Just Hammond: Difference between revisions
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This numbering scheme is consistent with the scheme used by [[Bill Sethares]]<ref>Sethares, William A. ''Tuning Timbre Spectrum Scale.'' London: Springer Verlag , 1999.</ref> : “''In general, the n<sup>th</sup> octave contains 2<sup>n-1</sup> pitches''.” | This numbering scheme is consistent with the scheme used by [[Bill Sethares]]<ref>Sethares, William A. ''Tuning Timbre Spectrum Scale.'' London: Springer Verlag , 1999.</ref> : “''In general, the n<sup>th</sup> octave contains 2<sup>n-1</sup> pitches''.” | ||
== Mapping | == Mapping Hammond’s Rational Intervals (cont.): Examples == | ||
The following examples illustrate how to map intervals or chords to the harmonic series. In the first example we map the combination of a Hammond Organ’s note E and a higher note A (a fourth up): | The following examples illustrate how to map intervals or chords to the harmonic series. In the first example we map the combination of a Hammond Organ’s note E and a higher note A (a fourth up): | ||
| Line 279: | Line 279: | ||
| style="text-align: center;" | 9.6 | | style="text-align: center;" | 9.6 | ||
|- | |- | ||
| colspan="3" | | colspan="3" | | ||
Multiply | Multiply --------><br> | ||
to find (D)'s least<br> | to find (D)'s least<br> | ||
common multiple<br> | common multiple<br> | ||
| Line 300: | Line 300: | ||
The resulting interval E-A appears between partial # 206 and partial # 275.<br> | The resulting interval E-A appears between partial # 206 and partial # 275.<br> | ||
The frequency ratio is (275:206), which equals 500.14 cents. | The frequency ratio is (275:206), which equals 500.14 cents. | ||
Amending an upper note B, the next example illustrates how to map the resulting sus4-chord to the harmonic series. | |||
{| class="wikitable" | |||
! rowspan="3" style="text-align: center;" |<br><br><br> | |||
Pitch<br> | |||
Class | |||
! colspan="2" style="text-align: center;" | | |||
HAMMOND<br> | |||
Gear Ratio<br> | |||
(canceled) | |||
! colspan="6" style="text-align: center;" | | |||
Prime<br> | |||
Factorization... | |||
! style="text-align: center;" | | |||
Ascending<br> | |||
Partial Numbers of<br> | |||
an Overtone Scale | |||
! style="text-align: center;" | | |||
Partial found<br> | |||
in Octave | |||
|- | |||
| style="text-align: center;" | (C) | |||
| style="text-align: center;" | (D) | |||
| rowspan="2" colspan="6" style="text-align: center; background-color:#cbcefb;" | | |||
...of Column (D)<br><br><br><br> | |||
| rowspan="2" style="text-align: center;" | | |||
Recalculated<br> | |||
Numerator =<br> | |||
Partial#<br> | |||
P=LCM *(C)/(D) | |||
| rowspan="2" style="text-align: center;" | | |||
Counted<br> | |||
from 1/1:<br> | |||
Octave# =<br> | |||
1+(ln(P)/ln(2)) | |||
|- | |||
| colspan="2" style="text-align: center;" | Fraction | |||
(C)/(D) | |||
|- | |||
| style="text-align: center;" | E | |||
| style="text-align: right; background-color:#ffffff;" | 103 | |||
| style="font-weight:bold; background-color:#ffffff;" | 100 | |||
| style="background-color:#cbcefb;" | 2 | |||
| style="background-color:#cbcefb;" | 2 | |||
| style="background-color:#cbcefb;" | | |||
| style="background-color:#cbcefb;" | 5 | |||
| style="background-color:#cbcefb;" | 5 | |||
| style="background-color:#cbcefb;" | | |||
| style="text-align: center;" | 1442 | |||
| style="text-align: center;" | 11.5 | |||
|- | |||
| style="text-align: center;" | A | |||
| style="text-align: right; background-color:#ffffff;" | 11 | |||
| style="font-weight:bold; background-color:#ffffff;" | 8 | |||
| style="background-color:#cbcefb;" | 2 | |||
| style="background-color:#cbcefb;" | 2 | |||
| style="background-color:#cbcefb;" | 2 | |||
| style="background-color:#cbcefb;" | | |||
| style="background-color:#cbcefb;" | | |||
| style="background-color:#cbcefb;" | | |||
| style="text-align: center;" | 1925 | |||
| style="text-align: center;" | 11.9 | |||
|- | |||
| style="text-align: center;" | B | |||
| style="text-align: right;" | 54 | |||
| style="font-weight:bold;" | 35 | |||
| style="background-color:#cbcefb;" | | |||
| style="background-color:#cbcefb;" | | |||
| style="background-color:#cbcefb;" | | |||
| style="background-color:#cbcefb;" | 5 | |||
| style="background-color:#cbcefb;" | | |||
| style="background-color:#cbcefb;" | 7 | |||
| style="text-align: center;" | 2160 | |||
| style="text-align: center;" | 12.1 | |||
|- | |||
| colspan="3" | | |||
Multiply --------><br> | |||
to find (D)'s least<br> | |||
common multiple<br> | |||
(LCM) | |||
| style="font-weight:bold; background-color:#cbcefb;" | 2<br><br><br><br> | |||
| style="font-weight:bold; background-color:#cbcefb;" | 2<br><br><br><br> | |||
| style="font-weight:bold; background-color:#cbcefb;" | 2<br><br><br><br> | |||
| style="font-weight:bold; background-color:#cbcefb;" | 5<br><br><br><br> | |||
| style="font-weight:bold; background-color:#cbcefb;" | 5<br><br><br><br> | |||
| style="font-weight:bold; background-color:#cbcefb;" | 7<br><br><br><br> | |||
| style="text-align: center; font-weight:bold;" | | |||
1400<br><br> | |||
(LCM)<br><br> | |||
| style="text-align: center;" | | |||
11.4<br><br> | |||
Decimal printed<br> | |||
for orientation only | |||
|} | |||