APPROXIMATION OF 5-LIMIT JUST INTONATION

Computer MIDI Modeling in Negative Systems of Equal Divisions of the Octave

Mykhaylo Khramov

Independent researcher, 53-2, Plekhanova street, 91005, Lugansk, Ukraine

Keywords: Intonation, just, music, score, tuning.

Abstract: The article matter is related with music processing by MIDI protocol during computer modeling of fixed

scales with non-traditional equal temperaments. Are touched negative temperaments, which based on closed

series of fifths, compressed relative conventional tuning. Is marked, that such systems, can better approach

to just intonation. They give sensation out of tuning during listening to music performed by scores with

mistaken using of accidentals, which inaccessible in a conventional temperament system. Is given a

subprogram example of the automatic Pitch Bend change in MIDI protocol for modeling of negative system

of equal divisions of the octave.

1 INTRODUCTION

1.1 Purpose of the Paper

Author imparts about applying of negative systems

of equal divisions of the octave and how they was

obtained by means of MIDI protocol.

1.2 5-limit Just Intonation (5LJI) and

Natural Scale (NS)

5LJI is an ideal infinite graded system of fixed

tuning which provide:

1. for all pairs of grades, correlation of sound

frequencies between grades of each pair with

rational number factorable to primes no more 5

by value;

2. for each grade existence of all other grades,

with anyone ratios of frequencies, which

satisfy the condition 1.

Everywhere in this paper a music which allow

5LJI is supposed. Compositions by J. S. Bach can be

such examples (Asmussen, 2001).

NS is a set of sounds of defined frequencies with

correlations, which proper to natural numbers.

Frequency of the 1-st element of NS is lowest, 2-nd

in 2 times higher of lowest, 3-rd in 3 times higher of

lowest, and so on.

In 5LJI is possible to build not full NS from any

grade. In such ones there will be no elements with

numbers which contain prime factors more 5 by

value. Numbers 7, 11, 13, 14, … will be absent.

1.3 System of Equal Divisions of the

Octave (EDO), Evaluations and

Initial Values

The EDO system is graded finite fixed tuning

ensuring:

1. for each grade possbility or existence of other

grades, with interval of a perfect octave up

and/or down;

2. for all intervals of a perfect octave existence

of an identical quantity of grades inside each

octave;

3. for all pairs of an identical interval between

grades of each pair.

Conventional 12EDO system has so called

semitone between adjacent grades. 1/100 of

semitone named cent, is adopted as a basis of

evaluations in this paper. The values cited without

verifying calculations, are obtained in the software

Scala (Op de Coul, 2004).

1.4 Negative EDO System

R. H. M. Bosanquet has offered a method of

deriving of other EDO systems from an accepted for

initial system 12EDO (E. T. at Bosanquet):

«…Let a regular system of fifths start from c. If they

181

Khramov M. (2008).

APPROXIMATION OF 5-LIMIT JUST INTONATION - Computer MIDI Modeling in Negative Systems of Equal Divisions of the Octave.

In Proceedings of the International Conference on Signal Processing and Multimedia Applications, pages 181-184

DOI: 10.5220/0001932001810184

Copyright

c

SciTePress

are positive, then at each step the pitch rises further

from E. T. It can only return to c by sharpening an E.

T. note.

Suppose that b is sharpened one E. T. semitone,

so as to become c; then the return may be effected at

the first; b in 5 fifths, at the second b in 17 fifths, at

the third b in 29 fifths; and so on. Thus we obtain the

primary positive systems…

If the fifths are negative, the return may be

effected by depressing c# a semitone in 7, 19, 31...

fifths; we thus obtain the primary negative

systems…» (Bosanquet, 1875)

The Figure 1 illustrates application of this

method from note A

2

as origin.

Initial system is created by uniform compressing

of a series of 12 perfect fifths with a small interval,

which is known as comma of Pythagoras (CP). In a

scale of Pythagoras (SP) are used fifths absolutely

coincident with interval between 2-nd and 3-rd

elements of NS or perfect fifth.

The octave of all EDO systems is exactly equal

to an interval between 2-nd and 4-th elements of NS

or perfect octave. 12 fifths of SP exceed 7 octaves

on CP by size 23.46 cent.

