5L 5s: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox MOS
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| Name = pentawood
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2015-11-06 11:49:01 UTC</tt>.<br>
| Periods = 5
: The original revision id was <tt>565461621</tt>.<br>
| nLargeSteps = 5
: The revision comment was: <tt></tt><br>
| nSmallSteps = 5
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
| Equalized = 1
<h4>Original Wikitext content:</h4>
| Collapsed = 0
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">There is only one significant harmonic entropy minimum with this MOS pattern: [[Archytas clan|blackwood]], in which intervals of the prime numbers 3 and 7 are all represented using steps of [[5edo]], and the generator gets you to intervals of 5 like 6/5, 5/4, or 7/5.
| Pattern = LsLsLsLsLs
}}
{{MOS intro}}


The true MOS, LsLsLsLsLs, is always proper because there is only one small step per period, but because there are 5 periods in an octave, there are a wealth of near-MOSes in which multiples of the period (that is, intervals of an even number of steps) are the only generic intervals that come in more than two different flavors. Specifically, there are 6 others: LLssLsLsLs, LLssLLssLs, LLsLssLsLs, LLsLssLLss, LLsLsLssLs, LLsLsLsLss. In the blackwood temperament, these are right on the boundary of being [[Rothenberg propriety|proper]] (because 1\15 is in the middle of the range of good blackwood generators).
There is only one significant [[harmonic entropy]] minimum with this MOS pattern: [[limmic temperaments#5-limit_.28blackwood.29|blackwood]], in which intervals of the prime numbers 3 and 7 are all represented using steps of [[5edo|5edo]], and the generator reaches intervals of 5 like 6/5, 5/4, or 7/5.
||||||||||~ Generator ||~ Cents ||~ Comments ||
|| 0\5 ||  ||  ||  ||  || 0 ||=  ||
||  ||  ||  ||  || 1\30 || 40 ||  ||
||  ||  ||  || 1\25 ||  || 48 ||  ||
||  ||  ||  ||  ||  || 240/(1+pi) ||  ||
||  ||  || 1\20 ||  ||  || 60 ||=  ||
||  ||  ||  ||  ||  || 240/(1+e) ||  ||
||  ||  ||  || 2\35 ||  || 68.57 ||  ||
||  ||  ||  ||  || 3\50 || 72 ||  ||
||  || 1\15 ||  ||  ||  || 80 ||= Blackwood is around here
Optimum rank range (L/s=2/1) for MOS ||
||  ||  ||  ||  ||  || 240/(1+sq&lt;span style="line-height: 1.5;"&gt;rt(3)&lt;/span&gt;) ||  ||
||  ||  ||  || 3\40 ||  || 90 ||=  ||
||  ||  ||  ||  || 5\65 || 92.31 ||= Golden blackwood ||
||  ||  ||  ||  ||  || 240/(1+pi/2) ||  ||
||  ||  || 2\25 ||  ||  || 96 ||=  ||
|| 1\10 ||  ||  ||  ||  || 120 ||=  ||</pre></div>
<h4>Original HTML content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;5L 5s&lt;/title&gt;&lt;/head&gt;&lt;body&gt;There is only one significant harmonic entropy minimum with this MOS pattern: &lt;a class="wiki_link" href="/Archytas%20clan"&gt;blackwood&lt;/a&gt;, in which intervals of the prime numbers 3 and 7 are all represented using steps of &lt;a class="wiki_link" href="/5edo"&gt;5edo&lt;/a&gt;, and the generator gets you to intervals of 5 like 6/5, 5/4, or 7/5.&lt;br /&gt;
&lt;br /&gt;
The true MOS, LsLsLsLsLs, is always proper because there is only one small step per period, but because there are 5 periods in an octave, there are a wealth of near-MOSes in which multiples of the period (that is, intervals of an even number of steps) are the only generic intervals that come in more than two different flavors. Specifically, there are 6 others: LLssLsLsLs, LLssLLssLs, LLsLssLsLs, LLsLssLLss, LLsLsLssLs, LLsLsLsLss. In the blackwood temperament, these are right on the boundary of being &lt;a class="wiki_link" href="/Rothenberg%20propriety"&gt;proper&lt;/a&gt; (because 1\15 is in the middle of the range of good blackwood generators).&lt;br /&gt;


In addition to the true MOS form (LsLsLsLsLs and sLsLsLsLsL), there are 6 near-MOS forms – LLssLsLsLs, LLssLLssLs, LLsLssLsLs, LLsLssLLss, LLsLsLssLs, LLsLsLsLss – in which the period and its multiples (intervals of 2, 4, 6, and 8 mossteps) have more than two varieties. These forms are proper if the bright generator is less than 160¢.


