The 45 basic meantone triads

I am calling a triad "basic" if it falls within the "basic" meantone
span of 10, meaning no goofy intervals such as diminished octaves in
it. Below I list them in a base two order, according to how large
they are if we interpret the numbers as exponents for two. I give the
numerical and standard notation form; the normalization is that the
lowest note on the chain of fifths is C. How many of these chords
have offical names, I wonder?

1: [0, 1, 2] [C, G, D]
2: [0, 1, 3] [C, G, A]
3: [0, 2, 3] [C, D, A]
4: [0, 1, 4] [C, G, E]
5: [0, 2, 4] [C, D, E]
6: [0, 3, 4] [C, A, E]
7: [0, 1, 5] [C, G, B]
8: [0, 2, 5] [C, D, B]
9: [0, 3, 5] [C, A, B]
10: [0, 4, 5] [C, E, B]
11: [0, 1, 6] [C, G, F#]
12: [0, 2, 6] [C, D, F#]
13: [0, 3, 6] [C, A, F#]
14: [0, 4, 6] [C, E, F#]
15: [0, 5, 6] [C, B, F#]
16: [0, 1, 7] [C, G, C#]
17: [0, 2, 7] [C, D, C#]
18: [0, 3, 7] [C, A, C#]
19: [0, 4, 7] [C, E, C#]
20: [0, 5, 7] [C, B, C#]
21: [0, 6, 7] [C, F#, C#]
22: [0, 1, 8] [C, G, G#]
23: [0, 2, 8] [C, D, G#]
24: [0, 3, 8] [C, A, G#]
25: [0, 4, 8] [C, E, G#]
26: [0, 5, 8] [C, B, G#]
27: [0, 6, 8] [C, F#, G#]
28: [0, 7, 8] [C, C#, G#]
29: [0, 1, 9] [C, G, D#]
30: [0, 2, 9] [C, D, D#]
31: [0, 3, 9] [C, A, D#]
32: [0, 4, 9] [C, E, D#]
33: [0, 5, 9] [C, B, D#]
34: [0, 6, 9] [C, F#, D#]
35: [0, 7, 9] [C, C#, D#]
36: [0, 8, 9] [C, G#, D#]
37: [0, 1, 10] [C, G, A#]
38: [0, 2, 10] [C, D, A#]
39: [0, 3, 10] [C, A, A#]
40: [0, 4, 10] [C, E, A#]
41: [0, 5, 10] [C, B, A#]
42: [0, 6, 10] [C, F#, A#]
43: [0, 7, 10] [C, C#, A#]
44: [0, 8, 10] [C, G#, A#]
45: [0, 9, 10] [C, D#, A#]

Can anyone provide comments, feedback, insight into
the following?

Mystery Music Ratio and Chord 7129 / 6105195

99.80.3 Edward Leedskalnin's ratio 7129 / 6105195
(Iuliano)

From: Jerryiuliano@aol.com
To: MetPhys@aol.com
Subject: Edward Leedskalnin's ratio 7129 / 6105195
Date: 04/01/03

A plaque was found in Ed's bedroom, at Coral Castle.

It read: THE SECRET TO THE UNIVERSE IS 7129 / 6105195.

Sir:

Ed Leedskalnin, the strange man who built the Coral
Gables castle in Florida had a plaque in his bedroom
that referred to a Theory of Everything as a ratio
between two integers...7129 / 6105195...as quoted at
the bottom of the following article:

......... One final note: A plaque was found in Ed's
bedroom. It read: THE SECRET TO THE UNIVERSE IS 7129 /
6105195. To those of you interested in deciphering
this additional mystery: Good Luck!.........end quote.

The integer 37 solves this mysterious ratio...cosine
in radians

(( 7129 / 6105195 / -37 ) - 3 ) / 10 = cos ( 6105195 /
7129 ) -0.30000315593 = -0.300003156397

The integer 37 also cracks the ratio to the Feigenbaum
constant, t ruler of the mandelbrot fractal, chaos to
order phase transitions...F=4.669201609 = Feigenbaum
constant...tangent in radians

( tan^-1 ( 6105195 / 7129 / 37 )) + Pi = 4.6692....

The Leedskalnin ratio also clicks to the Cheops
constructs through the amplitude for an electron to
emit or absorb a photon , the fine-structure
constant...a(em) = 137.03599976...1998 NIST

( 6105195 / 7129 / 37 ) ^ - ( ht / 2 / bl ) = cos
137.0359815

when ht = height of Cheops pyramid = 486.256 ft
bl = base leg Cheops pyramid = 763.81 ft

reference Churchill/Massey 1910 expedition to Egypt
.....

1/137.0359815 is 99.9999867% of 1998 NIST value

Interesting to note that when Leedskalnin died (1951)
the value of the fine-structure was thought to be
....a(em) = 1/137.035978....

Another interesting Cheops constructs form is as
follows:

6105195 / 7129 / ( 37^17 ) * ( 10^25) = 10 ^ ( 2*ht/bl
)

if ht = 486.2573394...bl = 763.81

J.Iuliano

Source Article omitted as irrelevant.

___________________________________
99.80.4 Edward Leedskalnin's ratio 7129 / 6105195
(Luigi)

From: luigi.di-martino@ntlworld.com
To: MetPhys@aol.com
Subject: Edward Leedskalnin's ratio 7129 / 6105195
Date: 04/01/03

<< Jerry's solution to Leedskalnin's numbers 7129 /
6105195: >>

Here's another take on it.
Any number can be broken down to single digit.
These two numbers break down to a 1 and an 8, mirror
number sequence partners, +1 and -1.
In the Vedic Square the 1 number sequence = 1 2 3 4 5
6 7 8 9. Cycling the 7129 will produce the same
sequence.

The Vedic Square and how it works (PDF)

7129 = 1
14258 = 2
21387 = 3 etc

The 8 number sequence is 8 7 6 5 4 3 2 1 9. Cycling
the 6105195 will produce this sequence.

The number 37 too expresses a 1. To find the mirror of
37 one could use my mirror number table where they
will see it equal to 44, which is the 8. Then add the
37 and 44 and it equals 81. All number sequence pairs
equal 9.

