Yahoo Tuning Groups Ultimate Backup tuning Re: Neo-Gothic "septimal comma" symbols (for Dave Keenan)

Hello, there, Dave Keenan, and thank you for raising an interesting
question of standardization regarding the notation of small intervals
such as the septimal comma in the setting of neo-Gothic tunings and
temperaments.

At the outset, I might suggest a distinction between two types of
"standardization" questions:

(1) Should people using a specific type of tuning system
such as 72-tone equal temperament (72-tET) or its
subsets seek a common spelling and notation of steps
and intervals?

(2) Should one attempt to apply such a "standard" over
a range of other tuning systems with often quite
different structures and interval sizes?

With 72-tET, we have a single tuning with a large set of notes and
intervals. Everyone agrees as to the size of these intervals, whatever
differences of opinion may obtain as to their most apt notation.

With neo-Gothic music, we have a wide range of tuning systems,
often based on a 24-note gamut.

Thus with 72-tET, there are obvious advantages in picking _some_
commonly agreed symbol for a specific size of interval, for example
the symbols ">" and "<" to show a note raised or lowered by 1/6-tone,
which serves among other things as the "septimal comma" in this
system.

In contrast, a neo-Gothic "septimal comma" symbol used across
different tuning systems may represent intervals of widely variable
sizes which would not map consistently to a 72-tET notation.

In what I hope may be amusing presentation, I would like to illustrate
this point.

Specifically, I focus on the following two neo-Gothic notational
symbols:

^ shows a note raised by a "comma" of ~15-35 cents, e.g. G^3
v shows a note lowered by a "comma" of ~15-35 cents, e.g. Ebv4

In tunings with such "comma-like" intervals smaller than 35 cents or
so, one purpose of these symbols is to identify "near-equivalent"
pitches which might be taken as two versions of the same basic note.
These small intervals typically define a "septimal comma" or
equivalent in a given tuning system.

Thank you very much for your stimulating remarks, which made me
consider a number of tunings and conclude that this symbol can indeed
generally be equated with a "septimal comma" role.

However, structurally defined categories of intervals such as a
"septimal comma" or a "diatonic semitone" may have widely varying
sizes in different neo-Gothic tunings and temperaments.

For example, consider the interval E4-F4, a routine diatonic
semitone. In Pythagorean intonation, this interval has a size of
256:243 or ~90.22 cents; in 17-tET, it is 1/17 octave, or ~70.59
cents; in 22-tET, it is 1/22 octave, or ~54.55 cents.

While the "diatonic semitone" category (e.g. E4-F4) is a very
convenient one across a range of neo-Gothic tunings, this category
does not consistently map to any one interval size in 72-tET or a
similar system.

If we wished to transcribe pieces in these different tunings into
72-tET, of course, then the solution would be in each case to
substitute an appropriate 72-tET interval notation -- for example, a
1/3-tone symbol to approximate the 70-cent semitone of 17-tET.

However, in approaching the neo-Gothic tuning systems themselves, I
find it very convenient to have a generic "diatonic semitone" concept
tied to a structural role rather than any specific interval size.

The generic "septimal comma" symbols "^" and "v" likewise serve to
identify a role played by intervals which may not consistently map to
a single 72-tET or similar category.

In illustrating this point, I would like to restrict the discussion to
regular neo-Gothic tunings in the "characteristic" range between
Pythagorean and 17-tET. While it is possible to come up with "exotic"
systems proving or disproving almost any notational point, I would
like to show how the size of the "septimal comma" varies within the
range of the more typical systems.

As you point out, in a Pythagorean-based system with just intonation
of "7-flavor" ratios (e.g. 9:7, 7:6, 12:7, 7:4, 8:7), the symbols "^"
and "v" will indeed signify the septimal comma in its most proper
sense, the interval of 64:63 (~27.26 cents). Here, for example, is the
Sesquisexta tuning we were discussing, with two keyboards a pure 7:6
apart (~266.87 cents), with intervals here shown in rounded cents:

381 561 879 1083 1263
C#}/Ev Eb}/Gbv F#}/Av G#}/Bv Bb}/Dbv
C}/Ebv D}/Fv E}/Gv F}/Abv G}/Bbv A}/Cv B}/Dv C}/Ebv
267 470 675 765 969 1173 1377 1467
-------------------------------------------------------------------
114 294 612 816 996
C# Eb F# G# Bb
C D E F G A B C
0 204 408 498 702 906 1110 1200

Here each note on the upper manual is given in two notations, for
example D}=Fv. The first notation tells us that we can find this note
at the D key of the upper manual, while the second tells us that this
note is a septimal comma lower than usual Pythagorean F.

For this tuning, the 64:63 septimal comma represented by "v" does
nicely map to "<" in 72-tET or the like.

Now let us consider the e-based temperament, which also features a set
of 7-flavor intervals and a step playing a "septimal comma" role. Here
the neo-Gothic asterisk symbol (*) shows a note raised by the diesis
of ~55.28 cents, the distance between the keyboards:

188 341 682 892 1046
C#}/Dv Eb*/D# F#*/Gv G#*/Av Bb*/A#
C*/Dbv D*/Ebv E*/Fv F*/Gbv G*/Abv A*/Bbv B*/Cv C*/Dbv
55 264 473 551 760 969 1178 1255
-------------------------------------------------------------------
132 286 628 837 991
C# Eb F# G# Bb
C D E F G A B C
0 209 418 495 705 914 1123 1200

Here, again the sign "v" usefully indicates a note lowered by a
"septimal comma," which I would define in this loose sense as the
difference between a regular major third and the best approximation of
9:7, or between a regular minor third and the best approximation of
7:6, etc.

