Some listening tests will use what I call 'inverse just intonation'. In this case, the pure interval will be as far away from just intonation, as equal temperament is from just intonation, but the other way. For example, if the major 3rd in equal temperament is about 1.26, and the JI 3rd is 1.25, then the 'inverse' will be about 1.24. This is included, because in theory, the interval should sound as off-tune from just intonation as equal temperament sounds from Just intonation, since they are equally as far apart.

To convert pitch interval to cents use this formula:

To convert cents to pitch interval, use this formula:

The following intervals were used for the sounds below:

Equal temperament_____ C = 1 (0 cents) ....... E = 1.25992 (400 cents) ..... G = 1.49831 (700 cents)

Just intonation_________ C = 1 (0 cents) ....... E = 1.25 (386.3 cents) ...... G = 1.5 (702 cents)

'Almost' Just intonation___ C = 1 (0 cents) ....... E = 1.251 (387.7 cents) ...... G = 1.502 (704 cents)

'Inverse' temperament___ C = 1 (0 cents) ....... E = 1.2402 (372.7 cents) .... G = 1.502 (704 cents)

Major third in Just intonation (f1 and f1.25) - first 5 harmonics.

Major third in Equal temperament (f1 and f1.25992) - first 5 harmonics.

As you can hear, compared to the JI version, the equal tempered version has a slight tremelo effect (quiet-loud-quiet-loud). However, if we exclude the fourth harmonic from the major third in both cases:

Major third in just intonation (f1 and f1.25) - first 5 harmonics, excluding 4th harmonic in major third.

Major third in equal temperament (f1 and f1.25992) - first 5 harmonics, excluding 4th harmonic in major third.

As you can hear, these two sound much more similar to each other than before! Therefore, it should be much easier to compare the raw pitch of the two diads if we exclude the 4th harmonic of the M3rd in this way.

The below section is taken from my 'diary' in the main Scale page and represents my latest views on the debate between equal temperament versus Just intonation.

An update at last! It's been over 3 years! I'm going to spend more time researching this, especially since I'm using this topic for my dissertation at university. First off, here's a new page with lots of listening tests based on the Major 3rd.

Secondly, I'm beginning to wonder again if two people who hear the same interval may perceive it differently in their 'mind's eye'. If this is the case, then it can take one of two forms:

In both of the cases above, the timbre (or feeling of the beats created by an interval) would remain the same for all people. However, I count this is a separate type of consonance to the consonance type created by the 12 intervals.

Just for the record, that doesn't necessarily mean there's something special about the number 1.25992 (2^(4/12)) for the major third. The thing that's special is the

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