37 cps = D
44 cps = F

The interval is that of a Minor 3rd

A chord can be built up using just minor 3rd intervals

D F Ab B = chord of D Diminished

This chord is perfectly symmetrical. It will mirror
the other way, from right to left. The center D is the
axis:

F Ab B (D) F Ab B

1 cps = C
8 cps = C
81 cps = E

This is a major 3rd interval. A chord can be built
using only major third intervals and it too will be
perfectly symmetrical.

E Ab (C) E Ab = An augmented triad mirroring to the
same augmented triad.

The mirror of 81 = 72 = 153

The mirror of 153 = 144 = 297

The 297 may as well be the 2nd number sequence and the
7th number sequence, mirror pairs in the Vedic square,
held together over the 9 axis.

Ab = 52 cps
B = 62 cps

Mirror of 52 = 47
mirror of 62 = 55

52+47 = 99
62+55 = 117 (99 +117 = 216)

47 cps = F#
55 cps = A

F# to A is another minor 3rd interval

99 cps = G
117 cps = Bb

G to Bb is a minor 3rd interval as well.

52 +47 = 99 = Ab F# G
62 + 55 =117 = B A Bb

And on it spews right?!

Lui

__________________________________________________
Do You Yahoo!?
Tired of spam? Yahoo! Mail has the best spam protection around
http://mail.yahoo.com

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> I am calling a triad "basic" if it falls within the "basic"
meantone
> span of 10, meaning no goofy intervals such as diminished octaves
in
> it.

If we define a q-odd-limit consonant chord as one where every
interval is a q-odd-limit consonance, we get the following list of
consonant triads up to the 9-limit in meantone. Does anyone know the
correct terminology for the three different kinds of diminished triad?

4: [0, 1, 4] [C, G, E]
6: [0, 3, 4] [C, A, E]

13: [0, 3, 6] [C, A, F#]
31: [0, 3, 9] [C, A, D#]
34: [0, 6, 9] [C, F#, D#]
37: [0, 1, 10] [C, G, A#]
40: [0, 4, 10] [C, E, A#]
42: [0, 6, 10] [C, F#, A#]
45: [0, 9, 10] [C, D#, A#]

1: [0, 1, 2] [C, G, D]
2: [0, 1, 3] [C, G, A]
3: [0, 2, 3] [C, D, A]
5: [0, 2, 4] [C, D, E]
12: [0, 2, 6] [C, D, F#]
14: [0, 4, 6] [C, E, F#]
23: [0, 2, 8] [C, D, G#]
25: [0, 4, 8] [C, E, G#]
27: [0, 6, 8] [C, F#, G#]
29: [0, 1, 9] [C, G, D#]
36: [0, 8, 9] [C, G#, D#]
38: [0, 2, 10] [C, D, A#]
44: [0, 8, 10] [C, G#, A#]

--- In tuning-math@yahoogroups.com, Dan Amateur <xamateur_dan@...>
wrote:
>
> Can anyone provide comments, feedback, insight into
> the following?
>
> Mystery Music Ratio and Chord 7129 / 6105195

The mystery is why the heck you think this has anything to do with
music.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:

> Does anyone know the
> correct terminology for the three different kinds of diminished triad?

> 13: [0, 3, 6] [C, A, F#]
This chord is only consonant in the seven limit if we use the
36:35/64:63 7-limit extension of meantone rather than the usual
126:125/225:224 extension. This is also the only chord which is
properly referred to as a diminished triad. Since it only contains 5-
prime-limit intervals, it might be called a 5-limit diminished triad.
> 31: [0, 3, 9] [C, A, D#]
This is an otonal diminished triad, which would be referred to as a
fully diminished 7th (no 3)
> 34: [0, 6, 9] [C, F#, D#]
This is a utonal diminished triad, which would be referred to as a
fully diminished 7th (no 5)

--- In tuning-math@yahoogroups.com, "speciman1729" <rperlner@...> wrote:

> > 13: [0, 3, 6] [C, A, F#]

> This chord is only consonant in the seven limit if we use the
> 36:35/64:63 7-limit extension of meantone rather than the usual
> 126:125/225:224 extension.

Remember, I defined a consonant chord as one whose intervals were all
consonant. Here C-A is a major sixth, F#-A is a minor third, and C-F#
is a tritone--all 7-limit consonances. Hence, the chord is 7-limit
consonance by the definition I gave.

A JI version of 1-7/5-5/3 has an interval of 25/21, but (6/5)/(25/21) =
126/125. Hence this chord belongs in any 126/125 tempering system, even
leaving aside the fact that 25/21, at 302 cents, is a better
approximation to 6/5 than 300 cents anyway. Other JI version of this
chord are plausible also; see below.

> This is also the only chord which is
> properly referred to as a diminished triad. Since it only contains 5-
> prime-limit intervals, it might be called a 5-limit diminished triad.

Well, if you want the 5-limit version, that would be 1-45/32-5/3 or 1-
25/18-5/3 I suppose. Both temper to the same thing in meantone.

> > 31: [0, 3, 9] [C, A, D#]
> This is an otonal diminished triad, which would be referred to as a
> fully diminished 7th (no 3)

> > 34: [0, 6, 9] [C, F#, D#]
> This is a utonal diminished triad, which would be referred to as a
> fully diminished 7th (no 5)

Where are these "fully diminished" names used? Who originated this
terminology?

>If we define a q-odd-limit consonant chord as one where every
>interval is a q-odd-limit consonance, we get the following list of
>consonant triads up to the 9-limit in meantone.

Can you paste your code for doing this into a post?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >If we define a q-odd-limit consonant chord as one where every
> >interval is a q-odd-limit consonance, we get the following list of
> >consonant triads up to the 9-limit in meantone.
>
> Can you paste your code for doing this into a post?

x5 := {1, 3, 4}:
x7 := {1, 3, 4, 6, 9, 10}:
x9 := {1, 2, 3, 4, 6, 8, 9, 10}:

tex3 := proc(v)
local u;
u:={v[2]-v[1],v[3]-v[1],v[3]-v[2]};
if u subset x5 then RETURN(5) fi;
if u subset x7 then RETURN(7) fi;
if u subset x9 then RETURN(9) fi;
0 end:

Hi Gene,

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "speciman1729" <rperlner@> wrote:
>
> > > 31: [0, 3, 9] [C, A, D#]
> >
> > This is an otonal diminished triad, which would be
> > referred to as a fully diminished 7th (no 3)
>
> > > 34: [0, 6, 9] [C, F#, D#]
> >
> > This is a utonal diminished triad, which would be
> > referred to as a fully diminished 7th (no 5)
>
> Where are these "fully diminished" names used?
> Who originated this terminology?