&lt;table class="wiki_table"&gt;
== Intervals ==
    &lt;tr&gt;
{{MOS intervals}}
        &lt;th colspan="5"&gt;Generator&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Cents&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Comments&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;0\5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;0&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1\30&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;40&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1\25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;48&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;240/(1+pi)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1\20&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;60&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;240/(1+e)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2\35&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;68.57&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3\50&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;72&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1\15&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;80&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Blackwood is around here&lt;br /&gt;
Optimum rank range (L/s=2/1) for MOS&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;240/(1+sq&lt;span style="line-height: 1.5;"&gt;rt(3)&lt;/span&gt;)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;3\40&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;90&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;5\65&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;92.31&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Golden blackwood&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;240/(1+pi/2)&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;2\25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;96&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1\10&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;120&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;/body&gt;&lt;/html&gt;</pre></div>
==Modes==
{{MOS mode degrees}}
 
== Scale tree ==
{{MOS tuning spectrum
| 6/5 = Qintosec&nbsp;↑
| 7/5 = Warlock
| 13/8 = Unnamed golden tuning
| 7/4 = Quinkee
| 2/1 = Blacksmith is optimal around here
| 9/4 = Trisedodge
| 13/5 = Unnamed golden tuning
| 6/1 = Cloudtone&nbsp;↓
}}
 
[[Category:Pentawood| ]]
[[Category:10-tone scales]]
<!-- main article -->

Latest revision as of 13:19, 13 March 2025

↖ 4L 4s ↑ 5L 4s 6L 4s ↗
← 4L 5s 5L 5s 6L 5s →
↙ 4L 6s ↓ 5L 6s 6L 6s ↘ 
┌╥┬╥┬╥┬╥┬╥┬┐
│║│║│║│║│║││
││││││││││││
└┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LsLsLsLsLs
sLsLsLsLsL
Equave 2/1 (1200.0 ¢)
Period 1\5 (240.0 ¢)
Generator size
Bright 1\10 to 1\5 (120.0 ¢ to 240.0 ¢)
Dark 0\5 to 1\10 (0.0 ¢ to 120.0 ¢)
TAMNAMS information
Name pentawood
Prefix pentawd-
Abbrev. pw
Related MOS scales
Parent none
Sister 5L 5s (self)
Daughters 10L 5s, 5L 10s
Neutralized 10edo
2-Flought 15L 5s, 5L 15s
Equal tunings
Equalized (L:s = 1:1) 1\10 (120.0 ¢)
Supersoft (L:s = 4:3) 4\35 (137.1 ¢)
Soft (L:s = 3:2) 3\25 (144.0 ¢)
Semisoft (L:s = 5:3) 5\40 (150.0 ¢)
Basic (L:s = 2:1) 2\15 (160.0 ¢)
Semihard (L:s = 5:2) 5\35 (171.4 ¢)
Hard (L:s = 3:1) 3\20 (180.0 ¢)
Superhard (L:s = 4:1) 4\25 (192.0 ¢)
Collapsed (L:s = 1:0) 1\5 (240.0 ¢)

5L 5s, named pentawood in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 5 large steps and 5 small steps, with a period of 1 large step and 1 small step that repeats every 240.0 ¢, or 5 times every octave. Generators that produce this scale range from 120 ¢ to 240 ¢, or from 0 ¢ to 120 ¢. Scales of the true MOS form, where every period is the same, are proper because there is only one small step per period.

There is only one significant harmonic entropy minimum with this MOS pattern: blackwood, in which intervals of the prime numbers 3 and 7 are all represented using steps of 5edo, and the generator reaches intervals of 5 like 6/5, 5/4, or 7/5.

In addition to the true MOS form (LsLsLsLsLs and sLsLsLsLsL), there are 6 near-MOS forms – LLssLsLsLs, LLssLLssLs, LLsLssLsLs, LLsLssLLss, LLsLsLssLs, LLsLsLsLss – in which the period and its multiples (intervals of 2, 4, 6, and 8 mossteps) have more than two varieties. These forms are proper if the bright generator is less than 160¢.

