FW: A Summary of "Theory of Intonation"

I dug this up; it's kind of interesting:
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From: rom
To: paule
Subject: mail
Date: Wednesday, March 08, 1995 7:32PM

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Date: Sun, 4 Dec 1994 19:21:21 +0800 (HKT)
From: "Mr. Linus ChiFai Liu"
To: rom@MIT.EDU
Subject: One Correct Tuning System
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Hello,

Please do not feel offended my sending you this message. Forgive me if
you are not interested, but I think you might.


The "One Correct Tuning" is not what I called it. I do not know who did.
It is just one that seem to be used by all major composers in the
mainstream traditional.


A Summary of "Theory of Intonation"
===================================

by
Linus Liu
of
Hong Kong

1. Scale

Below is the interval arrangements in a MAJOR SCALE, as the
writer has correctly found and identified:

DO RE ME FA SO LA TE DO'
10/9 10/9 27/25 9/8 10/9 10/9 27/25

There are two sizes of whole tone, the smaller 10/9 and the
larger 9/8. Their sum is major third (10/9 times 9/8 = 5/4).

Therefore, it is seen in the above string that between DO and
ME in a SCALE, there is NOT a major third interval (10/9 times
10/9 is 100/81 and not 5/4).

2. Chord

In a tonic major triad, the bass in root position is lowered by
the necessary interval (81/80, syntonic comma) so as to become a
major third beneath the median, which is a VERY IMPORTANT phenominon
in music. The dominant must similarily be lowered, unless this
dominant is in soprano.

Be surprised, therefore, that the octave between BASS and SOPRANO
in an SATB chord is not an octave, but an interval sized 81/40.

The Tonic Major Chord
=====================

Perfect 5th
__________________
| |
| |
DO ------ ME ------ SO DO
maj 3rd min 3rd
| |
|____________________________|

This is 81/40, not an octave, but bigger!

DO and ME in Scale = 10/9 times 10/9 = 100/81 # Major 3rd
DO and ME in Chord = 9/8 times 10/9 = 5 / 4 = Major 3rd
(DO lowered by 81/80)

ME and SO in Scale = 27/25 times 9/8 = 243/200 # Minor 3rd
ME and SO in Chord = 27/25 times 10/9 = 6 / 5 = Minor 3rd
(SO lowered by 81/80)

When DO happens to be in the alto, and ME in the Soprano: this
ME will not tolerate being comparatively so much lower (relative
to each other, or to the averaged pitch on the piano) than alto's
DO - when the major third is already small, now even smaller by a
comma.

Hence the soprano ME is adjusted upwards by the same sytonic
comma (81/80). For this reason, alto remains instead of being
lowered like the bass would have to. Now alto and soprano are in
tune as major third.

3. Cadence

It is eaily seen that in a subdominant chord of a major key, all
its component notes are already in tune. The tonic in the bass
need not be lowered as in the case of the tonic chord.

The DO in the subdominant chord need not be lowered. The DO in
the bass in a tonic chord must be.

Hence in the Plagal Cadence, the DO will contradict if remain in
a same lower part through both chords (unless this DO is in the
alto AND also the soprano is ME, which is raised).

This explains a very basic rule learned by every student studying
Harmony.

There are many other rules in traditional Harmony and Counterpoint
that may be similarly explained.

4. Modulation

When a major key modulates to its dominant, the fourth note raises
by a chromatic semitone of 25/24 becoming the seventh note.

The third note now becomes a bigger whole tone from the sharpened
fourth note than the required smaller whole tone needed between
the new sixth and seventh notes. It has to be raised by a comma
(81/80). So does the second note in becoming the fifth.

Hence in modulation of a major key to its dominant, three, not one,
notes need be adjusted in pitch.

5. Keys

The intonation of different keys is such that the largest
number of notes on the open strings of the violin can be
played in tune on the scale/chords.

Observe the following sequence that is derived from the major scale:

ME TE RE FA LA DO SO
P5 mr3 mr3 Mj3 mr3 P5

P5 = Perfect 5th
mr3= Minor 3rd
Mj3= Major 3rd

Because perfect fifths and minor thirds are larger than averaged
intervals (as on piano), notes on the right are also relatively
higher than those on their left (except FA and LA which is major
third).

