Yahoo Tuning Groups Ultimate Backup tuning Peppermint 24: Near-11:12:13:14 tuning

Hello, everyone, and I'd like to share a "bulletin" about a tuning
I've come to call "Peppermint 24," a kind of "near-superparticular"
temperament.

In September 2000, the young theorist Keenan Pepper proposed here a
"Noble Fifth" temperament with the ratio of sizes between whole-tone
and chromatic semitone, and also chromatic and diatonic semitone,
equal to the Golden Section, or Phi (~1.618034). This yields a fifth
of around 704.096 cents, or about 2.141 cents wide.

As I was soon noting, this temperament offers approximations of 14:11,
13:11, 17:14, and 21:17 all within 1.5 cents of just.

During this excitement, Kraig Grady duly observed that the generator
for this temperament also appears in Erv Wilson's Scale Tree, thus
suggesting to me the name "Wilson/Pepper Noble Fifth Tuning."

Early in October, I explored some of the possibilities of a 24-note
regular version of this tuning, a process which, at the time, led me
in some other directions.

A bit less than two weeks ago, however, an idea occurred to me: how
about two 12-note chains of Wilson/Pepper fifths at a distance
producing pure 7:6 thirds? For these fifths at about 704.096 cents,
that distance is around 58.68 cents, the difference between the
regular major second (~208.19 cents) and 7:6 (~266.87 cents).

To the excellent approximations of 14:11, 13:11, 17:14, and 21:17 in
the original temperament, this 24-note system adds tempered versions
of 12:14:18:21 or 14:18:21:24 in nine positions, with 7:6 or 12:7 pure
and the other intervals impure by ~2.141 cents, the amount by which
the fifth is tempered.

The main compromise is that the fifths are indeed tempered by a full
2.14 cents, and thus 9:8 and 16:9 by twice this amount, 4.28 cents.

In contrast, a "near-just" system such as George Secor's original
version of the Miracle tuning with a generator of (18/5)^(1/19) has
approximations of these and all other ratios in the specified set
within about 3.32 cents of pure.

However, "Peppermint 24," as I have come to call it, has a special
feature which I might term "near-superparticular," with some emphasis
on the engaging melodic as well as vertical possibilities. In the
regular Wilson/Pepper temperament, 14:11 and 13:11 are quite close to
pure, and 14:13 virtually just; in Peppermint 24, these ratios are
supplemented by a pure 14:12 or 7:6.

As a result, all steps and ratios included in 11:12:13:14 are within
1.5 cents of pure. A keyboard layout might make a discussion of
scales easier to follow, with an asterisk (*) showing a note on the
upper keyboard raised by an interval of about 58.68 cents:

187.349 346.393 683.253 891.445 1050.488
C#* Eb* F#* G#* Bb*
C* D* E* F* G* A* B* C*
58.680 266.871 475.062 554.584 762.775 970.967 1179.158 1258.680
7/6
-------------------------------------------------------------------------
128.669 287.713 624.574 832.765 991.809
C# Eb F# G# Bb
C D E F G A B C
0 208.191 416.382 495.904 704.096 912.287 1120.478 1200

Here the steps and intervals for a just realization of 11:12:13:14 are
shown, along with note spellings and interval sizes for a tempered
Peppermint 24 version:

150.637 138.573 128.298
12:11 13:12 14:13
0 150.637 289.210 417.508
1/1 12/11 13/11 14/11 Just
............................................................
C*4 D4 Eb*4 E*4 Peppermint 24
0 149.512 287.713 416.382
-1.126 -1.497 -1.126
149.512 138.202 128.669
-1.126 -0.371 +0.371

One source for the idea of 11:12:13:14 as a melodic or scale
progression was a tetrachord mentioned in a book on medieval Near
Eastern music, with a notable middle step of 7:6:

1/1 12/11 14/11 4/3
0 150.637 417.508 498.045
12:11 7:6 22:21
150.637 266.871 80.537