Ray is probably using "fully diminished" as opposed to
"half diminished": in this example the "half diminished"
chord would be (from the root up, in 3rds) D# : F# : A : C#,
and the "fully diminished" is D# : F# : A : C.

The "fully diminished" is a diminished triad with a
diminished-7th on top, whereas the "half diminished"
is a diminished triad with a minor-7th on top.
The "half diminished" also goes by the name of
"minor 7th flat 5".

The "half diminished" terminology is used regularly
in jazz theory, less in classical, and much less in pop.
In popular music the "minor 7th flat 5" description is
found much more often.

As for the origin of the word "diminished" in describing
intervals, that goes back to the medieval period. I'm
not sure which theorist used it first (Margo Schulter
probably knows), but i'd hazard a guess that it might
have been in the works attributed to Johannes de Garlandia,
early 1300's:

http://en.wikipedia.org/wiki/Johannes_de_Garlandia_(music_theorist)

-monz
http://tonalsoft.com
Tonescape microtonal music software

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> I am calling a triad "basic" if it falls within the
> "basic" meantone span of 10, meaning no goofy intervals
> such as diminished octaves in it. Below I list them in
> a base two order, according to how large they are if
> we interpret the numbers as exponents for two. I give
> the numerical and standard notation form; the
> normalization is that the lowest note on the chain of
> fifths is C. How many of these chords have offical names,
> I wonder?

1: [0, 1, 2] [C, G, D] - G suspended
2: [0, 1, 3] [C, G, A] - A minor-7th, no 5th
3: [0, 2, 3] [C, D, A] - (D 7th, no 3rd)
4: [0, 1, 4] [C, G, E] - C major triad
5: [0, 2, 4] [C, D, E] - C add9, no 5th
6: [0, 3, 4] [C, A, E] - A minor triad
7: [0, 1, 5] [C, G, B] - (C major-7th, no 3rd)
8: [0, 2, 5] [C, D, B] - (C major-9th, no 3rd, no 5th)
9: [0, 3, 5] [C, A, B] - A minor add9, no 5th
10: [0, 4, 5] [C, E, B] - C major-7th, no 5th
11: [0, 1, 6] [C, G, F#] - (C sharp-11, no 3rd, no 5th, no 7th)
12: [0, 2, 6] [C, D, F#] - D 7th, no 5th
13: [0, 3, 6] [C, A, F#] - F# diminished triad
14: [0, 4, 6] [C, E, F#] - implied D 9th with missing root
15: [0, 5, 6] [C, B, F#] - (C major-7th sharp-11, no 3rd, no 5th)
16: [0, 1, 7] [C, G, C#] - no official name: two C's
17: [0, 2, 7] [C, D, C#] - no official name: two C's
18: [0, 3, 7] [C, A, C#] - no official name: two C's
19: [0, 4, 7] [C, E, C#] - no official name: two C's
20: [0, 5, 7] [C, B, C#] - no official name: two C's
21: [0, 6, 7] [C, F#, C#] - no official name: two C's
22: [0, 1, 8] [C, G, G#] - no official name: two G's
23: [0, 2, 8] [C, D, G#] - (C augmented add9, no 3rd)
24: [0, 3, 8] [C, A, G#] - A minor, major-7th [no 5th]
25: [0, 4, 8] [C, E, G#] - C augmented triad
26: [0, 5, 8] [C, B, G#] - (C augmented, major-7th, no 3rd)
27: [0, 6, 8] [C, F#, G#] - (C augmented, sharp-11, no 3rd)
28: [0, 7, 8] [C, C#, G#] - no official name: two C's
29: [0, 1, 9] [C, G, D#] - (C sharp-9, no 3rd, no 7th)
30: [0, 2, 9] [C, D, D#] - no official name: two D's
31: [0, 3, 9] [C, A, D#] - implied B 7th flat-9, no root
32: [0, 4, 9] [C, E, D#] - (C sharp-9, no 7th)
33: [0, 5, 9] [C, B, D#] - (B, flat-9, no 7th)
34: [0, 6, 9] [C, F#, D#] - D# diminished-7th, no 5th
35: [0, 7, 9] [C, C#, D#] - no official name: two C's
36: [0, 8, 9] [C, G#, D#] - (C augmented, sharp 9, no 3rd, no 7th)
37: [0, 1, 10] [C, G, A#] - (C German-6th, but no 3rd)
38: [0, 2, 10] [C, D, A#] - (D 7, augmented-5th, no 3rd)
39: [0, 3, 10] [C, A, A#] - no official name: two A's
40: [0, 4, 10] [C, E, A#] - C Italian-6th
41: [0, 5, 10] [C, B, A#] - not likely in standard theory
42: [0, 6, 10] [C, F#, A#] - (C French-6th, no 3rd)
43: [0, 7, 10] [C, C#, A#] - no official name: two C's
44: [0, 8, 10] [C, G#, A#] - not likely in standard theory
45: [0, 9, 10] [C, D#, A#] - not likely in standard theory

Notes:

. in standard music-theory, chords will not have two
members which have the same name

. it is standard in jazz to omit the 5th of a chord
if it is a perfect-5th

. in standard practice, especially in jazz, the root,
3rd, and 7th are considered essential, and all other
chord members (assuming chords built in 3rds) are optional;
to consider the flat-9, sharp-9, 11th and 13th as
chord members, the 7th is generally required

. names in parentheses are only suggested chords,
missing essential members such as 3rds or 7ths

. number 14 can also be analyzed as an
"F# half-diminished-7th, no 3rd", but since the A
is missing, which is essential for the interpretation
of F# as root but unnecessary if D is the root, the
name given in the list is more likely; similarly for
number 31

It might surprise some folks how many of these actually
can be analyzed as regularly occurring chords.

