Porcupine family: Difference between revisions

The 5-limit parent comma for the '''porcupine family''' is [[250/243]], the maximal diesis or porcupine comma. Its [[monzo]] is {{monzo| 1 -5 3 }}, and flipping that yields {{multival| 3 5 1 }} for the [[wedgie]]. This tells us the [[generator]] is a minor whole tone, the [[10/9]] interval, and that three of these add up to a fourth, with two more giving the minor sixth. In fact, (10/9)³ = 4/3 × 250/243, and (10/9)⁵ = 8/5 × (250/243)². [[22edo|3\22]] is a very recommendable generator, and MOS of 7, 8 and 15 notes make for some nice scale possibilities.

= Porcupine =

Subgroup: 2.3.5

[[Mapping]]: [{{val| 1 2 3 }}, {{val| 0 -3 -5 }}]

[[POTE generator]]: ~27/25 = 163.950

* [[diamond monotone]] range: [150.000, 171.429] (1\8 to 1\7)

* [[diamond purer]] range: [157.821, 166.015]

* [[diamond nice]] range: [157.821, 166.015]

{{Val list|legend=1| 7, 15, 22, 95c, 117bc, 139bc, 161bc, 183bc }}

[[Badness]]: 0.0308

== Extensions ==

The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which [[7-limit]] family member we are looking at. That means

* [[49/48]], the slendro diesis, for [[#Nautilus|nautilus]].

Porcupine uses six of its minor tone generator steps to get to [[7/4]]. For this to work you need a small minor tone such as [[22edo]] provides, and once again 3\22 is a good tuning choice, though we might pick in preference 8\59, 11\81, or 19\140 for our generator.

Subgroup: 2.3.5.7

[[Mapping]]: [{{val| 1 2 3 2 }}, {{val| 0 -3 -5 6 }}]

{{Multival|legend=1| 3 5 -6 1 -18 -28 }}

 

[[POTE generator]]: ~10/9 = 162.880

* [[7-odd-limit|7-]] and [[9-odd-limit]] eigenmonzo: 9/7

* [[diamond monotone]] range: [160.000, 163.636] (2\15 to 3\22)

* [[diamond purer]] range: [157.821, 166.015]

* [[diamond nice]] range: [160.000, 163.636]

{{Val list|legend=1| 7, 15, 22, 59, 81bd, 140bbd }}

[[Badness]]: 0.0411

== 11-limit ==

Mapping: [{{val| 1 2 3 2 4 }}, {{val| 0 -3 -5 6 -4 }}]

POTE generator: ~11/10 = 162.747

* 11-odd-limit eigenmonzo: 9/7

* [[diamond monotone]] range: [160.000, 163.636] (2\15 to 3\22)

* [[diamond purer]] range: [150.637, 182.404]

* [[diamond nice]] range: [160.000, 163.636]

Vals: {{val list| 7, 15, 22, 37, 59 }}

Badness: 0.0217

Mapping: [{{val| 1 2 3 2 4 4 }}, {{val| 0 -3 -5 6 -4 -2 }}]

POTE generator: ~11/10 = 162.708

* 13- and 15-odd-limit eigenmonzo: 11/8

* [[diamond monotone]] range: [160.000, 163.636] (15 to 22f)

* [[diamond purer]] range: [138.573, 182.404]

* [[diamond nice]] range: [160.000, 163.636]

Vals: {{val list| 7, 15, 22f, 37f }}

Badness: 0.0213

=== Porcupinefish ===

{{see also| The Biosphere }}

Mapping: [{{val| 1 2 3 2 4 6 }}, {{val| 0 -3 -5 6 -4 -17 }}]

POTE generator: ~11/10 = 162.277

* 13- and 15-odd-limit eigenmonzo: 13/11

* [[diamond monotone]] range: [160.000, 162.162] (15 to 37)

* [[diamond purer]] range: [150.637, 182.404]

* [[diamond nice]] range: [160.000, 162.162]

Vals: {{val list| 15, 22, 37, 59, 96b }}

Badness: 0.0253

=== Pourcup ===

Mapping: [{{val| 1 2 3 2 4 1 }}, {{val| 0 -3 -5 6 -4 20 }}]

POTE generator: ~11/10 = 162.482

* 13- and 15-odd-limit eigenmonzo: 13/7

Vals: {{val list| 15f, 22f, 37 }}

Badness: 0.0351

=== Porkpie ===

Mapping: [{{val| 1 2 3 2 4 3 }}, {{val| 0 -3 -5 6 -4 5 }}]

POTE generator: ~11/10 = 163.688

* 13- and 15-odd-limit eigenmonzo: 9/7

 

Vals: {{val list| 7, 15f, 22 }}

Badness: 0.0260

Hystrix provides a less complex avenue to the 7-limit. Unfortunately in temperaments as in life you get what you pay for, and hystrix, for which a generator of 2\15 or 9\68 can be used, is a temperament for the adventurous souls who have probably already tried [[15edo]]. They can try the even sharper fifth of hystrix in [[68edo]] and see how that suits.

