View source for 833 cents scale

{{Short description|Musical tuning and scale}}
[[File:Golden ratio line.svg|thumb|Line segments in the golden ratio ((A + B) / A = A / B = 1.618).]]
[[File:Stack of golden ratio intervals in Hz.png|thumb|400px|Stack of golden ratio (φ) intervals, measured in Hz ((11.09 + 6.854) ÷ 11.09 = 11.09 ÷ 6.854 = 1.618).]]
[[File:833 cents scale in 36-tet on C one cycle.png|350px|thumb|833 cents scale in 36-tet notation, with slurs indicating a golden ratio]]

The '''833 cents scale''' is a [[musical tuning]] and [[scale (music)|scale]] proposed by [[Heinz Bohlen]]{{clarify|date=November 2018|reason=Later the article claims that other theorists thought of the idea earlier, so what date did Bohlen propose it? The webarchive has a [https://web.archive.org/web/20110722110611/http://www.huygens-fokker.org/bpsite/833cent.html capture] from 2011 that says, "Latest update: February 8, 2008".}} based on [[combination tone]]s, an [[interval (music)|interval]] of 833.09<!--833.090296356741--> [[cent (music)|cents]], and, coincidentally, the [[Fibonacci number|Fibonacci sequence]].<ref name="Experiment">Bohlen, Heinz (last updated 2012<!--The webarchive has a capture from 2011 that says, "Latest update: February 8, 2008".-->). "[http://www.huygens-fokker.org/bpsite/833cent.html An 833 Cents Scale: An experiment on harmony]", ''Huygens-Fokker.org''.</ref> The [[golden ratio]] is <math>\varphi = \frac{1+\sqrt{5}}{2} = 1.6180339887\ldots.</math>, which as a musical interval is 833.09<!--833.0902014...--> cents ({{audio|Golden ratio on C.mid|Play}}). In the 833 cents scale this interval is taken as an alternative to the [[octave]] as the interval of [[octave-repeating scale|repetition]],<ref name="Xen">"[https://en.xen.wiki/w/833_Cent_Golden_Scale_(Bohlen) 833 Cent Golden Scale (Bohlen)]", ''Xenharmonic Wiki''.</ref> however the golden ratio is not regarded as an [[equivalence class (music)|equivalent]] interval (notes 833.09 cents apart are not "the same" in the 833 cents scale the way notes 1200 cents apart are in traditional tunings). Other music theorists such as [[Walter O'Connell]], in his 1993 "The Tonality of the Golden Section",<ref>O'Connell, Walter (1993). "[http://anaphoria.com/oconnell.pdf The Tonality of the Golden Section]", ''Anaphoria.com''.</ref> and Lorne Temes in 1970,<ref>Temes, Lorne, "[https://www.anaphoria.com/temes.pdf Golden Tones]", University of Toronto, 1970. ''Anaphoria.com''</ref> appear to have also created this scale prior to Bohlen's discovery of it.

==Derivation==
[[File:833 cents scale lattice to 4 and -3.png|thumb|Lattice with golden ratios on the unison (center), on the perfect fourth (left), and the perfect fifth (right)]]

Starting with any interval, take the interval produced by the highest original tone and the closest combination tone. Then do the same for that interval. These intervals "[[limit (mathematics)|converge]] to a value close to 833 cents. That means nothing else than that for instance for an interval of 144:89 (833.11 cents) both the summation and the difference tone appear...again 833 cents distance from this interval".<ref name="Experiment"/>

{| class="wikitable" style="text-align:center"
! Base interval
! Closest combination<br/> tone (ratio)
! Closest combination<br/> tone (cents)
|-
| 2:1
| 3:2
| {{audio|help=no|Just perfect fifth on C.mid|701.955}}
|-
| 3:2
| 5:3
| {{audio|help=no|Just major sixth on C.mid|884.359}}
|-
| 5:3
| 8:5
| {{audio|help=no|Just minor sixth on C.mid|813.686}}
|-
| 8:5
| 13:8
| {{audio|help=no|Tridecimal neutral sixth on C.mid|840.528}}
|-
| 13:8
| 21:13
| 830.253
|-
| 21:13
| 34:21
| 834.175
|-
| 34:21
| 55:34
| 832.676
|-
| 55:34
| 89:55
| 833.248
|-
| 89:55
| 144:89
| 833.030
|-
| 144:89
| 233:144
| 833.113
|-
| 233:144
| 377:233
| 833.081
|-
| 377:233
| 610:377
| 833.094
|-
|colspan=3| ...
|}