Uniform compression of all 12 fifths SP with the

goal to eliminate CP gives 12 fifths of the most

spreaded 12EDO system. A fifth of this system

approximates perfect fifth with an error -1.955 cent,

and is adopted as initial for a further building.

19 fifths of a 12EDO system exceed 11 octaves

by one semitone of a 12EDO system. After uniform

compression (for elimination of this semitone) all 19

fifths of a 12EDO system, are obtained 19 fifths of a

19EDO system.

A fifth of 19EDO system has departure -5.263

cent from an initial value and approximates perfect

one with an error -7.218 cent.

A fifth of the next after 19EDO negative 31EDO

system approximates perfect one with an error -

5.181 cent.

The useful paradox of negative EDO systems:

the impairment of approximation of perfect fifth of

an initial 12EDO system, may automatically

improve approximation of major third of 5LJI

(interval between 8-th and 10-th elements of NS on

Figure 1).

«… For if we take 4 negative fifths up, we have a

third with negative departure (-4δ) which can

approximately represent the departure of the perfect

third. Thus c# is either the third to a, or four fifths

up from a, in accordance with the usage of

musicians…» (Bosanquet, 1875).

1.5 Approximation 5LJI in 19/31EDO

Systems and Simplicity of

Transposing of Music, from 12EDO

System to Them

The attractiveness of such approximation becomes

noticeable from a Table 1.

Natural

Scale of

5LJI

Octave

Scale

12 Fifts

of Pyhago-

rean Scale

12 Fifths of

12EDO

System

19 Fifths of

12EDO

System

19 Fifths of

19EDO

System

Figure 1: Scheme of Building of 19EDO System from Note A of Subcontraoctave (A

2

) as Origin.

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182

19EDO system is less preferable, as in

polyphony of instruments, rich by upper overtones

(harpsichord, piano), it can give noticeable sensation

out of tuning. It can not arise in same system, for

timbres of flute for example. The most probable

reason of it, is in a too large error of perfect fifth.

The Table 2 demonstrates simplicity of the

correspondences between all possible designations

of the notes and numbers of grades of these systems.

In it the accidental n indicates a lack or cancellation

of operation of other ones, i. e. neutral pitch of the

note.

Application of additional accidentals actually is

not required and direct using of scores for 12EDO

system is possible for playing in 19/31EDO ones.

2 SUBJECT OF RESEARCH

2.1 Possibility of Computer Modeling

of 19/31EDO Systems

Practically each modern computer has a sound

synthesizer, supporting protocol MIDI. Such

synthesizer has a set of channels.

Each channel is polyphony 12EDO system with

independent control, providing changed Pitch Bend

(PB).

PB works similarly to accidental, but shifts

pitches of all notes in the channel, as sounding, as

well consequent. Therefore its application for

retuning of the musical pieces from a 12EDO system

to other ones, requires in each channel only one

voice. A polyphony in each channel also is possible,

but the octaves and unisons only are admissible

between voices.

Any grades of 19EDO and 31EDO systems can

be obtained from suited 12EDO grades equipped by

an appropriate PB value from the Table 2.

PB value for each note can be added to software

Sibelius 4 (Eastwood and Others, 2005), allowing to

transpose conventional scores to MIDI versions:

//(c)Mykhaylo Y. Khramov, 2006

switch (NameOfNote)

{

case ("Gx"){PB = "-48¢~B 66,48";}

case ("G#"){PB = "-26¢~B 95,55";}

case ("Gn"){PB = "-3¢~B 124,62";}

case ("Gb"){PB = "+19¢~B 25,70";}

case ("Gbb"){PB = "+42¢~B 54,77";}

:

Table 1: Comparison of Three Variants of 5LJI Approximation by EDO systems.

5LJI Approximation

NS 12EDO 19EDO 31EDO

Cents Cents Cents

System

Interval

El-ts Cents

Size Abs. error Size Abs. error Size Abs. error

Major Third 10/8 386.314 400.000 13.686 378.947 7.367 387.096 0.782

Perfect Fifth 3/2 701.955 700.000 1.955 694.737 7.218 696.774 5.181

Octave 4/2 1200.000 1200.000 0.000 1200.000 0.000 1200.000 0.000

The average absolute error of approximation 5.214

4.862

1.988

Table 2: Modeling of 19EDO and 31EDO Systems by Evaluation for All Possible Notes of PB of Protocol MIDI.

