Dear Marcel and All,

Please let me begin by celebrating our opportunity to explore the
various possibilities of Pythagorean JI.

Although it wasn't evidently known to medieval European or Near
Eastern musicians, a 53-note Pythagorean set makes a fine
circulating tuning, and also a repository from which a range of
very useful subsets can be drawn for instruments which cannot
handle a full 53 notes per octave (a very large majority, even
today!). 

Let's look at a very practical 17-note Pythagorean subset for
medieval or modern Persian music -- and also, interestingly, for
a keyboard tuning for 14th-century Italian or possibly also
French music that implements some of the ideas recommended for
singers by Marcheto of Padua in 1318.

A big caution is that Marcheto did not address keyboard tunings,
and we have no evidence that any keyboards of the time attempted
to emulate his fine points of vocal intonation. However, it just
so happens that our Persian tuning provides one shading of the
kind of expressive intonation Marcheto evidently wanted from
vocalists singing directed progressions or cadences, and thus
gives us the opportunity to explore an intriguing if
controversial area of late medieval European practice.

At the end of this article, I'll add a brief mathematical aside
on how a 53-note circle of Pythagorean circulates nicely,
although from a mathematical point of view it's an open tuning
where would could add more fifths and get new notes rather than
octave duplications of existing ones.


1. A medieval or modern 17-note Persian tuning

Persian music, from at least the time of Ibn Sina (980-1037)
about a millennium ago, has been based on a few general types of
melodic steps, including the small and large neutral steps, often
at around 13/12 (138.573 cents) and 128/117 (155.562 cents). 
These two steps add up to a Pythagorean 32/27 minor third at
294.135 cents.

Our Pyth-53 subset for Persian music has many of these neutral
second divisions of the 32/27 third:

! persian-pyth53.scl
!
Persian 17-note set in Pythagorean 53
 17
!
 256/243
 137.14502
 9/8
 32/27
 341.05502
 81/64
 4/3
 544.96502
 635.19002
 3/2
 128/81
 839.10002
 27/16
 16/9
 1043.01002
 243/128
 2/1


As it happens, the Pythagorean steps of 137.145 cents and 156.990
cents are very close to Ibn Sina's 13/12 and 128/117, ratios
which Persian musicians such as Hormoz Farhat still recommend or
closely approximate in their model tunings for tar or setar a
thousand years or so later!

An advantage of a 17-note tuning is that it has a good deal of
symmetry, and provides enough notes for getting quite idiomatic
versions of the various Persian dastgah-s or modal families.

The best solution for Persian music is to use modern Persian
notation, which uses two symbols in addition to usual sharps and
flats (which, as in Western notation, may be taken to raise or
lower a note by a Pythagorean apotome or 2187/2048, 113.685
cents). 

The first extra symbol is the koron or ASCII p as in Dp, which
lowers a note by about a third of a tone, here often the
Pythagorean thirdtone formed by 17 fifths down, at 66.765
cents. Thus Dp here is at 137.145 cents, or very close to 13/12,
as we have noted. 

The other is the sori or ASCII >, which raises a note by an
amount often somewhere between 40 and 70 cents or so. Here it is
often by a small interval we might call a "double comma," equal
to 24 fifths up, or twice the regular 12-note comma at
531441/524288 or 23.460 cents. Thus F> is at 544.965 cents. This
is very close to a complex JI interval based on prime 13 which
occurs in Ibn Sina's tuning for the `oud (the Near Eastern
instrument from which the European lute is derived), 351/256 at
546.393 cents. 

For some idea of how our 17-note set supports Persian modes,
let's consider the leading modal familiar of the modern Persian
classical repertoire, or radif: Shur Dastgah.

To play a good Shur, we really need about 10 notes per octave. 
Let's see how, although a "textbook" version of Shur may indicate
only 7 main notes, we will really want at least 9 or 10.

Here we'll follow the Persian convention that often places Shur
on D, G, or A as the 1/1, picking G, the 3/2 of our 17-note set.
Let's start with the 7 "textbook" notes of Shur.

G-Ap-Bb-C-D-Eb-F-G

This standard version of Shur has a lower tetrachord of G-Ap-Bb-C
or 0-137-294-498 cents, with steps of 137-157-204 cents, the
smaller neutral second coming before the larger as often happens
in Persian modes. Then we have an upper tetrachord of C-D-Eb-F,
or the familiar 1/1-9/8-32/27-4/3 (0-204-294-498 cents), a type
of tetrachord or dang sometimes known as Nava (or Nawa). Finally,
there's a 9/8 step F-G to the 2/1 octave.

