View source for Nearest just interval

An irrational interval or ratio of frequencies given by a real number r has an infinite list of ''nearest just intervals''; if r is rational, the list is finite, terminating in r. For arbitrary (including negative) real numbers this corresponds to what number theorists call ''best rational approximations''. A ratio of integers p/q with q &gt; 0 and p and q relatively prime is a best rational approximation if there is no ratio m/n with n &lt; q which is a better approximation to r. If r is an interval of music it is positive, and both p and q are positive. Note that a nearest just interval is not necessarily nearest in logarithmic terms; 4/3 and 3/2 are the same distance in cents from √2 = 600 cents, but |4/3 - √2| = .08088 whereas |3/2 - √2| = 0.08479, which is larger.

Best rational approximations also arise in music theory logarithmically, as the best rational approximations to the logarithm base two of some number such as 3/2 or ∜5 is often of interest.

The [http://en.wikipedia.org/wiki/Continued_fraction#Semiconvergents semiconvergents] of the continued fraction for r include all of the best rational approximations. The convergents are equivalent with a stronger notion of best approximation, namely [http://en.wikipedia.org/wiki/Continued_fraction#Best_rational_approximations best relative approximation]. Here it is required that |qr - p| is less than |nr - m| for any n &lt; q.

== Examples ==

=== Approximations for Ratios (of Pure Intervals) ===
The best rational approximations to log2(3/2) define edos which have especially good approximations to the fifth (701.955000865... [[cent|cents]]):

{| class="wikitable"
|-
| | '''Step\EDO'''
| | '''log([[Tenney_Height|Tenney Height]])'''
| | '''size''' in cents
| | '''"error"''' in cents
|-
| | ...
| | ...
| | ...
| | ...
|-
| style="text-align:center;" | 1 \ 1
| | 0.0
| style="text-align:center;" | 1200.0
| style="text-align:center;" | 498.04
|-
| style="text-align:center;" | 1 \ 2
| | 1.0
| style="text-align:center;" | 600.00
| style="text-align:center;" | -101.96
|-
| style="text-align:center;" | 2 \ 3
| | 2.585
| style="text-align:center;" | 800.00
| style="text-align:center;" | 98.045
|-
| style="text-align:center;" | 3 \ [[5edo|5]]
| | 3.907
| style="text-align:center;" | 720.00
| style="text-align:center;" | 18.045
|-
| style="text-align:center;" | 4 \ [[7edo|7]]
| | 4.807
| style="text-align:center;" | 685.7143
| style="text-align:center;" | -16.2407
|-
| style="text-align:center;" | 7 \ [[12edo|12]]
| | 6.392
| style="text-align:center;" | 700.00
| style="text-align:center;" | -1.955
|-
| style="text-align:center;" | 17 \ [[29edo|29]]
| | 8.945
| style="text-align:center;" | 703.4483
| style="text-align:center;" | 1.4933
|-
| style="text-align:center;" | 24 \ [[41edo|41]]
| | 9.943
| style="text-align:center;" | 702.43902
| style="text-align:center;" | 0.48402
|-
| style="text-align:center;" | 31 \ [[53edo|53]]
| | 10.682
| style="text-align:center;" | 701.88679
| style="text-align:center;" | -0.06821
|}

<ul><li>''for approximations of the harmonic seventh see [[7/4#Approximations|7_4]]''</li></ul>

=== Approximation for Logarihmic Measures ===
The 600-cent interval sqrt(2) (6 steps of [[12edo|12edo]], "Tritone") approximates following ratios:

{| class="wikitable"
|-
| | '''freq. ratio'''
| | '''log2([[Tenney_Height|Tenney Height]])'''
| | '''size''' in cents
| | '''"error"''' in cents
|-
| | ...
| | ...
| | ...
| | ...
|-
| style="text-align:center;" | 1 / 1
| style="text-align:center;" | 0.0
| style="text-align:center;" | 0.0
| style="text-align:center;" | 600.0
|-
| style="text-align:center;" | [[3/2|3 / 2]]
| style="text-align:center;" | 2.585
| style="text-align:center;" | 701.96
| style="text-align:center;" | 101.96
|-
| style="text-align:center;" | [[4/3|4 / 3]]
| style="text-align:center;" | 3.585
| style="text-align:center;" | 498.04
| style="text-align:center;" | -101.96
|-
| style="text-align:center;" | [[7/5|7 / 5]]
| style="text-align:center;" | 5.129
| style="text-align:center;" | 582.51
| style="text-align:center;" | -17.49
|-
| style="text-align:center;" | [[17/12|17 / 12]]
| style="text-align:center;" | 7.672
| style="text-align:center;" | 603.000
| style="text-align:center;" | 3.000
|-
| style="text-align:center;" | 24 / 17
| | 
| style="text-align:center;" | 597.000
| style="text-align:center;" | -3.000
|-
| style="text-align:center;" | 99 / 70
| | 
| style="text-align:center;" | 600.0883
| style="text-align:center;" | 0.0883
|-
| style="text-align:center;" | 140 / 99
| | 
| style="text-align:center;" | 599.9117
| style="text-align:center;" | -0.0883
|-
| | ...
| | ...
| | ...
| | ...
|}

The 300-cent interval 2^(1/4) (3 steps of [[12edo|12edo]], "minor third") approximates following ratios:

{| class="wikitable"
|-
| | '''freq. ratio'''
| | '''log([[Tenney_Height|Tenney Height]])'''
| | '''size''' in cents
| | '''"error"''' in cents
|-
| | ...
| | ...
| | ...
| | ...
|-
| style="text-align:center;" | 1 / 1
| style="text-align:center;" | 0.0
| style="text-align:center;" | 0.0
| style="text-align:center;" | 300.0
|-
| style="text-align:center;" | [[6/5|6 / 5]]
| style="text-align:center;" | 4.907
| style="text-align:center;" | 315.64
| style="text-align:center;" | 15.64
|-
| style="text-align:center;" | [[13/11|13 / 11]]
| style="text-align:center;" | 7.160
| style="text-align:center;" | 289.21
| style="text-align:center;" | -10.79
|-
| style="text-align:center;" | [[19/16|19 / 16]]
| style="text-align:center;" | 8.248
| style="text-align:center;" | 297.51
| style="text-align:center;" | -2.49
|-
| style="text-align:center;" | [[25/21|25 / 21]]
| style="text-align:center;" | 9.036
| style="text-align:center;" | 301.84
| style="text-align:center;" | 1.84
|-
| | ...
| | ...
| | ...
| | ...
|}

[[Category:Elementary math]]
[[Category:Approximation]]
[[Category:Just intonation]]

{{todo|add examples}}