Yahoo Tuning Groups Ultimate Backup tuning Byzantine chant scales and frequency ratios?

Hi there. This is my first post to this list. I've been looking in for
a while, though, so I *think* it is an appropriate query.

My focus is on Byzantine chant, as it is explained and practiced
today. I'm learning the official explanation for the (several
different) scales in Byz chant, but I'm not clear on how the official
explanations for the intervals would translate to frequency ratios
(such as 2:3 for a perfect fifth).

Since it's possible that some folks here can construct such a
translation from the official explanation, I'll try to give a brief
account here. :-)

For the last ~120 years or so, the scales used in Byz chant have been
officially explained in terms of a structure of 72 equal "moiroi" or
micro-steps to the octave. (Since Byz chant's written notation
records intervals, not pitches, this focus on the size of the
intervals in scales emerged naturally in the Byz chant community.)

In this structure, one of the Byz chant scales actually corresponds to
the standard Western equal-temperament major scale:

The Byz enharmonic scale has these intervals:

12-12-6-12-12-12-6

Just like the standard major scale.

Now, the Byz diatonic scale has these intervals:

12-10-8-12-12-10-8

Some intervals match those in the standard major scale: the octave (72
moiroi), the fifths (42 moiroi), and the fourths (30 moiroi). Others
vary by 2 moiroi (the thirds, some seconds).

There are also two chromatic scales (soft & hard) with more dramatic
variations from Western scales, but I won't go into those in this
post. Here's my question:

Is there a way to translate these intervals (42 moiroi, etc) into the
frequency ratios that are used to assess how close an interval is to a
harmonically "perfect" interval? (And of course if so, what is it?)

Yours,

-- (Mr) Dana Netherton, dana 1 netherton 2 net, where "1" equals the
usual, and "2" equals the other usual
-----
I'm not a member of any organized religion.
I'm Eastern Orthodox.

🔗Paul Erlich <paul@stretch-music.com>

Hi Dana,

Thanks for your interesting question! You may have seen other
discussions here about Byzantine scales, with both the proportions
you cite (e.g., by John Chalmers) and with others that seem to
conform more closely to a "frequency ratio" derivation (e.g., by Rami
Vitale).

You might have also seen a number of posts raving about the ability
of 72-equal to approximate, within 4 cents at worst, all the
frequency ratios of numbers up to 12 (and their octave-equivalents) --
the twenty-nine intervals in an octave considered by Harry Partch to
possess 'consonance'.

If not, you might want to search the archives for 'vitale'
and 'byzantine' after you're done reading this.

The best way to assess a scale in the way you mention is to construct
a lattice and/or an interval matrix for the scale . . .

For now, i'll construct the interval matrix (since you mention the
Byzantine focus on intervals, which i find most fascinating) -- the
scale is, in moria from the tonic, 0 10 22 30 42 54 64 (72).
intervals are, of course, the differences between pitches, so the
table looks like this:

-------0....10....22....30....42....54....64

.0.....0...-10...-22...-30...-42...-54...-64

10....10.....0...-12...-20...-32...-44...-54

22....22....12.....0....-8...-20...-32...-42

30....30....20.....8.....0...-12...-24...-34

42....42....32....20....12.....0...-12...-22

54....54....44....32....24....12.....0...-10

64....64....54....42....34....22....10.....0

Here's the sorted list of intervals in moria, and the simple "pure"
intervals they approximate -- and what i think you're asking for, the
closeness of the approximation in cents:

0 1:1 (exact)
8 -
10 10:11 (2 cents narrow)
12 8:9 (4 cents narrow)
20 -
22 -
24 -
30 3:4 (2 cents wide)
32 -
34 -
42 2:3 (2 cents narrow)
44 -
54 -
64 -

If the range of the scale exceeds one octave, the negative intervals
are "modded" by 72 and become

8 -
18 -
28 -
30 3:2 (2 cents wide)
38 -
40 -
42 4:3 (2 cents narrow)
48 -
50 -
52 16:9 (4 cents wide)
60 20:11 (2 cents wide)
62 -
64 2:1 (exact)

As you can see from

http://www.72note.com/1/intervalliccontinuum.html

this is not a particularly efficient set of intervals for capturing
the many other fine approximations 72-equal makes to "pure" ratios --
you get only quite a minority of the twenty-nine.

But I personally don't think interval ratios more complex than 4:3
play a significant role in shaping or enhancing sung monodic musical
traditions, as I understand Byzantine chant is. So I certainly don't
feel this Byzantine diatonic scale is "inferior" in any way to the
one Rami Vitale presented or its 72-equal approximation.

