# Major third intervals in Just Intonation and Equal temperament

If anyone would like me to knock up any more listening tests, please email me giving me the specifications such as timbre (harmonics used, and in what strengths), chord wanted, vibrato etc. It's quite easy for me to create these, so I may put it up immediately.

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:
logBase2(interval) * 1200 ..... (some calculators, you may need to use instead: log(interval) / log(2) * 1200

To convert cents to pitch interval, use this formula:
2 ^ (cents / 1200)

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)

## Other notes:

• "Almost Just intonation" uses fractionally wider intervals than normal JI to create a pleasant phasing effect, whilst still maintaining the characteristic 'JI-ness'.
• Vibrato versions use a 10.4 cent (*/1.006) vibrato depth either way, and different vibrato speed on each pitch.
• In each case, the major third will be played without the fourth harmonic due to first order beating effects. This is a 'problem' with the design of the ear (discrimination areas on basilar membrane not fine enough), and may cause bias towards just intonation if one's not careful. To demonstrate this effect, listen to the following sounds:

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.

## Onto the sound comparisons....

### Saw wave (15 harmonics) - Major 3rd interval

• Equal temperament ..... with vibrato
• Just intonation ..... with vibrato
• Almost Just intonation
• Inverse

### Saw wave (15 harmonics) - Major triad

• Equal temperament ..... with vibrato
• Just intonation ..... with vibrato
• Almost Just intonation
• Inverse

### Octave partials (5 octaves) - Major triad

• Equal temperament
• Just intonation
• Inverse

### Saw wave broken chord (+-3 cent vibrato, 20 harmonics)

• Inverse ..... lower pitch
• Equal temperament ..... lower pitch
• Just intonation ..... lower pitch

### Saw wave broken chord + chord (+-3 cent vibrato, 5 harmonics?)

• Inverse ..... lower pitch
• Equal temperament ..... lower pitch
• Just intonation ..... lower pitch

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.

January 2007
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:

1: They literally hear the interval differently. For example, someone who hears an interval of 390 cents could hear it as 400 cents (or vice versa).

2: They hear the same 'pitch height difference', so in one sense, they are still hearing a '390 cents' interval. However, if we assume that the brain abstractly maps to 12 intervals, and that these intervals are being represented by different neuron clusters in the brain, then it may be the case that they are receiving no minor 3rd neuron pollution, and only major third neurons are being excited. This is in contrast to people like myself, where the 390 cents interval would excite not many, but a few neurons for the minor third as well as the major third, and thus cause interval ambiguity.

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 Major third *sensation* in the mind's eye. It's just that different people will need different input from the outside world to activate this interval as sweetly as possible.