Yahoo Tuning Groups Ultimate Backup tuning-math Pan-alphabetic notation applied to ET's and temperaments

I've been doing some calculations with the idea of using the golden meantone scale as a first approximation to a notation for octave- repeating temperaments. It seems reasonable to use the 19-ET notation for 19 steps of the temperament, but beyond that it isn't clear which steps should get the remaining letters of the alphabet. So I'll use the convention of using the adjacent letters to name the missing note: for instance, the note between D and S is called DS. Any note near DS that's closer to D than S will be notated Ds, and notes closer to S will be notated dS. Then if no ambiguity results, Ds can be notated as D, and dS can be notated as S.

This lets us go to 31 steps, which seems more reasonable than an arbitrary 25. We can decide later which four notes get the names W, X, Y, and Z.

D DS S L LE E M T F FU U N NG G GV V H HA A AP P I IB B J Q C CR R K KD

Notation of ET's from 5 to 26:

5-ET: D M nG Ap Q D
6-ET: D E U O I C D
7-ET: D E Fu G A iB C D
8-ET: D lE F Ng O aP B Cr D
9-ET: D Le T U Gv hA I J cR D
10-ET: D L M fU nG O Ap Ib Q R D
11-ET: D L M F N Gv hA P B Q R D
12-ET: D L E F U G O A I B C R D
13-ET: D S E T fU nG gV Ha Ap Ib J C K D
14-ET: D S E T Fu N G O A P iB J C K D
15-ET: D S lE M F U nG gV Ha Ap I B Q Cr K D
16-ET: D S lE M F U Ng G O A aP I B Q Cr K D
17-ET: D S Le E T Fu N G V H A P iB J C cR K D
18-ET: D S Le E T F U nG Gv O hA Ap I B J C cR K D
19-ET: D S L E T F U N G V H A P I B J C R K D
20-ET: D S L E M F fU N nG Gv O hA W P Ib B Q C R K D
21-ET: D dS L E M T Fu U Ng G V H A aP I iB J Q C R Kd D
22-ET: D dS L lE M T F U N G Gv O hA A P I B J Q Cr R Kd D
23-ET: D dS L lE E T F fU N nG G V H A Ap P Ib B J C Cr R Kd D
24-ET: D dS L lE E M F Fu U Ng G gV O Ha A aP I iB B Q C Cr R Kd D
25-ET: D dS S Le E M T Fu U N nG Gv V H hA Ap P I iB J Q C cR K Kd D
26-ET: D dS S Le E M T F fU N nG G gV O Ha A Ap P Ib B J Q C cR K Kd D

Note that in the case of 18-ET, 20-ET, and 22-ET, we could notate Gv as V and hA as H. But this won't help with 25-ET, if we need to give every note a letter name, since we need both G and V as distinct notes. Distinct names for nG and Ap are needed for 23-ET and 26-ET, while dS and Kd are required for 25-ET and 26-ET. On the other hand, 24-ET requires lE, Fu, iB, and Cr, and two of these (lE and Cr) are also required by 23-ET. So it's not yet clear which four (if any) should get the extra letter names.

One thing that's clear is that different tunings of a temperament can end up being notated differently if we just take the nearest note of golden meantone. Take the three common ET versions of mavila / pelogic temperament:

4/9: I Le hA D Gv cR U
7/16: I lE A D G Cr U
10/23: Ib lE A D G Cr fV

So we'll probably end up needing to use a "standard" tuning of a temperament, like TOP or poptimal. TOP mavila gives us

iB lE A D G Cr Fv,

which could be notated simply as B E A D G C F.

Here are some examples of notation of the temperaments from the early draft of Paul's paper sorted by generator/period ratio. Note that "negripent" and "negrisept" are notated differently! Two steps of negri temperament ends up between M and T on the scale, and since this is basically a 19-note temperament, it would be best to stick with either M or T for both (whichever one we end up using for 19-ET). Extra notes will be required for Superpyth (hA, Gv), Würschmidt (Gv, nG, iB, Ds, kD, Fu, Ap, hA), Myna (Gv, iB, kD, Ds, Fu, hA) and Cynder (Ds, Ap, nG, kD).

