++ which does weaken the connection. So you could
call this the "meantone-like neutral third temperament" as 7 will usually
be approximated that way.

> The 24-equal version may be better, I've never really decided. The
> fifths are better, and help reign in everything else. The neutral
> thirds are also good. The major and minor thirds are no worse than
> people are used to. The 11:8 is better, and the neutral second becomes
> a clear 12:11 instead of a wishy-washy compromise between 12:11 and
> 11:10. As I said above, you don't need 7.

I forgot the major seconds there. Moving towards 24-equal will give a
good 9:8 at the expense of the 10:9. 55-equal looks like a good
compromise. That's fine if you only want well-tuned 11-limit intervals
and aren't bothered about when 9:5 or 11:5 will be simpler.

Graham

Graham!
It seems that 53 would be best for a temperment as it is one they use to derive their 17 tone
subsets. It is also the temperment that they refer to. 24 ET -where is your reference

graham@microtonal.co.uk wrote:

> > I've been playing with the 31-et version of Arabic diatonic and it
> > seems pretty good, concordance-wise.
>
> The 24-equal version may be better, I've never really decided. The fifths
> are better, and help reign in everything else. The neutral thirds are
> also good. The major and minor thirds are no worse than people are used
> to. The 11:8 is better, and the neutral second becomes a clear 12:11
> instead of a wishy-washy compromise between 12:11 and 11:10. As I said
> above, you don't need 7.
>
> Graham
>

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

Kraig wrote:

> It seems that 53 would be best for a temperment as it is one they
> use to derive their 17 tone
> subsets. It is also the temperment that they refer to. 24 ET -where is
> your reference

I was talking about neutral third scales, which are this bit of the scale
tree (hit "view all messages" or select a monospaced font or whatever):

10 7

. 17
. 27 24
. 37 44 41 31
. 47 64 71 61 58 65 55 38

in particular, the bit on the right. I don't see 53 anywhere.

I'll guess you must be talking about Arabic music, in which case 53-equal
certainly does arise in theory. In fact, I do have references for
24-equal, from the New Grove entry on "Arabic Music", 6(ii) Theory
p.812

"In 1905-6 the Kitab al-musiqa al-sharqi ('The book of eastern music') by
Kamil al-Khula'i (1879-1938) established the equidistance of quartertones
in the octave. This scale of 24 quarter-tones was the subject of fierce
discussion at the Congress of Cairo in 1932, where the participants
divided into two opposing camps; the Egyptians supported the division of
the octave into 24 equal quarters, while the Turks (represented by Yekta
Bey) and the Syro-Lebanese (Sabra and Tawfiq al-Sabbagh) rejected the
system of equal division.

"In 1959 and 1964 the Egyptians organized two symposia to settle the
differences of opinion arising from the controversy at the 1932 Congress
over the equidistance of quarter-tones. The aim of these symposia was to
establish the principle of equal temperament on the basis of the
quarter-tone and give official sanction to its teaching. However,
theoretical thinking between 1949 and 1974 took a different direction in
Syria, especially in the works of Mikha'il Allawirdi(1904-81) and in
volume four of al-Sama' 'inda al-'Arab ('Music among the Arabs') by Majdi
al-Aqili (1917-83), published in Damascus (1969-79). Both writers
employed abstruse mathematics to reach conclusions contrary to those of
their Egyptian colleagues, theoretically reaffirming the existence of the
natural scale in opposition to the artificial quarter-tone scale
recommended in Egypt."

