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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | {{Infobox ET}} |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | {{ED intro}} |
| : This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2011-06-24 04:22:28 UTC</tt>.<br>
| |
| : The original revision id was <tt>238541723</tt>.<br>
| |
| : The revision comment was: <tt></tt><br>
| |
| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
| |
| <h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">The Cube Root of the [[Perfect fourth]] ([[4_3|4:3]]) is a nonoctave scale which divides the just perfect fourth (frequency ratio 4:3) into three steps of approximately 166.015[[cent|¢]] each.
| |
|
| |
|
| ==Intervals== | | == Theory == |
| | === Harmonics === |
| | {{Harmonics in equal|3|4|3|intervals=integer|columns=11}} |
| | {{Harmonics in equal|3|4|3|intervals=integer|columns=11|start=12|collapsed=true|title=Approximation of harmonics in 3ed4/3 (continued)}} |
|
| |
|
| || degrees of CRP4 || cents value || cents value octave-reduced || | | == Intervals == |
| || 0 || 0.00 || ||
| | {| class="wikitable" |
| || 1 || 166.01 || ||
| | |- |
| || 2 || 332.03 || ||
| | ! # |
| || 3 || 498.04 || ||
| | ! Cents |
| || 4 || 664.06 || ||
| | ! Approximate ratios |
| || 5 || 830.07 || ||
| | |- |
| || 6 || 996.09 || ||
| | | 0 |
| || 7 || 1162.10 || ||
| | | 0.000 |
| || 8 || 1328.12 || 128.12 ||
| | | [[1/1]] |
| || 9 || 1494.13 || 294.13 ||
| | |- |
| || 10 || 1660.15 || 460.15 ||
| | | 1 |
| || 11 || 1826.16 || 626.16 ||
| | | 166.015 |
| || 12 || 1992.18 || 792.18 ||
| | | [[11/10]] |
| || 13 || 2158.19 || 958.19 ||
| | |- |
| || 14 || 2324.21 || 1124.21 ||
| | | 2 |
| || 15 || 2490.22 || 90.22 ||
| | | 332.030 |
| || 16 || 2656.24 || 256.24 ||
| | | |
| || 17 || 2822.25 || 422.25 ||
| | |- |
| || 18 || 2988.27 || 588.27 ||
| | | 3 |
| || 19 || 3154.28 || 754.28 ||
| | | 498.045 |
| || 20 || 3320.30 || 920.30 ||
| | | [[4/3]] |
| || 21 || 3486.31 || 1086.31 ||
| | |- |
| || 22 || 3652.33 || 52.33 ||
| | | 4 |
| || 23 || 3818.34 || 218.34 ||</pre></div>
| | | 664.060 |
| <h4>Original HTML content:</h4>
| | | [[22/15]] |
| <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"><html><head><title>Cube Root of P4</title></head><body>The Cube Root of the <a class="wiki_link" href="/Perfect%20fourth">Perfect fourth</a> (<a class="wiki_link" href="/4_3">4:3</a>) is a nonoctave scale which divides the just perfect fourth (frequency ratio 4:3) into three steps of approximately 166.015<a class="wiki_link" href="/cent">¢</a> each.<br />
| | |- |
| <br />
| | | 5 |
| <!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Intervals"></a><!-- ws:end:WikiTextHeadingRule:0 -->Intervals</h2>
| | | 830.075 |
| <br />
| | | [[13/8]] |
| | |- |
| | | 6 |
| | | 996.090 |
| | | [[16/9]] |
| | |- |
| | | 7 |
| | | 1162.105 |
| | | 88/45 |
| | |- |
| | | 8 |
| | | 1328.120 |
| | | [[13/6]] |
| | |- |
| | | 9 |
| | | 1494.135 |
| | | [[64/27]] |
| | |- |
| | | 10 |
| | | 1660.150 |
| | | |
| | |- |
| | | 11 |
| | | 1826.165 |
| | | [[13/9]] |
| | |- |
| | | 12 |
| | | 1992.180 |
| | | |
| | |- |
| | | 13 |
| | | 2158.195 |
| | | |
| | |- |
| | | 14 |
| | | 2324.210 |
| | | |
| | |- |
| | | 15 |
| | | 2490.225 |
| | | [[135/32]] |
| | |- |
| | | 16 |
| | | 2656.240 |
| | | |
| | |- |
| | | 17 |
| | | 2822.255 |
| | | |
| | |- |
| | | 18 |
| | | 2988.270 |
| | | [[45/8]] |
| | |- |
| | | 19 |
| | | 3154.285 |
| | | |
| | |- |
| | | 20 |
| | | 3320.300 |
| | | [[17/5]] |
| | |- |
| | | 21 |
| | | 3486.315 |
| | | [[15/2]] |
| | |- |
| | | 22 |
| | | 3652.330 |
| | | |
| | |- |
| | | 23 |
| | | 3818.345 |
| | | [[68/15]] |
| | |- |
| | | 24 |
| | | 3984.360 |
| | | [[10/1]] |
| | |} |
|
| |
|
| | == Regular temperaments == |
| | 3ed4/3 tuning is related to temperaments which temper out [[4000/3993]] (wizardharry temperament). The unit step of 3ed4/3 is approximately a cent sharp of [[11/10]]. Tempering out 4000/3993 leads equating three 11/10s with 4/3, hence wizardharry temperaments split the fourth in three. |
|
| |
|
| <table class="wiki_table">
