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.

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

All these 7-limit extensions notably share the same 2.3.5.11 subgroup, ''porkypine''.  

Temperaments discussed elsewhere include [[opossum]] and [[Dicot family #Jamesbond|jamesbond]].

Subgroup: 2.3.5

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

[[POTE generator]]: ~10/9 = 163.950

* 5-odd-limit [[diamond monotone]]: ~10/9 = [150.000, 171.429] (1\8 to 1\7)

{{Val list|legend=1| 7, 15, 22, 95c }}

[[Badness]]: 0.030778

=== Porkypine ===

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

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

Gencom: [2 10/9; 55/54, 100/99]

POTE generator: ~11/10 = 164.0777

Vals: {{val list| 7, 15, 22, 37, 73ce, 95ce }}

{{main| Porcupine }}

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|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

: Eigenmonzos (unchanged intervals): 2, 5/4

: Eigenmonzos (unchanged intervals): 2, 9/7

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

[[Badness]]: 0.041057

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

POTE generator: ~10/9 = 162.747

* 11-odd-limit: ~10/9 = {{monzo| 1/6 -1/6 0 1/12 }}

: Eigenmonzos (unchanged intervals): 2, 9/7

* 11-odd-limit diamond monotone: ~10/9 = [160.000, 163.636] (2\15 to 3\22)

* 11-odd-limit diamond tradeoff: ~10/9 = [150.637, 182.404]

* 11-odd-limit diamond monotone and tradeoff: ~10/9 = [160.000, 163.636]

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

Badness: 0.021562

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

POTE generator: ~10/9 = 162.708

: Eigenmonzo (unchanged intervals): 2, 11/8

* 13-odd-limit diamond monotone: ~10/9 = [160.000, 163.636] (2\15 to 3\22)

* 15-odd-limit diamond monotone: ~10/9 = 163.636 (3\22)

* 13- and 15-odd-limit diamond tradeoff: ~10/9 = [138.573, 182.404]

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

Badness: 0.021276

{{see also| The Biosphere }}

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

POTE generator: ~10/9 = 162.277

: Eigenmonzos (unchanged intervals): 2, 13/11

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

Badness: 0.025314

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

POTE generator: ~10/9 = 162.482

* 13- and 15-odd-limit: ~10/9 = {{monzo| 1/14 0 0 -1/14 0 1/14 }}

: Eigenmonzos (unchanged intervals): 2, 14/13

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

Badness: 0.035130

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

POTE generator: ~10/9 = 163.688

* 13- and 15-odd-limit: ~10/9 = {{monzo| 1/6 -1/6 0 1/12 }}

: Eigenmonzos (unchanged intervals): 2, 9/7

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

Badness: 0.026043

== Hystrix ==

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|15EDO]]. They can try the even sharper fifth of hystrix in [[68edo|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]]: ~8/7 = 158.868

* [[7-odd-limit|7-]] and [[9-odd-limit]]: ~8/7 = {{monzo| 3/5 0 -1/5 }}

: Eigenmonzos (unchanged intervals): 2, 5/4

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

[[Badness]]: 0.044944

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

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

POTE generator: ~8/7 = 158.750

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

Badness: 0.026790

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

: Eigenmonzos (unchanged intervals): 2, 7/5

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

[[Badness]]: 0.054389

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

POTE generator: ~10/9 = 164.552

* 11-odd-limit: ~10/9 = {{monzo| 2/11 0 1/11 -1/11 }}

: Eigenmonzos (unchanged intervals): 2, 7/5

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

Badness: 0.027268

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

POTE generator: ~10/9 = 164.953

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

Badness: 0.026543

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

: Eigenmonzos (unchanged intervals): 2, 3

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

[[Badness]]: 0.118344

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

POTE generator: ~10/9 = 165.981

* 11-odd-limit: ~10/9 = {{monzo| 2/3 -1/3 }}

: Eigenmonzos (unchanged intervals): 2, 3

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

Badness: 0.049669

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

POTE generator: ~10/9 = 165.974

* 13- and 15-odd-limit: ~10/9 = {{monzo| 2/3 -1/3 }}

: Eigenmonzos (unchanged intervals): 2, 3

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

Badness: 0.030233

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]]: 0.043983

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

POTE generator: ~9/7 = 435.386

Vals: {{val list| 8d, 14c, 22, 58ce, 80ce, 102cde }}

Badness: 0.023095

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

POTE generator: ~9/7 = 435.861

Vals: {{val list| 8d, 14cf, 22 }}

Badness: 0.021516

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: 0.025233

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: 0.068406

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

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| 14c, 15, 29, 44d, 59d, 73cd, 102cd }}

[[Badness]]: 0.057420

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, 44d, 59d, 73cde, 102cde }}

Badness: 0.026023

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

POTE generator: ~21/20 = 82.530

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

Badness: 0.022285

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: 0.029816

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

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]]: 0.107686

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: 0.045694

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

POTE generator: ~13/10 = 454.529

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

Badness: 0.027168

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| 1c, 21c, 22 }}

[[Badness]]: 0.115304

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

POTE generator: ~36/35 = 54.376

Vals: {{val list| 1ce, 21ce, 22 }}

Badness: 0.051319

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

POTE generator: ~36/35 = 54.665

Vals: {{val list| 1ce, 21cef, 22 }}

Badness: 0.044739

[[Category:Regular temperament theory]]

[[Category:Temperament family]]

[[Category:Porcupine]]