That distant major sixth

Hi all,

The tables below represent some basic harmonic facts.
I'm sure there's nothing particularly new in this for
most of you. Still, every now and then I find it worth
my while to review basics.

I worked them out yesterday because I wanted to tune
up the Roland MIDI keyoard in JI. The only thing that
I found surprising was that (a good approximation to)
the major sixth doesn't appear in the harmonic series
until the 27th partial. However, something similar to
the tritones F# and Gb, and the minor sixths Ab and G#,
do appear much earlier in the overtone series.

Best viewed in a monospace font; dots added to align
the columns.

Regards,
Yahya
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.....Overtone......................Cents
Note..Integer....Cents.Semitone...adjust
C.1.........1.....0.00........0........0
C.2.........2..1200.00........0........0
G.2.........3..1901.96........7........2
C.3.........4..2400.00........0........0
E.3.........5..2786.31........4......-14
G.3.........6..3101.96........7........2
Bb.3-.......7..3368.83.......10......-31
C.4.........8..3600.00........0........0
D.4.........9..3803.91........2........4
E.4........10..3986.31........4......-14
F#.4-......11..4151.32........6......-49
G.4........12..4301.96........7........2
Ab.4+......13..4440.53........8.......41
Bb.4-......14..4568.83.......10......-31
B.4........15..4688.27.......11......-12
C.5........16..4800.00........0........0
Db.5.......17..4904.96........1........5
D.5........18..5003.91........2........4
Eb.5.......19..5097.51........3.......-2
E.5........20..5186.31........4......-14
F.5-.......21..5270.78........5......-29
F#.5-......22..5351.32........6......-49
Gb.5+......23..5428.27........6.......28
G.5........24..5501.96........7........2
G#.5-......25..5572.63........8......-27
Ab.5+......26..5640.53........8.......41
A.5........27..5705.87........9........6

Harmonic...Note...Semitone..Count
....1.......C........0........1
....3.......G........7........2
....5.......E........4........3
....7.......Bb-.....10........4
....9.......D........2........5
...11.......F#-......6........6
...13.......Ab+......8........7
...15.......B.......11........8
...17.......Db.......1........9
...19.......Eb.......3.......10
...21.......F........5.......11
...23.......Gb+......6.......12
...25.......G#-......8.......13
...27.......A........9.......14

+ or - indicates a change of between an eighth-tone
(25c) and a quarter-tone (50c) sharper or flatter
than the nominal.
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