from Pythagorean tuning to equal temperament

If it can still be useful to you, you can set up a spreadsheet using the formula

Semitone-1 = Semitone / twelfth root of 2

or in Excel

=CELL/2^(1/12)

Then copy the formula into the preceding cells

In your case, if you enter the value 40 in cell M1, in L1 you will write =40/2^(1/12) and you will get 37.755 (rounded to the thousandth); then copy the content of L1 into K1 and you will get 35.636 etc.. until you reach A1 where you will have 20 (the upper octave)

hi Zedef, thanks for your input!

forgive me, but... I am not very experienced with excel spreadsheets, so I find it difficult to follow your reasoning! ;-S

just to be clear.... given a long string e.g. 40 cm, assigning it by convention the sound MIDDLE C (C outside the staff, ledger line above, bottom, treble clef), what calculations do I need to make to derive the various string lengths to obtain a chromatic octave (C, C#, D, D#, etc.) in equal temperament?

MIDDLE C = 40cm

C# = ?

D = ?

D# = ?

etc.

I don't understand the wording "Semitone-1 = Semitone/twelfth root of 2"

thanks and have a good evening

ScalaQuaranta

Hi, I am writing you the approximate measurements to the hundredth.

C=40 cm; C#=37.75; D=35.64; Eb=33.64; E=31.75; F=29.97; F#=28.28; G=26.70; G#=25.20; A=23.78; Bb=22.45; B=21.19; C=20

The calculation seems complex but in reality it is trivial. To obtain the semitone above, divide the length of the semitone by the twelfth root of 2. With a spreadsheet it is immediate, otherwise use a calculator 🙂

For example, taking the 40 cm (C) to obtain C#, you need to do 40 divided by the twelfth root of 2 (which is 1.05463 etc..): it comes out to 37.75 and change.

Then from 37.75 divided by the twelfth root of 2, it comes out to 35.64 and change.. etc..

Thanks a million, Zedef, now I understand!

ScalaQuaranta

hi, Zedef, thanking you once again for your previous valuable contribution, I would like to ask you one more thing, still on the same subject.

As far as you know, is there a mathematical formula that allows, by substituting the appropriate parameters each time, the direct calculation of the string length in reference to a precise interval? I will try to be clearer with an example:

"given a free string of a monochord, 40cm long which, when excited, produces a certain base sound that we will call C.

Question: how long must the string be (shortening it with a movable bridge or pressing it with a finger) so that the new sound produced is exactly a PERFECT FIFTH above the initial C? and for a MINOR SEVENTH? and for an AUGMENTED FOURTH?"

I am going by intuition.... I have a feeling that one of the parameters of this hypothetical formula is the interval distance (quantified in semitones) between the initial C and the new sound to be achieved. consequently:

- PERFECT FIFTH: 7 semitones

- MINOR SEVENTH: 11 semitones

- AUGMENTED FOURTH: 6 semitones

then again, I imagine that, being a "general" formula, who knows if it could also calculate BACKWARDS, i.e., also considering DESCENDING intervals and, therefore, also considering only virtually an EXTENSION of the string, instead of a SHORTENING?!

thanks and have a good summer

ScalaQuaranta

Hi, sorry for the delay but I was on vacation. The formula exists, of course. Start from that base which gives you the length of the upper semitone:

=OPENSTRINGLENGTH/2^(1/12) [this is the Excel formulation, in which instead of OPENSTRINGLENGTH you will put the cell with the reference measurement, for example 40].

By replacing the number 1 in the parentheses with the number of semitones you need, you obtain the length.

For example:

for a perfect fifth (7 semitones), the formula is =OPENSTRINGLENGTH/2^(7/12); starting from 40 cm, you get 26.70 (rounded to the centimeter)

for a minor seventh (11 semitones), the formula is =OPENSTRINGLENGTH/2^(11/12); starting from 40 cm, you get 21.19 (rounded to the centimeter).

To calculate the lengths relative to descending semitones, just replace the division (/) with multiplication (*):

=OPENSTRINGLENGTH*2^(1/12).

I will try to attach an image with the calculations. In any case, you can send me a private message so we can talk directly.

hi, Zedef! infinitesimal thanks for the precious information and for your great availability!

have a good summer

ScalaQuaranta

Hello, everyone.

I have a specific question for Zedef, with whom I had opened a discussion that is currently work in progress.

The issue is as follows:

from what I have learned, the mathematical reasoning that leads to the Pythagorean diatonic scale (considering the parameter LENGTH of the vibrating string) is as follows:

1:2 = octave; 2:3 = perfect fifth; 3:4 = perfect fourth

let us consider a vibrating string L long, e.g., 60 cm

C1 = 60 cm

G1 = 60 x 2:3

D1 = 60 x (2:3)^2 x 2

E1 = etc.

F1 = 60 x (2:3)^(-1) x (1:2)
etc.

then, at a certain point, in many of the books read so far on the subject, ratios appear relative to:

- major third: 4:5; - minor third: 5:6; - major sixth: 3:5; - minor sixth: 5:8

my question is: are these latter ratios originated from subsequent mathematical calculations or do they arise from a direct observation derived from listening to the vibrating string, just as it is for the octave and the fifth?

i have read that the WHOLE TONE interval is calculated by subtracting the fourth interval from the fifth interval: (2:3) / (3:4) = 2:3 x 4:3 = 8:9

were the thirds/sixths also calculated with similar reasoning?

thanks

ScalaQuaranta