This suggested to me a "symmetrical pentachord scale" in Peppermint 24
with approximate ratios and tempered intervals in cents as follows:

|------------------------------| |--------------------------|
1/1 ~12/11 ~13:11 ~14:11 ~4:3 ~3:2 ~18/11 ~39/22 ~21/11 2/1
0 149.51 287.71 416.38 495.90 704.10 853.61 991.81 1120.48 1200
C*4 D4 Eb*4 E*4 F*4 G*4 A4 Bb*4 B*4 C*5
~12:11 ~13:12 ~14:13 ~22:21 ~9:8 ~12:11 ~13:12 ~14:13 ~22:21
149.51 138.20 128.67 79.52 208.19 149.51 138.28 128.67 79.52

This "nonatonic" scale -- nine notes to the octave -- is very charming
to my ears above a drone. Each "pentachord" divides a tempered fourth
into approximate ratios of 33:36:39:42:44, with the two pentachords
connected by a usual whole-tone near 9:8.

In conclusion, while Peppermint 24 is notably less accurate for some
vertical ratios (especially 9:8 or 16:9) than a near-just temperament
such as Secor's original Miracle tuning, its close approximation of
the steps 11:12:13:14 might suggest a "near-superparticular"
category.

Of course, a near-just temperament may also have near-superparticular
melodic patterns, as with a 43-note version of Miracle with 42 secors
of ~116.71559 cents, offering close approximations of 8:9:10:11:12 in
21 positions. This is a division mentioned by Kraig Grady as advocated
by Douglas Leedy, who finds 10:11:12 characteristic of unaccompanied
music for fiddle or voice.

My special thanks to Erv Wilson and Keenan Pepper, of course, and also
to such helpful people as George Secor, Alison Monteith, and Kraig
Grady, among others, for enriching my view of vertical and melodic
aspects of music and tunings.

Here's an unofficial Scala file for the curious, with the caution that
I use the name "Peppermint 24" subject to correction of the premise
that this name is not already in use for some other tuning:

! peprmint.scl
!
Peppermint 24: Wilson/Pepper apotome/limma=Phi, 2 chains spaced for pure 7:6
24
!
58.679693 cents
128.669246 cents
187.348938 cents
208.191213 cents
7/6
287.713180 cents
346.392873 cents
416.382426 cents
475.062119 cents
495.904393 cents
554.584086 cents
624.573639 cents
683.253332 cents
704.095607 cents
762.775299 cents
832.764852 cents
891.444545 cents
912.286820 cents
970.966512 cents
991.808787 cents
1050.488479 cents
1120.478033 cents
1179.157725 cents
2/1

Most appreciatively,

Margo Schulter
mschulter@value.net

🔗Kraig Grady <kraiggrady@anaphoria.com>

🔗M. Schulter <MSCHULTER@VALUE.NET>

Dear Kraig,

Thank you for your striking point that "Peppermint 24" suggests an
allusion to the sense of smell, and I might compare to it a kind of
aromatic herb as well as to the taste of some peppermint flavor.

If asked to develop this metaphor, I might say that there are many
"spicy" sonorities, with the "mint" suggesting to me especially the
near-7-based sonorities such as 12:14:18:21 or 14:18:21:24.

Here I'd like to share the comment, especially in view of some recent
remarks about music and mathematics, that Erv Wilson's Scale Tree,
which includes the generator for this temperament, is like a map of
stars, or even galaxies, in the universe of tuning systems.

To map a star or galaxy, and to visit and explore it, are two
different things, but by no means unrelated or incompatible ones;
neither activity is a substitute for the other, and any account of
Erv's musical creativity must consider not only his mapping activities
but his instrument-building and musicmaking.

Keenan Pepper, interestingly, proposed a temperament using "the other
noble fifth" as a mathematical and musical counterpart to Kornerup's
Golden Meantone, where the ratio of the larger diatonic semitone to
the narrower chromatic semitone is equal to Phi (fifth ~696.21 cents).
This "other" tuning has a ratio of Phi between the large chromatic and
small diatonic semitone (fifth ~704.10 cents) -- an idea which just
happened to fit my musical interests to a "T."

