Sqrtphi: Difference between revisions

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The '''sqrtphi''' is a temperament for the 7, 11, 13, 17, and 19 prime limits. It is a member of [[kleismic family]], [[mirkwai clan]] and [[Wizmic microtemperaments|wizmic temperaments]]. The name "sqrtphi" stands for "square root of phi", which means the positive square root of the [[golden ratio]] <math>(\sqrt{\varphi} = \sqrt{\frac{1+\sqrt{5}}{2}})</math> as a frequency ratio.
The '''sqrtphi''' is a [[regular temperament|temperament]] for the 7-, 11-, 13-, 17-, and 19-limit. It is a member of [[kleismic family]], [[mirkwai clan]] and [[wizmic microtemperaments|wizmic temperaments]]. The name ''sqrtphi'' stands for square root of phi, which means the positive square root of the [[golden ratio]] <math>(\sqrt{\varphi} = \sqrt{\frac{1+\sqrt{5}}{2}})</math> as a frequency ratio.


See [[Kleismic family #Sqrtphi|Kleismic family]] for more technical data.
See [[Kleismic family #Sqrtphi]] for technical data.


== Tuning spectrum ==
== Scales ==
Gencom: [2 14/11; 325/324 364/363 375/374 400/399 442/441 595/594]
=== Scala files ===
* [[Sqrtphi17]]
* [[Sqrtphi23]]
* [[Sqrtphi49]]


Gencom mapping: [{{val|1 12 11 16 17 28 27 -2}}, {{val|0 -30 -25 -38 -39 -70 -66 18}}]
== Tunings ==
 
=== Tuning spectrum ===
{| class="wikitable center-all"
{| class="wikitable center-all left-3"
|-
|-
! | [[eigenmonzo|eigenmonzo<br>(unchanged interval]])
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval]])
! | undecimal<br>major third (¢)
! Generator (¢)
! | comments
! Comments
|-
|-
| | 26/21
| 26/21
| | 415.12662
| 415.12662
| |
|  
|-
|-
| | 17/13
| 17/13
| | 416.10694
| 416.10694
| |
|  
|-
|-
| | 18/13
| 18/13
| | 416.33823
| 416.33823
| |
|  
|-
|-
| | 15/11
| 15/11
| | 416.44058
| 416.44058
| |
|  
|-
|-
| | 13/11
| 13/11
| | 416.47711
| 416.47711
| |
|  
|-
|-
| | 18/17
| 18/17
| | 416.49243
| 416.49243
| |
|  
|-
|-
| | 15/14
| 15/14
| | 416.50336
| 416.50336
| |
|  
|-
|-
| | 14/13
| 14/13
| | 416.50932
| 416.50932
| |
|  
|-
|-
| | 15/13
| 15/13
| | 416.51607
| 416.51607
| |
|  
|-
|-
| | 19/16
| 19/16
| | 416.52850
| 416.52850
| |
|  
|-
|-
| | 22/17
| 22/17
| | 416.53195
| 416.53195
| |
|  
|-
|-
| | 13/12
| 13/12
| | 416.53568
| 416.53568
| |
|  
|-
|-
| | 20/19
| 20/19
| | 416.53952
| 416.53952
| |
|  
|-
|-
| | 11/9
| 11/9
| | 416.54324
| 416.54324
| |
|  
|-
|-
| | (φ)
| (φ)
| | 416.54515
| 416.54515
| | square root of phi
| square root of phi
|-
|-
| | 5/4
| 5/4
| | 416.54745
| 416.54745
| |
|  
|-
|-
| | 26/19
| 26/19
| | 416.55665
| 416.55665
| |
|  
|-
|-
| | 16/13
| 16/13
| | 416.56389
| 416.56389
| |
|  
|-
|-
| | 19/15
| 19/15
| | 416.56499
| 416.56499
| |
|  
|-
|-
| | 17/14
| 17/14
| | 416.56680
| 416.56680
| |
|  
|-
|-
| | 22/21
| 22/21
| | 416.57024
| 416.57024
| |
|  
|-
|-
| | 13/10
| 13/10
| | 416.57302
| 416.57302
| | 13, 15, 17, 19 and 21-odd-limit minimax
| 13, 15, 17, 19 and 21-odd-limit minimax
|-
|-
| | 24/19
| 24/19
| | 416.57413
| 416.57413
| |
|  
|-
|-
| | 16/15
| 16/15
| | 416.57693
| 416.57693
| |
|  
|-
|-
| | 19/17
| 19/17
| | 416.57807
| 416.57807
| |
|  
|-
|-
| | 24/17
| 24/17
| | 416.58332
| 416.58332
| |
|  
|-
|-
| | 19/14
| 19/14
| | 416.58370
| 416.58370
| |
|  
|-
|-
| | 19/18
| 19/18
| | 416.58465
| 416.58465
| |
|  
|-
|-
| | 9/7
| 9/7
| | 416.58709
| 416.58709
| |
|  
|-
|-
| | 21/19
| 21/19
| | 416.58991
| 416.58991
| |
|  
|-
|-
| | 17/16
| 17/16
| | 416.59158
| 416.59158
| |
|  
|-
|-
| | 22/19
| 22/19
| | 416.59991
| 416.59991
| |
|  
|-
|-
| | 4/3
| 4/3
| | 416.60150
| 416.60150
| | 5-odd-limit minimax
| 5-odd-limit minimax
|-
|-
| | 21/16
| 21/16
| | 416.60616
| 416.60616
| |
|  
|-
|-
| | 8/7
| 8/7
| | 416.60984
| 416.60984
| | 7 and 9-odd-limit minimax
| 7 and 9-odd-limit minimax
|-
|-
| | 20/17
| 20/17
| | 416.61850
| 416.61850
| |
|  
|-
|-
| | 11/8
| 11/8
| | 416.63287
| 416.63287
| | 11-odd-limit minimax
| 11-odd-limit minimax
|-
|-
| | 10/9
| 10/9
| | 416.64011
| 416.64011
| |
|  
|-
|-
| | 21/20
| 21/20
| | 416.64030
| 416.64030
| |
|  
|-
|-
| | 7/6
| 7/6
| | 416.64114
| 416.64114
| |
|  
|-
|-
| | 17/15
| 17/15
| | 416.66485
| 416.66485
| |
|  
|-
|-
| | 7/5
| 7/5
| | 416.72983
| 416.72983
| |
|  
|-
|-
| | 12/11
| 12/11
| | 416.73745
| 416.73745
| |
|  
|-
|-
| | 11/10
| 11/10
| | 416.78541
| 416.78541
| |
|  
|-
|-
| | 6/5
| 6/5
| | 416.87174
| 416.87174
| |
|  
|-
|-
| | 21/17
| 21/17
| | 417.08725
| 417.08725
| |
|  
|-
|-
| | 14/11
| 14/11
| | 417.50796
| 417.50796
| |
|  
|}
|}


