Survey X

Not a very exciting list this time.

<64/63,875/864>

Minkowski reduction: <64/63,250/243>

Wedge invariant: [-3,-5,6,28,-18,-1] length = 34.3366

Ets: 7,8,15,22,37,59

Map:

[ 0 1]
[-3 2]
[-5 3]
[ 6 2]

Generators: a = .1353148451 = 5.006649269 / 37; b = 1

Errors and 37-et:

3: 10.91 11.56
5: 1.80 2.88
7: 5.44 4.15

Measures: 42.461 sc, 264.793 s2c

Pretty much 22+15=37 if someone wants to try sharp fifths.

<64/63, 4375/4374>

Wedge invariant: [4,9,-8,-44,24,5] length = 51.9423

Ets: 7, 27

Map:

[ 0 1]
[ 4 1]
[ 9 1]
[-8 4]

Generators: a = .14788728 = 3.99295636 / 27; b = 1

Errors and 27-et:

3: 7.90 9.16
5: 10.87 13.69
7: 11.46 8.95

Measures: 71.962 sc; 694.499 s2c

This would be excluded by Paul's criterion, and it doesn't seem like
a big loss.

<50/49, 81/80>

Wedge invariant: [2,8,8,-4,-7,8] length = 16.1555

Ets: 12, 26, 38

Map:

[0 2]
[1 2]
[4 0]
[4 1]

Generators: a = .578042519 (~3/2) = 15.02910557 / 26; b = 1/2

Errors and 26 et:

3: -8.30 -9.65
5: -11.71 -17.08
7: 5.78 0.40

Measures: 33.657 sc; 100.970 s2c

Paul has apparently played with this oddball temperament, and Graham
has it on his catalog page as "Double Negative".

<81/80, 875/864>

Minkowski reduction: <81/80, 525/512>

Wedge invariant: [1,4,-9,-32,17,4] length = 37.7757

Ets: 7,19,26,45

Map:

[ 0 1]
[ 1 1]
[ 4 0]
[-9 8]

Generators: a = .577880065 (~3/2) = 26.00460293 / 45; b = 1

Errors and 45-et:

3: -8.50 -8.62
5: -12.49 -12.98
7: -9.93 -8.83

Measures: 52.276 sc; 357.112 s2c

An alternative septimal meantone for those who like flat systems.

--- In tuning-math@y..., genewardsmith@j... wrote:

> Ets: 12, 26, 38
>
> Map:
>
> [0 2]
> [1 2]
> [4 0]
> [4 1]
>
> Generators: a = .578042519 (~3/2) = 15.02910557 / 26; b = 1/2
>
> Errors and 26 et:
>
> 3: -8.30 -9.65
> 5: -11.71 -17.08
> 7: 5.78 0.40
>
> Measures: 33.657 sc; 100.970 s2c
>
> Paul has apparently played with this oddball temperament, and
Graham
> has it on his catalog page as "Double Negative".

When I mentioned it before, as two interlaces diatonic scales, each
completing the other's triads into tetrads, you seemed interested and
said you'd print it out.

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:

> When I mentioned it before, as two interlaces diatonic scales, each
> completing the other's triads into tetrads, you seemed interested
and
> said you'd print it out.

My printer ran out of ink, and I haven't replaced it. I needed to
print out your 22-et paper mostly, and I presume this isn't in there.
What would I print out here?

--- In tuning-math@y..., genewardsmith@j... wrote:
> --- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
>
> > When I mentioned it before, as two interlaces diatonic scales,
each
> > completing the other's triads into tetrads, you seemed interested
> and
> > said you'd print it out.
>
> My printer ran out of ink, and I haven't replaced it. I needed to
> print out your 22-et paper mostly, and I presume this isn't in
there.

It is mentioned.

> What would I print out here?

>Date: Fri, 18 Dec 1998 17:30:43 -0500
>From: "Paul H. Erlich" <PErlich@A...>
>To: "'tuning@e...'" <tuning@e...>
>Subject: Assymmetrical 14-tone modes in 26-tET
>Message-ID: <85B74BA01678D211ACDE00805FBE3C050B6587@M...>
>
>I decided to look at the modes of the assymmetrical scale 0 2 4 6 8
10
>12 14 15 17 19 21 23 25 in 26-tET -- because all octave species of
this
>scale is constructed of two identical heptachords, each spanning a
~4/3,
>separated by either a ~4/3 or a ~3/2; and because the scale contains
10
>consonant 7-limit tetrads constructed by the generic scale pattern
1, 5,
>9, 12 (5 are otonal and 5 are utonal) with a maximum tuning error of
17
>cents.
>
>In my paper, I introduce the notion of a charateristic dissonance.
This
>is a dissonant interval which is the same generic size (same number
of
>scale steps) as a consonant interval. Allowing the 7-limit to define
>consonance and allowing errors up to 17 cents, the 14-out-of-26 scale
>has three characteristic dissonances (plus their octave inversions
and
>extensions). Two are "sevenths" of 554 cents instead of the usual 508
>cents, and one is an "eighth" of 554 cents instead of the usual 600
>cents.
>
>None of the modes of this scale satisfy all the properties for a
>strongly tonal mode according to my paper. But a few come close. The
>mode
>
>0 2 4 6 8 10 11 13 15 17 19 21 23 25
>
>or in cents,
>
>0 92 185 277 369 462 508 600 692 785 877 969 1062 1154
>
>has all characteristic dissonances disjoint from the tonic tetrad (0
8
>15 21), which is major. The only other mode with this property is the
>minor equivalent:
>
>0 2 4 5 7 9 11 13 15 16 18 20 22 24
>
>or in cents
>
>0 92 185 231 323 415 508 600 692 738 831 923 1015 1108.
>
>The following modes have one characteristic dissonance which shares a
>note with the tonic tetrad, but it approximates the 1 identity and
the
>11 identity when played along with the tetrad. Therefore the interval
>does not disturb the stability of the tonic too much, and the mode
can
>be considered tonal:
>
>major: 0 2 4 6 8 10 12 13 15 17 19 21 23 24
>
>or in cents
>
>0 92 185 277 369 462 554 600 692 785 877 969 1062 1108
>
>and
>
>minor: 0 2 3 5 7 9 11 13 15 17 18 20 22 24
>
>or in cents
>
>0 92 138 231 323 415 508 600 692 785 831 923 1015 1108.