Hence we have more chances of finding in performance the dominant,
tonic and subdominant notes out of tune sounding too flat.

In particular, if the note SO is tuned accurately to, say, A 440,
which is the D major, all the other notes on the violin will
become lower than those corresponding notes on the piano. Hence
the D major is obviously a low key.

Brahms made an exception in his violin concerto. By going through
the D minor, he caused the subsequent D major to raise by a comma
in pitch. By doing so correctly one should find no trouble with the
subsequent double stops, nor passages in the second movement when
it modulates.

When A 440 become the LA as in C major, the open E string is
higher than required inside the scale, but good in the soprano of
a tonic major chord.

The F major is a dilemma. If A 440 and E 660 are in tune as
respectively the ME and TE, the whole F major is high and the
open strings G 195.56 is too low for RE, and open D 293.33 too
low for LA.

Or the G and D may be followed, leaving the A and E strings too
high.

As performing musicians commonly know, the F major IS high.

Beethoven made some exceptions. When the violin and piano swapped
their respective roles of soloist and accompaniment in the opening
of the Spring Sonata, the violin played in the lower pitch in
the lower string. If this has not happened, the Spring Sonata may
probably not sound like spring. Nor will it sound like spring if
a violinist does not observe such fact.

Unit used below is 1 / 4096 of a half-tone. Or, octave is divided
into 12 x 4096 equal divisions. The figures shows deviations from
a equally divided scale, or keyboard. A negative value indicates
that the note when compared to a equally tempered keyboard would
be comparatively flat, positive would mean sharp.

These figures can be applied directly to the PitchWheel command in
MIDI music.

KEY A-maj C-maj D-maj E-maj F-maj G-maj Bb-maj Eb-maj Ab-maj

SO 80e 720 0a 160 1520 800 1440 1360 1280 x s
ME -560 80e -640 -480 880 160e 800 720 640 *
DO 0a 640 -80d 80e 1440 720 1360 1280 1200 + * s
LA -640 0a -720 -560 800 80a 720 640 560 s
FA -80d 560 -160g 0a 1360 640 1280 1200 1120 s
RE -720 -80d -800 -640 720 0d 640 560 480 s
TE -1360 -720 -1440 -1280 80e -640 0a -80d -160g s
SO -800 -160G -880 -720 640 -80g 560 480 400 +
ME -1440 -800 -1520 -1360 0a -720 -80d -160g -240 + * x s
DO -880 -240 -960 -800 560 -160 480 400 320 + * x

Two ways of a tonic major chord______|_| | |
and one tonic major triad _________| |
Notes played in a major scale____________|

F major is therefore the sharpest key, and D major, lowest.

The keys C, G, D and F major use all the four notes on the open
strings of the violin.

The D major can be modulated to A major quite easily, seen above,
the open A string being the note SO in D major becomes the DO in
the new key, A major. The same goes for A major to E major.

On the violin, the player only need to shift the fingering pattern
on each violin string up an adjacent string.

However, this is not true for G major going to D major, nor F major
to C major.

===============

The following analysis of excerpts from the Beethoven Violin
Concerto shows how Beethoven adhere closely to the above scheme
of intonation in relation to the different keys.

A. ---------

In bar 224 of the first movement, the last note played by the
soloist is A. This A is a high note in the key of A major.


The orchestra takes over this note and plays it as the third note,
a low note in the key of F major.

What is apparently an abrupt modulation achieved modulation to
F major which is to be played in the manner "commonly practiced",
as above table.

B. ---------

The key of C major can end up one of three ways:

1. As the dominant key of F major, the note E in the scale is in
tune on the open E string.

2. As the subdominant key of G major, the notes D and A in the
scale are in tune on these open strings.

3. As the subdominant of the subdominant key of D major, the note
G is in tune on the open G string.

The following analysis shows how Beethoven manipulated such
modulation:

At bar 239 of the first movement of the Violin Concerto, the
music is A major. At bar 247, it modulates to its simple A minor.
At bar 258, the note E in the chord C major is held by five
instruments. As previously explained, this top note being the
third of the (tonic) major chord, must be a comma higher.
However, this note is sustained from previously, and is not
possible to the open E string. The low E is in tune to the open G
string.

