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
13/12 138.5726609
14/13 128.2982447
15/14 119.4428083
16/15 111.7312853
17/16 104.9554095
18/17 98.95459223
19/18 93.6030144
20/19 88.80069773
21/20 84.46719347
22/21 80.53703503
23/22 76.9564049
24/23 73.6806536
By reversing the order of ratios we have:
Superparticular Cent
ratio
1: 24/23 73.681
2: 23/22 76.956
3: 22/21 80.537 undecimal minor semitone
4: 21/20 84.467 minor semitone
5: 20/19 88.801 small undevicesimal semitone
6: 19/18 93.603 undevicesimal semitone
7: 18/17 98.955 Arabic lute index finger
8: 17/16 104.955 17th harmonic
9: 16/15 111.731 minor diatonic semitone
10: 15/14 119.443 major diatonic semitone
11: 14/13 128.298 2/3-tone
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 :
- Schlesinger's harmoniai, Wilson's diaphonic cycles, and other similar constructs ( By john chalmers)
- Planetary scales and EDL system according to ptolemy ( By permission of Dr.john chalmers)
- Adventures in Scale Construction , By : X. J. Scott
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 Andreas Werckmeister 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
In http://launch.groups.yahoo.com/group/tuning/message/69724 , Gene ward smith , has a new glance to Werckmeister's mythical tuning wich is based on 1568-EDL :
Degree Ratio Cent
1568 1/1 0.000
1485 1568/1485 94.155
1400 28/25 196.198
1320 196/165 298.065
1248 49/39 395.169
1176 4/3 498.045
1112 196/139 594.923
1048 196/131 697.544
990 784/495 796.110
936 196/117 893.214
880 98/55 1000.020
832 49/26 1097.124
784 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 :
Degree Ratio Cent
120 ………… 1/1 ………… 0
114 ………… 20/19 ………88.80069773
108 ………… 10/9 ……… 182.4037121
102 ………… 20/17 ………281.3583044
96 ………… 5/4 …………386.3137139
90 ………… 4/3 …………498.0449991
85 ………… 24/17 …… 596.9995914
80 ………… 3/2 …………701.9550009
76 ………… 30/19 ………790.7556986
72 ………… 5/3 …………884.358713
68 ………… 30/17 ………983.3133052
64 ………… 15/8 ……… 1088.268715
60 ………… 2/1 …………1200
Here you can see the interval matrix of this tuning and its GRAPHIC DEVIATIONS FROM 12-EDO:
and here is rational form of the interval matrix (By scala ):
Here you can download an VISUAL TEMPERAMENT ANALYZER spreadsheet related to this well-Temperament.
Equal divisions of length and Harmonic Mean
For any two numbers like 'a1' and 'a2' , we have Harmonic Mean as:
_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:
Tanbour of baghdad
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:
1/1 0
40/39 43.83105123
20/19 88.80069773
40/37 134.9696751
10/9 182.4037121
8/7 231.1740935
20/17 281.3583044
40/33 333.0407706
5/4 386.3137139
40/31 441.2781414
4/3 498.0449991
40/29 556.7365197
10/7 617.4878074
40/27 680.4487113
20/13 745.7860521
8/5 813.6862861
5/3 884.358713
40/23 958.0393666
20/11 1034.995772
40/21 1115.532807
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: 1/1 0.000
2: 40/39 43.831
3: 20/19 88.801
4: 40/37 134.970
5: 10/9 182.404
6: 8/7 231.174
Second 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
Third string :
1: 16/9 996.090 Pythagorean minor seventh
2: 640/351 1039.921
3: 320/171 1084.891
4: 640/333 1131.060
5: 160/81 1178.494 octave - syntonic comma
6: 128/63 1227.264 septimal comma, Archytas' comma + 1 octave
It is very interesting reaching septimal comma. The compelet scale is :
0: 1/1 0.000
1: 40/39 43.831
2: 20/19 88.801
3: 40/37 134.970
4: 10/9 182.404
5: 8/7 231.174
6: 4/3 498.045
7: 160/117 541.876
8: 80/57 586.846
9: 160/111 633.015
10: 40/27 680.449
11: 32/21 729.219
12: 16/9 996.090
13: 640/351 1039.921
14: 320/171 1084.891
15: 640/333 1131.060
16: 160/81 1178.494 octave - syntonic comma
17: 128/63 1227.264 septimal comma, Archytas' comma + 1 octave
Now we are actually in 640-EDL.
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A Brief History of the Monochord
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