3L 7s: Difference between revisions

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Line 5:

| nSmallSteps = 7

| Equalized = 3

| ~~Paucitonic~~ = 1

| Collapsed = 1

| Pattern = LssLssLsss

}}

== Name ==

[[TAMNAMS]] suggests the temperament-agnostic name '''sephiroid''' for this scale, in reference to Kosmorsky's ''Tracatum de Modi Sephiratorum.''

~~'''3L 7s''' occupies the spectrum from 10edo (L~~ = ~~s) to 3edo (s~~ = ~~0).~~

== Scale properties ==

~~[[TAMNAMS]] calls this~~ MOS ~~pattern '''sephiroid''' (named after the abstract temperament [[sephiroth]]).~~

=== Intervals ===

~~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.

=== Generator chain ===

~~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 5/4 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]].~~

=== Modes ===

~~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~~.)

=== Proposed Names ===

Mode names are described by Kosmorsky, which use names from the [[wikipedia:Sefirot|Sefirot]] (or sephiroth). Kosmorsky describes the mode Keter to be akin to the lydian mode of 5L 2s, and the mode Malkuth like the locrian mode.

{{MOS modes

| Mode Names=

Malkuth $

Yesod $

Hod $

Netzach $

Tiferet $

Gevurah $

Chesed $

Binah $

Chokmah $

Keter $

}}

~~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"~~.

== Theory ==

=== The ''modi sephiratorum'' ===

This MOS can represent tempered-flat chains of the 13th harmonic, which approximates phi (~833 cents).

=~~= Modes ==~~

With sephiroid scales with a soft-of-basic step ratio (around {nowrap|L:s {{=}} 3:2}}, or 23edo), 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.

~~s s s~~ L s s L s ~~s L - Keter~~

Scales approaching an equalized step ratio ({{nowrap|L:s {{=}} 1:1}}, or [[10edo]]) contain a 13th harmonic that's nearly perfect. [[121edo]] seems to be the first to 'accurately' represent the comma{{Clarify}}. Scales approaching a collapsed step ratio ({{nowrap|L:s {{=}} 1:0}}, or [[3edo]]) have the comma [[65/64]] liable to be tempered out, thus equating [[8/5]] and [[13/8]]. Edos include [[13edo]], [[16edo]], [[19edo]], [[22edo]], [[29edo]], and others.

~~s s L s s L s s L s - Chesed~~

Harmonically, the arrangement forming a chord (degrees 0, 1, 4, 7, 10){{Clarify}} 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.

s ~~L s s L s s L s s - Netzach~~

There are MODMOS as well, but Kosmorsky has not explored them yet, as "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".

~~L s s L s s L s s s - Malkuth~~

== Scale tree ==

{{MOS tuning spectrum

~~s s L s s L s s s L - Binah~~

| 6/5 = [[Submajor (temperament)|Submajor]]

| 13/8 = Unnamed golden tuning

~~s L s s L s s s L s - Tiferet~~

| 5/2 = [[Sephiroth]]

| 13/5 = Golden sephiroth

~~L s s L s s s L s s - Yesod~~

| 11/3 = [[Muggles]]

| 4/1 = [[Magic]] / horcrux

~~s s L s s s L s s L - Chokmah~~

| 9/2 = Magic / witchcraft / necromancy

| 5/1 = Magic / telepathy

~~s L s s s L s s L s - Gevurah~~

| 6/1 = [[Würschmidt]] ↓

}}

~~L s s s L s s L s s - Hod~~

== ~~Scales tree~~ ==

== External links ==

~~{| class="wikitable center-all"~~

* [https://ia800703.us.archive.org/12/items/TractatumDeModiSephiratorum/ModiSephiratorum.pdf Tractatum de Modi Sephiratorum] by Kosmorsky

~~! colspan="6" | Generator~~

~~! Cents~~

~~! L~~

~~! s~~

~~! L~~/s

~~! Comments~~

|-

~~| 3\10 || || || || || || 360.000 || 1 || 1 || 1.000 ||~~

|-

~~| || || || || || 16\53 || 362.264 || 6 || 5 || 1.200 || Submajor~~

|-

~~| || || || || 13\43 || || 362.791 || 5 || 4 || 1.250 ||~~

|-

~~| || || || || || 23\76 || 363.158 || 9 || 7 || 1.286 ||~~

|-

~~| || || || 10\33 || || || 363.636 || 4 || 3 || 1.333 ||~~

|-

~~| || || || || || 27\89 || 364.045 || 11 || 8 || 1.375 ||~~

|-

~~| || || || || 17\56 || || 364.286 || 7 || 5 || 1~~.~~400 ||~~

|-

~~| || || || || || 24\79 || 364~~.~~557 || 10 || 7 || 1~~.~~428 ||~~

|-

~~| || || 7\23 || || || || 365.217 || 3 || 2 || 1.500 || L/s = 3~~/2

|-

~~| || || || || || 25\82 || 365.854 || 11 || 7 || 1.571 ||~~

|-

~~| || || || || 18\59 || || 366.102 || 8 || 5 || 1.600 ||~~

|-

~~| || || || || || 29\95 || 366.316 || 13 || 8 || 1.625 || Unnamed golden tuning~~