-monz
http://tonalsoft.com
Tonescape microtonal music software

--- In tuning-math@yahoogroups.com, "monz" <monz@...> wrote:
> > > > 34: [0, 6, 9] [C, F#, D#]
> > >
> > > This is a utonal diminished triad, which would be
> > > referred to as a fully diminished 7th (no 5)
> >
> > Where are these "fully diminished" names used?
> > Who originated this terminology?

> Ray is probably using "fully diminished" as opposed to
> "half diminished": in this example the "half diminished"
> chord would be (from the root up, in 3rds) D# : F# : A : C#,
> and the "fully diminished" is D# : F# : A : C.

I figured that part out, but wanted to know if saying "no 5" and
using "fully diminished" rather than "diminished seventh" was, in
combination, some sort of standard terminology by someone or from
somewhere.

> The "half diminished" terminology is used regularly
> in jazz theory, less in classical, and much less in pop.

I've seen it in classical--what else is it called there?

> As for the origin of the word "diminished" in describing
> intervals, that goes back to the medieval period.

Knowing what the intervals are won't help you much in figuring out
what the chord names mean. And for that matter, knowing what they
were in medieval times is rather misleading, as an augmented second
would be sharper than a minor third, for example.

--- In tuning-math@yahoogroups.com, "monz" <monz@...> wrote:

> 1: [0, 1, 2] [C, G, D] - G suspended
> 2: [0, 1, 3] [C, G, A] - A minor-7th, no 5th

Thanks, Monz, this is cool. Interesting to see most can be given
names.

> Notes:
>
> . in standard music-theory, chords will not have two
> members which have the same name

What does this mean? You don't have a fifth, a diminished fifth, and
an augmented fifth all in the same chord?

> . in standard practice, especially in jazz, the root,
> 3rd, and 7th are considered essential, and all other
> chord members (assuming chords built in 3rds) are optional;

You just got through saying exactly the opposite.

--- In tuning-math@yahoogroups.com, "monz" <monz@...> wrote:

> 29: [0, 1, 9] [C, G, D#] - (C sharp-9, no 3rd, no 7th)

I'd just call it a subminor triad; I regard it as a basic chord.
What's "sharp-9" mean?

> 45: [0, 9, 10] [C, D#, A#] - not likely in standard theory

I'd call it a supermajor triad. I think this does occur, even if
namelessly, in common practice as it's what you get in place of a major
triad in remote keys using a 12-note meantone compass.

>> 29: [0, 1, 9] [C, G, D#] - (C sharp-9, no 3rd, no 7th)
>
>I'd just call it a subminor triad; I regard it as a basic chord.
>What's "sharp-9" mean?

D# is a sharp ninth.

>> 45: [0, 9, 10] [C, D#, A#] - not likely in standard theory
>
>I'd call it a supermajor triad.

Why would you do that?

>I think this does occur, even if
>namelessly, in common practice as it's what you get in place of a major
>triad in remote keys using a 12-note meantone compass.

Octave registration matters here. Chord names are not
pitch class invariant (if I may) in jazz theory.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >> 29: [0, 1, 9] [C, G, D#] - (C sharp-9, no 3rd, no 7th)
> >
> >I'd just call it a subminor triad; I regard it as a basic chord.
> >What's "sharp-9" mean?
>
> D# is a sharp ninth.

It's a ninth and not a second, you figure?

> >> 45: [0, 9, 10] [C, D#, A#] - not likely in standard theory
> >
> >I'd call it a supermajor triad.
>
> Why would you do that?

Because I grew up on Helmholtz and Ellis. Why wouldn't I do that?

> Octave registration matters here. Chord names are not
> pitch class invariant (if I may) in jazz theory.

They aren't in standard music theory either, but they are in "set
theory" and they are in this.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> Remember, I defined a consonant chord as one whose intervals were
> all consonant. Here C-A is a major sixth, F#-A is a minor third, and
> C-F# is a tritone--all 7-limit consonances.

Probably a beginner's question: is there a clear and widely accepted
definition which 7-limits intervals are consonances?
--
Hans Straub

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:

> > >I'd call it a supermajor triad.
> >
> > Why would you do that?
>
> Because I grew up on Helmholtz and Ellis. Why wouldn't I do that?

Here's a notational thought--take "super" to mean a diesis up,
and "sub" to mean a diesis down. Hence not only would C-Fb be a
supermajor third, and C-D# be a subminor third, but C-D## would be a
submajor third, and C-Fbb would be a superminor third. In the 31&43
version of 11-limit meantone ("huyghens", over Paul's objection since
he offered no alternative) a submajor third is an 11/9 neutral third.
It's also a 49/40 neutral third. In the 31&50, "meanpop" version of 11-
limit meantone, a superminor third C-Fbb is an 11/9 neutral third, and
also a 60/49 neutral third. In 31-et, a submajor third and a superminor
third are the same, and have all of the above interpretations.

--- In tuning-math@yahoogroups.com, "hstraub64" <hstraub64@...> wrote:

> Probably a beginner's question: is there a clear and widely accepted
> definition which 7-limits intervals are consonances?

The standard definition of a 7-limit consonance is the tonality
diamond, hence anything octave reducible to 1, 8/7, 7/6, 6/5, 5/4, 4/3,
7/5, 10/7, 3/2, 8/5, 5/3, 12/7, 7/4.

>> >> 29: [0, 1, 9] [C, G, D#] - (C sharp-9, no 3rd, no 7th)
>> >
>> >I'd just call it a subminor triad; I regard it as a basic chord.
>> >What's "sharp-9" mean?
>>
>> D# is a sharp ninth.
>
>It's a ninth and not a second, you figure?

Like I said, octave registration matters in chord naming.
What it isn't is a third, as you seem to be treating it as.

>> >> 45: [0, 9, 10] [C, D#, A#] - not likely in standard theory
>> >
>> >I'd call it a supermajor triad.
>>
>> Why would you do that?
>
>Because I grew up on Helmholtz and Ellis. Why wouldn't I do that?

C, D#, A#... how do you figure?

>> Octave registration matters here. Chord names are not
>> pitch class invariant (if I may) in jazz theory.
>
>They aren't in standard music theory either, but they are in "set
>theory" and they are in this.

