page_title: Just intonation
section_title: Diatonic scale
hierarchical_section_title: Just intonation / Diatonic scale
caption_attribution_description: English: Just tuned diatonic scale derivation.
context_page_description: In music, just intonation or pure intonation is the tuning of musical intervals as whole number ratios of frequencies. Any interval tuned in this way is called a just interval. Just intervals consist of members of a single harmonic series of a implied fundamental. For example, in the diagram, the notes G and middle C, are both members of the harmonic series of the lowest C and their frequencies will be 3 and 4 times, respectively, the fundamental frequency; thus, their interval ratio will be 4:3. If the frequency of the fundamental is 64 Hertz, the frequencies of the two notes in question would be 192 and 256.
  To use a string as an example, it will simultaneously vibrate the full length of the string, with a nodal point in the middle, with two nodal points dividing the string in three, with three nodal points dividing the string in four, four nodal points dividing the string in five etc. Just intonation involves reproducing these exact pitches so that the resulting combination of frequencies resonate sympathetically, and the intervals have a stability and "ring" to the sound that results from this resonance.
  Instruments are not always tuned using these intervals.
context_section_description: The prominent notes of a given scale may be tuned so that their frequencies form (relatively) small whole number ratios.
  The 5-limit diatonic major scale is tuned in such a way that major triads on the tonic, subdominant, and dominant are tuned in the proportion 4:5:6, and minor triads on the mediant and submediant are tuned in the proportion 10:12:15. Because of the two sizes of wholetone – 9:8 (major wholetone) and 10:9 (minor wholetone) – the supertonic must be microtonally lowered by a syntonic comma to form a pure minor triad.
  5-limit diatonic major scale on C is shown in the table below: (Ptolemy's intense diatonic scale):
  In this example the interval from D up to A would be a wolf fifth with the ratio ​⁴⁰⁄₂₇, about 680 cents, noticeably smaller than the 702 cents of the pure ​³⁄₂ ratio.
  For a justly tuned harmonic minor scale, the mediant is tuned 6:5 and the submediant is tuned 8:5. Natural minor would include a tuning of 9:5 for the subtonic.
caption: Just tuned diatonic scale derivation.[4]
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