Subgroup: 2.3.5.7

[[Comma list]]: 36/35, 160/147

[[Mapping]]: [{{val| 1 2 3 3 }}, {{val| 0 -3 -5 -1 }}]

{{Multival|legend=1| 3 5 1 1 -7 -12 }}

[[POTE generator]]: ~10/9 = 158.868

* [[7-odd-limit|7-]] and [[9-odd-limit]] eigenmonzo: 5/4

{{Val list|legend=1| 7, 8d, 15d }}

[[Badness]]: 0.0449

== 11-limit ==

Comma list: 22/21, 36/35, 80/77

Mapping: [{{val| 1 2 3 3 4 }}, {{val| 0 -3 -5 -1 -4 }}]

POTE generator: ~11/10 = 158.750

Vals: {{val list| 7, 8d, 15d }}

Badness: 0.0268

Subgroup: 2.3.5.7

[[Mapping]]: [{{val| 1 2 3 5 }}, {{val| 0 -3 -5 -16 }}]

{{Multival|legend=1| 3 5 16 1 17 23 }}

 

[[POTE generator]]: ~10/9 = 164.412

* [[7-odd-limit|7-]] and [[9-odd-limit]] eigenmonzo: 7/5

{{Val list|legend=1| 7d, 15d, 22, 29, 51, 73c }}

[[Badness]]: 0.0544

== 11-limit ==

Mapping: [{{val| 1 2 3 5 4 }}, {{val| 0 -3 -5 -16 -4 }}]

POTE generator: ~11/10 = 164.552

* 11-odd-limit eigenmonzo: 7/5

Vals: {{val list| 7d, 15d, 22, 29, 51, 73ce }}

Badness: 0.0273

== 13-limit ==

Mapping: [{{val| 1 2 3 5 4 3 }}, {{val| 0 -3 -5 -16 -4 5 }}]

POTE generator: ~11/10 = 164.953

Vals: {{val list| 7d, 22, 29, 51f, 80cdeff }}

Badness: 0.0265

Subgroup: 2.3.5.7

[[Mapping]]: [{{val| 1 2 3 1 }}, {{val| 0 -3 -5 13 }}]

{{Multival|legend=1| 3 5 -13 1 -29 -44 }}

 

[[POTE generator]]: ~10/9 = 166.041

* [[7-odd-limit|7-]] and [[9-odd-limit]] eigenmonzo: 3/2

{{Val list|legend=1| 7, 29, 65c, 94cd }}

[[Badness]]: 0.1183

== 11-limit ==

Mapping: [{{val| 1 2 3 1 4 }}, {{val| 0 -3 -5 13 -4 }}]

POTE generator: ~11/10 = 165.981

* 11-odd-limit eigenmonzo: 3/2

Vals: {{val list| 7, 29, 65ce, 94cde }}

Badness: 0.0497

== 13-limit ==

Mapping: [{{val| 1 2 3 1 4 3 }}, {{val| 0 -3 -5 13 -4 5 }}]

POTE generator: ~11/10 = 165.974

* 13- and 15-odd-limit eigenmonzo: 3/2

Vals: {{val list| 7, 29, 65cef, 94cdef }}

Badness: 0.0302

= Hedgehog =

Hedgehog has a period 1/2 octave and a generator which can be taken to be 9/7 instead of 10/9. It also tempers out [[245/243]], the sensamagic comma. 22edo provides the obvious tuning, but if you are looking for an alternative, you could try the {{val| 146 232 338 411 }} val with generator 10\73, or you could try 164 cents if you are fond of round numbers. The 14 note MOS gives scope for harmony while stopping well short of 22.