|-

~~| || || || 11\36 || || || 366.667 || 5 || 3 || 1.667 ||~~

|-

~~| || || || || || 26\85 || 367.059 ||~~ 12 ~~|| 7 || 1.714 ||~~

|-

~~| || || || || 15\49 || || 367.347 || 7 || 4 || 1.750 ||~~

|-

~~| || || || || || 19\62 || 367.742 || 9 || 5 || 1.800 ||~~

|-

~~| || 4\13 || || || || || 369.231 || 2 || 1 || 2.000 || Basic sephiroid<br>(Generators smaller than this are proper)~~

|-

~~| || || || || || 17\55 || 370.909 || 9 || 4 || 2.250 ||~~

|-

~~| || || || || 13\42 || || 371.429 || 7 || 3 || 2.333 ||~~

|-

~~| || || || || || 22\71 || 371.831 || 12 || 5 || 2.400 ||~~

|-

~~| || || || 9\29 || || || 372.414 || 5 || 2 || 2.500 || Sephiroth~~

|-

~~| || || || || || 23\74 || 372.973 || 13 || 5 || 2.600 || Golden sephiroth~~

|-

~~| || || || || 14\45 || || 373.333 || 8 || 3 || 2.667 ||~~

|-

~~| || || || || || 19\61 || 373.770 || 11 || 4 || 2.750 ||~~

|-

~~| || || 5\16 || || || || 375.000 || 3 || 1 || 3.000 || L~~/~~s = 3~~/1

|-

~~| || || || || || 16\51 || 376.471 || 10 || 3 || 3.333 ||~~

|-

~~| || || || || 11\35 || || 377.143 || 7 || 2 || 3.500 ||~~

|-

~~| || || || || || 17\54 || 377.778 || 11 || 3 || 3.667 || Muggles~~

|-

~~| || || || 6\19 || || || 378.947 || 4 || 1 || 4.000 || Magic~~/~~horcrux~~

|-

~~| || || || || || 13\41 || 380~~.~~488 || 9 || 2 || 4.500 || Magic/witchcraft~~

|-

~~| || || || || 7\22 || || 381.818 || 5 || 1 || 5.000 || Magic/telepathy~~

|-

~~| || || || || || 8\25 || 384.000 || 6 || 1 || 6.000 || Würschmidt↓~~

|-

~~| 1\3 || || || || || || 400.000 || 1 || 0 || → inf ||~~

|}

[[Category:~~Scales]]~~

[[Category:10-tone scales]]

~~[[Category:Abstract MOS patterns~~]]

