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→Modi: Skalenbaum-Abschnitt erstellt. Auskommentiert, was ich nicht übersetzen konnte
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| Lokrisch
| Lokrisch
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== Skalenbaum ==
Wenn man das Intervall 4\7 (vier Schritte von 7-EDO) auf ein Ende setzt und 3\5 (three degrees of [[5-EDO]]) auf das andere, liegen alle andere möglichen 5L 2s-Skalen in einem Kontinuum zwischen den zweien Enden. Man kann dieses Kontinuum unterteilen, indem man die Zähler addiert und dann die Nenner addiert (diese Operation heißt die [[Mediant]]-Operation oder die Farey-Summe). Zwischen 4\7 und 3\5 entsteht also (4+3)\(7+5) = 7\12, also sieben Schritte von 12-EDO.
{| class="wikitable"
|-
| 4\7 ||
|-
| || 7\12
|-
| 3\5 ||
|}
Wenn man diese Mediant-Operation weiter ausführt, tauchen im Kontinuum größere EDOs auf.
{| class="wikitable center-all"
! colspan="7" | Generator
! Cents
! L
! s
! L/s
! Comments
|-
| 4\7 || || || || || || || 685.714 || 1 || 1 || 1.000 ||
|-
| || || || || || || 27\47 || 689.362 || 7 || 6 || 1.167 ||
|-
| || || || || || 23\40 || || 690.000 || 6 || 5 || 1.200 ||
|-
| || || || || || || 42\73 || 690.411 || 11 || 9 || 1.222 ||
|-
| || || || || 19\33 || || || 690.909 || 5 || 4 || 1.250 ||
|-
| || || || || || || 53\92 || 691.304 || 14 || 11 || 1.273 ||
|-
| || || || || || 34\59 || || 691.525 || 9 || 7 || 1.286 ||
|-
| || || || || || || 49\85 || 691.765 || 13 || 10 || 1.300 ||
|-
| || || || 15\26 || || || || 692.308 || 4 || 3 || 1.333 ||
|-
| || || || || || || 56\97 || 692.784 || 15 || 11 || 1.364 ||
|-
| || || || || || 41\71 || || 692.958 || 11 || 8 || 1.375 ||
|-
| || || || || || || 67\116 || 693.103 || 18 || 13 || 1.385 ||
|-
| || || || || 26\45 || || || 693.333 || 7 || 5 || 1.400 ||
|-
| || || || || || || 63\109 || 693.578 || 17 || 12 || 1.417 ||
|-
| || || || || || 37\64 || || 693.750 || 10 || 7 || 1.429 ||
|-
| || || || || || || 48\83 || 693.976 || 13 || 9 || 1.444 ||
|-
| || || 11\19 || || || || || 694.737 || 3 || 2 || 1.500 || L/s = 3/2
|-
| || || || || || || 51\88 || 695.455 || 14 || 9 || 1.556 ||
|-
| || || || || || 40\69 || || 695.652 || 11 || 7 || 1.571 ||
|-
| || || || || || || 69\119 || 695.798 || 19 || 12 || 1.583 ||
|-
| || || || || 29\50 || || || 696.000 || 8 || 5 || 1.600 ||
|-
| || || || || || || 66\131 || 696.183 || 21 || |13 || 1.615 || Goldene mitteltönige Stimmung
|-
| || || || || || 47\81 || || 696.296 || 13 || 8 || 1.625 ||
|-
| || || || || || || 65\112 || 696.429 || 18 || 11 || 1.636 ||
|-
| || || || 18\31 || || || || 696.774 || 5 || 3 || 1.667 || Mitteltönige Stimmungen sind in dieser Generatorregion
|-
| || || || || || || 61\105 || 697.143 || 17 || 10 || 1.700 ||
|-
| || || || || || 43\74 || || 697.297 || 12 || 7 || 1.714 ||
|-
| || || || || || || 68\117 || 697.436 || 19 || 11 || 1.727 ||
|-
| || || || || 25\43 || || || 697.674 || 7 || 4 || 1.750 ||
|-
| || || || || || || 57\98 || 697.959 || 16 || 9 || 1.778 ||
|-
| || || || || ||32\55 || || 698.182 || 9 || 5 || 1.800 ||
|-
| || || || || || || 39\67 || 698.507 || 11 || 6 || 1.833 ||
|-
| || 7\12 || || || || || || 700.000 || 2 || 1 || 2.000 || <!--Basic diatonic<br>(Generators smaller than this are proper)-->
