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| There are two notable [[Harmonic_Entropy|harmonic entropy]] minima with this [[MOSScales|MOS]] pattern. The first is [[Porcupine_family|porcupine]], in which two generators make a 6/5 and three make a 4/3. The range of porcupine tunings is about 2\15 to 3\22. Less well-known is [[Chromatic_pairs#Greeley|greeley]], in which two generators are still 6/5 but three fall quite short of a 4/3, but the scale happens to closely approximate a lot of higher-complexity intervals like 10/7, 11/7, etc.
| | {{Infobox MOS}} |
| | {{MOS intro}} |
| | == Name == |
| | {{TAMNAMS name}} |
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| Scales of this form are always [[Rothenberg_propriety|proper]], because there is only one small step.
| | == Scale properties == |
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| {| class="wikitable"
| | === Intervals === |
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| | {{MOS intervals}} |
| ! colspan="6" | [[generator|Generator]]
| | |
| ! | [[cent|Cent]]s
| | === Generator chain === |
| ! | Scale in [[EDO|EDO]] steps
| | {{MOS genchain}} |
| ! | Comments
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| | === Modes === |
| | | 1\7
| | {{MOS mode degrees}} |
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| | === Proposed names === |
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| | Mode names are from [[Porcupine Temperament Modal Harmony|Porcupine temperament modal harmony]]. Descriptive mode names are based on using {{dash|1, 4, 7}}, i.e. 3+3 triads as a basis for harmony. |
| | |
| | {{MOS modes |
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| | | Mode names = |
| | | 171.43
| | octopus $ |
| | style="text-align:center;" | 1 1 1 1 1 1 1 0
| | mantis $ |
| | style="text-align:center;" |
| | dolphin $ |
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| | crab $ |
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| | tuna $ |
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| | salmon $ |
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| | starfish $ |
| | | 4\29
| | whale $ |
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| | | Table Headers=Name Origin |
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| | | Table Entries= |
| | | 165.52
| | Bright quartal $ |
| | style="text-align:center;" | 4 4 4 4 4 4 4 1
| | Dark quartal $ |
| | style="text-align:center;" | L/s = 4
| | Bright major $ |
| |-
| | Middle major $ |
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| | Dark major $ |
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| | Bright minor $ |
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| | Middle minor $ |
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| | Dark minor $ |
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| | }} |
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| | | 163.97
| | == Theory == |
| | style="text-align:center;" | pi pi pi pi pi pi pi 1
| | === Low harmonic entropy scales === |
| | style="text-align:center;" | <span style="display: block; text-align: center;">L/s = pi</span>
| | There are three notable [[harmonic entropy]] minima with this [[mos]] pattern. |
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| | * The lowest accuracy one is [[porcupine]], in which two generators make a [[6/5]] and three make a [[4/3]]. The range of porcupine tunings is about 2\15 to 3\22. |
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| | * Less well-known and more accurate is [[greeley]], in which two generators are still 6/5 but three fall quite short of a 4/3, but the scale happens to closely approximate a lot of higher-complexity intervals like [[10/7]], [[11/7]], etc. |
| | | 3\22
| | * Thirdly and finally, [[tempering out]] [[4000/3993|S10/S11]] so that ([[4/3]])/([[11/10]])<sup>3</sup> is tempered out results in an unusually high accuracy and efficient rank-2 temperament in the 2.3.11/5 subgroup for which interpretation as a rank-3 temperament in 2.3.5.11 (the no-7's [[11-limit]]) is natural, making [[10/9]] and [[12/11]] [[square superparticular|equidistant from 11/10]] and offering many fruitful tempering opportunities. Note therefore that [[porkypine]] can be seen as a trivial tuning of [[4000/3993|pine]] tempering out {{nowrap|[[100/99]] {{=}} S10}} and {{nowrap|[[121/120]] {{=}} S11}}. |
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| | == Scale tree == |
| | |
| | {{MOS tuning spectrum |
| | | 163.64
| | | 5/2 = General range of porcupine |
| | style="text-align:center;" | 3 3 3 3 3 3 3 1
| | | 2/1 = Optimum rank range for porcupine |
| | style="text-align:center;" | L/s = 3
| | | 13/8 = Golden porcupine/hemikleismic |
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| | | 10/7 = General range of greeley |
| | style="text-align:center;" |
| | }} |
| | style="text-align:center;" |
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| | style="text-align:center;" |
| | [[Category:8-tone scales]] |
| | style="text-align:center;" |
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| | style="text-align:center;" |
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| | style="text-align:center;" | 162.87
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| | style="text-align:center;" | e e e e e e e e 1
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| | style="text-align:center;" | <span style="display: block; text-align: center;">L/s = e</span>
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| | | 8\59
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| | | 162,71
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| | style="text-align:center;" | <span style="display: block; text-align: center;">8 8 8 8 8 8 8 3</span>
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| | | 13\96
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| | | 162.5
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| | style="text-align:center;" | <span style="display: block; text-align: center;">13 13 13 13 13 13 13 5</span> | |
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| | | 5\37
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| | | 162.16
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| | style="text-align:center;" | 5 5 5 5 5 5 5 2
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| | style="text-align:center;" | Porcupine is in this general region
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| | | 7\52
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| | | 161.54
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| | style="text-align:center;" | 7 7 7 7 7 7 7 3
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| | style="text-align:center;" |
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| | | 2\15
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| | | 160
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| | style="text-align:center;" | 2 2 2 2 2 2 2 1
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| | style="text-align:center;" | Optimum rank range (L/s=2/1) porcupine
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| | | 158.37
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| | style="text-align:center;" | <span style="background-color: #ffffff;">√3 √3 √3 √3 √3 √3 √3 1</span>
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| | | 5\38
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| | | 157.89
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| | style="text-align:center;" | 5 5 5 5 5 5 5 3
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| | style="text-align:center;" |
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| | | 13\99
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| | | 157.58
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| | style="text-align:center;" | 13 13 13 13 13 13 13 8
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| | style="text-align:center;" | Golden porcupine / golden hemikleismic
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| | | 8\61
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| | | 157.38
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| | style="text-align:center;" | 8 8 8 8 8 8 8 5
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| | | (11\84)
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| | | 157.14
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| | style="text-align:center;" | <span style="display: block; text-align: center;">11 11 11 11 11 11 11 7 </span><span style="display: block; text-align: center;">pi pi pi pi pi pi pi 2</span>
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| | | 3\23
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| | | 156.52
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| | style="text-align:center;" | 3 3 3 3 3 3 3 2
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| | | 10\77
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| | | 155.84 | |
| | style="text-align:center;" | 10 10 10 10 10 10 10 7
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| | style="text-align:center;" | Greeley is around here
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| | | 7\54
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| | | 155.56
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| | style="text-align:center;" | 7 7 7 7 7 7 7 5
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| | style="text-align:center;" |
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| | | 4\31
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| | | 154.84
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| | style="text-align:center;" | 4 4 4 4 4 4 4 3
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| | style="text-align:center;" | | |
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| | | 1\8
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| | | 150
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| | style="text-align:center;" | 1 1 1 1 1 1 1 1
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| | style="text-align:center;" |
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| |}
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