Oh. I forgot this is your attempt at "set theory". What
useful acrobatics can come of it?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >> >> 29: [0, 1, 9] [C, G, D#] - (C sharp-9, no 3rd, no 7th)
> >> >
> >> >I'd just call it a subminor triad; I regard it as a basic
chord.
> >> >What's "sharp-9" mean?
> >>
> >> D# is a sharp ninth.
> >
> >It's a ninth and not a second, you figure?
>
> Like I said, octave registration matters in chord naming.
> What it isn't is a third, as you seem to be treating it as.

Why is C-D# is a sharp-9 and not a sharp-2?

> C, D#, A#... how do you figure?

Ack, you're right. It's an appoximation to 1-7/6-7/4 if you want to
look at it that way. Nice chord. Should have a name.

> Oh. I forgot this is your attempt at "set theory". What
> useful acrobatics can come of it?

It's the kind of analysis I've been doing before tackling a
temperament, and one thing which could come out of it, it seems to
me, is a general approach to chords in a given temperament.

>>> It's a ninth and not a second, you figure?
>>
>> Like I said, octave registration matters in chord naming.
>> What it isn't is a third, as you seem to be treating it as.
>
>Why is C-D# is a sharp-9 and not a sharp-2?

It could be #2 if you want, but the typical voicing in
jazz in #9.

>> Oh. I forgot this is your attempt at "set theory". What
>> useful acrobatics can come of it?
>
>It's the kind of analysis I've been doing before tackling a
>temperament, and one thing which could come out of it, it seems to
>me, is a general approach to chords in a given temperament.

I have my doubts about any chord-naming that is octave-equivalent.
Then again, I wasn't following this thread, just replying to
the names.

-Carl

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:

> Here's a notational thought--take "super" to mean a diesis up,
> and "sub" to mean a diesis down.

If we do this, then we get the following small intervals:

C-Db minor second/diatonic semitone -5
C-C# augmented unison/chromatic semitone/subminor second +7
C-Dbb diminished second/superunison/diesis -12
C-B## subaugmented unison/subchromatic semitone +19
C-A### subsubaugmented unison +31

In 31-et, the diesis is exactly half of a chromatic semitone, which
means that a nice elaboration of standard notation could be to add
half-flat and half-sharp symbols. However, this is only close to
correct when you are close to 31 equal. The various meantones can be
classed by a ratio (running from 1 to infinity) which is the
chromatic semitone over the diesis. This gives:

19: 1
88: 5/4
69: 4/3
50: 3/2
81: 5/3
31: 2
43: 3
55: 4
12: infinity

How hard would it be to get musicians trained in meantone already to
adopt the point of view that C-half-sharp was a diminished second, I
wonder? It would make the logic of the system easier to deal with and
perhaps to understand how to play, I would think.

>How hard would it be to get musicians trained in meantone already to
>adopt the point of view that C-half-sharp was a diminished second,

I shouldn't think it is any kind of second.

>I wonder? It would make the logic of the system easier to deal with
>and perhaps to understand how to play, I would think.

Howso?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >How hard would it be to get musicians trained in meantone already to
> >adopt the point of view that C-half-sharp was a diminished second,
>
> I shouldn't think it is any kind of second.
>
> >I wonder? It would make the logic of the system easier to deal with
> >and perhaps to understand how to play, I would think.

Generally intervals like the diminished second don't show up in 31
equal music until we start trying to incorporate the 11 limit. If we
extend meantone to the 11 limit by tempering out 121/120 rather than
385/384 or 176/175 (we incorporate a second independent spiral of
fifths in doing so,) we have a system wherein we can use hemisharps
and hemiflats more systematically, and we no longer have to worry
about heavily augmented and diminished intervals, except when making
sure our temperament actually closes at 31 pitches.

Hi Gene,

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "monz" <monz@> wrote:
>
> > 1: [0, 1, 2] [C, G, D] - G suspended
> > 2: [0, 1, 3] [C, G, A] - A minor-7th, no 5th
>
> Thanks, Monz, this is cool. Interesting to see most
> can be given names.

Yes, actually it also surprised me how many of them
do have "legal" names ... at first glance, i thought
there would be only a few.

> > . in standard music-theory, chords will not have
> > two members which have the same name
>
> What does this mean? You don't have a fifth,
> a diminished fifth, and an augmented fifth all
> in the same chord?

As soon as i clicked the "Send" button, i realized
that where i wrote "same name" my meaning would
probably not be clear. The only reason i didn't
clarify it was because it was way past bedtime.

By "same name" i really meant "same letter".

So in standard theory, you won't have C and C#,
or Bb and B, in the same chord. If the chord has
what would be those pairs of notes in 12-edo,
they'd be spelled C and Db, and A# and B
(or Bb and Cb) respectively.

The main exception i see to this is in rock
and jazz where the chord normally called
"dominant-7th sharp-9" is sometimes spelled
and referred to as "dominant-7th flat-10", so
that for example the one with C as the root
would have both the major-3rd "E" and the
flat-10 "Eb" in a higher octave.

But in general, the spelling of chords conforms
to the norm of building chords up in 3rds, so
for a C chord you'd get C - E - G - B - D - F - A
and optionally any necessary sharps or flats.
Thus the standard way of using "sharp-9" "D#"
for the above chord.

Along the same line of thinking, it's *very*
rare to find a dominant-11th chord which also
uses the 3rd. With C as the root, you'd have
C E G Bb D F, and the clashing of the E and F
is normally felt to "not work" in jazz, rock,
and pop, the exception being "alternative rock"
where the 3rd and 4th of a scale are often used
together in a chord. The "standard" 11th chord
is C Bb D F (the perfect-5th is always optional
in a chord which includes the 7th).

> > . in standard practice, especially in jazz, the root,
> > 3rd, and 7th are considered essential, and all other
> > chord members (assuming chords built in 3rds) are optional;
>
> You just got through saying exactly the opposite.

Not sure what you mean by that.