Gawel 569.05 / 1202.62, 19 steps
B T I E P L A S H D V K G R N C U J F

Mavila 521.52 / 1206.55, 7 steps
iB lE A D G Cr Fv

Flattone 507.14 / 1202.54, 7 steps
Meantone 504.13, 1201.7, 7 steps
F C G D A E B

Helmholtz 498.28 / 1200.07, 12 steps
(H) K U B E A D G C F I S (V)

Dominant 495.88 / 1195.23, 12 steps
Garibaldi 498.12 / 1200.76, 12 steps
(H) K N B E A D G C F P S (V)

Superpyth 489.43 / 1197.6, 22 steps
(H) R U B lE hA Kd N J M A D G Q T P dS Gv Cr F I L (V)

Sensisept 443.16 / 1198.39, 19 steps
Sensipent 442.99 / 1199.59, 19 steps
I dS G J E H R F P D N B L V C T A Kd U

Würschmidt 387.64 / 1199.69, 28 steps
(V) Q lE Gv J L G B S nG iB Ds N I D U P kD Fu Ap K F A R T hA Cr M (H)

Magic 380.8 / 1201.28, 19 steps
E gV J L G B dS N I D U P Kd F A R T Ha C

Beatles 354.72 / 1197.1, 10 steps
(H) Q Le G iB D Fv A cR M (V)

Dicot 353.22 / 1207.66, 7 steps
lE G iB D Fu A Cr

Amity 339.47 / 1199.85, 7 steps
E G iB D Fu A C

Keenan 317.84 / 1203.19, 11 steps
Hanson 317.07 / 1200.29, 11 steps
I K M V B D F H Q S U

Myna 309.89 / 1198.83, 27 steps
P R E G I K M Gv iB kD T V B D F H J Ds Fu hA Q S U A C L N

Orwell 271.49 / 1199.53, 9 steps
Orson 271.65 / 1200.24, 9 steps
L U hA J D T Gv I R

Semaphore 252.49 / 1203.67, 19 steps
Le U Ha B Kd E Ng A Q D M G aP C dS F gV I cR

Cynder 232.52 / 1201.7, 26 steps
(V) I Cr S F Gv P C Ds T G Ap Q D M nG A J kD E N hA B K lE U (H)

Tetracot 176.11 / 1199.03, 7 steps
hA iB C D E Fu Gv

Porcupine 162.32 / 1196.91, 7 steps
A B Cr D lE F G

Negripent 126.14 / 1201.82, 9 steps
A I J R D L T U G

Negrisept 124.84 / 1203.19, 9 steps
A I Q R D L M U G

Miracle 116.72 / 1200.63, 10 steps
(H) Ap iB Q R D L M Fu nG (V)

Superchrome 101.99 / 1203.32, 12 steps
(V) A I B C R D L E F U G (H)

Subchrome 99.3 / 1198.31, 12 steps
(H) A I B C R D L E F U G (V)

Nautilus 82.97 / 1202.66, 15 steps
H A P B Q Cr K D S lE M F N G V

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
> I've been doing some calculations with the idea of using the
golden
> meantone scale as a first approximation to a notation for octave-
> repeating temperaments.
...
> This lets us go to 31 steps, which seems more reasonable than an
> arbitrary 25. We can decide later which four notes get the names
W, X,
> Y, and Z.
>
> D DS S L LE E M T F FU U N NG G GV V H HA A AP P I IB B J Q C CR R
K KD
>

This is a brilliant approach to investigating it, although it still
seems an open question whether or not to swap M for T and J for Q.
Do you have an argument for why they should go this way round?

> Notation of ET's from 5 to 26:
...

Wow! Thanks for doing all the number-crunching on this.

> Note that in the case of 18-ET, 20-ET, and 22-ET, we could notate
Gv as
> V and hA as H. But this won't help with 25-ET, if we need to give
every
> note a letter name, since we need both G and V as distinct notes.