As soon as we go below G, our 1/1 of Shur, however, we'll want an
additional note: the neutral step Ep at 361 cents below G. We can
think of this lower Ep as part of a Shur tetrachord on the fourth
below the final or 1/1:

D-Ep-F-G-Ap-Bb-C-D-Eb-F-G 

So Shur often tends to have a minor sixth Eb, here at 128/81,
above the 1/1, but a neutral third consistently below it! On the
Persian santur, it is in fact customary when playing Shur to tune
Ep in the lower octave but Eb in the upper one, since this
instrument typically only has room for around 8 notes per octave,
but allows different octaves to make distinctions like this.

Also, at times, the usual upper minor sixth of Shur may become a
neutral sixth, so that instruments which can support it, as our
17-note tuning can, we will find it helpful to have this note
available also: 

D-Ep-F-G-Ap-Bb-C-D-Eb-Ep-F-G 

With this upper Ep at 839.1 cents available -- very close to a
just 13/8 (840.5 cents) -- we can also have an upper Shur
tetrachord on D, the 3/2 step of the mode, D-Ep-F-G.

At this point we have eight notes per octave, but a very common
melodic figure calls for a ninth note very characteristic of at
least modern Persian music: a koron fifth, or Dp, lowered by
about a third of a tone from 3/2, here at 635.2 cents, quite
close to a just ratio of 13/9 (636.6 cents).

This Dp step often comes up when a melody is descending from the
upper portion of the octave back toward the 1/1. For example, we
might have

C-Dp-C-Bb-Ap-A (498-635-498-294-137-0 cents)

or Eb-Dp-C-Dp-Bb-Ap-A (792-635-498-294-137-0 cents).

Thus we have arrived at a 9-note set for Shur:

D-Ep-F-G-Ap-Bb-C-Dp-D-Eb-Ep-F-G 

One additional note becomes very useful to have if our melody gets
into the high range above the 2/1 of Shur: a minor ninth step Ab
at 512/243 or 1290 cents, which often occurs in this upper range,
and contrasts with the neutral second Ep at around a 13/12 above
the final. So we have:

D-Ep-F-G-Ap-Bb-C-Dp-D-Eb-Ep-F-G-Ab-Bb-C...

Again, santur players often tune Ap at 13/12, but Ab at 512/243
or so in the higher octave. With an instrument supporting 17
notes per octave, however, we can have both forms in all
octaves.

Getting into the Persian modal system, for Shur alone or for the
range of dastgah-s or modal families, would be a lengthy process;
but our 17-note tuning provides a good basis for exploring this
system, and also the often related but sometimes different
medieval Persian system, or what we know of it from theoretical
texts and a few preserved examples of notated music from around
the 13th century.

For example, a very characteristic medieval Persian mode is Ibn
Sina's Mustaqim, known by the 13th century as one tuning of Rast
-- both names meaning, respectively in Arabic and Persian, the
"right, correct, or usual" mode. Here we'll place this Mustaqim
or Rast (with lower neutral steps than in a typical modern Arab
or Turkish Rast) on C, the 1/1 for the Scala file:

C-D-Ep-F-G-Ap-Bb-C

This is 0-204-341-498 cents for the lower tetrachord, with steps
of 204-137-157 cents, and then another Mustaqim tetrachord on the
4/3 step, F-G-Ap-Bb, followed by a 9/8 step, Bb-C, to the 2/1
octave. We are very close to Ibn Sina's suggested tuning of
1/1-9/8-39/32-4/3 or 0-204-342-498 cents.

An interesting feature of Mustaqim is that while the mode is
built from two conjunct Mustaqim tetrachords on the 1/1 and 4/3,
we can also find a Shur tetrachord on the 3/2 step, 0-137-294-498
cents, with the steps of smaller and larger neutral second and
then a regular 9/8 tone. While I'm not aware of any pieces in
this mode that have come down to us from medieval times, one
modern possibility for improvisations or compositions is to
explore both the two conjunct Mustaqim tetrachords and the upper
Shur tetrachord on the 3/2 step of the mode.