Since you mentioned in another thread some tunings and timbres like
Chowning's based on the interval ratio of Phi (~833.09 cents), I would
add that a really neat touch of Wilson/Pepper is that the augmented
fifth is about 832.76 cents, only about 0.33 cents narrow of Phi.

One aspect of Peppermint 24 I'm looking into are patterns like the
_harmoniai_ of Kathleen Schlesinger, with 14:13:12:11 taken as an
arithmetic division providing a fine example. Schlesinger gives ratios
for a Hindustani _Rag Malkos_ reported by A. H. Fox Strangeways,
which I offer with a query as to whether, if these ratios are
accurate, they might reflect the influence of some Near Eastern
tradition:

1:1 14:12 14:11 14:9 14:8 14:7
0 267 418 765 969 1200
7:6 12:11 11:9 9:8 8:7
267 151 347 204 231

In Peppermint 24, using an asterisk (*) to show a note raised by a
"diesis" defined by the distance between the two keyboards, ~58.68
cents, this scale could be mapped (with a rounding discrepancy here
and there):

C4 D*4 E4 G*4 A*4 C5
0 267 416 763 971 1200
267 150 346 208 229

As Schlesinger observes, this fits the kind of arithmetic division by
which she defines the _harmoniai_, here 14:12:11:9:8:7.

Some similar mixtures of intervals occur in a scale from the Scala
archives with the name harmd-conmix.scl including some harmonic
divisions (7:8:9, 11:12:13:14):

1/1 8/7 9/7 3/2 11/7 12/7 13/7 2/1
0 231 435 702 782 933 1072 1200
8:7 9:8 7:6 22:21 12:11 13:12 14:13
231 204 267 81 151 139 128

Here's one mapping in Peppermint 24 of this 14:16:18:21:22:24:26:28
pattern:

F#*4 A4 B4 C#*5 D*5 E5 F5 F#*5
0 229 437 704 784 933 1071 1200
229 208 267 80 150 138 129

Needless to say, I'd be curious about any responses you might have to
these scales, or similar ones you may have sung or played in or
otherwise encountered.

Thank you again for your encouraging post, and for the opportunity of
sharing these scales (about which I learned in the process of
replying) and celebrating the contributions of Erv Wilson and Keenan
Pepper.

Most appreciatively,

Margo Schulter
mschulter@value.net

🔗Kraig Grady <kraiggrady@anaphoria.com>

Hello Margo!
I don't know, but cannot remember if i pointed out the keyboard at

http://www.anaphoria.com/key.PDF
top of Page 8

shows what i understand to be the two MOS 12 tone scales that involve Phi as a relation between sizes where the generator is a fourth.

BTW one could place your scale on the Bosanquet by taking this pattern and instead of it being placed on adjacent keys, it could be spaced (each 12 tone scale) on every other key with the other series being placed in between. If this is not clear i can draw it up for you. Someday we will all have one and the music will flow from under our fingertips, at least easier than now!

>
> From: "M. Schulter" <MSCHULTER@VALUE.NET>
> Subject: Peppermint 24: Near-11:12:13:14 tuning
>
>
> In September 2000, the young theorist Keenan Pepper proposed here a
> "Noble Fifth" temperament with the ratio of sizes between whole-tone
> and chromatic semitone, and also chromatic and diatonic semitone,
> equal to the Golden Section, or Phi (~1.618034). This yields a fifth
> of around 704.096 cents, or about 2.141 cents wide.
>
> As I was soon noting, this temperament offers approximations of 14:11,
> 13:11, 17:14, and 21:17 all within 1.5 cents of just.
>
> During this excitement, Kraig Grady duly observed that the generator
> for this temperament also appears in Erv Wilson's Scale Tree, thus
> suggesting to me the name "Wilson/Pepper Noble Fifth Tuning."
>

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

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

🔗M. Schulter <MSCHULTER@VALUE.NET>

[Please note that people reading this on Yahoo's Web site should select
the "expand messages" option to see the diagrams formatted as intended]

Hello, everyone, and this is a response to a recent post by Kraig Grady
most generously sharing more manuscripts of Erv Wilson, including what
might be a generalized keyboard mapping for a Phi-based scale:

> Hello Margo!

