How to Calculate Octaves

[Russell Smith]

Good morning All,

The topic of this morning has prompted me to formulate the following, in regards to answering, “How many ways can you calculate an octave?”

#1: If the octave is a Musical Octave, that is, if the bottom DO doubles to the Top DO, do this:
Multiply the Bottom DO by 9/8, 5/4, 4/3, 3/2, 5/3, and 7/8 to find the RE, MI, FA, SO, LA, and TI of that Musical octave.

#2: If the octave is not Musical, that is, if the bottom DO does NOT double to the top DO, but you know the values of the bottom DO and the top DO, do this:
Step 1 – subtract the bottom DO from the top DO, to discover the LENGTH of the octave. That is, its totality.
Step 2 – to find Re (1/8) divide that length by 8 and add that quotient to the value of the bottom DO.
Step 3 – to find Mi (1/4) divide that length by 4 and add that quotient to the value of the bottom DO.
Step 4 – to find Fa (1/3) divide that length by 3 and add that quotient to the value of the bottom DO.
Step 5 – to find So (1/2) divide that length by 2 and add that quotient to the value of the bottom DO.
Step 6 – to find La (2/3) divide that length by 3, multiply the quotient by 2, and then add that product to the value of the bottom DO.
Step 7 – to find Ti (7/8) divide that length by 8, multiply the quotient by 7, and then add that product to the value of the bottom DO.

#3: If you only know the value of ONE note in some octave, assume that the octave is Musical, and do this:
Ratio the note you are given note to the same note that is found in your objective recipe (divide the value of the note that you are given by the value of the same note that is found in your objective recipe, which is DO=24, RE=27, MI=30, FA=32, SO=36, LA=40, TI=45, and DO=48). Once you discover the ratio, you will have a MULTIPLIER. Then, simply apply that MULTIPLIER to the other notes in your objective recipe to discover what they are. NOTE: if the ratio value of the notes gives you a number that has a recurring decimal, try inverting the notes (that is, divide the value of the same note that is found in your objective into the value of the note that you were given) to see if the ratio of the inverted notes is a number that does not have a recurring decimal; if the answer does not have a recurring decimal, it becomes a DIVISOR instead of a multiplier. Then, simply apply that DIVISOR to the other notes in your objective recipe to discover what the new notes will be.

#4: If you only know the value of ONE interval in some octave, also assume that the octave is Musical, and do this:
Ratio the interval you are given to the same interval that is found your objective recipe (divide the value of the interval that you are given by the value of the same interval that is found in your objective recipe, which is DO-RE=3, RE-MI=3, MI-FA=2, FA-SO=4, SO-LA=4, LA-TI=5, and TI-DO=3. Once you discover the ratio, you will have a MULTIPLIER. Then, to make the octave Musical, simply add the intervals together, the total of the intervals will be the length of the octave, which in Musical octaves, is always the value as the bottom DO … then simply add the intervals, starting at the bottom DO to each other to find the other notes. OR: Once you discover the MULTIPLIER (based of the fact that you are going to make the octave a Musical octave) simple multiply all the numbers that are found in you objective recipe by that Multiplier, and you will have all the answers in the Musical octave you seek to construct. Again NOTE: if the ratio value of the intervals gives you a number that has a recurring decimal, try inverting the intervals (that is, divide the value of the same interval that is found in your objective into the value of the interval that you were given) to see if the ratio of the inverted intervals is a number that does not have a recurring decimal; if the answer does not have a recurring decimal, it becomes a DIVISOR, instead of a multiplier. Then, simply apply that DIVISOR to the other intervals (and to the notes, as stated above) in your objective recipe to discover what they are.

#5: If you are given TWO notes in some octave, do this:
Simply subtract one note from the other note, to reveal the distance between them, and then, ratio that distance to the distance that is found between the same TWO of your objective recipe. Thus, finding the MULTIPLIER (or the DIVISOR, as described in #3 and #4 above). Next, apply the MULTIPLIER (or the DIVISOR, as stated previously) to the INTERVALS that are found in your objective, revealing the INTERVALS in the new proposed octave. Then, simply add or subtract those INTERVALS to or from the notes that you are given.

#6: If you are given ONE note and ONE interval in some octave, trust the INTERVAL … again, trust the INTERVAL, and do this:
Ratio that interval to the same interval in your objective recipe to find the MULTIPLIER (or the DIVISOR) and apply that MULTIPLIER or DIVISOR to the INTERVALS in your objective recipe, revealing the INTERVALS in the new proposed octave. Then, simply add or subtract those INTERVALS to or from the note that you are given.

I think that covers them all.

Love,

Russ

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