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1. Functional Tuning Construction Table
See TABLE-1 and use the BACK button to return.>/hlink>
Each value in this table represents the distance of a note from "D",
the central note of the natural diatonic system.
These distances are indicated in thousandths of an octave
(Milli-Logarithms base 2, abbreviated ML2).
Each note is situated in the octave between "D" 000.0000 and "D" 1000.0000.
The traditional measure for these distances is the cent,
defined as a hundredth of an equi-tempered semi-tone.
There are therefore 1,200 cents in the octave.
For those desiring to transpose to this form of measure,
the following formula must be applied
Distance in ML2 X 1.2 = Distance in cents.
Since we are here interested in Just Intonation and not in Equi-temperament,
measure in ML2 seems more appropriate than that in cents.
(see USE OF COLOR)
First column
The first column contains the notes of Trunk Tuning (Pythagorean),
generated uniquely with the perfect fifth (proportion - 3/2)
around the central note "D", colored as a COMMON TONE.
The proportion 3/2 is represented as 1.5.
The logarithm base 2 of 1.5 is .5849625.
Multiplying by 1000 to get ML2s, we have 584.9625
which is found at the note "A", colored as a PROPER TONE.
If we add 584.9625 we will get the ML2 distance of all the notes
above D in this column.
If we subtract 584.9625 we will get all the notes
below D in this column.
To keep everything between 0 and 1000,
we must keep adding or subtracting 1000 whenever required.
Second column
The second column contains the notes of Short Branch Tuning (natural)
generated with the major third (proportion - 5/4) used only once.
The proportion 5/4 is represented as 1.25.
The logarithm base 2 of 1.25 is .3219281.
Multiplying by 1000 to get ML2s, we have 321.9281
which is found at the note "F#-", colored as a MEDIAN.
If we add 321.9281 to the bottom note of a FRAME (in column 1),
we will get the major MEDIAN of the FRAME (indicated "-").
If we subtract 321.9281 from the top note of the FRAME, (in column 1),
we will get the minor MEDIAN of the FRAME (indicated "+").
Third column
The third column contains the notes of Long Branch Tuning (natural)
generated with the major third used twice.
Multiplying the size of the major third in ML2s by 2
(321.9281 X 2 = 643.8562)
gives us the size of the augmented fifth
which is found at the note "A#--", colored as a MOTRIX.
If we add 643.8562 to any note in column 1,
we will get the augmented fifth higher (indicated "--").
If we subtract 643.8562 from any note in column 1,
we will get the augmented fifth lower (indicated "++").
2. Table of Progressive Pitch
See TABLE-2 and use the BACK button to return.>/hlink>
The elements in this table are presented (from bottom to top)
in order of increasing pitch from D to D.
The column(s) titled -
TT(Trunk Tuning),
SB (Short Branch Tuning),
LB (Long Branch Tuning),
contain the notes of the Funcional Tuning CONSTRUCTION TABLE;
T-12 contains the notes of Equi-tempered tuning,
with the octave divided into 12 equal parts;
T-7 contains the notes of an imaginary system without Chrominicism,
with the octave divided into 7 equal parts.
The following columns indicate the position of each note given in
Frequency (vitrations per second) in Hertz,
Proportion (ratio) of the frequency divided by the frequency of D (293.6648 Hz);
the distance between this note and the note D, as a proportion, and,
ML2 (Milli-Logarithm base 2 of this proportion)
the distance between this note and the note D in thousandths of an octave.
The last 2 columns indicate the distance (in ML2) of each note
from certain points of reference
Diff. T-7, from the note of the same name in 7-part tuning
which indicates the precise Chrominic Position of the note;
Diff. T-12, from the note of the same name in Equi-trempered tuning
which indicates "how out of tune" each note is in Equi-tempered tuning.
To see the other sections of Just Intonation
Preface
Diagrams
Performance
Natural Harmonics
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