| | Tempering out both [[55/54]] and [[100/99]] (equating 10/9 with 11/10 and 12/11) leads to [[porcupine]] (2.3.5.11 subgroup) or [[sonic]] (full 11-limit). Sonic temperaments include [[porcupine]], [[hystrix]], [[porky]], [[coendou]], [[hedgehog]], [[nautilus]], [[ammonite]], [[ceratitid]], and [[opossum]]. |
| <tr>
| |
| <td>degrees of CRP4<br />
| |
| </td>
| |
| <td>cents value<br />
| |
| </td>
| |
| <td>cents value octave-reduced<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>0<br />
| |
| </td>
| |
| <td>0.00<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>1<br />
| |
| </td>
| |
| <td>166.01<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>2<br />
| |
| </td>
| |
| <td>332.03<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>3<br />
| |
| </td>
| |
| <td>498.04<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>4<br />
| |
| </td>
| |
| <td>664.06<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>5<br />
| |
| </td>
| |
| <td>830.07<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>6<br />
| |
| </td>
| |
| <td>996.09<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>7<br />
| |
| </td>
| |
| <td>1162.10<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>8<br />
| |
| </td>
| |
| <td>1328.12<br />
| |
| </td>
| |
| <td>128.12<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>9<br />
| |
| </td>
| |
| <td>1494.13<br />
| |
| </td>
| |
| <td>294.13<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>10<br />
| |
| </td>
| |
| <td>1660.15<br />
| |
| </td>
| |
| <td>460.15<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>11<br />
| |
| </td>
| |
| <td>1826.16<br />
| |
| </td>
| |
| <td>626.16<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>12<br />
| |
| </td>
| |
| <td>1992.18<br />
| |
| </td>
| |
| <td>792.18<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>13<br />
| |
| </td>
| |
| <td>2158.19<br />
| |
| </td>
| |
| <td>958.19<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>14<br />
| |
| </td>
| |
| <td>2324.21<br />
| |
| </td>
| |
| <td>1124.21<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>15<br />
| |
| </td>
| |
| <td>2490.22<br />
| |
| </td>
| |
| <td>90.22<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>16<br />
| |
| </td>
| |
| <td>2656.24<br />
| |
| </td>
| |
| <td>256.24<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>17<br />
| |
| </td>
| |
| <td>2822.25<br />
| |
| </td>
| |
| <td>422.25<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>18<br />
| |
| </td>
| |
| <td>2988.27<br />
| |
| </td>
| |
| <td>588.27<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>19<br />
| |
| </td>
| |
| <td>3154.28<br />
| |
| </td>
| |
| <td>754.28<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>20<br />
| |
| </td>
| |
| <td>3320.30<br />
| |
| </td>
| |
| <td>920.30<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>21<br />
| |
| </td>
| |
| <td>3486.31<br />
| |
| </td>
| |
| <td>1086.31<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>22<br />
| |
| </td>
| |
| <td>3652.33<br />
| |
| </td>
| |
| <td>52.33<br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td>23<br />
| |
| </td>
| |
| <td>3818.34<br />
| |
| </td>
| |
| <td>218.34<br />
| |
| </td>
| |
| </tr>
| |
| </table>
| |
|
| |
|
| </body></html></pre></div>
| | Other wizardharry temperaments include [[octoid]], [[harry]], [[tritikleismic]], [[wizard]], [[Porwell temperaments #Septisuperfourth|septisuperfourth]], [[unthirds]], [[supers]], [[alphaquarter]], [[quincy]], [[stearnscape]], [[pogo]], [[marvolo]], [[cotritone]], [[echidna]], [[marvo]], [[mystery]], [[zarvo]], [[escaped]], [[thuja]], and [[escapade]]. |
| | |
| | [[Category:Equal-step tuning]] |
| | [[Category:Nonoctave]] |
| | [[Category:Perfect fourth]] |
| Prime factorization
|
3 (prime)
|
| Step size
|
166.015 ¢
|
| Octave
|
7\3ed4/3 (1162.1 ¢)
(semiconvergent)
|
| Twelfth
|
11\3ed4/3 (1826.16 ¢)
(semiconvergent)
|
| Consistency limit
|
4
|
| Distinct consistency limit
|
4
|
3 equal divisions of 4/3 (abbreviated 3ed4/3) is a nonoctave tuning system that divides the interval of 4/3 into 3 equal parts of about 166 ¢ each. Each step represents a frequency ratio of (4/3)1/3, or the 3rd root of 4/3.
