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Golden sequences and tuning: Difference between revisions

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== Golden operations and MOS height ==
== Golden operations and MOS height ==
Any MOS may be constructed from 1L 1s using two "golden operations": chromaticizing (taking the soft child of the MOS, thus continuing the golden sequence) and inverting (inverting the number of large and small steps, switching to a new golden sequence). In effect, this makes taking the hard child "cost" 2 instead of 1. This table shows the number of such operations required to reach any MOS under 15 notes (that MOS' "height"): The total number of MOSes at each height corresponds to the Fibonacci sequence.
Any MOS may be constructed from 1L 1s using two "golden operations": chromaticizing (taking the soft child of the MOS, thus continuing the golden sequence) and inverting (inverting the number of large and small steps, switching to a new golden sequence). In effect, this makes taking the hard child "cost" 2 instead of 1. This table shows the number of such operations required to reach any MOS under 15 notes (that MOS' "height"): The total number of MOSes at each height corresponds to the Fibonacci sequence.
 
{| class="wikitable sortable mw-collapsible mw-collapsed"
{| class="wikitable sortable mw-collapsible mw-collapsed"
|+MOS height for MOSes under 15 notes
|+ style="font-size: 105%; white-space: nowrap;" | MOS height for MOSes under 15 notes
!Height
|-
!MOS
! Height
!Total notes
! MOS
! Total notes
|-
|-
|1
| 1
|1L 1s
| 1L 1s
|2
| 2
|-
|-
|2
| 2
|2L 1s
| 2L 1s
|3
| 3
|-
|-
|3
| 3
|1L 2s
| 1L 2s
|3
| 3
|-
|-
|3
| 3
|3L 2s
| 3L 2s
|5
| 5
|-
|-
|4
| 4
|3L 1s
| 3L 1s
|4
| 4
|-
|-
|4
| 4
|2L 3s
| 2L 3s
|5
| 5
|-
|-
|4
| 4
|5L 3s
| 5L 3s
|8
| 8
|-
|-
|5
| 5
|4L 3s
| 4L 3s
|7
| 7
|-
|-
|5
| 5
|1L 3s
| 1L 3s
|4
| 4
|-
|-
|5
| 5
|5L 2s
| 5L 2s
|7
| 7
|-
|-
|5
| 5
|3L 5s
| 3L 5s
|8
| 8
|-
|-
|5
| 5
|8L 5s
| 8L 5s
|13
| 13
|-
|-
|6
| 6
|7L 4s
| 7L 4s
|11
| 11
|-
|-
|6
| 6
|3L 4s
| 3L 4s
|7
| 7
|-
|-
|6
| 6
|4L 1s
| 4L 1s
|5
| 5
|-
|-
|6
| 6
|2L 5s
| 2L 5s
|7
| 7
|-
|-
|6
| 6
|7L 5s
| 7L 5s
|12
| 12
|-
|-
|6
| 6
|8L 3s
| 8L 3s
|11
| 11
|-
|-
|6
| 6
|5L 8s
| 5L 8s
|13
| 13
|-
|-
|7
| 7
|4L 7s
| 4L 7s
|11
| 11
|-
|-
|7
| 7
|7L 3s
| 7L 3s
|10
