Overtones and Accidentals
Main Page
Charles W. London
Music is God's best gift to man, the only art of heaven given to
earth, the only art of earth that we take to heaven. But music, like all
our gifts, is given us in the germ. It is for us to unfold and develop it
by instruction and cultivation.
I would like to consider the alterations of melodies and chords in
the relevency of the tonal `frequencies' played; its divisions and
multiplications. This section discusses the occurances of comparitive
tones by their frequencies and a manner inwhich one may possibly modulate
chords and for melodic lines a variance of mode without leaving the
intended tension. The harmonic series is not the only means to which a
system of music can be based on. Twelve-tone tonality and Atonality of
pitch classes is also a very valid writing and performing source. It is
merely something that needs to be accustomed to, but I must admit
improvisation becomes nearly impossible to exercised because they are
microstructural systems. This is my position that it also can be
available in modality, not that I'm trying to do away with improvisation,
but instead, what I'm getting at is that modality could allie in the
homogeneous.
Overtones encompass tonal music in that it represents what tones
are resonant or are resounding in frequency to each other. In this system
I shall not expound into microtones and fractals, but rather rounding just
intonations to the closest 12tet tones, since it depends heavily on mode
and key changes. In spite of just intonation having a better consonance,
it is preferred here the twelve spans vectors by diatonic dispensation and
cannot be delimited in free modulations. On the means of the established
tonal methods, partials to the seventh from the fundamental has been
acquired as the basis and essential concept. Irregardless, sonority's
structure may be accepted in other fashions of their applicability. Unless
a means concentrating enharmonic equivalence can be generated, it would
seem impractical to adapt just intonation. However, it is allowable to
assimilate overtones on open strings or brass as deem purposeful. For
more information on just intonation, see Just Intonation
Explained by Kyle Gann, and John Starrett's
Micro-tonal Music Page.
12
The make up of the sounds in an instrument are portions of the
harmonic series and its identification is due to the attack the instrument
has. The well-tempered tuning system was something to adjust to, but the
liberation to free key changes is worth the difference to precise
partials. I view the harmonic series as a guidepost to sampled tones.
The modal exchange of chordal play in this set of formulas does recognize
partials, yet not in the conventional manner. The seven modes referred to
by this report do yield to plenty of pliability and a means to
chromaticism as will be explained.
On working with partials, a tonal center is always referred to and
that being the tonic. So off from there we observe the frequencies and
identify what notes are associated. Certain fractions were developed in
the just intonation and in the even tempered scale is derived 2^(1/12).
Here is the list using C as the tonic:
Just Intonation Even Tempered
Tones Derivations Fractions Decimals
C Tonic 1/1 1.00000
C#/Db Min 2d 16/15 1.05946
D Maj 2d 9/8 1.12246
D#/Eb Min 3d 6/5 1.18921
E Maj 3d 5/4 1.25992
F Perfect 4th 4/3 1.33484
F#/Gb Dim 5th 17/12 1.41421
G Perfect 5th 3/2 1.49831
G#/Ab Aug 5th 8/5 1.58740
A 6th 5/3 1.68179
A#/Bb Dim 7 9/5 1.78180
B Maj 7 15/8 1.88775
C Tonic 2/1 2.00000
Now, when we compare two tones it may be determined what other
notes are in common. Using for example E and G, we have:
Related Tones Formulas (example):
5/4 * 3/2 =15/8 which is B or
1.25992 * 1.49831 =1.88775 again B and
5/4 * 2/3 =10/12 =5/3 which is A or
1.25992 * (1/1.49831) =0.84089 * 2 =1.68179 is A and
4/5 * 3/2 =12/10 =6/5 which is Eb or
(1/1.25992) * 1.49831 =1.18921 is Eb and
4/5 * 2/3 =8/15 =16/15 which is C# or
(1/1.25992) * (1/1.49831) =0.52973 * 2 =1.05946 is C#
The minds ear would hear those notes which are even reciprocated as