Right. But of course ultimately the only reason we're wanting to
give unique letters to all steps of ETs, is so we can use max 10 of
them in notating good linear temps supported by that ET. It may turn
out that you never actually need those non-unique ones, or you don't
need to use the ones they clash with.

> Distinct names for nG and Ap are needed for 23-ET and 26-ET, while
dS
> and Kd are required for 25-ET and 26-ET. On the other hand, 24-ET
> requires lE, Fu, iB, and Cr, and two of these (lE and Cr) are also
> required by 23-ET. So it's not yet clear which four (if any)
should get
> the extra letter names.
>
> One thing that's clear is that different tunings of a temperament
can
> end up being notated differently if we just take the nearest note
of
> golden meantone. Take the three common ET versions of mavila /
pelogic
> temperament:
>
> 4/9: I Le hA D Gv cR U
> 7/16: I lE A D G Cr U
> 10/23: Ib lE A D G Cr fV

That final fV should of course be fU.
>
> So we'll probably end up needing to use a "standard" tuning of a
> temperament, like TOP or poptimal. TOP mavila gives us

I'll go for TOP. I assume you're using gen/period and not just
gen/1200 cents.

>
> iB lE A D G Cr Fv,
>
> which could be notated simply as B E A D G C F.

Seems fair.

You know, we ought to look at the other noble fifth too, Keenan
Pepper's at around 704.1 cents. This is closer to a just fifth than
golden meantone. It is at the limit of the ET series 5, 12, 17, 29,
46, .... 29-ET is 15-limit consistent.

> Here are some examples of notation of the temperaments from the
early
> draft of Paul's paper sorted by generator/period ratio. Note that
> "negripent" and "negrisept" are notated differently!
...

I'd go for the higher prime limit one.

Why did you take some of these out to so many steps? I though we
were aiming for max 10?

It would be great to see a number line showing generator/period from
1/12 to 1/2 with the TOP generators for these temperaments plotted
and regions shaded to show where each notation applies, i.e.
divisions where the notation changes. Or better still, the full 2D
plot with g/p on the vertical axis and octave-equivalent pitch (from
D) on the horizontal axis, and the nominals plotted as lines that
zig-zag and start and stop.

🔗Herman Miller <hmiller@IO.COM>

Dave Keenan wrote:
>>D DS S L LE E M T F FU U N NG G GV V H HA A AP P I IB B J Q C CR R > > K KD
> > > This is a brilliant approach to investigating it, although it still > seems an open question whether or not to swap M for T and J for Q. > Do you have an argument for why they should go this way round?

The notes H-N form a diatonic scale, as do P-V. This seems to be useful if we're going with a fourth-based tuning.

>>So we'll probably end up needing to use a "standard" tuning of a >>temperament, like TOP or poptimal. TOP mavila gives us
> > > I'll go for TOP. I assume you're using gen/period and not just > gen/1200 cents.

Right.

> You know, we ought to look at the other noble fifth too, Keenan > Pepper's at around 704.1 cents. This is closer to a just fifth than > golden meantone. It is at the limit of the ET series 5, 12, 17, 29, > 46, .... 29-ET is 15-limit consistent.

I'd also like to look at golden versions of kleismic and magic, and possibly some others. I'm not entirely convinced that meantone is the best choice, but I'm willing to give it a try and see how it compares.

> Why did you take some of these out to so many steps? I though we > were aiming for max 10?

It's not obvious on some of these scales where the best place to stop is.

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
> You know, we ought to look at the other noble fifth too, Keenan
> Pepper's at around 704.1 cents. This is closer to a just fifth
than
> golden meantone. It is at the limit of the ET series 5, 12, 17,
29,
> 46, .... 29-ET is 15-limit consistent.

And for that matter, maybe we should look at the _just_ fifth as
generator of nominals, and for investigative purposes taking it out
to 29 with either digits or double-letters as you have done with
golden meantone out to 31.

Here's how FCGDAEB fits in 29-ET.
D....E.F....G....A....B.C....D

And a possible way of filling it in, at least temporarily.