For example, 220 [[Hertz|Hz]] and 220&nbsp;Hz (unison) produce combination tones at 0 and 440&nbsp;Hz. 440&nbsp;Hz is an octave above 220&nbsp;Hz. 220&nbsp;Hz and [[A440 (pitch standard)|440]] Hz produce combination tones at 220&nbsp;Hz and 660&nbsp;Hz. 660&nbsp;Hz is a perfect fifth (3:2) above 440&nbsp;Hz, and produce combination tones at 220&nbsp;Hz and 1,100&nbsp;Hz. 1,100&nbsp;Hz is a major sixth (5:3) above 660&nbsp;Hz, and produce combination tones at 440&nbsp;Hz and 1,760&nbsp;Hz. 1100&nbsp;Hz and 1760&nbsp;Hz are a minor sixth (8:5), and so on. "It is by the way unimportant which interval we choose as a starting point for the above exercise; the result is always 833 cent."<ref name="Experiment"/>

Once the interval of 833.09 cents is determined, a stack of them is produced:
{| class="wikitable" style="text-align:center"
! Tone
| −5
| −4
| −3
| −2
| −1
| 0
| 1
| 2
| 3
| 4
| 5
|-
! Cents
| 634.55
| 267.64
| 1100.73
| '''733.82'''
| '''366.91'''
| '''0'''
| '''833.09'''
| '''466.18'''
| '''99.27'''
| 932.36
| 565.45
|-
! Ratio
| 1.443
| 1.167
| 1.889
| 1.528
| 1.236
| 1.000
| 1.618
| 1.309
| 1.059
| 1.713
| 1.386
|-
! Scale step
| 
| 
| 
| 6
| 3
| 0
| 7
| 4
| 1
| 8
| 15
|}

Two stacks are also produced on 3:2 and its inverse 4:3 to provide steps 2 and 5, creating a two dimensional [[lattice (music)|lattice]]. Given that the golden ratio is an [[irrational number]] there are three infinite stacks of possible golden ratios which never return exactly back to the unison or octave. Scale step 5 is 597.32 cents and scale step -5 is 602.68 cents (5.37 cents apart).

==Scale==
Bohlen describes a [[symmetric scale|symmetrical]] [[heptatonic scale|seven tone]] scale, with the pitches of steps 0, 1, 3, 4, & 6 derived from the stack of golden ratio intervals. {{audio|833 cents scale on C.mid|Play}}

{| class="wikitable" style="text-align:center"
! Scale step
|colspan=2| 0
|colspan=2| 1
|colspan=2| 2
|colspan=2| 3
|colspan=2| 4
|colspan=2| 5
|colspan=2| 6
|colspan=2| 7
|colspan=2| ...
|-
! Cents
|colspan=2| 00.00
|colspan=2| 99.27
|colspan=2| 235.77
|colspan=2| 366.91
|colspan=2| 466.18
|colspan=2| 597.32
|colspan=2| 733.82
|colspan=2| 833.09
|colspan=2| ...
|-
! Step width
|
|colspan=2| 99.27
|colspan=2| 136.5
|colspan=2| 131.14
|colspan=2| 99.27
|colspan=2| 131.14
|colspan=2| 136.5
|colspan=2| 99.27
|colspan=2| ...
|
|-
! Stacked <math>\varphi</math> tone
|colspan=2| 0
|colspan=2| 3
|colspan=2| 
|colspan=2| −1
|colspan=2| 2
|colspan=2| 
|colspan=2| −2
|colspan=2| 1
|colspan=2| ...
|}

{{Listen
| filename = 833 cents scale Root - GRI - GR "minor" chord.mid
| title    = Trichord, steps: 0, 3, 7<ref name="Xen"/>
| filename2= 833 cents scale Root - GRD - GR "major" chord.mid
| title2   = Trichord, steps: 0, 4, 7<ref name="Xen"/>
| filename3= 833 cents scale Root - GRI - GRD - GR "seventh" chord.mid
| title3   = Tetrachord, steps: 0, 3, 4, 7<ref name="Xen"/>
| plain    = yes
| style    = float:right
}}
[[File:833 cents scale generator circle w P5 and P4 bnw 7.png|thumb|Three stacks of 13 golden ratios on the unison, perfect fourth, and perfect fifth]]

This is comparable to the derivation of the major scale from a [[circle of fifths|stack of perfect fifths]] (FCGDAEB = CDEFGAB). See: [[Generated collection]].

