Info for those interested in pitch changing - Cents and all

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Michael Kazmierski
Posts: 72
Joined: Wed Aug 17, 2016 1:06 am
Operating System: Windows 10

Info for those interested in pitch changing - Cents and all

Post by Michael Kazmierski » Wed Oct 26, 2016 12:27 am

Hi there,
I thought I could offer some help for those of you who are really interested in pitch or speed changing with Audacity. I wasn't sure where to post this since none of the subcategories seemed to fit. I discovered this this past summer, so I'd like to share it with anybody who needs help in this regard.

Converting Frequencies to percentages

Going higher

High divided by low
Result times 100, take percentage
Subtract 100 from the result given

Going lower
Low divided by high
Repeat - result times 100, take percentage
Subtract 100 from the result given

Frequencies of piano notes
Determine the frequency of the A in the octave you choose (e.g. 55, 27.5, 110, etc). It should be the note A, as this is the only value without any ridiculous decimals.
Higher
Multiply by (2root12) to the power of the number of half steps
Lower
Divide by (2root12) to the power of the number of half steps. Do not use negative half steps, this is why we divide. For the Root button on Windows Calculator, it's Control-Y. For the Power button, press Y.

Cents equals 2root1200
50 cents equals 2root24

Frequency ratios to the number of cents or half steps
(high divided by low) Log (automatically displays result when you press Log) divided by 2Log, enter for equals
Take this result and multiply by 12 for half steps, 1200 for cents.
Example: Suppose you want to figure out how many cents it takes for the ratio of 45/32. Here are the steps.
(45/32)Log, you don't do anything and it says "Result: 0.14806253545543770330762243816327" - divide by 2Log to get the log base of 2. When you press Log, it automatically displays the result: 0.30102999566398119521373889472449. Multiply by 12 for half steps (which is 5.9022371559560965293335678086203) or 1200 for cents (which is 590.22371559560965293335678086203). Almost a Devil interval, as a Devil interval is exactly 6 cents.

Does anybody on here play the bagpipes? I do, and so I'd like to tell you what I have discovered regarding cents and ratios. I learned this from a guy called Ewan MacPherson, and I also analyzed the tuning in very old bagpipe recordings. If any of you are pipers I hope you find this very useful.

Relationships of bagpipe chanter notes in both cents and precise fractions to Low A (the tonic note)
Low G: -231.174093530875071069636819321803 (or 7/8)
B: 203.91000173077483548897346547476 (or +3.91000173077483548897346547476 cents from a true equal-tempered major second, or 9/8, this is what D. A. Smith circa 1917 uses unlike many other pipers of his day)
C: 386.31371386483481744438331538727 (or -13.68628613516518255561668461273 cents from a true equal-tempered major third, or 5/4)
D (modern tuning): 498.04499913461258225551326726262 (or -1.95500086538741774448673273738 cents from a true equal-tempered fourth, or 4/3)
E: 701.95500086538741774448673273738 (or +1.95500086538741774448673273738 cents from a true equal-tempered fifth, or 3/2)
F: 884.35871299944739969989658264989 (or -15.64128700055260030010341735011 cents from a true equal-tempered major sixth, or 5/3)
High G: 968.8259064691249289303631806782 (or -31.1740935308750710696368193218 cents from a true equal-tempered minor seventh, or 7/4)

Old tuning relationships (only the notes which have different tuning)
Low G used by D. A. Smith and most other pipers in the record era: -203.91000173077483548897346547476 (or -3.91000173077483548897346547476 cents from a true equal-tempered major-second below, or 8/9) used from about 1975 onwards to the beginning of the bagpipe era. From 1975 to present day, the Low G has been flattening to the point of 7/8.

Low G (Willie Ross): -182.40371213405998195540984991251 (or +17.59628786594001804459015008749 cents from a true equal-tempered major second below, or 9/10)
B: approximately 200 (D. A. Smith uses the 9/8 B)
D: 519.55128873132743578907688282487 (or +519.55128873132743578907688282487 cents from a true equal-tempered fourth, or 27/20)
F used by D. A. Smith: 905.86500259616225323346019821214 (or +5.86500259616225323346019821214 cents from a true equal-tempered major sixth, or 27/16)
High G for most pipers in the record era including Willie Ross: 1049.3629414993692896947489233336 (or +49.3629414993692896947489233336 cents from a true equal-tempered minor seventh, or 11/6)
High G used by Henry Forsyth: 1017.5962878659400180445901500875 (or +17.5962878659400180445901500875 cents from a true equal-tempered minor seventh, or 9/5)
High G used by D. A. Smith: 1088.2687147302222351888700481246 (or -11.7312852697777648111299518754 cents from a true equal-tempered major seventh, or 15/8)
From about 1950 to 1970, the High G tuning changed from ubiquitously 9/5 to 16/9, that is 996.08999826922516451102653452524, or -3.91000173077483548897346547476 from a true equal-tempered minor-seventh. From 1970 onwards, the High G tuning stuck to 7/4.

Octaves higher or lower than a pitch change
Octave higher
50 subtract absolute value of negative number - multiply this result by 2
Octave lower
High value divided by 2. Subtract 50 from the result

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