Theory
Harmonics
Approximation of harmonics in 3ed4/3
| Harmonic
|
2
|
3
|
4
|
5
|
6
|
7
|
8
|
9
|
10
|
11
|
12
|
| Error
|
Absolute (¢)
|
-37.9
|
-75.8
|
-75.8
|
+35.9
|
+52.3
|
-48.5
|
+52.3
|
+14.4
|
-2.0
|
-0.9
|
+14.4
|
| Relative (%)
|
-22.8
|
-45.7
|
-45.7
|
+21.6
|
+31.5
|
-29.2
|
+31.5
|
+8.7
|
-1.2
|
-0.6
|
+8.7
|
Steps
(reduced)
|
7
(1)
|
11
(2)
|
14
(2)
|
17
(2)
|
19
(1)
|
20
(2)
|
22
(1)
|
23
(2)
|
24
(0)
|
25
(1)
|
26
(2)
|
Approximation of harmonics in 3ed4/3 (continued)
| Harmonic
|
13
|
14
|
15
|
16
|
17
|
18
|
19
|
20
|
21
|
22
|
23
|
| Error
|
Absolute (¢)
|
+41.9
|
+79.6
|
-39.8
|
+14.4
|
+75.5
|
-23.5
|
+49.0
|
-39.8
|
+41.7
|
-38.8
|
+50.2
|
| Relative (%)
|
+25.2
|
+47.9
|
-24.0
|
+8.7
|
+45.5
|
-14.1
|
+29.5
|
-24.0
|
+25.1
|
-23.4
|
+30.3
|
Steps
(reduced)
|
27
(0)
|
28
(1)
|
28
(1)
|
29
(2)
|
30
(0)
|
30
(0)
|
31
(1)
|
31
(1)
|
32
(2)
|
32
(2)
|
33
(0)
|
Intervals
| #
|
Cents
|
Approximate ratios
|
| 0
|
0.000
|
1/1
|
| 1
|
166.015
|
11/10
|
| 2
|
332.030
|
|
| 3
|
498.045
|
4/3
|
| 4
|
664.060
|
22/15
|
| 5
|
830.075
|
13/8
|
| 6
|
996.090
|
16/9
|
| 7
|
1162.105
|
88/45
|
| 8
|
1328.120
|
13/6
|
| 9
|
1494.135
|
64/27
|
| 10
|
1660.150
|
|
| 11
|
1826.165
|
13/9
|
| 12
|
1992.180
|
|
| 13
|
2158.195
|
|
| 14
|
2324.210
|
|
| 15
|
2490.225
|
135/32
|
| 16
|
2656.240
|
|
| 17
|
2822.255
|
|
| 18
|
2988.270
|
45/8
|
| 19
|
3154.285
|
|
| 20
|
3320.300
|
17/5
|
| 21
|
3486.315
|
15/2
|
| 22
|
3652.330
|
|
| 23
|
3818.345
|
68/15
|
| 24
|
3984.360
|
10/1
|
Regular temperaments
3ed4/3 tuning is related to temperaments which temper out 4000/3993 (wizardharry temperament). The unit step of 3ed4/3 is approximately a cent sharp of 11/10. Tempering out 4000/3993 leads equating three 11/10s with 4/3, hence wizardharry temperaments split the fourth in three.
Tempering out both 55/54 and 100/99 (equating 10/9 with 11/10 and 12/11) leads to porcupine (2.3.5.11 subgroup) or sonic (full 11-limit). Sonic temperaments include porcupine, hystrix, porky, coendou, hedgehog, nautilus, ammonite, ceratitid, and opossum.
Other wizardharry temperaments include octoid, harry, tritikleismic, wizard, septisuperfourth, unthirds, supers, alphaquarter, quincy, stearnscape, pogo, marvolo, cotritone, echidna, marvo, mystery, zarvo, escaped, thuja, and escapade.