Spreadsheet for calculating chord frequencies *REVISED*

[IaNerd]

NAME: Spreadsheet for calculating chord frequencies


BASIC CONCEPT:
Recently I was retuning a chord harmonica. I found myself getting a little mixed up with "cents". I understand why musicians use them, but the physicist in me just wants to calculate absolute frequencies and be done with it.

What I present below is A DRAFT. I DON'T KNOW IF IT IS CORRECT. I ask for construcvtive input from the experts.

Note that if I got this basically correct, values other than 443 can be substituted. Users can customize this spreadsheet easily.


WHEN/HOW: February of 2018.


LAYOUT/DETAILS:

There are several schools of thought around "just" intonation.

In pretty much all styles of "just", the ratio between the root and perfect fifth is 2:3. This happens to be within 2 cents of where the fifth falls in equal temperament (ET). This makes for an extremely smooth dyad. The only dyad smoother than this would be an octave (ratio 1:2).

In pretty much all styles of "just", the ratio between the root and major third is 4:5. This is about 14 cents flatter than where the major third falls in ET.

When it comes to sevenths, there are several schools of thought. I will here focus on three different ways to define the minor seventh.

[Gnarly]

Boy, I just use a tuner app on my iOS device and my ears.
Numbers are great, but this is more than I have time for.
Have fun tho!

[IaNerd]

BIG NEWS ....

To test the various versions of the minor 7th, I opened up http://onlinetonegenerator.com/ in four separate windows on my computer. Then I entered the follwing values (one per window):

For the root of C: 263 Hz
For the major third (E): 329 Hz
For the perfect fifth (G): 395 Hz

And for the minor seventh, I tried three different values for the fourth window: 474 Hz, 468 Hz and then 461 Hz.

To my ear, one of these tetrads came out MUCH SMOOTHER than the other two. The difference between it and the other two was HUGE.

I urge you to try this. I will reveal my personal favorite after some of you have done so, as replies to this topic.

[IaNerd]

Here is an interesting and well-controlled demonstration of different intonations: https://youtu.be/hC8RPFIaTzU

[IaNerd]

Forum members may message me to request a copy of this Excel document.

[IaNerd]

https://en.m.wikipedia.org/wiki/7-limit_tuning

[karl.nilsson]

Quite clearly in my opinion, the best sounding tetrad of the above is the one where the frequencies are related 4:5:6:7. These are pretty small numbers, which make the harmonics smooth. But look at the end of this discussion for the reason why I think this chord sounds so nice.

From the root to the major third we have 4:5. The ratio between the major third and the fifth is 5:6, which is a minor third. And from the root to the fifth we have 4:6, which is, well, a perfect fifth. So the bottom triad major chord is very smooth and harmonious in all aspects.

Enter the debated (sub)minor 7th: Its relation to the root is 7/4 = 1.75, to the major 3rd it is 7/5 = 1.4, and to the 5th it is 7/6 = 1.1666... .

As you have stated, IaNerd, the 7/4 = 1.75 is the subminor 7th, smaller than both the "lesser minor 7th" (16/9=1.777...) and the "greater minor 7th" (9/5=1.8).

Also, the ratio from the major 3rd to our subminor 7th is 7/5 = 1.4, which is the simplest (and therefore the smoothest sounding) ratio for the infamous tritonus, "the devil's interval".

The last interval here would be from the 5th to our subminor 7th. The ratio is 7/6 = 1.1666..., which is half way between the "greater major 2nd" and the minor 3rd. Not too big numbers in the ratio, though, so should be a well behaved dissonance.

The other (maybe more) important aspect of this tetrad is that, when looking at the frequencies of the individual notes, you will see that the differences between adjacent notes are all equal. For instance, if you start with 440 Hz for the root, the tetrad will look like this: 440 - 550 - 660 - 770 (Hz).

These notes will cause sums and differences products, all of which will be multiples of the smallest difference 110 Hz. Meaning that when this chord is played, you will also hear notes such as 110, 220, 330 Hz, etc. The 2 lowest are "A" notes just as the 440 Hz A, only lower down bass notes, and the 330 Hz is the octave down from our 5th in the tetrad.

To expand this even further you could say that a single note of 110 Hz with many overtones (harmonics), would contain all those notes present in this tetrad discussed here.

[IaNerd]

Wow, Karl! I will study your thoughtful response deeply!

Thank you so much!

[IaNerd]

http://www.flutopedia.com/pitch_to_frequency.htm

[karl.nilsson]

Anyone interested in chords and frequencies mathematics, harmonics, etc. should read this article:

https://en.wikipedia.org/wiki/Harmonic

Here I will present a table, that is similar to one found in the above mentioned article, but here it will show what notes are actually contained in a single note with many harmonics:

Harmonics

overtones.jpg (57.11 KiB) Viewed 17448 times

When playing two or more notes with different pitches, these notes will produce a chord. If several of the (lower) frequencies of the harmonics coincide, they will form a consonance (sweet chord), otherwise a dissonance (harsh chord or even noise).