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Herz Well Tempered
Fifth Cycle
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Herz difference between
Perfect Fifth & Well Tempered Fifth Cycles
i.e. the 'Pythagorean Comma'
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Pythagorean tuning preserves the mathematically given fractions for the
different 'intervals' within the octave. This makes 'modulating' away from
the home key more and more out of tune. Well Tempered tuning adjusts
for this by making each of the 12 'semi-tone' steps 'equal in the octave'.
The PFC starts with an 'octave' based on 'G-G1' at 100-200 Hz and plots
the Perfect Fifth [PF] 'D' at 150 Hz. It then takes D-D1 based on 150-300
Hz and plots PF 'A' at 225 Hz, and so on . . . The WTFC starts with an
octave based on 'G-G1' at 100-200 Hz and plots the Well Tempered Fifth
[WTF] 'D' at 149.831 Hz. It then takes D-D1 based on 149.831-299.6614
Hz and plots PF 'A' at 224.492 Hz, and so on . . .
When you complete the 12 note fifth-cycle at the these Hz, the Perfect
Fifth Cycle continues through G-G8, going from 100 - 12974.63 Hz as
an 'imperfect fit' with multiples of 100Hz. But when you complete this
12 note cycle at the these Hz, the Well Tempered Fifth Cycle completes
through G-G8 as you go from G 100 to G8 12800 Hz as a 'perfect fit' with
multiples of 100Hz.
The differences between the values for perfect and well-tempered fifths
in the 'fifth cycle' is shown in columns 1 and 3 where the 'values' of the
emerging 'Pythagorean Comma' is shown column 5 of the table.
This is a significant, audible and increasingly obvious difference of pitch
[as it is a very out-of-tune]. This is why 'Well-Tempered' tuning was formalized
[for example Bach's Well-Tempered Clavier - 48 Preludes and
Fugues in all twelve keys both major and minor] in the 18th Century and
has been widely adopted as the basis of all music-making since that time.
Pre-tuning keyboard instruments in what is also called 'Equal Temperament', makes
'modulation' between all keys possible, while remaining 'in-tune' in all
keys. In other words, by slightly 'compressing' the perfect fifth so we
can have twelve exactly equal semi-tone steps per perfect octave, 'well-tempered'
tuning makes this twelve-note 'chromatic' division of that octave
into a perfect internally consistent whole that is consistently 'in-tune'
based at any Hz rates or pitch.
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Some Responses to CBAT so far
Some Support for the Classic C&C Concept over the years.