The scale "contains a network of harmonic relationships with the property to match harmonic interval cycles of 833 cents."<ref name="Self-Similarity">Pareyon, Gabriel (2011). ''On Musical Self-Similarity'', p.398. {{ISBN|9789525431322}}.</ref> Steps 2 and 5 were presumably chosen to fill in the gaps between what become steps 1 and 3 and 4 and 6 (267.64 cents). The value of step 2 (235.77) was chosen to create a perfect twelfth (compound perfect fifth) between steps 16 (235.77+833.09+833.09) and step 0, and once chosen determined the value of step 5 due to the symmetry of the scale. Step 10 and 0 form an octave. All notes 7 steps apart form the golden ratio with each other, for example 16 & 9 and 10 & 3.

The repetition of frequencies and the coincidence of higher steps with consonances such as the perfect fifth and octave may be seen (the step number of intervals that coincide with the stack of golden ratios are in bold, while the ratios of repeated intervals are in bold):
{| class="wikitable" style="text-align:center"
! Step
! Ratio
! Ratio<br/>(dec.)
! Ratio<br/>(cents)
! Width<br/>(cents)
! Audio
|-
|rowspan=2| '''0'''
|rowspan=2| {{frac|<math>\varphi</math>|<math>\varphi</math>}}
|rowspan=2| '''1.0000'''
|rowspan=2| '''0'''
| 
|rowspan=2| {{audio-nohelp|Unison on C.mid|Play}}
|-
|rowspan=2| {{overline|99.27}}
|-
|rowspan=2| 1
|rowspan=2| {{frac|<math>\varphi</math>{{sup|4}}|4<math>\varphi</math>}}
|rowspan=2| '''1.0590'''
|rowspan=2| '''99.27'''
|rowspan=2| {{audio-nohelp|833 cents scale step 1.mid|Play}}
|-
|rowspan=2| 136.50
|-
|rowspan=2| 2
|rowspan=2| {{frac|3<math>\varphi</math>|<math>\varphi</math>{{sup|3}}}}
|rowspan=2| 1.1459
|rowspan=2| 235.77
|rowspan=2| {{audio-nohelp|833 cents scale step 2.mid|Play}}
|-
|rowspan=2| 131.14
|-
|rowspan=2| '''3'''
|rowspan=2| {{frac|2<math>\varphi</math>|<math>\varphi</math>{{sup|2}}}}
|rowspan=2| '''1.2361'''
|rowspan=2| '''366.91'''
|rowspan=2| {{audio-nohelp|833 cents scale step 3.mid|Play}}
|-
|rowspan=2| 99.27
|-
|rowspan=2| 4
|rowspan=2| {{frac|<math>\varphi</math>{{sup|3}}|2<math>\varphi</math>}}
|rowspan=2| '''1.3090'''
|rowspan=2| '''466.18'''
|rowspan=2| {{audio-nohelp|833 cents scale step 4.mid|Play}}
|-
|rowspan=2| 131.14
|-
|rowspan=2| 5
|rowspan=2| {{frac|<math>\varphi</math>|{{sfrac|3<math>\varphi</math>|<math>\varphi</math>{{sup|3}}}}}}
|rowspan=2| 1.4120
|rowspan=2| 597.32
|rowspan=2| {{audio-nohelp|833 cents scale step 5.mid|Play}}
|-
|rowspan=2| 136.50
|-
|rowspan=2| '''6'''
|rowspan=2| {{frac|4<math>\varphi</math>|<math>\varphi</math>{{sup|3}}}}
|rowspan=2| 1.5279
|rowspan=2| 733.82
|rowspan=2| {{audio-nohelp|833 cents scale step 6.mid|Play}}
|-
|rowspan=2| {{underline|99.27}}
|-
|rowspan=2| '''7 (0)'''
|rowspan=2| {{frac|<math>\varphi</math>{{sup|2}}|<math>\varphi</math>}}
|rowspan=2| 1.6180
|rowspan=2| 833.09
|rowspan=2| {{audio-nohelp|833 cents scale step 7.mid|Play}}
