just intonation vs equal temperament

[mastershake]

I just wanted to get all of your thoughts on this

Skytopia : Tuning & Music Scales Theory - Why are there 12 notes in Equal temperament?
(someone's take or opinion on the subject) in favor of equal temperament

Just Intonation vs. Equal Temperament - YouTube
(JI vs ET) in favor of just intonation

It's a bit of info I know. I just thought some of you would find it interesting.

[fdeck]

I consider the 12-tone scale to be a technology -- something that was either developed or discovered to work. It's the simplest scale with good approximations for the lowest order rational intervals.

Why bother with simple? Because it takes effort to construct and tune instruments. Prior to modern times, an instrument had to be practical for artisans to make, and for the musicians themselves to maintain and tune.

Why rational intervals? Because they exploit the interplay of harmonics, and early Western culture latched onto harmony as a desirable component of music.

Could a 19-tone scale have worked? Yes, with considerably more effort devoted to the technology of instruments. With some instruments such as flutes, you run out of fingers.

Could anharmonic music have developed? Sure, we could have developed music based on pure rhythm and no tonality.

I suspect that extensive exposure to the 12-tone system affects the wiring of our brains.

[Stick_Player]

Originally Posted by fdeck

I consider the 12-tone scale to be a technology -- something that was either developed or discovered to work. It's the simplest scale with good approximations for the lowest order rational intervals.

Why bother with simple? Because it takes effort to construct and tune instruments. Prior to modern times, an instrument had to be practical for artisans to make, and for the musicians themselves to maintain and tune.

Why rational intervals? Because they exploit the interplay of harmonics, and early Western culture latched onto harmony as a desirable component of music.

Could a 19-tone scale have worked? Yes, with considerably more effort devoted to the technology of instruments. With some instruments such as flutes, you run out of fingers.

Could anharmonic music have developed? Sure, we could have developed music based on pure rhythm and no tonality.

I suspect that extensive exposure to the 12-tone system affects the wiring of our brains.

I can't say this any better so I'll cut n' paste it:

The twelve-tone equal-tempered scale is the only equal-tempered scale that contains all seven of the basic consonant intervals to a good approximation — within one percent — and contains more consonant intervals than dissonant intervals.

Basic Consonant Intervals
2/1 octave harmonic inverse of 1/1
3/2 perfect fifth harmonic inverse of 4/3
4/3 perfect fourth harmonic inverse of 3/2
5/3 major sixth harmonic inverse of 6/5
5/4 major third harmonic inverse of 8/5
6/5 minor third harmonic inverse of 5/3
8/5 minor sixth harmonic inverse of 5/4

[JimmyM]

Basses is pretty.

[longfinger]

Nice links. The second one sounds a bit like an advertisement.

check out these 5 videos. click the links to get to each one..

Violin Masterclass

[bass12]

Hansford Rowe is my teacher and we've talked quite a bit about just intonation. It's pretty cool to hear the results of intervals and chords played on his just intonation basses and I can see why people would want to explore that realm. For my needs, however, the twelve tone system works just fine.

[James Judson]

Monochord. If you pluck an open E string you get a strong fundamental note and (in theory) an infinite number of harmonics. If the harmonics are analyzed you'll find a third harmonic is also a harmonic of a B note (hmmmm a 1 and a 5 here) at a lesser amplitude. Also a fifth harmonic or a G# (hhmmmm a 1 - 5 - 3). See the pattern here. The whole major scale is being played everytime you play one string (called a MONOCHORD). Now if your instrument is tuned exactly to these 8 notes you have JUST INTONATION. And it sounds very sweet as long as you stay in the key of E. Play in another key and it may sound aweful.

Did you ever tune comparing every string to another string and you strum a chord and it sounds amazing then strum a different chord and yuk!!! That is probably a result of your ear trying to tune to JUST INTONATION.

Along comes Bach and he wants to play an instument in all keys and sound in tune in all keys. His solution is EQUAL TEMPERMENT. All notes are equally out of tune. All keys are equally out of tune. Not out of tune enough to make you go "yuk". But now we can have a piano that plays all keys well etc. etc. etc.

Todays technology (tuners) will have you in EQUAL TEMPERMENT in seconds. Any musician that thinks they can tune by ear better is just flat wrong. Cuz his ear will try to tune to JUST INTONATION and the fight is on. The fight with himself and his instrument that is.

Just my $.02 for why everyone needs to use a tuner.

I will say that ear tuning comes in real handy when you detect a string out of tune in the middle of a song and you simply reach up and put it in tune so you can get thru the rest of the song.

[Stick_Player]

Originally Posted by James Judson

Monochord. If you pluck an open E string you get a strong fundamental note and (in theory) an infinite number of harmonics. If the harmonics are analyzed you'll find a third harmonic is also a harmonic of a B note (hmmmm a 1 and a 5 here) at a lesser amplitude. Also a fifth harmonic or a G# (hhmmmm a 1 - 5 - 3). See the pattern here. The whole major scale is being played everytime you play one string (called a MONOCHORD). Now if your instrument is tuned exactly to these 8 notes you have JUST INTONATION. And it sounds very sweet as long as you stay in the key of E. Play in another key and it may sound aweful.