For example, if we wish to build an approximation of 12:14:18:21 or
14:18:21:24, our notation points to sonorities such as these:

Cv4 (=B*3) F4
A3 Ebv3 (=D*4)
Fv3 (=E*3) C4
D3 Abv3 (=G*3)

~12:14:18:21 ~14:18:21:24

Here the minor third and seventh D3-Fv3 and D3-Cv4 in the first
sonority are a "septimal comma" narrower than the regular intervals,
and likewise the major third and sixth Abv3-C4 and Abv3-F4 a "septimal
comma" wider.

Thus the symbol "v" serves the same musical role in either this tuning
or the previous one.

However, in the e-based tuning, this interval is actually equal to a
"subdiesis" of about 21.68 cents, the difference between the regular
diatonic semitone at ~76.97 cents and the diesis at ~55.28 cents.

In 72-tET, we would more closely approximate this interval by a
1/12-tone step rather than the 1/6-tone step of our previous example.

Similarly, let us consider the Pythagorean "tricomma tuning" where the
two manuals are a "tricomma" apart, an interval equal to three
Pythagorean commas, around 70.38 cents:

184 365 682 886 1066
C#*3/Dv Eb*3/Fb3v F#*3/Gv G#*3/Av3 Bb*3/Cb4v
C*3/ D*3/ E*3/ F*3/ G*3/ A*3/ B*3/ C*4/
Db3v Ebv3 Fv Gb3v Abv3 Bbv3 C4v Db4v
70 274 478 568 772 976 1180 1270
----------------------------------------------------------------------
114 294 612 816 996
C#3 Eb3 F#3 G#3 Bb3
C3 D3 E3 F3 G3 A3 B3 C4
0 204 408 498 702 906 1110 1200

Again, the "v" symbol shows a note lowered by a "septimal comma," and
helps in identifying 7-flavor sonorities such as D3-Fv3-A3-Cv4
(D3-E*3-A3-B*3) or Gb3v-Bb3-Db4v-Eb3 (F*3-Bb3-C*4-Eb4).

Here, the "septimal comma" is equal to around 19.84 cents, the
difference between the 90.22-cent diatonic semitone and the 70.38-cent
tricomma.

In 72-tET, this interval would most closely map to a step of
1/12-tone, as in our last example.

Finally, let us consider a 24-note tuning based on 17-tET, with two
12-note keyboards in 12-out-of-17-tET at the distance of an
"artificial diesis" of around 55.11 cents, the difference between the
usual 17-tET major second at ~211.76 cents and a pure 7:6 minor third:

184 337 690 902 1043
C#*3/Dv Eb*3/Fb3v F#*3/Gv G#*3/Av3 Bb*3/Cb4v
C*3/ D*3/ E*3/ F*3/ G*3/ A*3/ B*3/ C*4/
Db3v Ebv3 Fv Gb3v Abv3 Bbv3 C4v Db4v
55 267 479 568 772 973 1184 1255
----------------------------------------------------------------------
141 282 635 847 988
C#3 Eb3 F#3 G#3 Bb3
C3 D3 E3 F3 G3 A3 B3 C4
0 212 424 494 706 917 1129 1200

Again, "*" represents a "diesis" and "v" a "septimal comma" -- with
the latter interval here equal to the difference between the 17-tET
diatonic semitone at ~70.59 cents, and our artificial diesis of ~55.11
cents -- or about 15.48 cents.

In 72-tET terms, this "v" now represents an interval slightly smaller
than 1/12-tone, and very closely approximated by such a step.

To conclude, the type of neo-Gothic notation I generally prefer
permits writing a diatonic semitone as "E4-F4," or a 7-flavor sonority
approximating 7:9:12 as "Abv3-C4-F4," across a range of tunings where
semitones or "septimal commas" may vary considerably in size.

Of course, were we to transcribe pieces from these various tunings
into 72-tET, then it would be necessary to map these intervals to
appropriate 72-tET equivalents, using whatever notation might be most
convenient, elegant, or widely understood (at least among the intended
audience).

If we are discussing neo-Gothic tuning systems in their own terms,
however, a "nonstandard" system may have the distinct advantage of
avoiding certain inapplicable inferences about scale structure or
interval sizes which a "familiar" set of symbols might invite.

For example, in the tuning with two "12-of-17-tET" keyboards at a
distance of around 55 cents, the "septimal comma" defines the
difference, between the regular 17-tET major third at ~423.53 cents,
and the ~9:7 major third at ~439.01 cents.

Using a distinctive neo-Gothic notation may help to communicate that
we have a scale structure quite different from 72-tET, or from 7-limit
just intonation for that matter.

This last point, however, raises another issue: if neo-Gothic notation
is to be distinctive, would it not be best to _avoid_ where possible
symbols which might be familiar from a 72-tET or other "standard"
notation?

Given the limited number of ASCII symbols available, this last
question might be problematic; but, at least, there is the wise
precaution of explaining, for example, that a neo-Gothic "septimal
comma" might have a size of anywhere from around 15 cents to 33 cents
depending on the tuning system.

Most appreciatively,

Margo Schulter
mschulter@value.net

Re: Neo-Gothic "septimal comma" symbols (for Dave Keenan)

🔗mschulter <MSCHULTER@VALUE.NET>

🔗jpehrson@rcn.com