The 17-note tuning is deep enough to support lots of Persian
music, while also large enough to have a more symmetrical and
thus intuitive structure for a keyboardist. The 17 steps would
be, in Persian notation:

C-Db-Dp-D-Eb-Ep-E-F-F>-Gp-G-Ab-Ap-A-Bb-Bp-B-C

at 

0-90-137-204-294-341-408-498-545-635-702-792-839-906-992-1043-1110-1200
cents.


2. A keyboard version of Marcheto's vocal intonation (1318)

For medieval European music, Pythagorean is standard -- but how
about an unconventional view of vocal intonation which might have
reflected a widespread 14th-century reality in Italy and quite
possibly also France: the view of Marcheto of Padua (1318)?

As it happens, our 17-note Persian tuning also includes the notes
for a "Marchetan" keyboard version of his vocal recommendations,
although Marcheto himself, it must be repeatedly emphasized, did
not himself address keyboard tunings! We are emulating one of the
shadings which singers might have adopted for the accentuated
tuning of sharps he recommended for vocal music in two or more
parts.

What Marcheto suggested was that when a directed two-voice
progression involved sharps, e.g. the major third or tenth E-G#
expanding stepwise to the fifth or 12th D-A, then the sharp
should be raised rather more than the standard Pythagorean
apotome or major semitone (e.g. G-G#) at 2187/2048 or 113.7
cents. This means that the unstable third E-G# would be somewhat
wider than a usual 81/64 or 408 cents; and that the leading tone
G#-A in the upper voice would be somewhat narrower than a usual
limma or diatonic semitone at 256:243 or 90.2 cents.

Just how much or how little of a difference he desired, remains
an open question. Marcheto speaks of a division of the 9/8 tone
into "five parts," achieved with the help of the number 9. The
modern scholar Jay Rahn, looking to the monochord, proposes a
string division of 81-79-77-76-74-72 or into four intervals each
about half size of a usual 256/243 limma or diatonic semitone,
plus a middle comma of 77:76. This solution would make Marcheto's
E-G# somewhere around 450 cents, or almost two commas larger than
a usual 81/64; and G#-A as narrow as 37/36 (74/72 on the
monochord), around 47.4 cents as opposed to 256/243 at 90 cents.

In our 17-note tuning, the solution is a bit more moderate, and
may be especially appealing to modern musicians who enjoy
septimal or 7-based intervals such as 7/6, 7/4, and 9/7.

Here it generally makes sense to use spellings as in medieval
Pythagorean notation, which by the early 15th century had been
extended to 17 notes (Gb-A#). Our 17 notes are:

C Db C# D Eb D# E F Gb F# G Ab G# A Bb A# B C

0-90-139-204-294-341-408-498-545-635-702-792-839-906-996-1043-1110-1200
cents.

In the range of Db (256/243 or 90.2 cents) to B (243/128 or
1110.8 cents), we have regular Pythagorean steps. This agrees
with the system of Marcheto for singers, who assumed standard
Pythagorean intonation for naturals and flats (with 3/2 fifths,
4/3 fourths, and 9/8 tones, etc.).

Starting with F#, however, sharps are raised beyond their regular
positions, as Marcheto suggests for singers: the difference is
equal to a Pythagorean comma or 531441/524288 or 23.460 cents.

Let us again consider the progression E-G# to D-A. Here E is at
its usual position at 81/64, but G# is raised from 6561/4096 or
816.4 cents (equal to precisely a tetratone, or four 9/8 tones)
to 839.1 cents. Thus E-G# is 431.3 cents, or quite close to a
just 9/7 (435.1 cents).

From E-G#, the voices cadence to D-A at their regular
positions. Thus the lower voice descends E-D, a usual 9/8, but
the upper voice ascends G#-A, here reduced from a usual 256/243
or 90.225 cents to 66.765 cents, close to a just 28/27 or 62.961
cents, the famous thirdtone much favored by Archytas.

Although Marcheto didn't propose and evidently wasn't seeking a
Near Eastern kind of system where neutral intervals are a
standard part of the vocabulary of melodic modes, he did describe
certain progressions including a direct neutral step. For
example, he mentions the progression C-G to E-G# to D-A, with the
upper voice moving G-G#-A or 137.1-67.8 cents, dividing the 9/8
tone G-A into what, in this tuning, are realized as approximate
2/3-tone and 1/3-tone steps. Here C-G is a just 3/2, E-G# around
431.3 cents or a near-9/7, and D-A again a pure 3/2. The contrast
between G in the first sonority C-G, and G# in E-G#, is typical
of much medieval and Renaissance music, but especially dramatic
here because of the the neutral step G-G# and the wide major
third E-G#.

