Editing Harmonic series (music) (section)

== Harmonics and tuning ==
If the harmonics are octave displaced and compressed into the span of one [[octave]], some of them are approximated by the notes of what the [[Western world|West]] has adopted as the chromatic scale based on the fundamental tone. The Western chromatic scale has been modified into twelve equal [[minor second|semitone]]s, which is slightly out of tune with many of the harmonics, especially the 7th, 11th, and 13th harmonics. In the late 1930s, composer [[Paul Hindemith]] ranked musical intervals according to their relative [[consonance and dissonance|dissonance]] based on these and similar harmonic relationships.<ref name="Hindemith">[[Paul Hindemith|Hindemith, Paul]] (1942). [http://noty-naputi.info/sites/default/files/Musical%20Composition-Hindemit.pdf ''The Craft of Musical Composition: Book 1 – Theoretical Part''], pp. 15ff. Translated by [[Arthur Mendel]] (London: Schott & Co; New York: Associated Music Publishers. {{ISBN|0901938300}}). {{Webarchive|url=https://web.archive.org/web/20140701100137/http://noty-naputi.info/sites/default/files/Musical%20Composition-Hindemit.pdf |date=2014-07-01 }}.</ref>

Below is a comparison between the first 31 harmonics and the intervals of [[12-tone equal temperament]] (12TET), octave displaced and compressed into the span of one octave. Tinted fields highlight differences greater than 5 [[Cent (music)|cents]] ({{frac|1|20}} of a semitone), which is the human ear's "[[just noticeable difference]]" for notes played one after the other (smaller differences are noticeable with notes played simultaneously).

{| class="wikitable" align="right"
! colspan="5" | Harmonic
! Interval as a ratio
! Interval in binary
! 12TET interval
! Note
! Variance [[Cent (music)|cents]]
|-
| 1 || 2 ||4 ||8 || 16 || 1, 2 || 1 || prime (octave) || C || 0
|-
| || || || || 17 ||  17/16 (1.0625) || 1.0001 || minor second ||C{{music|#}}, D{{music|b}} || +5
|-
| || || || 9 || 18 || 9/8 (1.125) || 1.001  || major second ||D || +4
|-
| || || || || 19 || 19/16 (1.1875) || 1.0011 || minor third || D{{music|#}}, E{{music|b}} || &minus;2
|-
| || || 5 || 10 || 20 || 5/4 (1.25) || 1.01 || major third || E || bgcolor="yellow" | &minus;14
|-
| || || || || 21 || 21/16 (1.3125) || 1.0101 || fourth || F || bgcolor="yellow" |&minus;29
|-
| || || || 11 || 22 || 11/8 (1.375) || 1.011 || rowspan="2"| tritone || rowspan="2"|F{{music|#}}, G{{music|b}} || bgcolor="yellow" | &minus;49
|-
| || || || || 23 || 23/16 (1.4375) || 1.0111 || bgcolor="yellow" | +28
|-
| || 3 || 6 || 12 || 24 || 3/2 (1.5) || 1.1 || fifth || G || +2
|-
| || || || || 25 || 25/16 (1.5625) || 1.1001 || rowspan="2"| minor sixth || rowspan="2"|G{{music|#}}, A{{music|b}} || bgcolor="yellow" | &minus;27
|-
| || || || 13 || 26 || 13/8 (1.625) || 1.101 || bgcolor="yellow" | +41
|-
| || || || || 27 || 27/16 (1.6875) || 1.1011 || major sixth || A || bgcolor="yellow" | +6
|-
| || || 7 || 14 || 28 || 7/4 (1.75) || 1.11 || rowspan="2"| minor seventh || rowspan="2"|A{{music|#}}, B{{music|b}} || bgcolor="yellow" | &minus;31
|-
| || || || || 29 || 29/16 (1.8125) || 1.1101 || bgcolor="yellow" | +30
|-
| || || || 15 || 30 || 15/8 (1.875) || 1.111 || rowspan="2"| major seventh || rowspan="2"| B || bgcolor="yellow" | &minus;12
|-
| || || || || 31 || 31/16 (1.9375) || 1.1111 || bgcolor="yellow" | +45
|}

The frequencies of the harmonic series, being integer multiples of the fundamental frequency, are naturally related to each other by whole-numbered ratios and small whole-numbered ratios are likely the basis of the consonance of musical intervals (see [[just intonation]]). This objective structure is augmented by psychoacoustic phenomena. For example, a perfect fifth, say 200 and 300&nbsp;Hz (cycles per second), causes a listener to perceive a [[combination tone]] of 100&nbsp;Hz (the difference between 300&nbsp;Hz and 200&nbsp;Hz); that is, an octave below the lower (actual sounding) note. This 100&nbsp;Hz first-order combination tone then interacts with both notes of the interval to produce second-order combination tones of 200 (300&nbsp;&minus;&nbsp;100) and 100 (200&nbsp;&minus;&nbsp;100) Hz and all further nth-order combination tones are all the same, being formed from various subtraction of 100, 200, and 300. When one contrasts this with a dissonant interval such as a [[tritone]] (not tempered) with a frequency ratio of 7:5 one gets, for example, 700&nbsp;&minus;&nbsp;500&nbsp;=&nbsp;200 (1st order combination tone) and 500&nbsp;&minus;&nbsp;200&nbsp;=&nbsp;300 (2nd order). The rest of the combination tones are octaves of 100&nbsp;Hz so the 7:5 interval actually contains four notes: 100&nbsp;Hz (and its octaves), 300&nbsp;Hz, 500&nbsp;Hz and 700&nbsp;Hz. The lowest combination tone (100&nbsp;Hz) is a seventeenth (two octaves and a [[major third]]) below the lower (actual sounding) note of the [[tritone]]. All the intervals succumb to similar analysis as has been demonstrated by [[Paul Hindemith]] in his book ''The Craft of Musical Composition'', although he rejected the use of harmonics from the seventh and beyond.<ref name="Hindemith"/>

The [[Mixolydian mode]] is consonant with the first 10 harmonics of the harmonic series (the 11th harmonic, a tritone, is not in the Mixolydian mode). The [[Ionian mode]] is consonant with only the first 6 harmonics of the series (the seventh harmonic, a minor seventh, is not in the Ionian mode). The [[Rishabhapriya|Rishabhapriya ragam]] is consonant with the first 14 harmonics of the series.