One way to read the table posted by Karl:
In ET, the major third would be 400 cents (or 4 semitones) above the tonic (zero). But in JI, it is only 386 cents above the tonic; therefore, 14 cents flatter, or "offset" by 14 cents. A "compromise" tuning might flatten a major third only half that amount, e.g. 7 cents.
In ET, the perfect fifth would be 700 cents (or 7 semitones) above the tonic (zero). But in JI, it is 702 cents above the tonic; therefore, 2 cents sharper, or "offset" by +2 cents. So the major fifths of ET, JI and compromise tuning are nearly identical.
Spreadsheet for calculating chord frequencies *REVISED*
Re: Spreadsheet for calculating chord frequencies *REVISED*
Clearly, our auditory system can distinguish and appreciate polyphony that is built on certain ratios of integers.
Karl said: "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"."
7/5 is a fine fraction, and I should probably just let it go at that. But 7/5 is also darn close to the square root of the smallest prime number. So I have to wonder if we can also distinguish and perceive square roots.
Karl said: "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"."
7/5 is a fine fraction, and I should probably just let it go at that. But 7/5 is also darn close to the square root of the smallest prime number. So I have to wonder if we can also distinguish and perceive square roots.

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Re: Spreadsheet for calculating chord frequencies *REVISED*
Update: Changed the base frequency to 100 (from 440) for easier reading.
Let's look at the frequencies of the notes involved in the tetrad 4:5:6:7 discussed above. Here is a table with the example frequencies 100, 125, 150 and 175 Hz.
Each note has a fundamental (e.g. 100 Hz). This note has overtones/harmonics, (e.g. 200, 300 Hz etc.). So we have four series of harmonics. Some of the harmonics coincide between notes. In the table, coinciding frequencies are color coded. For example, if there is a coincidence between frequencies in column 4 and 7 they are marked red, between 5 and 7 they are blue, etc. Only in one case there is a coincidence between 3 columns, at 1500 Hz (tricolor). Note that only the first 4 octaves are displayed of each note's frequency spectrum. Going further up there would of course be more coincidences.
Generally, more coincidences, especially in the lower part of the spectrum, will make the chord sound more pleasing to the ear. This is another way of looking at the reasons why it sounds sweet.
Let's look at the frequencies of the notes involved in the tetrad 4:5:6:7 discussed above. Here is a table with the example frequencies 100, 125, 150 and 175 Hz.
Each note has a fundamental (e.g. 100 Hz). This note has overtones/harmonics, (e.g. 200, 300 Hz etc.). So we have four series of harmonics. Some of the harmonics coincide between notes. In the table, coinciding frequencies are color coded. For example, if there is a coincidence between frequencies in column 4 and 7 they are marked red, between 5 and 7 they are blue, etc. Only in one case there is a coincidence between 3 columns, at 1500 Hz (tricolor). Note that only the first 4 octaves are displayed of each note's frequency spectrum. Going further up there would of course be more coincidences.
Generally, more coincidences, especially in the lower part of the spectrum, will make the chord sound more pleasing to the ear. This is another way of looking at the reasons why it sounds sweet.
Last edited by karl.nilsson on Tue Feb 20, 2018 3:00 pm, edited 2 times in total.
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Re: Spreadsheet for calculating chord frequencies *REVISED*
So, what conclusions can you draw from this table? Well, first of all, the 3 lower notes cause a major triad: root, major third and fifth. They sound very sweet together; they are a base in all Western music. All intervals are consonances. There are several frequency coincidences between these three notes, and the ratios between base frequencies use small numbers (3/2, 5/4, 6/5).
Now, the 7th, it doesn't sound that great with any of the other present notes, even when fine tuned. The 7th makes a dissonance with any of the other 3 notes. At most, there is one frequency coincidence with either note in any given octave, and ratios use higher numbers (7/4, 7/5, 7/6).
But it's the dissonances that are giving interest and spice to the other consonances. And with one frequency per octave in common to either one of the other notes, the whole chord is kept together, while still showing a desire to be resolved into something else.
Now, the 7th, it doesn't sound that great with any of the other present notes, even when fine tuned. The 7th makes a dissonance with any of the other 3 notes. At most, there is one frequency coincidence with either note in any given octave, and ratios use higher numbers (7/4, 7/5, 7/6).
But it's the dissonances that are giving interest and spice to the other consonances. And with one frequency per octave in common to either one of the other notes, the whole chord is kept together, while still showing a desire to be resolved into something else.
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Re: Spreadsheet for calculating chord frequencies *REVISED*
I want to add that by consonance and dissonance I mean in the relative sense such as mentioned in this very interesting article:
https://en.wikipedia.org/wiki/Consonance_and_dissonance
We must also remember that few musical intruments produce perfectly harmonic overtones. So depending on the instrument, the quality of harmonies may differ.
https://en.wikipedia.org/wiki/Consonance_and_dissonance
We must also remember that few musical intruments produce perfectly harmonic overtones. So depending on the instrument, the quality of harmonies may differ.
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