This 17-note system, in a way, combines one possible shading of
the vocal intonation recommended for part-singing by Marcheto
with an early 15th-century development: the 17-note Pythagorean
system recommended by Prosdocimo of Beldomandi and Ugolino of
Orvieto. In this colorful version, unlike the regular version of
Prosdocimo and Ugolino, there is one flaw as it might reasonably
be viewed: the lack of a perfect fifth B-F#.

For pieces calling for a simple and stable fifth B-F#, one might
either try transpositions in this 17-note tuning, or else
consider a different tuning system -- possibly a 24-note
Pythagorean set with two 12-note chains of fifths at 17 fourths
or 66.767 cents apart, which includes our 17-note set plus a 3/2
fifth at B-F#, for example if we take the 1/1 of such a system as
D on the upper chain of fifths.

However, if a smooth fifth B-F# is not required, then our 17-note
set supports adventurous performances of lots of 14th-century
Italian and French music, in addition to Near Eastern modes where
the neutral steps become as routine as tones or semitones.

Incidentally, in a cadence like D-F#-B to C-G-C, where F#-B at a
narrow 474.6 cents (close to 21/16 or 470.7 cents) is part of an
unstable sonority and occurs between two upper voices, there may
not be in practice be a "problem." Our D-F#-B is an unstable
cadential sonority, and already somewhat tense by definition, so
that the narrow and impure fourth between the upper voices might
be heard more as an additional element of color than as a
clashing dissonance where pure concord was expected, as with the
simple B-F# in many situations and with usual harmonic timbres.

However, narrow fourths at 475 cents can also be used as a
valuable resource in such tunings as Erv Wilson's 1-3-7-9 hexany
or 1/1-9/8-7/6-21/16-3/2-7/4-2/1 (0-204-267-471-702-969-1200
cents). We can find a Pythagorean approximation of this at
F#-G#-A-B-C#-E-F# or 0-204-271-475-702-973-1200 cents. A sonority
like 16:21:24:28 (0-471-702-969 cents, or here F#-B-C#-E at
0-475-702-973 cents) can project a very striking kind of
"energetic concord" featuring a mixture of stable 4:3 and 3:2
fourths with the relatively concordant 7:6 and 7:4 and the highly
active 21:16. This exciting sonority can then resolve to the pure
fifth G#-D#, releasing the creative tension of the near 21:16.

This modernistic kind of treatment is yet another approach this
17-note Pythagorean system opens to us, along with the medieval
or modern Persian approach and a medieval European approach based
more or less on the ideas of Marcheto of Padua.


3. A brief mathematical aside on circulating Pyth-53

As textbooks and articles often correctly observe, from a
strictly mathematical point of view a series of 3/2 fifths will
never exactly produce a 2/1 octave, so all Pythagorean tunings
are mathematically open -- unlike in an equal division of the 2/1
octave, we can also add another 3/2 fifth and get another new
note rather than an octave duplication of an existing note.

However, from a musical point of view, a chain of 52 pure fifths
plus a pure 2/1 octave makes a circle or circulating system in
which all fifths will be pure except the one completing the
octave, which will be narrow by 3.615 cents, the comma of
Mercator. This amount of tempering is quite mild by comparison
with 1/4-comma meantone (5.376 cents), for example, and only
affects one fifth out of 53. So Pythagorean 53 can and does
circulate, and quite nicely!

The main complication this introduces, again at a certain
mathematical level of theory rather than in practice, is that
when we speak of Pyth-53 as dividing an octave into "53 commas,"
we must note that these commas are not all equal.

The usual Pythagorean comma of 531441/524288 or 23.460 cents
occurs 41 times as we move through the 53-note circle. However,
in 12 locations, we find instead a smaller "41-note comma," the
difference between 41 pure fourths down and 17 pure octaves, at
19.845 cents. The regular or 12-note and small or 41-note commas
differ, again, by the comma of Mercator at 3.615 cents (the
amount by which 53 pure fifths up fall short of 31 pure octaves).

Having noted these mathematical niceties, we reach the conclusion
that for most practical purposes, it's quite possible to apply
the Turkish or Syrian comma system, in which the octave is
divided into 53 equal parts, to Pyth-53, and count our commas as
if they were equal.

Most appreciatively,

Margo Schulter
mschulter@calweb.com
29 September 2013