Subgroup: 2.3.5.7

[[Mapping]]: [{{val| 2 1 1 2 }}, {{val| 0 3 5 5 }}]

{{Multival|legend=1| 6 10 10 2 -1 -5 }}

[[POTE generator]]: ~9/7 = 435.648

{{Val list|legend=1| 8d, 14c, 22, 146bccdd }}

[[Badness]]: 0.0440

== 11-limit ==

Mapping: [{{val| 2 1 1 2 4 }}, {{val| 0 3 5 5 4 }}]

POTE generator: ~9/7 = 435.386

Vals: {{val list| 14c, 22, 58ce, 80ce, 102cde }}

Badness: 0.0231

=== 13-limit ===

Mapping: [{{val| 2 1 1 2 4 3 }}, {{val| 0 3 5 5 4 6 }}]

POTE generator: ~9/7 = 435.861

Vals: {{val list| 14cf, 22 }}

Badness: 0.0215

=== Urchin ===

Mapping: [{{val| 2 1 1 2 4 6 }}, {{val| 0 3 5 5 4 2 }}]

POTE generator: ~9/7 = 437.078

Vals: {{val list| 14c, 22f }}

Badness: 0.0252

== Hedgepig ==

Mapping: [{{val| 2 1 1 2 12 }}, {{val| 0 3 5 5 -7 }}]

POTE generator: ~9/7 = 435.425

Vals: {{val list| 22, 80c, 102cd, 124cd }}

Badness: 0.0684

[http://micro.soonlabel.com/22-ET/20120207-phobos-light-hedgehog14.mp3 Phobos Light] by [[Chris Vaisvil]] in Hedgehog[14] [[hedgehog14|tuned]] to 22edo.

= Nautilus =

Subgroup: 2.3.5.7

[[Mapping]]: [{{val| 1 2 3 3 }}, {{val| 0 -6 -10 -3 }}]

{{Multival|legend=1| 6 10 3 2 -12 -21 }}

[[POTE generator]]: ~21/20 = 82.505

{{Val list|legend=1| 15, 29, 43cd, 44d, 59d, 73cd, 102cd }}

== 11-limit ==

Mapping: [{{val| 1 2 3 3 4 }}, {{val| 0 -6 -10 -3 -8 }}]

POTE generator: ~21/20 = 82.504

Vals: {{val list| 14c, 15, 29, 43cde, 44d, 59d, 73cde, 102cde }}

=== 13-limit ===

Mapping: [{{val| 1 2 3 3 4 5 }}, {{val| 0 -6 -10 -3 -8 -19 }}]

POTE generator: ~21/20 = 62.530

Vals: {{val list| 15f, 29, 43cde, 44d, 59df, 73cde, 102cde }}

Badness: 0.0223

=== Belauensis ===

Mapping: [{{val| 1 2 3 3 4 4 }}, {{val| 0 -6 -10 -3 -8 -4 }}]

POTE generator: ~21/20 = 81.759

Vals: {{val list| 14c, 15, 29f, 44df }}

Badness: 0.0298

[http://micro.soonlabel.com/gene_ward_smith/Others/Igs/NautilusReverie.mp3 Nautilus Reverie] by [[Igliashon Jones|Igliashon Calvin Jones-Coolidge]]

= Ammonite =

Subgroup: 2.3.5.7

[[Mapping]]: [{{val| 1 5 8 10 }}, {{val| 0 -9 -15 -19 }}]

{{Multival|legend=1| 9 15 19 3 5 2 }}

[[POTE generator]]: ~9/7 = 454.448

{{Val list|legend=1| 29, 37, 66 }}

[[Badness]]: 0.1077

== 11-limit ==

Mapping: [{{val| 1 5 8 10 8 }}, {{val| 0 -9 -15 -19 -12 }}]

POTE generator: ~9/7 = 454.512

Vals: {{val list| 29, 37, 66 }}

Badness: 0.0457

== 13-limit ==

Mapping: [{{val| 1 5 8 10 8 9 }}, {{val| 0 -9 -15 -19 -12 -14 }}]

POTE generator: ~13/10 = 454.429

Vals: {{val list| 29, 37, 66 }}

Badness: 0.0272

Subgroup: 2.3.5.7

[[Mapping]]: [{{val| 1 2 3 3 }}, {{val| 0 -9 -15 -4 }}]

[[POTE generator]]: ~36/35 = 54.384

{{Val list|legend=1| 22 }}

[[Badness]]: 0.115

== 11-limit ==

Comma list: 55/54, 100/99, 5324/5145

Mapping: [{{val| 1 2 3 3 4 }}, {{val| 0 -9 -15 -4 -12 }}]

POTE generator: ~36/35 = 54.376

Vals: {{val list| 22 }}

Badness: 0.0513

== 13-limit ==

Mapping: [{{val| 1 2 3 3 4 4 }}, {{val| 0 -9 -15 -4 -12 -7 }}]

POTE generator: ~36/35 = 54.665

Vals: {{val list| 22 }}

Badness: 0.0447

[[Category:Regular temperament theory]]

[[Category:Temperament family]]

[[Category:Porcupine]]