| 10
|-
|-
|7
| 7
|5L 4s
| 5L 4s
|9
| 9
|-
|-
|7
| 7
|1L 4s
| 1L 4s
|5
| 5
|-
|-
|7
| 7
|7L 2s
| 7L 2s
|9
| 9
|-
|-
|7
| 7
|5L 7s
| 5L 7s
|12
| 12
|-
|-
|7
| 7
|3L 8s
| 3L 8s
|11
| 11
|-
|-
|8
| 8
|3L 7s
| 3L 7s
|10
| 10
|-
|-
|8
| 8
|4L 5s
| 4L 5s
|9
| 9
|-
|-
|8
| 8
|9L 5s
| 9L 5s
|14
| 14
|-
|-
|8
| 8
|5L 1s
| 5L 1s
|6
| 6
|-
|-
|8
| 8
|2L 7s
| 2L 7s
|9
| 9
|-
|-
|8
| 8
|11L 3s
| 11L 3s
|14
| 14
|-
|-
|9
| 9
|10L 3s
| 10L 3s
|13
| 13
|-
|-
|9
| 9
|9L 4s
| 9L 4s
|13
| 13
|-
|-
|9
| 9
|5L 9s
| 5L 9s
|14
| 14
|-
|-
|9
| 9
|1L 5s
| 1L 5s
|6
| 6
|-
|-
|9
| 9
|6L 5s
| 6L 5s
|11
| 11
|-
|-
|9
| 9
|9L 2s
| 9L 2s
|11
| 11
|-
|-
|9
| 9
|3L 11s
| 3L 11s
|14
| 14
|-
|-
|10
| 10
|3L 10s
| 3L 10s
|13
| 13
|-
|-
|10
| 10
|4L 9s
| 4L 9s
|13
| 13
|-
|-
|10
| 10
|6L 1s
| 6L 1s
|7
| 7
|-
|-
|10
| 10
|5L 6s
| 5L 6s
|11
| 11
|-
|-
|10
| 10
|2L 9s
| 2L 9s
|11
| 11
|-
|-
|11
| 11
|1L 6s
| 1L 6s
|7
| 7
|-
|-
|11
| 11
|7L 6s
| 7L 6s
|13
| 13
|-
|-
|11
| 11
|11L 2s
| 11L 2s
|13
| 13
|-
|-
|12
| 12
|7L 1s
| 7L 1s
|8
| 8
|-
|-
|12
| 12
|6L 7s
| 6L 7s
|13
| 13
|-
|-
|12
| 12
|2L 11s
| 2L 11s
|13
| 13
|-
|-
|13
| 13
|1L 7s
| 1L 7s
|8
| 8
|-
|-
|14
| 14
|8L 1s
| 8L 1s
|9
| 9
|-
|-
|15
| 15
|1L 8s
| 1L 8s
|9
| 9
|-
|-
|16
| 16
|9L 1s
| 9L 1s
|10
| 10
|-
|-
|17
| 17
|1L 9s
| 1L 9s
|10
| 10
|-
|-
|18
| 18
|10L 1s
| 10L 1s
|11
| 11
|-
|-
|19
| 19
|1L 10s
| 1L 10s
|11
| 11
|-
|-
|20
| 20
|11L 1s
| 11L 1s
|12
| 12
|-
|-
|21
| 21
|1L 11s
| 1L 11s
|12
| 12
|-
|-
|22
| 22
|12L 1s
| 12L 1s
|13
| 13
|-
|-
|23
| 23
|1L 12s
| 1L 12s
|13
| 13
|-
|-
|24
| 24
|13L 1s
| 13L 1s
|14
| 14
|-
|-
|25
| 25
|1L 13s
| 1L 13s
|14
| 14
|}
|}


Line 353: Line 355:
Here are some golden generators for many rank-2 octave-periodic temperaments on [[Bird's eye view of temperaments by accuracy|this page]] as well as some other well-known temperaments. Buzzard and trienstonian were omitted because of their very awkward tuning ranges (both extremely close to 5edo) for the purposes of generating scales.
Here are some golden generators for many rank-2 octave-periodic temperaments on [[Bird's eye view of temperaments by accuracy|this page]] as well as some other well-known temperaments. Buzzard and trienstonian were omitted because of their very awkward tuning ranges (both extremely close to 5edo) for the purposes of generating scales.