related. So, what has been accomplished here is, if you would notice that
the outcomes have two pairs of harmonies: C# B, and Eb A. Referring back
to the key changes chart these paired tones share the same
concordance/discordance value. These figure well in both calculations:
Tones No. of Just Even
the Intonation Tempered
Tones
C 0 1/1 * 1/1 =1/1 or 2/1 1*1 =1 1*2 =2
F/G 5|7 4/3 * 3/2 =12/6 = 2/1 1.33484*1.49831 =2
E/Ab 4|8 5/4 * 8/5 =40/20 =2/1 1.25992*1.58740 =2
Eb/A 3|9 6/5 * 5/3 =30/15 =2/1 1.18921*1.68179 =2
D/Bb 2|10 9/8 * 9/5 =81/40 ~2/1 1.12246*1.78180 =2
C#/B 1|11 16/15 * 15/8 =240/120 =2/1 1.05946*1.88775 =2
F# 6 17/12 * 17/12 =289/144 ~2/1 1.41421^2 =2
Further relations may be established in chords to exhibit what accidentals
can be usable in certain fields of chords. If, for example, we examine G B D
and C E G, we have to compare all the vectors in the individual chords:
(This example is considered as Ionian)
Compared
Tones Effects
G B F# F# E Ab
G D F G Eb A
B D C# B Eb A
C E E Ab E Ab
C G G F G F
E G B C# Eb A
This displays the accidentals available for the sequence GBD/CEG.
Melodies mand chords may be altered to these fixed tones, that is to say,
over GBD: one may play a tune Ab Eb F# and over CEG: one may play C# Eb
Ab or in the general field of the chords. Oddly enough even in this
extreme example it is possible to feel the relations, but this should not
be over done so that the melodic mode will become indistinguishable.
Here
are the tables of comparisons.
As mentioned earlier concerning mode swings certain modes
coordinate well in that they produce the same evaluations by relating the
tones. It is therefore suggestive that tonal evaluations need not follow
a series; instead it may merely indicate the scheme of the harmonic
influences.
A very informative book named `A Theory of Evolving Tonality'
[Author: Yasser, Joseph - Publisher: Da Capo Press] that talks about
what one may ask is the cause of the tonal scales, or why are the diatonic
tones arranged in their manner. There is an evolution to scales starting
with the smallest number of tones and that being 2 diatonics and 3
accidentals. Where the initial tonic note is F and the next note is ~C
and the next is the return to the tonic F; between F and ~C is equivalent
to ~G and ~A, and between ~C and F is equivalent to ~D. The ratio of this
scale is 2^(1/5). By cycles of fifths it is possible to interlay more
tones to produce a higher scale. So far we have F ~C ~G ~D ~A and now
primarily where there are tones in the scale that are adjacent to the
other tones by half steps - one may add another tone between them, that
becomes the new accidentals. Where we have after (~A) then ~E and then
~B, forming a new ratio of 2^(1/7). The succeeding scale is our diatonic
scale where F# C# G# D# A# are added between the semitones first.
So the formula is an addition by a golden ratio: A: accidentals T: tones
C: chromatics
A T C
3 + 2 = 5
A T C
2 + 5 = 7
A T C
5 + 7 = 12 etc...
This can go on indefinitely, where the subsequent scales are 19,
31, 50 etc. chromatics per octave. When attending these grander scales a
range is required beyond the human hearing, and overtones become pushed
way high to find a match. It seems for now best to stick with diatonics.
The golden ratio is found often in nature, for example counting the rows
of sunflower seeds leading to the left and right or the snail's shell
spiral rings measures, taking from the center point out, to values by this
addition. This might prove convincing that the diatonic tones are a
natural foundation and may have any starting location. Also it is
interesting to note that by the cycling fifths one is directed from the
brightest mode to the darkest, ie. F lydian, C Ionian, G mixolydian, D
dorian, A aeolian, E phrygian and B locrian.
Even though heavily triadic, another great dissertation (this one
online!) that talks on modality and Schenker's theory: Aspects of Early
Major-Minor Tonality, and his thesis The Development of
the Concept of `Line' - by Doug Anderson.
Continue