> I don't know, but cannot remember if i pointed out the keyboard at

> http://www.anaphoria.com/key.PDF
> top of Page 8

> shows what i understand to be the two MOS 12 tone scales that
> involve Phi as a relation between sizes where the generator is a
> fourth.

Dear Kraig,

Thank you very much for these diagrams, which I downloaded, but am not
sure how to read because of unfamiliarity with the notation. Is it the
"5/12" keyboard to which you might be referring (possibly a generator
defined by 5 steps of the 12-note MOS)?

Any explanation of the notation would be much appreciated; I might
guess that it could involve the number of generators in a chain for a
given note. That fits with a set of signed numbers which seem to me
clearly to follow a cycle of fifths -- or here of ascending fourths:

Generators Signed number

0 0
1 + 5 (fourth)
2 + 10 (minor 7th)
3 + 3 (minor 3rd)
4 + 8 (minor 6th)
5 + 1 (minor 2nd)
6 + 6 (dim. 5th)
7 + 11 (dim. 8ve)
8 + 4 (dim. 4th)
9 + 9 (dim. 10th)
10 + 2 (dim. 3rd)
11 + 7 (dim. 6th)

The notation is a bit of a mystery to me, but I'll make a beginner's
attempt to follow it both as a first exercise, and as a possibly
amusing example of my admitted naivete. The idea I get is a kind of
two-dimensional grid, with the x,y coordinates (if that's the right
term) added to define the number of generators for a given note.

My tendency is to take 12 notes as a chain from Eb to G#, or here down
by fourths from G# to Eb, but maybe just giving values in cents from
an arbitrary "root" (0 cents) might be the best way of communicating
my (mis)reading at first blush.

Here's I'll rotate Erv's diagram, or rather my (mis)interpretation of
it, so that there are seven notes to a row and five to a column (the
original orientation is diagonal, and has lots more visual information
that I'm not attempting to translate):

0 1 2 3 4 5 6 7

0 0 +5 +10 +3 +8 +1 +6 +11
0 495.90 991.81 287.71 783.62 79.52 575.43 1071.33

1 +5 +10 +3 +8 +1 +6 +11 +4
495.90 991.81 287.71 783.62 79.52 575.43 1071.33 367.24

2 +10 +3 +8 +1 +6 +11 +4 +9
991.81 287.71 783.62 79.52 575.43 1071.33 367.24 863.14

3 +3 +8 +1 +6 +11 +4 +9 +2
287.71 783.62 79.52 575.43 1071.33 367.24 863.14 159.04

4 +8 +1 +6 +11 +4 +9 +2 +7
783.62 79.52 575.43 1071.33 367.24 863.14 159.04 654.95

5 +1 +6 +11 +4 +9 +2 +7 octave
79.52 575.43 1071.33 367.24 863.14 159.04 654.95 1200

In my orientation, the columns would ascend in fifths and the rows
move to the right in fourths. Does this bear any resemblance to a
correct interpretation?

> BTW one could place your scale on the Bosanquet by taking this
> pattern and instead of it being placed on adjacent keys, it could be
> spaced (each 12 tone scale) on every other key with the other series
> being placed in between. If instead of it being placed on adjacent
> keys, it could be spaced (each 12 tone scale) on every other key
> with the other series being placed in between. If this is not clear
> i can draw it up for you. Someday we will all have one and the music
> will flow from under our fingertips, at least easier than now!

A diagram with cents could be very helpful in understanding both Erv's
document and the arrangement you propose here; I'm really interested
in comparing these two arrangements.