Later,
Paul

🔗monz <monz@attglobal.net>

hi again Dana,

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> hi Dana,
>
>
> --- In tuning@yahoogroups.com, "Dana Netherton" <dana@n...> wrote:
>
> > For the last ~120 years or so, the scales used in Byz chant
> > have been officially explained in terms of a structure of
> > 72 equal "moiroi" or micro-steps to the octave. (Since Byz
> > chant's written notation records intervals, not pitches,
> > this focus on the size of the intervals in scales emerged
> > naturally in the Byz chant community.)
> >
> > <etc.>

this was just a quickie, but hopefully it might help you a bit ...

here are the closest rational approximations to all of
the 72edo degrees, where the denominators have 1, 2, and 3
digits.

you can see, for example, that 14 moria are close to the
8:7 ratio, and that 21 moria are close to the 11:9 ratio, etc.

72edo 1-digit 2-digit 3-digit

72 2/1 52/51 512/507
71 2/1 53/52 512/507
70 2/1 53/52 717/710
69 2/1 54/53 922/913
68 2/1 54/53 205/203
67 2/1 55/54 205/203
66 17/9 56/55 205/203
65 15/8 56/55 205/203
64 13/7 57/56 205/203
63 11/6 57/56 923/914
62 9/5 58/57 718/711
61 9/5 58/57 513/508
60 16/9 59/58 513/508
59 7/4 59/58 821/813
58 7/4 60/59 308/305
57 7/4 61/60 308/305
56 12/7 61/60 308/305
55 5/3 62/61 308/305
54 5/3 62/61 719/712
53 5/3 63/62 719/712
52 5/3 63/62 411/407
51 13/8 64/63 411/407
50 13/8 65/64 411/407
49 8/5 65/64 925/916
48 8/5 66/65 514/509
47 11/7 67/66 514/509
46 14/9 67/66 617/611
45 3/2 68/67 617/611
44 3/2 69/68 720/713
43 3/2 69/68 823/815
42 3/2 70/69 926/917
41 3/2 70/69 103/102
40 3/2 71/70 103/102
39 3/2 72/71 103/102
38 13/9 73/72 103/102
37 10/7 73/72 103/102
36 7/5 74/73 103/102
35 7/5 75/74 103/102
34 7/5 75/74 103/102
33 11/8 76/75 103/102
32 4/3 77/76 103/102
31 4/3 78/77 103/102
30 4/3 78/77 103/102
29 4/3 79/78 103/102
28 4/3 80/79 103/102
27 9/7 81/80 103/102
26 9/7 81/80 103/102
25 5/4 82/81 928/919
24 5/4 83/82 825/817
23 5/4 84/83 722/715
22 5/4 85/84 722/715
21 11/9 85/84 619/613
20 6/5 86/85 619/613
19 6/5 87/86 619/613
18 6/5 88/87 516/511
17 7/6 89/88 516/511
16 7/6 90/89 929/920
15 7/6 90/89 929/920
14 8/7 91/90 413/409
13 9/8 92/91 413/409
12 9/8 93/92 413/409
11 10/9 94/93 413/409
10 10/9 95/94 723/716
9 1/1 96/95 723/716
8 1/1 97/96 310/307
7 1/1 98/97 310/307
6 1/1 99/98 310/307
5 1/1 99/98 310/307
4 1/1 100/99 310/307
3 1/1 1/1 310/307
2 1/1 1/1 310/307
1 1/1 1/1 827/819
0 1/1 1/1 827/819

-monz

🔗Carl Lumma <ekin@lumma.org>

🔗monz <monz@attglobal.net>

🔗Dana Netherton <dana@netherton.net>

On 31 Oct 2003 at 21:02, Paul Erlich wrote:

> Hi Dana,
>
> Thanks for your interesting question! You may have seen other
> discussions here about Byzantine scales, with both the proportions you
> cite (e.g., by John Chalmers) and with others that seem to conform
> more closely to a "frequency ratio" derivation (e.g., by Rami Vitale).
>
> You might have also seen a number of posts raving about the ability of
> 72-equal to approximate, within 4 cents at worst, all the frequency
> ratios of numbers up to 12 (and their octave-equivalents) -- the
> twenty-nine intervals in an octave considered by Harry Partch to
> possess 'consonance'.
>
> If not, you might want to search the archives for 'vitale'
> and 'byzantine' after you're done reading this.

Well, ding dang ding! I've just learned something:

Yahoo archive searches are case-sensitive.

Search on "byzantine" (as I had done): zilch

Search on "Byzantine" (as I just did, from your hint) -- jackpot.
Messages dating back to at least July 2000. *sigh*

Before I start digging into the archives big-time (now that I know
how to search them!) ...

> The best way to assess a scale in the way you mention is to construct
> a lattice and/or an interval matrix for the scale . . .
>
> For now, i'll construct the interval matrix (since you mention the
> Byzantine focus on intervals, which i find most fascinating) -- the
> scale is, in moria from the tonic, 0 10 22 30 42 54 64 (72). intervals
> are, of course, the differences between pitches, so the table looks
> like this:

Thanks very much for taking the time to compose the table for this
post.

