Equal divisions of length           Back to tuning models

For an intervallic system with n divisions , EDL is considered as equal divisions of length by dividing string length to n equal divisions ( So , we have n/2 divisions per octave).If the first division is L1 and the last, Ln , we have:

L1 = L2 = L3 = …… = Ln

So sum of the divisions is L or the string length. Note that the number of divisions in octave is half of the string length.By dividing string length of L to n division we have :

n : n-1 : n-2 : n-3 : ……. : n-m : ….. : n-n

which n-m is n/2. for example, by dividing string length to 12 equal divisions we have a series as:

12:11:10:9:8:7:6:5:4:3:2:1:0  or  12 11 10 9 8 7 6 5 4 3 2 1 0 which shows 12-EDL:

12:12 means 12 from 12 divisions,12:11 means 11 from 12 divisions and so on.Ratios as 12:11 shows active string length for each degree, which is vibrating.EDL system shows ascending trend of divisions sizes due to its inner structure and if compared with EDO :

For any C-EDL system with cardinality of C, we have ratios related to different degrees as :

(C/C-m)

For example in 24-EDL we have , C=24 and the third degree has a ratio of 24/(24-3) = 24/21 = 8/7

Division sizes of 24-EDL :

## 1:         73.681 cents        2:         76.956 cents        3:         80.537 cents        4:         84.467 cents        5:         88.801 cents        6:         93.603 cents        7:         98.955 cents        8:        104.955 cents       9:        111.731 cents      10:       119.443 cents      11:       128.298 cents      12:       138.573 cents

Relation between Utonality and EDL system

We can consider EDL system as Utonal system .Utonality is a term introduced by Harry Partch to describe chords whose notes are the  "undertones" (divisors) of a given fixed tone.
In the other hand , an utonality is a collection of pitches which can be expressedin ratios that have the same nominators. For example, 7/4, 7/5, 7/6 form an utonality which 7 as nominator is called "
Numerary nexus".
If a string is divided into equal parts, it will produce an utonality and so we have EDL system.

EDL systems are classified as systems with unequal epimorios (Superparticular) divisions which show descending series with ascending sizes.

- Fret position calculator (excel sheet ) based on EDL system and string length

- How to approximate EDO and EDL systems with each other?Download this file

Relation between harmonics and EDL system

Harmonic of a wave is a component frequency that is an integer multiple of the fundamental frequency. For a sine wave, it is an integer multiple of the frequency of the wave. For example, if the frequency is f, the harmonics have frequency 2f, 3f, 4f, etc.

In musical terms, harmonics are component pitches ("partials", "partial waves" or "constituent frequencies") of a harmonic tone which sound at whole number multiples above the named note being played on a musical instrument. Non-integer multiples are called inharmonic overtones. It is the amplitude and placement of harmonics (and partials in general) which give different instruments different timbre (despite not usually being detected separately by the untrained human ear), and the separate trajectories of the overtones of two instruments playing in unison is what allows one to perceive them as separate. Bells have more clearly perceptible partials than most instruments. Antique singing bowls are well known for their unique quality of producing multiple harmonic overtones or multiphonics.Harmonics may be used as the basis of just intonation systems or considered as the basis of all just intonation systems.The following table displays the first 10 harmonics which are transposed into the span of one octave (From :http://en.wikipedia.org/wiki/Harmonic):

EDL system , as one kind of just intonation systems is based on harmonics. Divisions of this system are consecutive superparticular ratios between two harmonics , but with reverse order.Consider harmonics 12 and 24 . The consecutive superparticular ratios between them are :

Superparticular                 Cent

ratio

#### 24/23                     73.6806536

By reversing the order of ratios we have:

Superparticular     Cent

ratio

#### 12:        13/12            138.573   tridecimal 2/3-tone

Which are divisions of 24-EDL.(Calculated by Scala Version 2.21g , Copyright Manuel Op de Coul, 2004)

- An spreadsheet showing relation between harmonics , superparticular ratios and EDL system

The Works of Kathleen Schlesinger and Elsie Hamilton in Naked Light

Elsie Hamilton

Kathleen Schlesinger was a musicologist, the author of "The Greek Aulos" (Methuen 1939) now out of print, where she analyses the tuning systems used in Greece and other ancient and indigenous cultures to deduce a system known as the Harmoniai. This method of tuning was used by her friend, the composer Elsie Hamilton in her music. To see reprints of some rare articles and scores by Schlesinger and Hamilton such as :

- Kathleen Schlesinger and Elsie Hamilton - Pioneers of Just Intonation"
- The Modes of Ancient Greece
- The New Language of Music
- The Language of Music
- The Return of the Planetary Modes

- The Nature of Musical Experience in the Light of Anthroposophy
- Appendix III from The Greek Aulos by Kathleen Schlesinger

and scores such as :
-Hymn to Ra.
For voice 2 lyres and bass lyre
(Sun scale 22, Venus species 12).
- Olaf Åsteson.
For male and female choirs and lyres.
- Wenn der Mensch warm in Liebe.
For voices and lyres.
Words by Rudolf Steiner
(Venus scale)

click here to go to website of Brian Lee and Naked Light:
Other related things :

196-EDL and The Septenarius, Werckmeister's mythical tuning

Message http://launch.groups.yahoo.com/group/tuning/message/68343 of tuning groupe is a link to The Septenarius: Werckmeister's mythical tuning.