Maybe I should show my arrangement of Peppermint 24 on two
conventional keyboards in approximate cents, with the chains at a
distance of about 58.68 cents:

187.35 346.39 683.25 891.44 1050.49
C#*4 Eb*4 F#*4 G#*4 Bb*4
C*4 D*4 E*4 F*4 G*4 A*4 B*4 C*5
58.68 266.87 475.96 554.90 762.78 970.97 1179.16 1258.68
7/6
---------------------------------------------------------------
128.67 288.19 624.57 832.76 991.81
C#4 Eb4 F#4 G#4 Bb4
C4 D4 E4 F4 G4 A4 B4 C5
0 208.19 416.38 495.90 704.096 912.29 1120.48 1200

This is what I call a "regularized keyboard," with each manual having
an identical arrangement of steps and intervals. For moderate passages
typically involving sonorities for three or four voices, I find this
approach user-friendly. An obvious complication, however, is quickly
and fluently playing scales moving back and forth between the manuals,
as with this approximation of Ptolemy's syntonic chromatic:

0 80.54 231.17 498.04 701.96 782.49 933.13 1200
1/1 22/21 8/7 4/3 3/2 11/7 12/7 2/1

D*4 Eb*4 F4 G*4 A*4 Bb*4 C5 D*5
0 79.52 229.03 495.90 704.10 783.62 933.13 1200

How might this kind of a scale map on the two Bosanquet arrangements
you discuss? -- maybe a helpful comparison for showing some of the
advantages of a generalized keyboard, although I realize that actually
playing one could be my best introduction to these advantages.

Thank you both for making Erv's manuscripts available, and for any
guidance you might lend a beginner in understanding both these
documents and your suggested alternative Bosanquet mapping.

Most appreciatively,

Margo Schulter
mschulter@value.net

🔗Kraig Grady <kraiggrady@anaphoria.com>

>
> From: "M. Schulter" <MSCHULTER@VALUE.NET>
> Subject: Re: Peppermint 24: Near-11:12:13:14 tuning

Hello Margo!
Quite a bit was touched upon in the last post of which this only covers a part of. So this post concern only the keyboard mapping of
peppermint 24 of which i came up with a simpler one than i originally suggested. I feel it is more important for both of us to be clear
on this before we go into the two most simple Phi scales that can be mapped to this scale. Hopefully we will be able to a picture or two
instead of a thousand words or two to clear up what the keyboard 5/12 shows.
this is again on page 8 of
http://www.anaphoria.com/key.PDF

just to remind everyone for reference

>
>
>
> Hello, everyone, and this is a response to a recent post by Kraig Grady
> most generously sharing more manuscripts of Erv Wilson, including what
> might be a generalized keyboard mapping for a Phi-based scale:
>
> > Hello Margo!
>
> > I don't know, but cannot remember if i pointed out the keyboard at
>
> > http://www.anaphoria.com/key.PDF
> > top of Page 8
>
> > shows what i understand to be the two MOS 12 tone scales that
> > involve Phi as a relation between sizes where the generator is a
> > fourth.
>
> Dear Kraig,
>
> Thank you very much for these diagrams, which I downloaded, but am not
> sure how to read because of unfamiliarity with the notation. Is it the
> "5/12" keyboard to which you might be referring (possibly a generator
> defined by 5 steps of the 12-note MOS)?

That is correct!

>
>
> Any explanation of the notation would be much appreciated; I might
> guess that it could involve the number of generators in a chain for a
> given note. That fits with a set of signed numbers which seem to me
> clearly to follow a cycle of fifths -- or here of ascending fourths:

That is exactly what it is. Most of the Xenharmonikon articles use this same notation. When he is referring to the number of steps in
the scale (as in low to high he uses a number followed by a period as 1. or 0. or 5.
what you have below i think you are confusing scale steps with generator series
+5 would mean going up 5 generator steps which would be a minor second or whereabouts

>
>
> Generators Signed number
>
> 0 0
> 1 + 5 (fourth)
> 2 + 10 (minor 7th)
> 3 + 3 (minor 3rd)
> 4 + 8 (minor 6th)
> 5 + 1 (minor 2nd)
> 6 + 6 (dim. 5th)
> 7 + 11 (dim. 8ve)
> 8 + 4 (dim. 4th)
> 9 + 9 (dim. 10th)
> 10 + 2 (dim. 3rd)
> 11 + 7 (dim. 6th)