@@ Line 5: / Line 5: @@
 | nSmallSteps = 7
 | Equalized = 3
-| Paucitonic = 1
+| Collapsed = 1
 | Pattern = LssLssLsss
 }}
+{{MOS intro}}
+== Name ==
+[[TAMNAMS]] suggests the temperament-agnostic name '''sephiroid''' for this scale, in reference to Kosmorsky's ''Tracatum de Modi Sephiratorum.''
-'''3L 7s''' occupies the spectrum from 10edo (L = s) to 3edo (s = 0).
+== Scale properties ==
+{{TAMNAMS use}}
-[[TAMNAMS]] calls this MOS pattern '''sephiroid''' (named after the abstract temperament [[sephiroth]]).
+=== Intervals ===
+{{MOS intervals}}
-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.
+=== Generator chain ===
+{{MOS genchain}}
-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 5/4 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]].
+=== Modes ===
+{{MOS mode degrees}}
-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.)
+=== Proposed Names ===
+Mode names are described by Kosmorsky, which use names from the [[wikipedia:Sefirot|Sefirot]] (or sephiroth). Kosmorsky describes the mode Keter to be akin to the lydian mode of 5L 2s, and the mode Malkuth like the locrian mode.
+{{MOS modes
+| Mode Names=
+Malkuth $
+Yesod $
+Hod $
+Netzach $
+Tiferet $
+Gevurah $
+Chesed $
+Binah $
+Chokmah $
+Keter $
+}}
-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".
+== Theory ==
+=== The ''modi sephiratorum'' ===
+This MOS can represent tempered-flat chains of the 13th harmonic, which approximates phi (~833 cents).
-== Modes ==
+With sephiroid scales with a soft-of-basic step ratio (around {nowrap|L:s {{=}} 3:2}}, or 23edo), 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.
-s s s L s s L s s L - Keter
+Scales approaching an equalized step ratio ({{nowrap|L:s {{=}} 1:1}}, or [[10edo]]) contain a 13th harmonic that's nearly perfect. [[121edo]] seems to be the first to 'accurately' represent the comma{{Clarify}}. Scales approaching a collapsed step ratio ({{nowrap|L:s {{=}} 1:0}}, or [[3edo]]) have the comma [[65/64]] liable to be tempered out, thus equating [[8/5]] and [[13/8]]. Edos include [[13edo]], [[16edo]], [[19edo]], [[22edo]], [[29edo]], and others.
-s s L s s L s s L s - Chesed
+Harmonically, the arrangement forming a chord (degrees 0, 1, 4, 7, 10){{Clarify}} 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.
-s L s s L s s L s s - Netzach
+There are MODMOS as well, but Kosmorsky has not explored them yet, as "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".
-L s s L s s L s s s - Malkuth
+== Scale tree ==
+{{MOS tuning spectrum
-s s L s s L s s s L - Binah
+| 6/5 = [[Submajor (temperament)|Submajor]]
+| 13/8 = Unnamed golden tuning
-s L s s L s s s L s - Tiferet
+| 5/2 = [[Sephiroth]]
+| 13/5 = Golden sephiroth
-L s s L s s s L s s - Yesod
+| 11/3 = [[Muggles]]
+| 4/1 = [[Magic]] / horcrux
-s s L s s s L s s L - Chokmah
+| 9/2 = Magic / witchcraft / necromancy
+| 5/1 = Magic / telepathy
-s L s s s L s s L s - Gevurah
+| 6/1 = [[Würschmidt]] ↓
+}}
-L s s s L s s L s s - Hod
-== Scales tree ==
+== External links ==
-{| class="wikitable center-all"
+* [https://ia800703.us.archive.org/12/items/TractatumDeModiSephiratorum/ModiSephiratorum.pdf Tractatum de Modi Sephiratorum] by Kosmorsky
-! colspan="6" | Generator
-! Cents
-! L
-! s
-! L/s
-! Comments
-|-
-| 3\10 || || || || || || 360.000 || 1 || 1 || 1.000 ||
-|-
-| || || || || || 16\53 || 362.264 || 6 || 5 || 1.200 || Submajor
-|-
-| || || || || 13\43 || || 362.791 || 5 || 4 || 1.250 ||
-|-
-| || || || || || 23\76 || 363.158 || 9 || 7 || 1.286 ||
-|-
-| || || || 10\33 || || || 363.636 || 4 || 3 || 1.333 ||
-|-
-| || || || || || 27\89 || 364.045 || 11 || 8 || 1.375 ||
-|-
-| || || || || 17\56 || || 364.286 || 7 || 5 || 1.400 ||
-|-
-| || || || || || 24\79 || 364.557 || 10 || 7 || 1.428 ||
-|-
-| || || 7\23 || || || || 365.217 || 3 || 2 || 1.500 || L/s = 3/2
-|-
-| || || || || || 25\82 || 365.854 || 11 || 7 || 1.571 ||
-|-
-| || || || || 18\59 || || 366.102 || 8 || 5 || 1.600 ||
-|-
-| || || || || || 29\95 || 366.316 || 13 || 8 || 1.625 || Unnamed golden tuning
-|-
-| || || || 11\36 || || || 366.667 || 5 || 3 || 1.667 ||
-|-
-| || || || || || 26\85 || 367.059 || 12 || 7 || 1.714 ||
-|-
-| || || || || 15\49 || || 367.347 || 7 || 4 || 1.750 ||
-|-
-| || || || || || 19\62 || 367.742 || 9 || 5 || 1.800 ||
-|-
-| || 4\13 || || || || || 369.231 || 2 || 1 || 2.000 || Basic sephiroid<br>(Generators smaller than this are proper)
-|-
-| || || || || || 17\55 || 370.909 || 9 || 4 || 2.250 ||
-|-
-| || || || || 13\42 || || 371.429 || 7 || 3 || 2.333 ||
-|-
-| || || || || || 22\71 || 371.831 || 12 || 5 || 2.400 ||
-|-
-| || || || 9\29 || || || 372.414 || 5 || 2 || 2.500 || Sephiroth
-|-
-| || || || || || 23\74 || 372.973 || 13 || 5 || 2.600 || Golden sephiroth
-|-
-| || || || || 14\45 || || 373.333 || 8 || 3 || 2.667 ||
-|-
-| || || || || || 19\61 || 373.770 || 11 || 4 || 2.750 ||
-|-
-| || || 5\16 || || || || 375.000 || 3 || 1 || 3.000 || L/s = 3/1
-|-
-| || || || || || 16\51 || 376.471 || 10 || 3 || 3.333 ||
-|-
-| || || || || 11\35 || || 377.143 || 7 || 2 || 3.500 ||
-|-
-| || || || || || 17\54 || 377.778 || 11 || 3 || 3.667 || Muggles
-|-
-| || || || 6\19 || || || 378.947 || 4 || 1 || 4.000 || Magic/horcrux
-|-
-| || || || || || 13\41 || 380.488 || 9 || 2 || 4.500 || Magic/witchcraft
-|-
-| || || || || 7\22 || || 381.818 || 5 || 1 || 5.000 || Magic/telepathy
-|-
-| || || || || || 8\25 || 384.000 || 6 || 1 || 6.000 || Würschmidt↓
-|-
-| 1\3 || || || || || || 400.000 || 1 || 0 || → inf ||
-|}
-[[Category:Scales]]
+[[Category:10-tone scales]]
-[[Category:Abstract MOS patterns]]