Note Designation System Grade from Origin at Cn

12EDO, initial 19EDO 31EDO

Cents Cents

Name Accidental

Number PB

Cents

(Pitch)

PB

Depart. Pitch

Number PB

Depart. Pitch

Number

x 9 8192 900.000 4959 -78.947 821.053 13 6210 -48.387 851.613 22

# 8 8192 800.000 6467 -42.105 757.895 12 7135 -25.806 774.194 20

n 7 8192 700.000 7976 -5.263 694.737 11 8060 -3.226 696.774 18

b 6 8192 600.000 9485 31.579 631.579 10 8985 19.355 619.355 16

G

bb 5 8192 500.000 10994 68.421 568.421 9 9910 41.935 541.935 14

:

: : : : : : : : : : : :

x 11 8192 1100.000 4527 -89.474 1010.526 16 5946 -54.839 1045.161 27

# 10 8192 1000.000 6036 -52.632 947.368 15 6871 -32.258 967.742 25

n 9 8192 900.000 7545 -15.789 884.211 14 7796 -9.677 890.323 23

b 8 8192 800.000 9054 21.053 821.053 13 8720 12.903 812.903 21

A

bb 7 8192 700.000 10563 57.895 757.895 12 9645 35.484 735.484 19

APPROXIMATION OF 5-LIMIT JUST INTONATION - Computer MIDI Modeling in Negative Systems of Equal

Divisions of the Octave

183

Figure 2: Prepared MIDI Version of Score.

case ("Ax"){PB = "-55¢~B 58,46";}

case ("A#"){PB = "-32¢~B 87,53";}

case ("An"){PB = "-10¢~B 116,60";}

case ("Ab"){PB = "+13¢~B 17,68";}

case ("Abb"){PB = "+35¢~B 45,75";}

default {PB = "±00¢~B 0,64";}

}

return PB;

Such simple subprogram allows to retune

automatically the musical pieces and to get audible

models of their sounding. Some from them are freely

offered for listening to and discussion (Khramov,

2004, 2006).

On Figure 2, the subprogram has added above

each note in ossias the text, which is seen from an

initial quotation marks up to a tilde, and is hidden,

since a tilde and up to a completing quotation marks.

The visible text shows to observer the departure of

each note pitch, and hidden one transmits

appropriate PB to a MIDI device. It calls a required

departure of each note pitch for a selected 31EDO

2.2 Modeling of Negative Systems,

Indicates Mistakes in Scores

If in a context 12EDO major third A-C# for example

is in score mistakenly designated as A-Db, the

playing by this score in 31EDO will cause obvious

sensation out of tuning, completely inaccessible

during the playing in a system 12EDO. The

respelling of mistakenly notation removes sensation

out of tuning also during the playing in 31EDO.

The verification of scores by hearing plays the

important role in the process of transposing of

12EDO scores for the performance of them in

positive by Bosanquet (22EDO, 53EDO) systems

(Khramov, 2008).

REFERENCES

Asmussen, R., 2001. Periodicity of sinusoidal frequencies

as a basis for the analysis of Baroque and Classical

harmony: a computer based study. The University of

Leeds. School of Music.

http://www.terraworld.net/c-jasmussen/

Bosanquet, R. H. M., 1875. On the theory of the division

of the octave. In Proceedings of the Royal Society. No.

161. http://geocities.com/kmy180753/downloads/

EnhrmArticleOfBosnquet.pdf, http://geocities.com/

kmy180753/downloads/EnhrmArticleOfBosnquet.doc

Op de Coul, M., 2004. Scala Software. Netherlands,

version 2.2p. http://www.xs4all.nl/~huygensf/scala

Eastwood, M., Spreadbury, D., Finn, B., Finn, J., 2005.

Sibelius 4. Reference. http://www.sibelius.com/buy

Khramov, M., 2004. My music. From http://

members.sibeliusmusic.com/commator

Khramov, M., 2006. Another tuning. Audio CD. http://

geocities.com/commator/CDPages/AnotherTuning.htm

Khramov, M., 2008. Project Commator and Sonantometry.

In Proceedings of the FRSM-2008 International

Symposium Frontiers of Research on Speech and

Music. Jadavpur University, Kolkata.

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