Although the word seventh in the name suggests the seventh note in a
scale and the seventh pitch up from the
tonic is indeed used to form a harmonic seventh in a few tuning systems, the harmonic seventh is a pitch relation to the tonic,
not an ordinal note position in a scale. As a pitch relation (968.826
cents up from the reference or tonic note) rather than a scale-position note, a harmonic seventh is produced by different notes in different tuning systems: • In
equal temperament, the harmonic seventh is about 32 cents smaller than the equal tempered
minor seventh. • In 5-limit
just intonation the harmonic 7th is very near
precisely an acute diminished seventh: {{efn| A
just acute diminished seventh is a
just seventh flattened twice (first flat is
min 7, second flat is
dim 7, each just flat lowers the pitch by ) sharpened by a
syntonic comma ("acute") (raises pitch by about ), hence: : = \tfrac{ 15 }{\ 8\ } \times \left( \tfrac{\ 24\ }{ 25 }\right)^2 \times \tfrac{\ 81\ }{ 80 } = \tfrac{ 3^7 }{\ 2\times5^4\ } = \frac{ 2187 }{\ 1250\ } ~. : \operatorname{cents}\!\left( \tfrac{ 7 }{\ 4\ } \right) \quad = 968.826; compare this to : \operatorname{cents}\!\left( \tfrac{ 2187 }{\ 1250\ } \right) = 968.430, only flat. Regardless of how accurately it reproduces the interval of a seventh harmonic, a 5-limit justly intoned
acute diminished seventh is only a theoretical pitch: The pitch's position in the just
tone net is too far separated from its tonic for both to be played together in the same chord without many more notes in the tone network. It is a correctly specified note that does exist among the extended network of
just intonation pitches, but the theoretical note cannot be put to practical use: An acute diminished seventh cannot be reached from its tonic in any feasible justly intoned octave made up of only 12 notes. }} • In multiple slight variations of
quarter comma meantone, the harmonic seventh is accurately rendered by the
augmented sixth interval (rather than a seventh).{{efn| A small modification of meantone – the fifth about one seventh of a comma flat, slightly sharper than exactly one quarter of a comma flat – adjusts the tuning to exactly reproduce the seventh harmonic as an augmented sixth: The adjusted quarter comma uses a fifth that is \ \left( 56 \right)^{1/10}\ , about 696.883musical cents| instead of \ \left( 5 \right)^{1/4}\ , or about 696.578 , used for conventional
quarter comma meantone (which produces pure major thirds by letting fifths fall a quarter-comma flat). }} • In
31 tone equal temperament, the harmonic seventh is quite accurately rendered as 25 steps out of 31 that make up the
octave,{{efn|\ 2^{25/31} \approx 1.749 \approx \tfrac{ 7 }{\ 4\ } which is 967.742 cents so only flat}} while several other
just intervals are as relatively well approximated as they are in
quarter comma meantone. ==In musical practice==