3L 7s (5/2-equivalent): Difference between revisions

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{{Infobox MOS
{{Infobox MOS}}
| Name = sephiroid
{{MOS intro}}
| Periods = 1
== Scale properties ==
| nLargeSteps = 3
{{TAMNAMS use}}
| nSmallSteps = 7
| Equalized = 3
| Paucitonic = 1
| Pattern = LssLssLsss
}}


{{Infobox MOS
=== Intervals ===
| Name = Greater sephiroid
{{MOS intervals}}
| Periods = 1
| nLargeSteps = 3
| nSmallSteps = 7
| Equalized = 3
| Paucitonic = 1
| Pattern = LssLssLsss
| Equave = 5/2}}


'''3L 7s(<5/2>)''' occupies the spectrum from 10edo (L = s) to 3edo (s = 0).
=== Generator chain ===
{{MOS genchain}}


[[TAMNAMS]] calls this MOS pattern '''sephiroid''' (named after the abstract temperament [[sephiroth]]).
=== Modes ===
{{MOS mode degrees}}


This MOS can represent tempered-flat chains of the 13th harmonic, which approximates phi (~833 cents). In the region of the spectrum around [[23edo]] (L = 3, s = 2) , the 17th and 21st harmonics are tempered toward most accurately, which together are a stable harmony. This is the major chord of the modi sephiratorum. Temperament using phi directly approximates the higher Fibonacci harmonics best.
== Scale tree ==
{{MOS tuning spectrum}}


If L = s, i.e. multiples of 10edo, the 13th harmonic becomes nearly perfect. [[121edo]] seems to be the first to 'accurately' represent the comma (which might as well be represented accurately as it is quite small). Towards the other end, where the large and small steps are more contrasted, the comma [[65/64]] is liable to be tempered out, equating [[8/5]] and [[13/8]]. In this category fall [[13edo]], [[16edo]], [[19edo]], [[22edo]], [[29edo]], and so on. This ends at s = 0 which gives multiples of [[3edo]].
{{stub}}
 
Harmonically, the arrangement forming a chord (degrees 0, 1, 4, 7, 10) is symmetrical – not ascending but rather descending, and so reminiscent of ancient Greek practice. These scales, and their truncated heptatonic forms referenced below, are strikingly linear in several ways and so seem suited to a similar outlook as traditional western music (modality, baroque tonality, classical tonality, etc. progressing to today) but with higher harmonics. For more details see Kosmorsky's [https://ia600706.us.archive.org/23/items/TractatumDeModiSephiratorum/ModiSephiratorum.pdf ''Tractatum de Modi Sephiratorum''] (Kosmorsky knows it should be "tractatus", but considers changing it is nothing but a bother.)
 
There are MODMOS as well, but Kosmorsky has not explored them yet. There's enough undiscovered harmonic resources already in these to last me a while! Taking this approach to the 13th harmonic also yields heptatonic MOS with similar properties: [[3L 4s|4s+3L "mish"]] in the form of modes of ssLsLsL "led".
 