== Scales ==
== Music ==
* [[Sqrtphi17]]
; [[Vito Sicurella]]
* [[Sqrtphi23]]
* [https://web.archive.org/web/20201127014110/http://micro.soonlabel.com/gene_ward_smith/Others/Sicurella/A%20Fight%20For%20Phi.mp3 ''A Fight for Phi'']
* [[Sqrtphi49]]


== Music ==
; [[Chris Vaisvil]]
* [http://micro.soonlabel.com/sqrt_phi/daily20111123a-sqrt-phi-17.mp3 Prelude for Piano in Square root of Phi Tuning] by [[Chris Vaisvil]]
* [https://web.archive.org/web/20201127012408/http://micro.soonlabel.com/sqrt_phi/daily20111123a-sqrt-phi-17.mp3 ''Prelude for Piano in Square root of Phi Tuning'']  
* [http://micro.soonlabel.com/gene_ward_smith/Others/Sicurella/A%20Fight%20For%20Phi.mp3 A Fight for Phi] by [[Vito Sicurella]]


[[Category:Sqrtphi| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Kleismic family]]
[[Category:Kleismic family]]
{{IoT}}
[[Category:Canopic clan]]
[[Category:Wizmic microtemperaments]]
[[Category:Golden ratio]]
[[Category:Golden ratio]]

Latest revision as of 10:30, 6 June 2026

The sqrtphi is a temperament for the 7-, 11-, 13-, 17-, and 19-limit. It is a member of kleismic family, mirkwai clan and wizmic temperaments. The name sqrtphi stands for square root of phi, which means the positive square root of the golden ratio [math]\displaystyle{ (\sqrt{\varphi} = \sqrt{\frac{1+\sqrt{5}}{2}}) }[/math] as a frequency ratio.

See Kleismic family #Sqrtphi for technical data.

Scales

Scala files

Tunings

Tuning spectrum

Eigenmonzo
(unchanged-interval) 		Generator (¢) Comments
26/21 415.12662
17/13 416.10694
18/13 416.33823
15/11 416.44058
13/11 416.47711
18/17 416.49243
15/14 416.50336
14/13 416.50932
15/13 416.51607
19/16 416.52850
22/17 416.53195
13/12 416.53568
20/19 416.53952
11/9 416.54324
(φ) 416.54515 square root of phi
5/4 416.54745
26/19 416.55665
16/13 416.56389
19/15 416.56499
17/14 416.56680
22/21 416.57024
13/10 416.57302 13, 15, 17, 19 and 21-odd-limit minimax
24/19 416.57413
16/15 416.57693
19/17 416.57807
24/17 416.58332
19/14 416.58370
19/18 416.58465
9/7 416.58709
21/19 416.58991
17/16 416.59158
22/19 416.59991
4/3 416.60150 5-odd-limit minimax
21/16 416.60616
8/7 416.60984 7 and 9-odd-limit minimax
20/17 416.61850
11/8 416.63287 11-odd-limit minimax
10/9 416.64011
21/20 416.64030
7/6 416.64114
17/15 416.66485
7/5 416.72983
12/11 416.73745
11/10 416.78541
6/5 416.87174
21/17 417.08725
14/11 417.50796

Music

Vito Sicurella
Chris Vaisvil
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