|-
|rowspan=2| {{overline|99.27}}
|-
|rowspan=2| '''8 (1)'''
|rowspan=2| {{frac|<math>\varphi</math>{{sup|5}}|4<math>\varphi</math>}}
|rowspan=2| 1.7135
|rowspan=2| 932.36
|rowspan=2| {{audio-nohelp|833 cents scale step 8.mid|Play}}
|-
|rowspan=2| 136.50
|-
|rowspan=2| 9 (2)
|rowspan=2| {{frac|3<math>\varphi</math>|<math>\varphi</math>{{sup|2}}}}
|rowspan=2| 1.8541
|rowspan=2| 1,068.86
|rowspan=2| {{audio-nohelp|833 cents scale step 9.mid|Play}}
|-
|rowspan=2| 131.14
|-
|rowspan=2| '''10 (3)'''
|rowspan=2| {{frac|<math>\varphi</math>|<math>\varphi</math>}}
|rowspan=2| '''1.0000'''
|rowspan=2| '''0'''
|rowspan=2| {{audio-nohelp|Perfect octave on C.mid|Play}}
|-
|rowspan=2| 99.27
|-
|rowspan=2| '''11 (4)'''
|rowspan=2| {{frac|<math>\varphi</math>{{sup|4}}|4<math>\varphi</math>}}
|rowspan=2| '''1.0590'''
|rowspan=2| '''99.27'''
|rowspan=2| {{audio-nohelp|833 cents scale step 11.mid|Play}}
|-
|rowspan=2| 131.14
|-
|rowspan=2| 12 (5)
|rowspan=2| {{frac|<math>\varphi</math>|{{sfrac|6<math>\varphi</math>|<math>\varphi</math>{{sup|4}}}}}}
|rowspan=2| 1.1424
|rowspan=2| 230.41
|rowspan=2| {{audio-nohelp|833 cents scale step 12.mid|Play}}
|-
|rowspan=2| 136.50
|-
|rowspan=2| '''13 (6)'''
|rowspan=2| {{frac|2<math>\varphi</math>|<math>\varphi</math>{{sup|2}}}}
|rowspan=2| '''1.2361'''
|rowspan=2| '''366.91'''
|rowspan=2| {{audio-nohelp|833 cents scale step 13.mid|Play}}
|-
|rowspan=2| {{underline|99.27}}
|-
|rowspan=2| '''14 (0)'''
|rowspan=2| {{frac|<math>\varphi</math>{{sup|3}}|2<math>\varphi</math>}}
|rowspan=2| '''1.3090'''
|rowspan=2| '''466.18'''
|rowspan=2| {{audio-nohelp|833 cents scale step 14.mid|Play}}
|-
|rowspan=2| {{overline|99.27}}
|-
|rowspan=2| '''15 (1)'''
|rowspan=2| {{frac|<math>\varphi</math>{{sup|6}}|8<math>\varphi</math>}}
|rowspan=2| 1.3863
|rowspan=2| 565.45
|rowspan=2| {{audio-nohelp|833 cents scale step 15.mid|Play}}
|-
|rowspan=2| 136.50
|-
|rowspan=2| 16 (2)
|rowspan=2| {{frac|3<math>\varphi</math>|2<math>\varphi</math>}}
|rowspan=2| 1.5
|rowspan=2| 701.96
|rowspan=2| {{audio-nohelp|Compound just perfect fifth on C.mid|Play}}
|-
| 
|-
|colspan=6| ...
|}

The scale contains .83333 × 12 steps per octave (≈10).<ref name="Self-Similarity"/> While ideally untempered, the scale may be approximated by [[36 equal temperament]], one advantage being that 36-TET includes traditional 12-TET.<ref name="Xen"/>

==See==
*[[Kepler triangle]]
*[[Zipf's law|Zipf's distribution]]

==Sources==
{{reflist}}

==External links==
*"[https://web.archive.org/web/20071028075125/http://www.ptank.com/catsynth/index.php?m=09&y=06&d=22&entry=entry060922-151816 Fun with Emulator X: Bohlen 833 cents scale and harmonics]", ''CatSynth''.
*"[https://en.xen.wiki/w/Golden_Ratio Golden Ratio]", ''Xenharmonic Wiki''.

{{Musical tuning}}

[[Category:Non–octave-repeating scales]]
[[Category:Golden ratio]]