|-
| || || || || || || 38\65 || 701.539 || 11 || 5 || 2.200 ||
|-
| || || || || || 31\53 || || 701.887 || 9 || 4 || 2.250 || Das EDO-Generatorintervall, das unter EDOs <= 200 zum reinen [[3/2]] am nächsten ist
|-
| || || || || || || 55\94 || 702.128 || 16 || 7 || 2.286 ||
|-
| || || || || 24\41 || || || 702.409 || 7 || 3 || 2.333 ||
|-
| || || || || || || 65\111 || 702.703 || 19 || 8 || 2.375 ||
|-
| || || || || || 41\70 || || 702.857 || 12 || 5 || 2.400 ||
|-
| || || || || || || 58\99 || 703.030 || 17 || 7 || 2.428 ||
|-
| || || || 17\29 || || || || 703.448 || 5 || 2 || 2.500 ||
|-
| || || || || || || 61\104 || 703.846 || 18 || 7 || 2.571 ||
|-
| || || || || || 44\75 || || 704.000 || 13 || 5 || 2.600 ||
|-
| || || || || || || 71\121 || 704.132 || 21 || 8 || 2.625 || Goldene "Neogotisch"-Stimmung
|-
| || || || || 27\46 || || || 704.348 || 8 || 3 || 2.667 || "Neogotisch"-Generatorbereich
|-
| || || || || || || 64\109 || 704.587 || 19 || 7 || 2.714 ||
|-
| || || || || || 37\63 || || 704.762 || 11 || 4 || 2.750 ||
|-
| || || || || || || 47\80 || 705.000 || 14 || 5 || 2.800 ||
|-
| || || 10\17 || || || || || 705.882 || 3 || 1 || 3.000 || L/s = 3/1
|-
| || || || || || || 43\73 || 706.849 || 13 || 4 || 3.250 ||
|-
| || || || || || 33\56 || || 707.143 || 10 || 3 || 3.333 ||
|-
| || || || || || || 56\95 || 707.368 || 17 || 5 || 3.400 ||
|-
| || || || || 23\39 || || || 707.692 || 7 || 2 || 3.500 ||
|-
| || || || || || || 59\100 || 708.000 || 18 || 5 || 3.600 ||
|-
| || || || || || 36\61 || || 708.197 || 11 || 3 || 3.667 ||
|-
| || || || || || || 49\83 || 708.434 || 15 || 4 || 3.750 ||
|-
| || || || 13\22 || || || || 709.091 || 4 || 1 || 4.000 || [[Archy]]-Generatorbereich
|-
| || || || || || || 42\71 || 709.859 || 13 || 3 || 4.333 ||
|-
| || || || || || 29\49 || || 710.204 || 9 || 2 || 4.500 ||
|-
| || || || || || || 45\76 || 710.526 || 14 || 3 || 4.667 ||
|-
| || || || || 16\27 || || || 711.111 || 5 || 1 || 5.000 ||
|-
| || || || || || || 35\59 || 711.864 || 11 || 2 || 5.500 ||
|-
| || || || || || 19\32 || || 712.500 || 6 || 1 || 6.000 ||
|-
| || || || || || || 22\37 || 713.514 || 7 || 1 || 7.000 ||
|-
| 3\5 || || || || || || || 720.000 || 1 || 0 || → inf ||
|}
<!--
Tunings above 7\12 on this chart are called "negative tunings" (as they lessen the size of the fifth) and include meantone systems such as 1/3-comma (close to 11\19) and 1/4-comma (close to 18\31). As these tunings approach 4\7, the majors become flatter and the minors become sharper.
Tunings below 7\12 on this chart are called "positive tunings" and they include Pythagorean tuning itself (well approximated by 31\53) as well as superpyth tunings such as 10\17 and 13\22. As these tunings approach 3\5, the majors become sharper and the minors become flatter. Around 13\22 through 16\27, the thirds fall closer to 7-limit than 5-limit intervals: 7:6 and 9:7 as opposed to 6:5 and 5:4.
[[File:5L2s.jpg|alt=5L2s.jpg|5L2s.jpg]]
5L 2s contains the pentatonic MOS [[2L_3s|2L 3s]] and (with the sole exception of the 5L 2s of 12EDO) is itself contained in a dodecaphonic MOS: either [[7L_5s|7L 5s]] or [[5L_7s|5L 7s]], depending on whether the fifth is flatter than or sharper than 7\12 (700c).-->