My point is that the perfect-5th can always be
left out for the three main types of chords. In jazz,
there are five basic types of chords, and the three
main ones can all be defined by 3 notes: the root,
3rd, and 7th:

major: C E B

dominant: C E Bb

minor: C Eb Bb

The other two need the 5th, because it is "altered":

half-diminished: C Eb Gb Bb

diminished: C Eb Gb A (technically Bbb)

-monz
http://tonalsoft.com
Tonescape microtonal music software

Hi Gene,

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >>> It's a ninth and not a second, you figure?
> >>
> >> Like I said, octave registration matters in chord naming.
> >> What it isn't is a third, as you seem to be treating it as.
> >
> > Why is C-D# is a sharp-9 and not a sharp-2?
>
> It could be #2 if you want, but the typical voicing in
> jazz in #9.

Carl is right, a chord with these notes would typically
have the D# voiced in the next octave above the C.

But jazz chord voicings can put chord members anywhere,
and even if the D# was truly an augmented-2nd above the C,
the sheet music would still call it a "sharp-9".
Trust me on that one -- i could cite thousands of
examples if i wanted to, but i don't want to.

-monz
http://tonalsoft.com
Tonescape microtonal music software

Hi Gene,

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:

> In 31-et, the diesis is exactly half of a
> chromatic semitone, which means that a nice
> elaboration of standard notation could be to
> add half-flat and half-sharp symbols. However,
> this is only close to correct when you are
> close to 31 equal.

That's exactly what Fokker proposed in his new
notation for 31-edo. It can be seen in the diagrams
in this paper:

http://www.xs4all.nl/~huygensf/doc/terp31.html

and in HTML variants there's an explanation of it
here, with a comparison to the regular meantone notation:

http://www.bikexprt.com/music/notation.htm

> The various meantones can be classed by a
> ratio (running from 1 to infinity) which is
> the chromatic semitone over the diesis. This gives:
>
> 19: 1
> 88: 5/4
> 69: 4/3
> 50: 3/2
> 81: 5/3
> 31: 2
> 43: 3
> 55: 4
> 12: infinity

Wow, now that's cool! Reminds me of Blackwood's "R"
ratio, which is kind of funny considering that all of
a sudden in the last two days i've gotten deeply back
into Blackwood again.

> How hard would it be to get musicians trained in
> meantone already to adopt the point of view that
> C-half-sharp was a diminished second, I wonder?
> It would make the logic of the system easier to
> deal with and perhaps to understand how to play,
> I would think.

The problem with that is that musicians understand
any kind of "second" to be a traversal from one
letter-name to the next, so your diminished-2nd
would have to be some kind of "D" in relation to "C".

The interval names Prime [= 1st], 2nd, 3rd, 4th, etc.,
are strictly determined by counting the letters
subtended by lower and upper notes of the interval,
inclusively, regardless of what the actual pitch size is.
So in 12-edo it's very easy to have a diminished-4th C#:F
which is smaller than an augmented-3rd C:E#.

I know it doesn't make sense, but that's how it works.

-monz
http://tonalsoft.com
Tonescape microtonal music software

--- In tuning-math@yahoogroups.com, "speciman1729" <rperlner@...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <genewardsmith@> wrote:
>
> > Does anyone know the
> > correct terminology for the three different kinds of diminished
triad?
>
> > 13: [0, 3, 6] [C, A, F#]
> This chord is only consonant in the seven limit if we use the
> 36:35/64:63 7-limit extension of meantone rather than the usual
> 126:125/225:224 extension. This is also the only chord which is
> properly referred to as a diminished triad. Since it only contains
5-
> prime-limit intervals, it might be called a 5-limit diminished
triad.
> > 31: [0, 3, 9] [C, A, D#]
> This is an otonal diminished triad, which would be referred to as a
> fully diminished 7th (no 3)
> > 34: [0, 6, 9] [C, F#, D#]
> This is a utonal diminished triad, which would be referred to as a
> fully diminished 7th (no 5)
>
I need to read through all the posts of course, but could you please
"show the fractions" or something that would give me a baseline,
I'm trying to figure out why you are using D# as a root in 31: and 34:

Using Polya, there are 19 different triads 1/12(C{12,3}+phi(12/4)*C
(4,1)) Gene's 45 is based on C(10,2) of course. We might want to talk
about meantone's effect on transposed triads, especially since they
are based on a chains of fifths, this would be meaningful I think.
Like the above diminished triads which are the only 3 in the list of
45, are at T0, T6 and T9

Thanks

>> > . in standard practice, especially in jazz, the root,
>> > 3rd, and 7th are considered essential, and all other
>> > chord members (assuming chords built in 3rds) are optional;
>>
>> You just got through saying exactly the opposite.
>
>Not sure what you mean by that.

I wasn't sure what he meant, either.

-Carl

> I need to read through all the posts of course, but could you please
> "show the fractions" or something that would give me a baseline,
> I'm trying to figure out why you are using D# as a root in 31: and 34:

The choice of root is quite simple. It is based on the 7-note based
notation usually used for meantone. You only need to look at the
letter names, not the accidentals. Chords are built on thirds, hence
the bottom note of the chord is the note with the letter name such
that you require the fewest steps in the pattern:

C->E->G->B->D->F->A->C

to list all the letter names of the chord.

consider the chord F# C D#:
If the root were F#, we'd have to traverse all seven steps to get to
the D#, and the chord would be some kind of a 13th with lots of
missing notes. If the root were C, we'd have to traverse through six
of the notes to get to the F#, hence we'd have an 11th. If we chose
D#, F# is a minor third above D#, and C is two minor thirds above F,
so we only need to make the chord a seventh with one missing note to
describe it with D# as the root. This is therefore the most economical
representation of the chord as a tertian (third-based) harmony, and
invariably the one which will be used. As far as the ratios go: just
remember that a minor third is 6/5 and an augmented second is 7/6 in
the most common seven limit mapping for meantone.

We'll use a different choice for the root note here, than we do for
the naming of the chord
C:D#:F# = 1/7:1/6:1/5. Thus this is a utonal triad.