At bar 260, these two notes remain unchanged. The low E is then
sustained in the second violin until bar 264. Here, the chord
becomes a C major seventh. All notes must be in tune. The note C
in the bass is in tune with the sustained low E in the second
violin, and is also in tune to the open G string. This C is
sustained in the bass until bar 268. The chord becomes tonic C
major. The top note C must be higher than the C in the bass by a
comma. Hence the tonic C major is the one named (2) above, where
the note C is in tune to the open E string.

C. ---------

When proceeding from the first movement, ending in D major,
into the second movement, in G major, Beethoven took the necessary
steps to manipulated the intonation of his notes so that the
second movement will be played in adherence to the above table,
and when going back from G major to D major from the second to
the third movement.

Beethoven violin concerto 2nd mvt chord and melody analysis,
Bar 89 to 91.
__________ ___________ _________ ______
G / Dim5 36/25 C#/chr.st 25/24 D / xn3 32/27 F \ ______ G#/ chr.st A
| | | | mj6 5/3 | 25/24 |
B.otv 81/80 dim5 36/25 B.mj6 27/16 mn3 6/5 dim5 36/25 B.mn6
81/50
| __________ | | | | |
G /B.otv 81/80 G \ ___________ F \ _________ D --------- \ _______ C#
| | big wt 9/8 | mn3 6/5 | semitone 27/25 |
unison mn3 6/5 mn3 6/5 otv 2/1 mj6 5/3
| __________ | | | ________________ |
G /B.mj6 27/16 E \ ___________ D \ _________ D / wt 10/9 E
| | B.wt 9/8 | otv 2/1 | |
mj6 5/3 P5 3/2 mj3 5/4 mj3 5/4 P5 3/2
| | ___________ | | |
Bb \__________ A / mn2 16/15 Bb ---------- Bb \ _________________ A
mn2 16/15 remain semitone 27/25

The chords put in their "up-right" positions reveal the intervals
required:
1. 2. 3. 4. 5.

G G D (G#) A
B.mj6< mn3< big mn3< mn3< Big4<
Bb E F F E
mn3< mn3< mn3< mn3< mn3<
G C# D D C#
unison< mj3< mj3< mj3< mj3<
G A Bb Bb A

Note:

1. In the end, the note A on the soprano is lowered by the targeted
interval of a comma, 81/80.

Since the started note G on soprano is per above table, tonic in G
major scale, a comma higher than the open G string, the final A on
the soprano is 36/25 x 25/24 x 32/27 / 5/3 x 25/24 = 10/9. This A
is the same on the open A string, satisfying the D major key on the
above table.

2. The soprano melody when singing from the third into the fourth
chord, need to sing the interval marked "xn3", an interval small
than the minor third by a comma. The F on the fourth chord may
therefore has a temptation of getting resolved, i.e., sung a comma
higher, and the "Big octave" formed. Beethoven lessens the tendency
of this happening by having the F (3rd) doubled. The F cannot be
played as resolved, since there is a G#, which participates as the
seventh of the chord, on top of the soprano F forming a minor third
interval.

===============

Re-written on 4 Dec, 1994

The writer can be reached at e-mail , or
fax +852 471 7557

BTW, my sister, Dilys Liu, is also an MBA from MIT.

Regards,
Linus Liu





------------------------------

Topic No. 2

Date: Fri, 12 Jul 96 14:38 EST
From: PAULE
To: tuning
Subject: FW: Darreg Disc Error: track 3 mislabell
Message-ID: <12960712193821/0005695065PK4EM@MCIMAIL.COM>


A second attempt at posting this message:
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I wrote:

>The lower note of the minor
>third is about a middle d#, and a little calculation shows that the beat
>frequency for a 22-tet minor third in this register is 2.5 Hz.

A little erroneous calculation, that is. The mystery deepens! Could Ivor
have been using 17 unequally-spaced tones? (He's laughing at us from the
beyond.) I just got a micro-tunable synth so I'm going to try to solve this
mystery.