This is the corrected series
Generators Signed number

0 0
1 + 5 (minor 2nd)
2 + 10 (major 2nd)
3 + 3 (minor 3rd)
4 + 8 (major 3rd)
5 + 1 (fourth)
6 + 6 (dim. 5th)
7 + 11 (5th)
8 + 4 (minor 6th)
9 + 9 (major 6th)
10 + 2 (minor 7th)
11 + 7 ((major 7th)

>
>
> The notation is a bit of a mystery to me, but I'll make a beginner's
> attempt to follow it both as a first exercise, and as a possibly
> amusing example of my admitted naivete. The idea I get is a kind of
> two-dimensional grid, with the x,y coordinates (if that's the right
> term) added to define the number of generators for a given note.

as i understand you this is correct

>
>
> My tendency is to take 12 notes as a chain from Eb to G#, or here down
> by fourths from G# to Eb, but maybe just giving values in cents from
> an arbitrary "root" (0 cents) might be the best way of communicating
> my (mis)reading at first blush.
>
> Here's I'll rotate Erv's diagram, or rather my (mis)interpretation of
> it, so that there are seven notes to a row and five to a column (the
> original orientation is diagonal, and has lots more visual information
> that I'm not attempting to translate):

something is not right here in the diagram
So i put up how the scale would be mapped
see
http://www.anaphoria.com/margoscale1.gif

next i put up your peppermint scale as you have it notated below with * used to show the other series. (it seemed the cents were not
necessary but can put them in if too confusing.
http://www.anaphoria.com/margoscale2.gif

and underline the scale you mentioned-approximation of Ptolemy's syntonic chromatic. You have to imagine the scale extended out pass
where it is pictured. But i wanted to show the relationship to the scale before. It might look like it is running down hill but one
could rotate the keyboard slightly to have it run more horizontally and least this is what is possible with the starr keyboard.
Please let me know if this helps to solve the keyboard layout part of our discussion.

>
>
> Maybe I should show my arrangement of Peppermint 24 on two
> conventional keyboards in approximate cents, with the chains at a
> distance of about 58.68 cents:
>
> 187.35 346.39 683.25 891.44 1050.49
> C#*4 Eb*4 F#*4 G#*4 Bb*4
> C*4 D*4 E*4 F*4 G*4 A*4 B*4 C*5
> 58.68 266.87 475.96 554.90 762.78 970.97 1179.16 1258.68
> 7/6
> ---------------------------------------------------------------
> 128.67 288.19 624.57 832.76 991.81
> C#4 Eb4 F#4 G#4 Bb4
> C4 D4 E4 F4 G4 A4 B4 C5
> 0 208.19 416.38 495.90 704.096 912.29 1120.48 1200
>
> This is what I call a "regularized keyboard," with each manual having
> an identical arrangement of steps and intervals. For moderate passages
> typically involving sonorities for three or four voices, I find this
> approach user-friendly. An obvious complication, however, is quickly
> and fluently playing scales moving back and forth between the manuals,
> as with this approximation of Ptolemy's syntonic chromatic:
>
> 0 80.54 231.17 498.04 701.96 782.49 933.13 1200
> 1/1 22/21 8/7 4/3 3/2 11/7 12/7 2/1
>
> D*4 Eb*4 F4 G*4 A*4 Bb*4 C5 D*5
> 0 79.52 229.03 495.90 704.10 783.62 933.13 1200
>
> How might this kind of a scale map on the two Bosanquet arrangements
> you discuss? -- maybe a helpful comparison for showing some of the
> advantages of a generalized keyboard, although I realize that actually
> playing one could be my best introduction to these advantages.
>
> Thank you both for making Erv's manuscripts available, and for any
> guidance you might lend a beginner in understanding both these
> documents and your suggested alternative Bosanquet mapping.
>
> Most appreciatively,
>
> Margo Schulter
> mschulter@value.net
>
>

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

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