Intervals

Intervals of 5L 5s
Intervals Steps
subtended 		Range in cents
Generic Specific Abbrev.
0-pentawdstep Perfect 0-pentawdstep P0pws 0 0.0 ¢
1-pentawdstep Minor 1-pentawdstep m1pws s 0.0 ¢ to 120.0 ¢
Major 1-pentawdstep M1pws L 120.0 ¢ to 240.0 ¢
2-pentawdstep Perfect 2-pentawdstep P2pws L + s 240.0 ¢
3-pentawdstep Minor 3-pentawdstep m3pws L + 2s 240.0 ¢ to 360.0 ¢
Major 3-pentawdstep M3pws 2L + s 360.0 ¢ to 480.0 ¢
4-pentawdstep Perfect 4-pentawdstep P4pws 2L + 2s 480.0 ¢
5-pentawdstep Minor 5-pentawdstep m5pws 2L + 3s 480.0 ¢ to 600.0 ¢
Major 5-pentawdstep M5pws 3L + 2s 600.0 ¢ to 720.0 ¢
6-pentawdstep Perfect 6-pentawdstep P6pws 3L + 3s 720.0 ¢
7-pentawdstep Minor 7-pentawdstep m7pws 3L + 4s 720.0 ¢ to 840.0 ¢
Major 7-pentawdstep M7pws 4L + 3s 840.0 ¢ to 960.0 ¢
8-pentawdstep Perfect 8-pentawdstep P8pws 4L + 4s 960.0 ¢
9-pentawdstep Minor 9-pentawdstep m9pws 4L + 5s 960.0 ¢ to 1080.0 ¢
Major 9-pentawdstep M9pws 5L + 4s 1080.0 ¢ to 1200.0 ¢
10-pentawdstep Perfect 10-pentawdstep P10pws 5L + 5s 1200.0 ¢

Modes

Scale degrees of the modes of 5L 5s
UDP Cyclic
order 		Step
pattern 		Scale degree (pentawddegree)
0 1 2 3 4 5 6 7 8 9 10
5|0(5) 1 LsLsLsLsLs Perf. Maj. Perf. Maj. Perf. Maj. Perf. Maj. Perf. Maj. Perf.
0|5(5) 2 sLsLsLsLsL Perf. Min. Perf. Min. Perf. Min. Perf. Min. Perf. Min. Perf.

Scale tree

Scale tree and tuning spectrum of 5L 5s
Generator(edo) Cents Step ratio Comments(always proper)
Bright Dark L:s Hardness
1\10 120.000 120.000 1:1 1.000 Equalized 5L 5s
6\55 130.909 109.091 6:5 1.200 Qintosec ↑
5\45 133.333 106.667 5:4 1.250
9\80 135.000 105.000 9:7 1.286
4\35 137.143 102.857 4:3 1.333 Supersoft 5L 5s
11\95 138.947 101.053 11:8 1.375
7\60 140.000 100.000 7:5 1.400 Warlock
10\85 141.176 98.824 10:7 1.429
3\25 144.000 96.000 3:2 1.500 Soft 5L 5s
11\90 146.667 93.333 11:7 1.571
8\65 147.692 92.308 8:5 1.600
13\105 148.571 91.429 13:8 1.625 Unnamed golden tuning
5\40 150.000 90.000 5:3 1.667 Semisoft 5L 5s
12\95 151.579 88.421 12:7 1.714
7\55 152.727 87.273 7:4 1.750 Quinkee
9\70 154.286 85.714 9:5 1.800
2\15 160.000 80.000 2:1 2.000 Basic 5L 5s
Blacksmith is optimal around here
9\65 166.154 73.846 9:4 2.250 Trisedodge
7\50 168.000 72.000 7:3 2.333
12\85 169.412 70.588 12:5 2.400
5\35 171.429 68.571 5:2 2.500 Semihard 5L 5s
13\90 173.333 66.667 13:5 2.600 Unnamed golden tuning
8\55 174.545 65.455 8:3 2.667
11\75 176.000 64.000 11:4 2.750
3\20 180.000 60.000 3:1 3.000 Hard 5L 5s
10\65 184.615 55.385 10:3 3.333
7\45 186.667 53.333 7:2 3.500
11\70 188.571 51.429 11:3 3.667
4\25 192.000 48.000 4:1 4.000 Superhard 5L 5s
9\55 196.364 43.636 9:2 4.500
5\30 200.000 40.000 5:1 5.000
6\35 205.714 34.286 6:1 6.000 Cloudtone ↓
1\5 240.000 0.000 1:0 → ∞ Collapsed 5L 5s
Retrieved from "https://en.xen.wiki/index.php?title=5L_5s&oldid=185675"