D7LWME8FXNYOGRHZPASITJB1CUKV2D

This is based on the following correspondence

AxBxCxDxExFxGxA#B#C#D#E#F#G#A B C D E F G AbBbCbDbEbFbGb
5 6 7 8 9 R S T U V W X Y Z A B C D E F G H I J K L M N

AbbBbbCbbDbbEbbFbbGbb
O P Q 1 2 3 4

The list of 29 steps uses 1, 2, 7, 8, and omits Q.
1 2 7 8 could be replaced with BC VD DL EF.

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@I...>
wrote:
> Dave Keenan wrote:
> > This is a brilliant approach to investigating it, although it
still
> > seems an open question whether or not to swap M for T and J for
Q.
> > Do you have an argument for why they should go this way round?
>
> The notes H-N form a diatonic scale, as do P-V. This seems to be
useful
> if we're going with a fourth-based tuning.

Agreed. Although you will have seen in my previous message that I'm
now proposing that since we're planning to use the whole alphabet, T-
Z should be a diatonic scale instead of P-V thereby treating the
alphabet as circular (A comes after Z), and then breaking it at the
point opposite D, namely at Q.

> I'd also like to look at golden versions of kleismic and magic,
and
> possibly some others.

But these will not include FCGDAEB as a chain of fifths, will they?

> I'm not entirely convinced that meantone is the
> best choice, but I'm willing to give it a try and see how it
compares.
>

I'm not convinced meantone is the best either. I think that just
fifths (or fourths) have the best claim to providing the central
locations for FCGDAEB for notational purposes, simply because
everything else is a temperament, and one can argue 'til the cows
come home about which temperament is best.

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Dave Keenan" <d.keenan@b...>
wrote:
>
> > You know, we ought to look at the other noble fifth too, Keenan
> > Pepper's at around 704.1 cents. This is closer to a just fifth
than
> > golden meantone. It is at the limit of the ET series 5, 12, 17,
29,
> > 46, .... 29-ET is 15-limit consistent.
>
> Is there some reason 118 pops up in this connection?
>
> I get this:
>
> Pepper fifth (67+sqrt(5))/118 octave

This is correct, but I'm not sure why. Can this be called a noble
schismic generator?

Here's how I obtain a noble generator, take any two consective
convergents of the optimal generator (use semiconvergents only if
you have to) that form a Fibonacci-like series with at least one
other (semi-) convergent. Call them n1/d1 and n2/d2, where n2/d2 is
the better approximation. Then your noble generator is the noble
mediant of the two, defined as

n1 + n2*phi
-----------
d1 + d2*phi

where phi = (sqrt(5)+1)/2

...
> 31, 41 Golden secor: (105+sqrt(5))/1102

or (3+4*phi)/(31+41*phi)

or (1+3*phi)/(10+31*phi)

> 31, 72 Golden secor: (389-sqrt(5))/3982

or (3+7*phi)/(31+72*phi)

or (4+3*phi)/(41+31*phi)

That's one reason I use the term "noble" since you can have more
than one noble generator for the same temperament, and then we can
argue about which one should be gold and which silver etc. :-)

🔗Gene Ward Smith <gwsmith@svpal.org>

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

Note that all of these golden generators have another golden generator
connected to them, the algebraic conjugate. It doesn't have to be a
generator for the same temperament!

> > > Pepper fifth (67+sqrt(5))/118 octave

The conjugate is (67-sqrt(5))/118; the convergents for this give us

9, 11, 20, 31, 51, 82, 133...

Seems to be a contorted version of miracle--twice the generator, down
an octave, gives a secor.

> Garigolden fifth: 245/418 - sqrt(5)/2090

Conjugating gives 17, 46, 63, 109, 172... which suggests
<1 -25 15 -42 21 105|

> 53, 118, 171 gives
>
> Golden Pontiac: (5683 + sqrt(5))/9722

Conjugating gives 12, 77, 89, 166, 255 ... which suggests grackle.

> Golden Schismic: (2375 - sqrt(5))/4058

Conjugating gives 29, 70, 99, 169... which could be
<<1 -37 -43 -61 -71 4||.