==Modes==
 
s s s L s s L s s L - Keter
 
s s L s s L s s L s - Chesed
 
s L s s L s s L s s - Netzach
 
L s s L s s L s s s - Malkuth
 
s s L s s L s s s L - Binah
 
s L s s L s s s L s - Tiferet
 
L s s L s s s L s s - Yesod
 
s s L s s s L s s L - Chokmah
 
s L s s s L s s L s - Gevurah
 
L s s s L s s L s s - Hod
 
==Scale tree==
{| class="wikitable center-all"
! colspan="6" |Generator
!Cents
!Normalized Cents
!''ed13\11''
!L
!s
!L/s
!Comments
|-
|3\10|| || || || || ||360.000
|400.000
|''390.000''||1||1||1.000||
|-
| || || || || ||16\53||362.264
|408.511
|''392.453''||6||5||1.200||Submajor
|-
| || || || ||13\43|| ||362.791
|410.526
|''393.023''||5||4||1.250||
|-
| || || || || ||23\76||363.158
|411.940
|''393.421''||9||7||1.286||
|-
| || || ||10\33|| || ||363.636
|413.793
|''393.939''||4||3||1.333||
|-
| || || || || ||27\89||364.045
|415.385
|''394.382''||11||8||1.375||
|-
| || || || ||17\56|| ||364.286
|416.2365
|''394.643''||7||5||1.400||
|-
| || || || || ||24\79||364.557
|417.319
|''394.937''||10||7||1.428||
|-
| || ||7\23|| || || ||365.217
|420.000
|''395.652''||3||2||1.500||L/s = 3/2
|-
| || || || || ||25\82||365.854
|422.535
|''396.3415''||11||7||1.571||
|-
| || || || ||18\59|| ||366.102
|423.529
|''396.610''||8||5||1.600||
|-
| || || || || ||29\95||366.316
|424.390
|''396.842''||13||8||1.625||Unnamed golden tuning
|-
| || || ||11\36|| || ||366.667
|425.8065
|''396.667''||5||3||1.667||
|-
| || || || || ||26\85||367.059
|427.397
|''397.647''||12||7||1.714||
|-
| || || || ||15\49|| ||367.347
|428.571
|''397.959''||7||4||1.750||
|-
| || || || || ||19\62||367.742
|430.189
|''398.387''||9||5||1.800||
|-
| ||4\13|| || || || ||369.231
|436.364
|''400.000''||2||1||2.000||Basic sephiroid<br>(Generators smaller than this are proper)
|-
| || || || || ||17\55||370.909
|443.478
|''401.818''||9||4||2.250||
|-
| || || || ||13\42|| ||371.429
|445.714
|''402.381''||7||3||2.333||
|-
| || || || || ||22\71||371.831
|447.458
|''402.817''||12||5||2.400||
|-
| || || ||9\29|| || ||372.414
|450.000
|''403.448''||5||2||2.500||Sephiroth
|-
| || || || || ||23\74||372.973
|452.459
|''404.054''||13||5||2.600||Golden sephiroth
|-
| || || || ||14\45|| ||373.333
|454.054
|''404.444''||8||3||2.667||
|-
| || || || || ||19\61||373.770
|456.000
|''404.981''||11||4||2.750||
|-
| || ||5\16|| || || ||375.000
|461.5385
|''406.250''||3||1||3.000||L/s = 3/1
|-
| || || || || ||16\51||376.471
|468.293
|''407.843''||10||3||3.333||
|-
| || || || ||11\35|| ||377.143
|471.429
|''408.571''||7||2||3.500||
|-
| || || || || ||17\54||377.778
|474.419
|''409.259''||11||3||3.667||Muggles
|-
| || || ||6\19|| || ||378.947
|480.000
|''410.526''||4||1||4.000||Magic/horcrux
|-
| || || || || ||13\41||380.488
|487.500
|''412.195''||9||2||4.500||Magic/witchcraft
|-
| || || || ||7\22|| ||381.818
|500.000
|''413.636''||5||1||5.000||Magic/telepathy
|-
| || || || || ||8\25||384.000
|505.263
|''416.000''||6||1||6.000||Würschmidt↓
|-
|1\3|| || || || || ||400.000
|600.000
|''433.333''||1||0||→ inf||
|}

Latest revision as of 14:19, 5 May 2025

↖ 2L 6s⟨5/2⟩ ↑ 3L 6s⟨5/2⟩ 4L 6s⟨5/2⟩ ↗
← 2L 7s⟨5/2⟩ 3L 7s (5/2-equivalent) 4L 7s⟨5/2⟩ →
↙ 2L 8s⟨5/2⟩ ↓ 3L 8s⟨5/2⟩ 4L 8s⟨5/2⟩ ↘ 
┌╥┬┬╥┬┬╥┬┬┬┐
│║││║││║││││
││││││││││││
└┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LssLssLsss
sssLssLssL
Equave 5/2 (1586.3 ¢)
Period 5/2 (1586.3 ¢)
Generator size(ed5/2)
Bright 3\10 to 1\3 (475.9 ¢ to 528.8 ¢)
Dark 2\3 to 7\10 (1057.5 ¢ to 1110.4 ¢)
Related MOS scales
Parent 3L 4s⟨5/2⟩
Sister 7L 3s⟨5/2⟩
Daughters 10L 3s⟨5/2⟩, 3L 10s⟨5/2⟩
Neutralized 6L 4s⟨5/2⟩
2-Flought 13L 7s⟨5/2⟩, 3L 17s⟨5/2⟩
Equal tunings(ed5/2)
Equalized (L:s = 1:1) 3\10 (475.9 ¢)
Supersoft (L:s = 4:3) 10\33 (480.7 ¢)
Soft (L:s = 3:2) 7\23 (482.8 ¢)
Semisoft (L:s = 5:3) 11\36 (484.7 ¢)
Basic (L:s = 2:1) 4\13 (488.1 ¢)
Semihard (L:s = 5:2) 9\29 (492.3 ¢)
Hard (L:s = 3:1) 5\16 (495.7 ¢)
Superhard (L:s = 4:1) 6\19 (500.9 ¢)
Collapsed (L:s = 1:0) 1\3 (528.8 ¢)