Werckmeister IV (the Septenarius tuning) is a good reason that EDL systems which are based on length divisions are used not only by greek musicians like Ptolemy as john Chalmers mentioned , but other music theorists like had a glance at it.

Septenarius is a subset of 196-EDL , which shows usage of EDL systems to show irregular tunings.

You can see this excel spreadsheet for Septenarius tuning to see :

* Interval matrix

* Deviation of interval matrix with 12-EDO

* Graphical presentation of interval matrix

* Frequency of notes in a range of one octave (from C)

* Frequency of beats

* Circle of fifths and fourths

1568-EDL and The Septenarius, Werckmeister's mythical tuning

## Degree       Ratio             Cent1568             1/1                  0.000 1485        1568/1485           94.1551400           28/25              196.198 1320         196/165             298.0651248          49/39               395.1691176            4/3                 498.045 1112         196/139             594.9231048         196/131             697.544990           784/495             796.110936           196/117             893.214880            98/55               1000.020 832            49/26               1097.124784              2/1                 1200.00

Here you can see the interval matrix of this tuning and its GRAPHIC DEVIATIONS from just intervals:

And now Here you can see its GRAPHIC DEVIATIONS from just intervals:

120-EDL and Ganassi's Well-Temperament

In Ganassi's Well-Temperament ,  monzo writes about a chromatic scale with rational form of a well-temperament as below :

0        89     182     281    386   498    597    702    791     884    983    1088   1200
C     C#/Db   D    D#/Eb   E      F    F#/Gb   G     G#/Ab    A     A#/Bb    B        C
120   114    108     102     96     90     85      80      76        72      68       64       60

He , then says that this scale is two fully chromatic tetrachords with the proportions 20:19:18:17:16:15, connected by a "tone of disjunction" which is divided 18:17:16.

Now considering EDL system , this shows that This well-temperament is based on 120-EDL :

## and here is rational form of the interval matrix (By scala ):

Equal divisions of length and Harmonic Mean

For any two numbers like 'a1' and 'a2' , we have Harmonic Mean as:

Intervallic structure of EDL system is based on harmonic means , that is , for any 2 ratios we can find the third by harmonic mean :

_Consider 1/1 and 2/1 , harmonic mean of these 2 ratio is  4/3.

_Consider 1/1 and 4/3 , harmonic mean of these 2 ratio is 8/7.

_Consider 4/3 and 2/1 , harmonic mean of these 2 ratio is 8/5.

All these intervals are parts of 8-EDL .This mean which is one of the three classical Pythagorean means ( arithmetic mean (A), the geometric mean (G), and the harmonic mean (H)) , was called the “subcontrary mean” in Pythagoras’ time, but its name was changed to “harmonic mean” by Archytas and Hippasus (Heath 1981). (From "Making scales by mathematical means" by Lydia Ayers , sent by john chalmers in a private mail)

256-EDL and  Well-Temperament

Here you can see a well-temperament based on tetrachord of 256-EDL as :

0:         1/1                0.000
1:        256/241         104.533
2:        64/57             200.532
3:        256/215         302.169
4:        256/203         401.597
5:        4/3                498.045

Two equal semitone (resulted from (9/8)^(1/2)) are used between 4/3 and 3/2 to make disjunctive tone:

0:        1/1               0.000
1:        256/241        104.533
2:        64/57            200.532
3:        256/215        302.169
4:        256/203        401.597
5:        4/3               498.045
6:        600.0            600.0
7:        3/2               701.955
8:        384/241        806.488
9:        32/19            902.487
10:       384/215        1004.124
11:       384/203        1103.552
12:       2/1               1200.000

Here you can see the interval matrix of this tuning and its GRAPHIC DEVIATIONS FROM 12-EDO:

Farabi in his great book Kitab al-Musiqi al-Kabir Talks about 2 kind of Tanbour , tanbour of khorasan and tanbour of baghdad ( Which is known also as Arabic or Iraqi tambour). Tanbour of baghdad was played in old iraq (western and centeral parts), There were 5 fret on the neck of instrument accoding to 40-EDL system:

#### 2/1     1200.0

This instrument had 2 strings which the Second was mainly tuned unison or a fourth above:

First string      -------1/1    40/39    20/19     40/37    10/9     8/7

Second string ------ 4/3  160/117  80/57   160/111 40/27  32/21

First string :

#### 1:          4/3             498.045   2:        160/117           541.876  3:         80/57            586.846  4:        160/111           633.015  5:         40/27            680.449   6:         32/21            729.219

Now assuming a tanbour with a third strings tuned a fourth above 2nd string :

First string      -------1/1    40/39    20/19     40/37    10/9     8/7

Second string ------ 4/3  160/117  80/57   160/111 40/27  32/21

Third string ----------16/9   640/351   320/171   640/333   160/81   128/63

#### 6:        128/63           1227.264 septimal comma, Archytas' comma + 1 octave

It is very interesting reaching septimal comma. The compelet scale is :

#### 17:        128/63           1227.264 septimal comma, Archytas' comma + 1 octave

Now we are actually in 640-EDL.

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