{| class="wikitable"
{| class="wikitable"
|+Common temperaments
|+ style="font-size: 105%; white-space: nowrap;" | Common temperaments
!Temperament
|-
!Generator
! Temperament
!Golden tuning
! Generator
!MOS
! Golden tuning
! MOS
|-
|-
|Schismic/Garibaldi
| Schismic/Garibaldi
|3/2
| 3/2
|702.75c
| 702.75{{c}}
|12L 17s
| 12L 17s
|-
|-
|Kleismic/Cata
| Kleismic/Cata
|6/5
| 6/5
|317.17c
| 317.17{{c}}
|4L 11s
| 4L 11s
|-
|-
|Sensi
| Sensi
|9/7
| 9/7
|440.59c
| 440.59{{c}}
|3L 5s
| 3L 5s
|-
|-
|Wurschmidt
| Wurschmidt
|5/4
| 5/4
|387.82c
| 387.82{{c}}
|3L 28s
| 3L 28s
|-
|-
|MIRACLE
| MIRACLE
|16/15
| 16/15
|115.59c
| 115.59{{c}}
|10L 11s
| 10L 11s
|-
|-
|Meantone
| Meantone
|3/2
| 3/2
|696.21c
| 696.21{{c}}
|2L 3s
| 2L 3s
|-
|-
|Mothra/Slendric
| Mothra/Slendric
|8/7
| 8/7
|231.74c
| 231.74{{c}}
|5L 21s
| 5L 21s
|-
|-
|Didacus
| Didacus
|28/25
| 28/25
|191.13c
| 191.13{{c}}
|6L 13s
| 6L 13s
|-
|-
|Superpyth
| Superpyth
|3/2
| 3/2
|708.05c
| 708.05{{c}}
|5L 12s
| 5L 12s
|-
|-
|Mohajira
| Mohajira
|11/9
| 11/9
|348.91c
| 348.91{{c}}
|7L 17s
| 7L 17s
|-
|-
|Semaphore
| Semaphore
|8/7
| 8/7
|254.04c
| 254.04{{c}}
|5L 9s
| 5L 9s
|-
|-
|Negri
| Negri
|16/15
| 16/15
|124.77c
| 124.77{{c}}
|1L 8s
| 1L 8s
|-
|-
|Flattone
| Flattone
|3/2
| 3/2
|693.06c
| 693.06{{c}}
|7L 12s
| 7L 12s
|-
|-
|Porcupine
| Porcupine
|10/9
| 10/9
|162.56c
| 162.56{{c}}
|7L 8s
| 7L 8s
|-
|-
|Magic
| Magic
|5/4
| 5/4
|380.82c<ref group="note">Also has another tuning of 377.6 cents for 3L 13s. This results in a fifth almost as flat as in 7edo, but is a simpler scale of 16 notes rather than 19.</ref>
| 380.82{{c}}<ref group="note">Also has another tuning of 377.6 cents for 3L&nbsp;13s. This results in a fifth almost as flat as in 7edo, but is a simpler scale of 16 notes rather than 19.</ref>
|3L 16s
| 3L&nbsp;16s
|}
|}
Exotemperaments are tuned to the simplest MOS that has a golden generator in the correct range.
Exotemperaments are tuned to the simplest MOS that has a golden generator in the correct range.
{| class="wikitable"
{| class="wikitable"
|+Exotemperaments
|+ style="font-size: 105%; white-space: nowrap;" | Exotemperaments
!Temperament
|-
!Generator
! Temperament
!Golden tuning
! Generator
!MOS
! Golden tuning
! MOS
|-
|-
|Father
| Father
|8/5
| 8/5
|741.64c
| 741.64{{c}}
|1L 1s
| 1L&nbsp;1s
|-
|-
|Mavila
| Mavila
|3/2
| 3/2
|672.85c
| 672.85{{c}}
|2L 5s
| 2L&nbsp;5s
|-
|-
|Bug
| Bug
|5/3
| 5/3
|940.15c
| 940.15{{c}}
|1L 3s
| 1L&nbsp;3s
|-
|-
|Dicot
| Dicot
|5/4
| 5/4
|354.83c
| 354.83{{c}}
|3L 4s
| 3L&nbsp;4s
|}
|}
Yo shares sensi's golden generator, so it is excluded here.
Yo shares sensi's golden generator, so it is excluded here.


== Big table of golden generators ==
== Big table of golden generators ==
This table originated on the page "Golden MOS" and has moved here. It shows the golden generators for a large number of simple MOSes. Many have the same generator, so they've been placed in the same row.
This table originated on the page "Golden MOS" and has moved here. It shows the golden generators for a large number of simple MOSes. Many have the same generator, so they've been placed in the same row.