3L 7s⟨5/2⟩ is a 5/2-equivalent (non-octave) moment of symmetry scale containing 3 large steps and 7 small steps, repeating every interval of 5/2 (1586.3 ¢). Generators that produce this scale range from 475.9 ¢ to 528.8 ¢, or from 1057.5 ¢ to 1110.4 ¢.

Scale properties

This article uses TAMNAMS conventions for the names of this scale's intervals and scale degrees. The use of 1-indexed ordinal names is reserved for interval regions.

Intervals

Intervals of 3L 7s⟨5/2⟩
Intervals Steps
subtended 		Range in cents
Generic Specific Abbrev.
0-mosstep Perfect 0-mosstep P0ms 0 0.0 ¢
1-mosstep Minor 1-mosstep m1ms s 0.0 ¢ to 158.6 ¢
Major 1-mosstep M1ms L 158.6 ¢ to 528.8 ¢
2-mosstep Minor 2-mosstep m2ms 2s 0.0 ¢ to 317.3 ¢
Major 2-mosstep M2ms L + s 317.3 ¢ to 528.8 ¢
3-mosstep Diminished 3-mosstep d3ms 3s 0.0 ¢ to 475.9 ¢
Perfect 3-mosstep P3ms L + 2s 475.9 ¢ to 528.8 ¢
4-mosstep Minor 4-mosstep m4ms L + 3s 528.8 ¢ to 634.5 ¢
Major 4-mosstep M4ms 2L + 2s 634.5 ¢ to 1057.5 ¢
5-mosstep Minor 5-mosstep m5ms L + 4s 528.8 ¢ to 793.2 ¢
Major 5-mosstep M5ms 2L + 3s 793.2 ¢ to 1057.5 ¢
6-mosstep Minor 6-mosstep m6ms L + 5s 528.8 ¢ to 951.8 ¢
Major 6-mosstep M6ms 2L + 4s 951.8 ¢ to 1057.5 ¢
7-mosstep Perfect 7-mosstep P7ms 2L + 5s 1057.5 ¢ to 1110.4 ¢
Augmented 7-mosstep A7ms 3L + 4s 1110.4 ¢ to 1586.3 ¢
8-mosstep Minor 8-mosstep m8ms 2L + 6s 1057.5 ¢ to 1269.1 ¢
Major 8-mosstep M8ms 3L + 5s 1269.1 ¢ to 1586.3 ¢
9-mosstep Minor 9-mosstep m9ms 2L + 7s 1057.5 ¢ to 1427.7 ¢
Major 9-mosstep M9ms 3L + 6s 1427.7 ¢ to 1586.3 ¢
10-mosstep Perfect 10-mosstep P10ms 3L + 7s 1586.3 ¢

Generator chain

Generator chain of 3L 7s⟨5/2⟩
Bright gens Scale degree Abbrev.
12 Augmented 6-mosdegree A6md
11 Augmented 3-mosdegree A3md
10 Augmented 0-mosdegree A0md
9 Augmented 7-mosdegree A7md
8 Major 4-mosdegree M4md
7 Major 1-mosdegree M1md
6 Major 8-mosdegree M8md
5 Major 5-mosdegree M5md
4 Major 2-mosdegree M2md
3 Major 9-mosdegree M9md
2 Major 6-mosdegree M6md
1 Perfect 3-mosdegree P3md
0 Perfect 0-mosdegree
Perfect 10-mosdegree		 P0md
P10md
−1 Perfect 7-mosdegree P7md
−2 Minor 4-mosdegree m4md
−3 Minor 1-mosdegree m1md
−4 Minor 8-mosdegree m8md
−5 Minor 5-mosdegree m5md
−6 Minor 2-mosdegree m2md
−7 Minor 9-mosdegree m9md
−8 Minor 6-mosdegree m6md
−9 Diminished 3-mosdegree d3md
−10 Diminished 10-mosdegree d10md
−11 Diminished 7-mosdegree d7md
−12 Diminished 4-mosdegree d4md