Since even this group might not have encountered it, I'll offer a
link to a friend's web site that has a table of the interval-signs:

<http://users.forthnet.gr/ath/frc/Aless2.html>

In that table, the first sign represents zero change in pitch.

The next several signs represent changes upward -- by 1 step in the
scale (denoted by the letter "alpha"), by 2 steps (a jump, denoted by
the letter "beta"), and by 4 steps (a jump, denoted by the letter
"delta").

And the last group of signs represent changes downward -- by 1 step
("alpha"), by a succession of 2 steps (in two notes, denoted by the
Arabic numeral "2"), by a jump of 2 steps ("beta"), and a jump of 4
steps ("delta").

The Greek letters are simply the conventional way of writing numbers,
in Greek. The use of the Arabic numeral was a quirk of the person
who composed the table in the early 1800s, and is not routinely used
today.

However, the signs *are* used routinely today.

Why several signs for 1 step upward? Because each sign represents a
slightly different way of moving up. The Oligon simply moves up. The
Petaste moves up with a bit of a quaver (and there is a wide
variation in performance practice on this sign from one cantor to
another). The Kentimata moves up on a weak syllable in the text
(remember, all this music is sung to a text; none is performed
instrumentally).

And these signs are combined with one another in certain ways to
produce the rest of the intervals one needs to write down a melody.
Along with other signs that indicate tempo and rhythm ... and the
scale being used, and various shadings of sharps and flats, and and
and. :-)

But I thought you might find the "interval" signs especially
interesting.

Oh, and here's an example on the web:

<http://cgi.di.uoa.gr/~gbelis/sample_text.html>

(The text is from the opening of Vespers on Saturday night; the music
is in Echos/Tone/Mode One.)

For something with an English text, here's a URL with links to some
PDF files. :-)

<http://sgpm.goarch.org/e_services/music.html>

As you'll see from the sampling on this page, the repertoire is rich
... and is being used actively today.

<snip>

You went on to say ...

> As you can see from
>
> http://www.72note.com/1/intervalliccontinuum.html
>
> this is not a particularly efficient set of intervals for capturing
> the many other fine approximations 72-equal makes to "pure" ratios --
> you get only quite a minority of the twenty-nine.
>
> But I personally don't think interval ratios more complex than 4:3
> play a significant role in shaping or enhancing sung monodic musical
> traditions, as I understand Byzantine chant is. So I certainly don't
> feel this Byzantine diatonic scale is "inferior" in any way to the one
> Rami Vitale presented or its 72-equal approximation.

Thanks. You're right, "harmonies" in the Western sense don't matter
much in Byzantine chant. Though my path has crossed those of a (very)
few people in the Byz chant world who enjoy debating the fine points
of the theory in terms such as ... "ah, a 68-step octave would
produce just-temperament-like 'beautiful intervals' so much better
than a 72-step octave", that sort of thing.

But me, I'm thoroughly in listening-mode.

Again, thanks.

Yours,

-- (Mr) Dana Netherton, dana@netherton.net
-----
I'm not a member of any organized religion.
I'm Eastern Orthodox.

🔗Dana Netherton <dana@netherton.net>

On 31 Oct 2003 at 22:16, monz wrote:

> hi Dana,
>
> --- In tuning@yahoogroups.com, "Dana Netherton" <dana@n...> wrote:
>
> > For the last ~120 years or so, the scales used in Byz chant
> > have been officially explained in terms of a structure of
> > 72 equal "moiroi" or micro-steps to the octave. (Since Byz
> > chant's written notation records intervals, not pitches,
> > this focus on the size of the intervals in scales emerged
> > naturally in the Byz chant community.)
> >
> > <etc.>
>
> i don't really have anything to add to paul's response, other
> than to direct you to my webpages on 72edo and on "morion":
>
> http://sonic-arts.org/dict/72edo.htm
>
> http://sonic-arts.org/dict/moria.htm
>
> you'll probably find some useful data there.

Thanks, Monz. Actually, you helped me in another way as well: in
vocabulary. Somewhere along the line, I had seized upon the Greek
word "moira", which means "a piece, portion". Probably because my
eye fell upon it first, in my Greek lexicon, while I was looking up
the term actually used: the term *you* used, "morion".

Which *also* means "a piece, portion"! (And both words have their
root in the same word, meiromai, "to receive one's share".)

After some confusion on my part, I looked around, found my source ...
and discovered my error. Thanks for the chance to get corrected!

Yours,

(Mr) Dana Netherton, dana@netherton.net
-------------
Polonius: My Lord, I will use them according to their desert.
Hamlet: God's bodykins, man, much better: use every man after
his desert, and who should 'scape whipping?
(Act II, scene ii)