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highly significant harmonic entropy minima not discussed
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{{MOS intro}}
{{MOS intro}}
Scales of this form are always [[Rothenberg_propriety|proper]], because there is only one small step.
== Name ==
[[TAMNAMS]] suggests the temperament-agnostic name '''pine''', in reference to porcupine temperament.
== Theory ==
=== Low harmonic entropy scales ===
There are three notable [[Harmonic_Entropy|harmonic entropy]] minima with this [[MOSScales|MOS]] pattern. The lowest accuracy one 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 and more accurate 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. Thirdly and finally, tempering [[4000/3993|S10/S11]] so that ([[4/3]])/([[11/10]])<sup>3</sup> is tempered results in an unusually high accuracy & efficient rank 2 temperament in the 2.3.11/10 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 [[Porcupine family#2.3.5.11 subgroup .28porkypine.29|porkypine]] can be seen as a trivial tuning of [[4000/3993|pine]] tempering [[100/99]] = S10 and [[121/120]] = S11.)
There are three notable [[Harmonic_Entropy|harmonic entropy]] minima with this [[MOSScales|MOS]] pattern. The lowest accuracy one 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 and more accurate 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. Thirdly and finally, tempering [[4000/3993|S10/S11]] so that ([[4/3]])/([[11/10]])<sup>3</sup> is tempered results in an unusually high accuracy & efficient rank 2 temperament in the 2.3.11/10 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 [[Porcupine family#2.3.5.11 subgroup .28porkypine.29|porkypine]] can be seen as a trivial tuning of [[4000/3993|pine]] tempering [[100/99]] = S10 and [[121/120]] = S11.)


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== Scale tree ==
== Scale tree ==
Scales of this form are always [[Rothenberg_propriety|proper]], because there is only one small step.
{{Scale tree|Comments=5/2: General range of porcupine;
 
2/1: Optimum rank range for porcupine;
{| class="wikitable"
13/8: Golden porcupine/hemikleismic
|-
10/7: General range of greeley}}
! colspan="6" | [[generator|Generator]]
! |[[Cent]]s
! |Scale in [[EDO|EDO]] steps
! |Comments
|-
| |1\7
| |
| |
| |
| |
| |
| | 171.43
| style="text-align:center;" |1 1 1 1 1 1 1 0
| style="text-align:center;" |
|-
|
|
|
|
|
|6\43
|167.44
| style="text-align:center;" |6 6 6 6 6 6 6 1
|
|-
|
|
|
|
| 5\36
|
|166.67
| style="text-align:center;" |5 5 5 5 5 5 5 1
| style="text-align:center;" |pine is around here
|
|-
| |
| |
| |
| | 4\29
| |
| |
| |165.52
| style="text-align:center;" |4 4 4 4 4 4 4 1
| style="text-align:center;" |L/s = 4
|-
| |
| |
| |
| |
| |
| |
| |163.97
| style="text-align:center;" |π π π π π π π 1
| style="text-align:center;" |<span style="display: block; text-align: center;">L/s = π</span>
|-
| |
| |
| |3\22
| |
| |
| |
| |163.64
| style="text-align:center;" |3 3 3 3 3 3 3 1
| style="text-align:center;" |L/s = 3
|-
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |
| style="text-align:center;" |162.87
| style="text-align:center;" | e e e e e e e e 1
| style="text-align:center;" |<span style="display: block; text-align: center;">L/s = e</span>
|-
| |
| |
| |
| |
| |8\59
| |
| |162,71
| style="text-align:center;" |<span style="display: block; text-align: center;">8 8 8 8 8 8 8 3</span>
| |
|-
| |
| |
| |
| |
| |
| |13\96
| |162.5
| style="text-align:center;" |<span style="display: block; text-align: center;">13 13 13 13 13 13 13 5</span>
| |
|-
| |
| |
| |
| |5\37
| |
| |
| |162.16
| style="text-align:center;" |5 5 5 5 5 5 5 2
| style="text-align:center;" |Porcupine is in this general region
|-
| |
| |
| |
| |
| |7\52
| |
| |161.54
| style="text-align:center;" | 7 7 7 7 7 7 7 3
| style="text-align:center;" |
|-
| |
| |2\15
| |
| |
| |
| |
| |160
| style="text-align:center;" |2 2 2 2 2 2 2 1
| style="text-align:center;" |Optimum rank range (L/s=2/1) porcupine
|-
| |
| |
| |
| |
| |
| |
| | 158.37
| style="text-align:center;" |<span style="background-color: #ffffff;">√3 √3 √3 √3 √3 √3 √3 1</span>
| |
|-
| |
| |
| |
| |5\38
| |
| |
| |157.89
| style="text-align:center;" |5 5 5 5 5 5 5 3
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| |13\99
| |157.58
| style="text-align:center;" |13 13 13 13 13 13 13 8
| style="text-align:center;" |Golden porcupine / golden hemikleismic
|-
| |
| |
| |
| |
| |8\61
| |
| | 157.38
| style="text-align:center;" |8 8 8 8 8 8 8 5
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| |(11\84)
| |157.14)
| 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;">π π π π π π π 2</span>
| |
|-
| |
| |
| | 3\23
| |
| |
| |
| |156.52
| style="text-align:center;" |3 3 3 3 3 3 3 2
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| | 10\77
| | 155.84
| style="text-align:center;" |10 10 10 10 10 10 10 7
| style="text-align:center;" |Greeley is around here
|-
| |
| |
| |
| |
| |7\54
| |
| | 155.56
| style="text-align:center;" |7 7 7 7 7 7 7 5
| style="text-align:center;" |
|-
| |
| |
| |
| |4\31
| |
| |
| |154.84
| style="text-align:center;" |4 4 4 4 4 4 4 3
| style="text-align:center;" |
|-
|
|
|
|
|5\39
|
|153.85
| style="text-align:center;" |5 5 5 5 5 5 5 4
|
|-
|
|
|
|
|
|6\47
|153.19
| style="text-align:center;" |6 6 6 6 6 6 6 5
|
|-
| |1\8
| |
| |
| |
| |
| |
| |150
| style="text-align:center;" |1 1 1 1 1 1 1 1
| style="text-align:center;" |
|}
 
[[Category:8-tone scales]]
[[Category:8-tone scales]]
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