Graham

Graham!
Thanks for that reference as it shows 24 use in Egypt and environs!
i have been using a scale based on a generator around 523 for neutral thirds which forms
MOS at 7, 9, 16

graham@microtonal.co.uk wrote:

> Kraig wrote:
>
> > It seems that 53 would be best for a temperment as it is one they
> > use to derive their 17 tone
> > subsets. It is also the temperment that they refer to. 24 ET -where is
> > your reference
>
> I was talking about neutral third scales, which are this bit of the scale
> tree (hit "view all messages" or select a monospaced font or whatever):
>
> 10 7
>
> . 17
> . 27 24
> . 37 44 41 31
> . 47 64 71 61 58 65 55 38
>
> in particular, the bit on the right. I don't see 53 anywhere.
>
> I'll guess you must be talking about Arabic music, in which case 53-equal
> certainly does arise in theory. In fact, I do have references for
> 24-equal, from the New Grove entry on "Arabic Music", 6(ii) Theory
> p.812
>
> "In 1905-6 the Kitab al-musiqa al-sharqi ('The book of eastern music') by
> Kamil al-Khula'i (1879-1938) established the equidistance of quartertones
> in the octave. This scale of 24 quarter-tones was the subject of fierce
> discussion at the Congress of Cairo in 1932, where the participants
> divided into two opposing camps; the Egyptians supported the division of
> the octave into 24 equal quarters, while the Turks (represented by Yekta
> Bey) and the Syro-Lebanese (Sabra and Tawfiq al-Sabbagh) rejected the
> system of equal division.
>
> "In 1959 and 1964 the Egyptians organized two symposia to settle the
> differences of opinion arising from the controversy at the 1932 Congress
> over the equidistance of quarter-tones. The aim of these symposia was to
> establish the principle of equal temperament on the basis of the
> quarter-tone and give official sanction to its teaching. However,
> theoretical thinking between 1949 and 1974 took a different direction in
> Syria, especially in the works of Mikha'il Allawirdi(1904-81) and in
> volume four of al-Sama' 'inda al-'Arab ('Music among the Arabs') by Majdi
> al-Aqili (1917-83), published in Damascus (1969-79). Both writers
> employed abstruse mathematics to reach conclusions contrary to those of
> their Egyptian colleagues, theoretically reaffirming the existence of the
> natural scale in opposition to the artificial quarter-tone scale
> recommended in Egypt."
>
> Graham
>

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

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

> What I said was that they seemed a tad more dissonant that 12/7s,
>but actually I tend to treat 11/9, 12/7 and 11/8 all as more or less
>consonant. How about you?

They don't do much for me outside a larger otonal context. To tune
them by ear, one has to listen for some very weak partials lining up -
- this doesn't really feel like "minimizing dissonance" to me.

--- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:
> Graham!
> Thanks for that reference as it shows 24 use in Egypt and
environs!
> i have been using a scale based on a generator around 523 for
neutral thirds which forms
> MOS at 7, 9, 16

523 cents? Strange, that's what we called the "pelog" generator on
tuning-math. Where do you see neutral thirds?

Paul!
That is one of the possible pelog generators. It is related to the tuning of Mavila/Chopi
xylophones, which is why some thought there was a connections with these people with the
indonesians
yes i guess 369 is stretching it. I use some variations in this size which gets me lower at
times.

paulerlich wrote:

>
> 523 cents? Strange, that's what we called the "pelog" generator on
> tuning-math. Where do you see neutral thirds?
>

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

--- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:
> Paul!
> That is one of the possible pelog generators.

It happens to be the 'optimal' generator implied by tempering out
135:125 . . . just as Woolhouse meantone fifth/fourth is
the 'optimal' generator implied by tempering out 81:80

> It is related to the tuning of Mavila/Chopi
> xylophones, which is why some thought there was a connections with
> these people with the
> indonesians

Eastern Africa? Makes sense . . .

> yes i guess 369 is stretching it.

Sounds like a 5:4 to me . . . not much worse than the 12-tET M3.

> I use some variations in this size which gets me lower at
> times.

OK . . . then I guess I would hear that more as "Thai" than "Arabic".

I wrote,

> [523¢] happens to be the 'optimal' generator implied by tempering
> out 135:125 . . .

Sorry -- that should be 135:128.