--- In tuning-math@yahoogroups.com, "speciman1729" <rperlner@...>
wrote:
>
>
> > I need to read through all the posts of course, but could you
please
> > "show the fractions" or something that would give me a baseline,
> > I'm trying to figure out why you are using D# as a root in 31:
and 34:
>
> The choice of root is quite simple. It is based on the 7-note based
> notation usually used for meantone. You only need to look at the
> letter names, not the accidentals. Chords are built on thirds, hence
> the bottom note of the chord is the note with the letter name such
> that you require the fewest steps in the pattern:
>
> C->E->G->B->D->F->A->C
>
> to list all the letter names of the chord.
>
> consider the chord F# C D#:
> If the root were F#, we'd have to traverse all seven steps to get to
> the D#, and the chord would be some kind of a 13th with lots of
> missing notes. If the root were C, we'd have to traverse through six
> of the notes to get to the F#, hence we'd have an 11th. If we chose
> D#, F# is a minor third above D#, and C is two minor thirds above F,
> so we only need to make the chord a seventh with one missing note to
> describe it with D# as the root. This is therefore the most
economical
> representation of the chord as a tertian (third-based) harmony, and
> invariably the one which will be used. As far as the ratios go: just
> remember that a minor third is 6/5 and an augmented second is 7/6 in
> the most common seven limit mapping for meantone.
>
> We'll use a different choice for the root note here, than we do for
> the naming of the chord
> C:D#:F# = 1/7:1/6:1/5. Thus this is a utonal triad.
>

Thanks! god, I wish Yahoo hadn't got rid of "Up Thread" - it was
invaluable for beaming back to the message being responded to.
Especially in these complicated threads. I take it 5:6:7 would
be otonal, which was the proceeding triad.

Now these three triads are related by transposition and they have
different expressions, it would be fun to go through and see (for
example) if this is true of some of the other 19 triad types,
obviously major is major is major (0,4,7), (0,3,8), (0,5,9).

[I guess I don't see the point of leaving out 11. I see it as a major
seventh, not a diminished octave?]

Triads only have one "mode of limited transposition" the Augmented
(0,4,8) and of course there is only one here. (It's mapping-to-self
is the (phi(12/4)(C(4,1))/12 part of the Polya expression, but of
course who really cares.)

Paul Hj

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >How hard would it be to get musicians trained in meantone already to
> >adopt the point of view that C-half-sharp was a diminished second,
>
> I shouldn't think it is any kind of second.

If C-half-sharp = Dbb, why not?

> >I wonder? It would make the logic of the system easier to deal with
> >and perhaps to understand how to play, I would think.
>
> Howso?

If you have learned how big a chromatic semitone is supposed to be,
splitting it in half seems a lot easier than splitting it any other
way. Half of a chromatic semitone is something I would think you could
learn to play, along the same lines as the idea that half of a 12edo
semitone is easier to manage than other ways of dividing it.

--- In tuning-math@yahoogroups.com, "speciman1729" <rperlner@...> wrote:

> Generally intervals like the diminished second don't show up in 31
> equal music until we start trying to incorporate the 11 limit. If we
> extend meantone to the 11 limit by tempering out 121/120 rather than
> 385/384 or 176/175 (we incorporate a second independent spiral of
> fifths in doing so,) we have a system wherein we can use hemisharps
> and hemiflats more systematically, and we no longer have to worry
> about heavily augmented and diminished intervals, except when making
> sure our temperament actually closes at 31 pitches.

I've put this temperament on my 11-limit list under the
name "speciman"; if you have an objection or alternative, now is a good
time to make it known. Of course as you suggest since the fifth is
divisible in two in 31-et, it's the obvious means of tuning speciman.
I'm not clear why using a neutral third generator makes things any more
systematic, though. People looking for ways to tune 17 or 24 notes
could certainly give it some thought; 7 is a little complex in this
system but with 24 notes you get some 7s, and of course more fives and
even more 3s and 11s, lots of neutral triads, etc. It's possible that
this could be an approach to harmonizing Arabic music, which seems to
be fond of both neutral thirds and the diatonic scale.

--- In tuning-math@yahoogroups.com, "monz" <monz@...> wrote:

> http://www.bikexprt.com/music/notation.htm

According to this web page

"Fokker's "half-sharps" and "half-flats," replacing double sharps and
double flats, make it possible for a musician to read Fokker's notation
with almost no training. About all the musician needs to know
is: "flats are sharper than the enharmonically equivalent sharps; half-
sharps are a shade sharper than the corresponding naturals, and half-
flats, a shade flatter." The purer harmonies of the 31-tone system
provide a further guide to correct intonation in this system."

I wonder how much this is based on experience in practice?

Hi Gene,

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "monz" <monz@> wrote:
>
> > http://www.bikexprt.com/music/notation.htm
>
> According to this web page
>
> "Fokker's "half-sharps" and "half-flats," replacing
> double sharps and double flats, make it possible for
> a musician to read Fokker's notation with almost no
> training. About all the musician needs to know is:
> "flats are sharper than the enharmonically equivalent
> sharps; half-sharps are a shade sharper than the
> corresponding naturals, and half-flats, a shade flatter."
> The purer harmonies of the 31-tone system provide a
> further guide to correct intonation in this system."
>
> I wonder how much this is based on experience in practice?

I can't speak for John S. Allen (the "bikexprt" of the URL),
but he does seem to have made a very thorough study of
31-edo.

But Fokker's work did spawn a whole "school" of 31-toners
in the Netherlands in the 1960-80s, and i'd bet that
Allen speaks for them a bit too. The 31-tone scene
seems to have been flourishing quite well during that
period (well, "quite well" for the tiny niche which is
the worldwide microtonal community, anyway), and there
were many concerts of 31-edo works which were *performed*
on instruments, not created on a computer.

PS -- On a totally unrelated note: i was already familiar
with John S. Allen years before i found his microtonal
webpages, because he used to write very technical articles
about bicycling science which were published in _Bike Tech_
in the 1980s, and i was a subscriber. I found it fascinating
that the two of us shared both of these interests so
intensely.

-monz
http://tonalsoft.com
Tonescape microtonal music software

>> >How hard would it be to get musicians trained in meantone already to
>> >adopt the point of view that C-half-sharp was a diminished second,
>>
>> I shouldn't think it is any kind of second.
>
>If C-half-sharp = Dbb, why not?

Because the nomenclature is based on diatonic scale degrees
(encoded by letters), not pitch height.