{| class="wikitable sortable"
{| class="wikitable sortable"
!MOS
!Generator (¢)
|-
|-
|1L&nbsp;1s, 2L 1s, 3L 2s, 5L 3s, 8L 5s
! MOS
|741.6408
! Generator (¢)
|-
| 1L&nbsp;1s, 2L&nbsp;1s, 3L&nbsp;2s, 5L&nbsp;3s, 8L&nbsp;5s
| 741.6408
|-
|-
|1L&nbsp;2s, 3L 1s, 4L 3s, 7L 4s
| 1L&nbsp;2s, 3L&nbsp;1s, 4L&nbsp;3s, 7L&nbsp;4s
|868.3282
| 868.3282
|-
|-
|1L&nbsp;3s, 4L 1s, 5L 4s
| 1L&nbsp;3s, 4L&nbsp;1s, 5L&nbsp;4s
|940.1492
| 940.1492
|-
|-
|2L&nbsp;3s, 5L 2s, 7L 5s
| 2L&nbsp;3s, 5L&nbsp;2s, 7L&nbsp;5s
|503.7855
| 503.7855
|-
|-
|1L&nbsp;4s, 5L 1s, 6L 5s
| 1L&nbsp;4s, 5L&nbsp;1s, 6L&nbsp;5s
|986.4021
| 986.4021
|-
|-
|1L&nbsp;5s, 6L 1s, 7L 6s
| 1L&nbsp;5s, 6L&nbsp;1s, 7L&nbsp;6s
|1018.6773
| 1018.6773
|-
|-
|3L&nbsp;4s, 7L 3s
| 3L&nbsp;4s, 7L&nbsp;3s
|354.8232
| 354.8232
|-
|-
|2L&nbsp;5s, 7L 2s
| 2L&nbsp;5s, 7L&nbsp;2s
|527.1497
| 527.1497
|-
|-
|1L&nbsp;6s, 7L 1s
| 1L&nbsp;6s, 7L&nbsp;1s
|1042.4790
| 1042.4790
|-
|-
|3L&nbsp;5s, 8L 3s
| 3L&nbsp;5s, 8L&nbsp;3s
|759.4078
| 759.4078
|-
|-
|1L&nbsp;7s, 8L 1s
| 1L&nbsp;7s, 8L&nbsp;1s
|1060.7571
| 1060.7571
|-
|-
|4L&nbsp;5s, 9L 4s
| 4L&nbsp;5s, 9L&nbsp;4s
|273.8497
| 273.8497
|-
|-
|2L&nbsp;7s, 9L 2s
| 2L&nbsp;7s, 9L&nbsp;2s
|541.3837
| 541.3837
|-
|-
|1L&nbsp;8s, 9L 1s
| 1L&nbsp;8s, 9L&nbsp;1s
|1075.2344
| 1075.2344
|-
|-
|3L&nbsp;7s, 10L 3s
| 3L&nbsp;7s, 10L&nbsp;3s
|366.2564
| 366.2564
|-
|-
|1L&nbsp;9s, 10L 1s
| 1L&nbsp;9s, 10L&nbsp;1s
|1086.9847
| 1086.9847
|-
|-
|5L&nbsp;6s
| 5L&nbsp;6s
|222.9668
| 222.9668
|-
|-
|4L&nbsp;7s
| 4L&nbsp;7s
|877.7318
| 877.7318
|-
|-
|3L&nbsp;8s
| 3L&nbsp;8s
|768.8815
| 768.8815
|-
|-
|2L&nbsp;9s, 11L 2s
| 2L&nbsp;9s, 11L&nbsp;2s
|550.9646
| 550.9646
|-
|-
|1L&nbsp;10s, 11L 1s
| 1L&nbsp;10s, 11L&nbsp;1s
|1096.7123
| 1096.7123
|-
|-
|5L&nbsp;7s
| 5L&nbsp;7s
|704.0956
| 704.0956
|-
|-
|1L&nbsp;11s, 12L 1s
| 1L&nbsp;11s, 12L&nbsp;1s
|1104.8980
| 1104.8980
|-
|-
|6L&nbsp;7s
| 6L&nbsp;7s
|188.0298
| 188.0298
|-
|-
|5L&nbsp;8s
| 5L&nbsp;8s
|465.0841
| 465.0841
|-
|-
|4L&nbsp;9s
| 4L&nbsp;9s
|280.6103
| 280.6103
|-
|-
|3L&nbsp;10s
| 3L&nbsp;10s
|373.0714
| 373.0714
|-
|-
|2L&nbsp;11s
| 2L&nbsp;11s
|557.8535
| 557.8535
|-
|-
|1L&nbsp;12s
| 1L&nbsp;12s
|1111.8816
| 1111.8816
|}
|}


Line 562: Line 569:


{{todo|inline=1|categorize}}
{{todo|inline=1|categorize}}
[[Category:Golden ratio]]
[[Category:Golden ratio]]
[[Category:Tuning]]
[[Category:Tuning]]
Retrieved from "https://en.xen.wiki/w/Golden_sequences_and_tuning"