Modes

Scale degrees of the modes of 3L 7s⟨5/2⟩
UDP Cyclic
order 		Step
pattern 		Scale degree (mosdegree)
0 1 2 3 4 5 6 7 8 9 10
9|0 1 LssLssLsss Perf. Maj. Maj. Perf. Maj. Maj. Maj. Aug. Maj. Maj. Perf.
8|1 4 LssLsssLss Perf. Maj. Maj. Perf. Maj. Maj. Maj. Perf. Maj. Maj. Perf.
7|2 7 LsssLssLss Perf. Maj. Maj. Perf. Min. Maj. Maj. Perf. Maj. Maj. Perf.
6|3 10 sLssLssLss Perf. Min. Maj. Perf. Min. Maj. Maj. Perf. Maj. Maj. Perf.
5|4 3 sLssLsssLs Perf. Min. Maj. Perf. Min. Maj. Maj. Perf. Min. Maj. Perf.
4|5 6 sLsssLssLs Perf. Min. Maj. Perf. Min. Min. Maj. Perf. Min. Maj. Perf.
3|6 9 ssLssLssLs Perf. Min. Min. Perf. Min. Min. Maj. Perf. Min. Maj. Perf.
2|7 2 ssLssLsssL Perf. Min. Min. Perf. Min. Min. Maj. Perf. Min. Min. Perf.
1|8 5 ssLsssLssL Perf. Min. Min. Perf. Min. Min. Min. Perf. Min. Min. Perf.
0|9 8 sssLssLssL Perf. Min. Min. Dim. Min. Min. Min. Perf. Min. Min. Perf.

Scale tree

Scale tree and tuning spectrum of 3L 7s⟨5/2⟩
Generator(ed5/2) Cents Step ratio Comments
Bright Dark L:s Hardness
3\10 475.894 1110.420 1:1 1.000 Equalized 3L 7s⟨5/2⟩
16\53 478.887 1107.427 6:5 1.200
13\43 479.583 1106.730 5:4 1.250
23\76 480.069 1106.245 9:7 1.286
10\33 480.701 1105.613 4:3 1.333 Supersoft 3L 7s⟨5/2⟩
27\89 481.241 1105.072 11:8 1.375
17\56 481.560 1104.754 7:5 1.400
24\79 481.918 1104.396 10:7 1.429
7\23 482.791 1103.523 3:2 1.500 Soft 3L 7s⟨5/2⟩
25\82 483.632 1102.681 11:7 1.571
18\59 483.960 1102.354 8:5 1.600
29\95 484.243 1102.071 13:8 1.625
11\36 484.707 1101.607 5:3 1.667 Semisoft 3L 7s⟨5/2⟩
26\85 485.225 1101.088 12:7 1.714
15\49 485.606 1100.707 7:4 1.750
19\62 486.128 1100.185 9:5 1.800
4\13 488.097 1098.217 2:1 2.000 Basic 3L 7s⟨5/2⟩
Scales with tunings softer than this are proper
17\55 490.315 1095.999 9:4 2.250
13\42 491.002 1095.312 7:3 2.333
22\71 491.534 1094.780 12:5 2.400
9\29 492.304 1094.009 5:2 2.500 Semihard 3L 7s⟨5/2⟩
23\74 493.043 1093.270 13:5 2.600
14\45 493.520 1092.794 8:3 2.667
19\61 494.098 1092.216 11:4 2.750
5\16 495.723 1090.591 3:1 3.000 Hard 3L 7s⟨5/2⟩
16\51 497.667 1088.647 10:3 3.333
11\35 498.556 1087.758 7:2 3.500
17\54 499.395 1086.919 11:3 3.667
6\19 500.941 1085.373 4:1 4.000 Superhard 3L 7s⟨5/2⟩
13\41 502.978 1083.336 9:2 4.500
7\22 504.736 1081.578 5:1 5.000
8\25 507.620 1078.693 6:1 6.000
1\3 528.771 1057.542 1:0 → ∞ Collapsed 3L 7s⟨5/2⟩
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