>If you have learned how big a chromatic semitone is supposed to be,
>splitting it in half seems a lot easier than splitting it any other
>way. Half of a chromatic semitone is something I would think you could
>learn to play, along the same lines as the idea that half of a 12edo
>semitone is easier to manage than other ways of dividing it.

Yes, I think this is one reason why 19, 31, and 12 are the most
important meantones.

-Carl

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
> I've put this temperament on my 11-limit list under the
> name "speciman"; if you have an objection or alternative, now is a good
> time to make it known. Of course as you suggest since the fifth is
> divisible in two in 31-et, it's the obvious means of tuning speciman.
> I'm not clear why using a neutral third generator makes things any more
> systematic, though. People looking for ways to tune 17 or 24 notes
> could certainly give it some thought; 7 is a little complex in this
> system but with 24 notes you get some 7s, and of course more fives and
> even more 3s and 11s, lots of neutral triads, etc. It's possible that
> this could be an approach to harmonizing Arabic music, which seems to
> be fond of both neutral thirds and the diatonic scale.
>
No objections. My real name is Ray Perlner, but either way, I'd be
honored to have a temperament named after me. I think that splitting
accidentals in half is an underutilized trick. Tempering out 81/80 and
121/120 (243/242 also) doesn't really make the use of 31-equal any
more systematic, but it does justify using hemisharps and hemiflats
fairly systematically. Another temperament which I think is worthy of
being named (and which uses a similar trick at the 13 limit) is the 23
-limit pajara variant I've been discussing as a tuning for the blues
on another thread.

It tempers out
50/49 64/63 85/84 99/98 169/168 247/245 and 300/299.

--- In tuning-math@yahoogroups.com, "speciman1729" <rperlner@...> wrote:

> Another temperament which I think is worthy of
> being named (and which uses a similar trick at the 13 limit) is the 23
> -limit pajara variant I've been discussing as a tuning for the blues
> on another thread.
>
> It tempers out
> 50/49 64/63 85/84 99/98 169/168 247/245 and 300/299.

I'm dubious about the utility of naming 23-limit temperaments, which
exist in vast profusion and are often similar. Pajarner temperament?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> I'm dubious about the utility of naming 23-limit temperaments, which
> exist in vast profusion and are often similar. Pajarner temperament?

Finding a 23-limit temperament that you can actually use to write music
is fairly nontrivial. This one has especially nice properties. For
example, whenever the generator is between 245.7c and 246.8c,
ALL 19-limit intervals are within half a tritonic diesis (50/49) of
just (17.5c). This is the best you can do with a pajara temperament
since the tritonic diesis is a unison vector. There's also some nice 5,
10, 12, 22, and 34 note scales you can use. The 34 note scale contains
a full set of harmonics up through the 27 limit. That said, Pajarner
sounds a bit goofy. You could just call it split pajara or double
pajara or something. What is the etymolygy of pajara anyway?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "hstraub64" <hstraub64@> wrote:
>
> > Probably a beginner's question: is there a clear and widely accepted
> > definition which 7-limits intervals are consonances?
>
> The standard definition of a 7-limit consonance is the tonality
> diamond, hence anything octave reducible to 1, 8/7, 7/6, 6/5, 5/4,
> 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 12/7, 7/4.
>

Quite simple. Thanks.

I just notice that most of these intervals are approximated quite well
in 22EDO - the ratio between the number of consonant intervals and the
number of dissonant intervals is exceptionally high.

Now I am weondering about this quantity (ratio between the number of
consonant intervals and the number of dissonant intervals). Has this
already been systematically investigated for different EDOs?
--
Hans Straub

--- In tuning-math@yahoogroups.com, "hstraub64" <hstraub64@...> wrote:

> > The standard definition of a 7-limit consonance is the tonality
> > diamond, hence anything octave reducible to 1, 8/7, 7/6, 6/5, 5/4,
> > 4/3, 7/5, 10/7, 3/2, 8/5, 5/3, 12/7, 7/4.

> I just notice that most of these intervals are approximated quite well
> in 22EDO - the ratio between the number of consonant intervals and the
> number of dissonant intervals is exceptionally high.

"Quite well" is a bit of a stretch. I would say 22 is the first et to
do a halfway decent job; the first to do a good job being 31.

> Now I am weondering about this quantity (ratio between the number of
> consonant intervals and the number of dissonant intervals). Has this
> already been systematically investigated for different EDOs?

Could you give a precise definition of what you mean by this ratio?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:

> "Quite well" is a bit of a stretch. I would say 22 is the first et to
> do a halfway decent job; the first to do a good job being 31.

It might be worth noting that the weak point is 7/5, which 22,
tempering out 50/49, equates to 600 cents. The same is true of 26 and
(if we use the pajara tuning) 34. However, 27 improves on this, and 31
improves even more. To my ear 27 still isn't quite good enough for me
to call it anything better than "halfway decent", but 31 is. I'd put 41,
46, 50 and 53 in the "good" category also, so maybe my implicit
definition of "good" for a particular limit is that the maximum error
should be under 9 cents or something of that sort.

Anyone else want to give a value for "good"?

On 07/02/07, Gene Ward Smith <genewardsmith@coolgoose.com> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <genewardsmith@...> wrote:
>
> > "Quite well" is a bit of a stretch. I would say 22 is the first et to
> > do a halfway decent job; the first to do a good job being 31.
>
> It might be worth noting that the weak point is 7/5, which 22,
> tempering out 50/49, equates to 600 cents. The same is true of 26 and
> (if we use the pajara tuning) 34. However, 27 improves on this, and 31
> improves even more. To my ear 27 still isn't quite good enough for me
> to call it anything better than "halfway decent", but 31 is. I'd put 41,
> 46, 50 and 53 in the "good" category also, so maybe my implicit
> definition of "good" for a particular limit is that the maximum error
> should be under 9 cents or something of that sort.
>
> Anyone else want to give a value for "good"?

It sounds to hopelessly subjective to expect agreement. But I did
feel the worst error being within 10 cents was close enough to JI a
while back. I even remember 9 cents having something to do with it,
but I don't know why I'd be so precise, and now I see it agrees with
you.

31 is good in the 7-limit, but not beyond.

Graham