Version vom 1. Juli 2021, 02:40 Uhr

Eine mögliche Definition für die diatonische Skala ist als der durch das Übereinanderschichten von "Quinten" (bzw. "Quarten") erzeugte MOS-Skalentyp namens 5L 2s, also mit 5 großen Schritten und 2 kleinen Schritten. Das entsprechende Schrittmuster lautet LLLsLLs (wenn man die verschiedene Modi ignoriert).

Modi

Die diatonischen Modi haben bekannte Namen aus der klassischen westlichen Musiktheorie:

Modus UDP Name
LLLsLLs 6|0 Lydisch
LLsLLLs 5|1 Ionisch
LLsLLsL 4|2 Mixolydisch
LsLLLsL 3|3 Dorisch
LsLLsLL 2|4 Äolisch
sLLLsLL 1|5 Phrygisch
sLLsLLL 0|6 Lokrisch

Skalenbaum

Wenn man das Intervall 4\7 (vier Schritte von 7-EDO) auf ein Ende setzt und 3\5 (three degrees of 5-EDO) auf das andere, liegen alle andere möglichen 5L 2s-Skalen in einem Kontinuum zwischen den zweien Enden. Man kann dieses Kontinuum unterteilen, indem man die Zähler addiert und dann die Nenner addiert (diese Operation heißt die Mediant-Operation oder die Farey-Summe). Zwischen 4\7 und 3\5 entsteht also (4+3)\(7+5) = 7\12, also sieben Schritte von 12-EDO.

4\7
7\12
3\5

Wenn man diese Mediant-Operation weiter ausführt, tauchen im Kontinuum größere EDOs auf.

Generator Cents L s L/s Comments
4\7 685.714 1 1 1.000
27\47 689.362 7 6 1.167
23\40 690.000 6 5 1.200
42\73 690.411 11 9 1.222
19\33 690.909 5 4 1.250
53\92 691.304 14 11 1.273
34\59 691.525 9 7 1.286
49\85 691.765 13 10 1.300
15\26 692.308 4 3 1.333
56\97 692.784 15 11 1.364
41\71 692.958 11 8 1.375
67\116 693.103 18 13 1.385
26\45 693.333 7 5 1.400
63\109 693.578 17 12 1.417
37\64 693.750 10 7 1.429
48\83 693.976 13 9 1.444
11\19 694.737 3 2 1.500 L/s = 3/2
51\88 695.455 14 9 1.556
40\69 695.652 11 7 1.571
69\119 695.798 19 12 1.583
29\50 696.000 8 5 1.600
66\131 696.183 21 13 1.615 Goldene mitteltönige Stimmung
47\81 696.296 13 8 1.625
65\112 696.429 18 11 1.636
18\31 696.774 5 3 1.667 Mitteltönige Stimmungen sind in dieser Generatorregion
61\105 697.143 17 10 1.700
43\74 697.297 12 7 1.714
68\117 697.436 19 11 1.727
25\43 697.674 7 4 1.750
57\98 697.959 16 9 1.778
32\55 698.182 9 5 1.800
39\67 698.507 11 6 1.833
7\12 700.000 2 1 2.000
38\65 701.539 11 5 2.200
31\53 701.887 9 4 2.250 Das EDO-Generatorintervall, das unter EDOs <= 200 zum reinen 3/2 am nächsten ist
55\94 702.128 16 7 2.286
24\41 702.409 7 3 2.333
65\111 702.703 19 8 2.375
41\70 702.857 12 5 2.400
58\99 703.030 17 7 2.428
17\29 703.448 5 2 2.500
61\104 703.846 18 7 2.571
44\75 704.000 13 5 2.600
71\121 704.132 21 8 2.625 Goldene "Neogotisch"-Stimmung
27\46 704.348 8 3 2.667 "Neogotisch"-Generatorbereich
64\109 704.587 19 7 2.714
37\63 704.762 11 4 2.750
47\80 705.000 14 5 2.800
10\17 705.882 3 1 3.000 L/s = 3/1
43\73 706.849 13 4 3.250
33\56 707.143 10 3 3.333
56\95 707.368 17 5 3.400
23\39 707.692 7 2 3.500
59\100 708.000 18 5 3.600
36\61 708.197 11 3 3.667
49\83 708.434 15 4 3.750
13\22 709.091 4 1 4.000 Archy-Generatorbereich
42\71 709.859 13 3 4.333
29\49 710.204 9 2 4.500
45\76 710.526 14 3 4.667
16\27 711.111 5 1 5.000
35\59 711.864 11 2 5.500
19\32 712.500 6 1 6.000
22\37 713.514 7 1 7.000
3\5 720.000 1 0 → inf
Abgerufen von „https://de.xen.wiki/index.php?title=5L2s&oldid=2560“