No. Actually the first 7 overtones plus the fundamental, are found in a Dominant Seventh chord.

Starting with an open E-string (fundamental);
the first overtone is: E;
the second overtone is: B;
the third overtone is: E;
the fourth overtone is: G#;
the fifth overtone is: B;
the sixth overtone is: D;
the seventh overtone is: E;

Continuing:
the eight overtone is: F#;
the ninth overtone is: G#;
the tenth overtone is: A#;
the eleventh overtone is: B;
the twelfth overtone is: C;
the thirteenth overtone is: D;
the fourteenth overtone is: D#;
the fifteenth overtone is: E;

The tenth overtone is very flat, almost exactly in between A and A#. The twelfth overtone is very sharp, not quite in between C and C# - closer to C.

First eight pitches: E, E, B, E, G#, B, D, E - Dominant Seventh

Next four pitches: F#, G#, A#, B - Dominant Seventh with a 9th and +11th

These pitches don't quite create a Major scale, starting on E.

[mambo4]

It's interesting, but unless everyone in your band tunes with it, Just intonation is largely irrelevant. Besides, doesn't just intonation need to be specific to a key? \

I suspect that extensive exposure to the 12-tone system affects the wiring of our brains.

so that'swhat made me this way ...

[fdeck]

Every tuning system is a compromise. Strings are tuned in perfect intervals. Woodwinds follow no strict temperament. Most instruments go sharp when played louder. Players adjust on the fly.

[James Judson]

Originally Posted by Stick_Player

No. Actually the first 7 overtones plus the fundamental, are found in a Dominant Seventh chord.

Starting with an open E-string (fundamental);
the first overtone is: E;
the second overtone is: B;
the third overtone is: E;
the fourth overtone is: G#;
the fifth overtone is: B;
the sixth overtone is: D;
the seventh overtone is: E;

Continuing:
the eight overtone is: F#;
the ninth overtone is: G#;
the tenth overtone is: A#;
the eleventh overtone is: B;
the twelfth overtone is: C;
the thirteenth overtone is: D;
the fourteenth overtone is: D#;
the fifteenth overtone is: E;

The tenth overtone is very flat, almost exactly in between A and A#. The twelfth overtone is very sharp, not quite in between C and C# - closer to C.

First eight pitches: E, E, B, E, G#, B, D, E - Dominant Seventh

Next four pitches: F#, G#, A#, B - Dominant Seventh with a 9th and +11th

These pitches don't quite create a Major scale, starting on E.

Thanks for the correction. Looking to prove you right or me wrong I could not verify your statement but I did find three things I got wrong.

1.) Monochord is a one string instrument (I thought it was an explanation of overtones generated by one string).

2.) One string does not ring the whole major scale (but close)

3.) Some guy named Andres Werkmeister invented equal temperament 1690. (not Bach)

So lets back up to where we agree.

1st harmonic E Tonic
2nd harmonic E Octave Tonic
3rd harmonic B Dominant
4th harmonic E Tonic
5th harmonic G# Mediant

Its easy to see the major triad here and if you calculate the supertonic, subdominant, submediant and leading tone (the rest of the major scale) you are now tuned to JUST INTONATON. Great as long as you stay in the key of E. Here is the break out if you will. Key of A now.

JUST INTONATION
A B C D E F G A
Tonic Supertonic Mediant Submediant Dominant Submediant Leading tone Tonic
HZ 440 495 550 586 660 733 825 880
Ratio 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1
Intval 0 204 386 498 702 884 1088 1200

EQUAL TEMPURAMENT
A B C D E F G A
Tonic Supertonic Mediant Submediant Dominant Submediant Leading tone Tonic
HZ 440 493 554 587 659 734 830 880
Ratio 1 1.1225 1.2599 1.3348 1.4983 1.6818 1.8877 2
Intval 0 200 400 500 700 900 1100 1200

Tlhe easy explaination is. Just Intonation is tuning using the math on frequencies. Equal Temperament is tuning using the math on all notes evenly spaced.

[Stick_Player]

Originally Posted by James Judson

Thanks for the correction. Looking to prove you right or me wrong I could not verify your statement but I did find three things I got wrong.

I wasn't trying to disprove anyone...

Here's an illustration of the overtone series based off of a Low C (33Hz):



An interesting demonstration using the above ratios (1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1).

[projectMalamute]

Originally Posted by James Judson

Thanks for the correction. Looking to prove you right or me wrong I could not verify your statement but I did find three things I got wrong.

1.) Monochord is a one string instrument (I thought it was an explanation of overtones generated by one string).

2.) One string does not ring the whole major scale (but close)

3.) Some guy named Andres Werkmeister invented equal temperament 1690. (not Bach)

So lets back up to where we agree.

1st harmonic E Tonic
2nd harmonic E Octave Tonic
3rd harmonic B Dominant
4th harmonic E Tonic
5th harmonic G# Mediant

Its easy to see the major triad here and if you calculate the supertonic, subdominant, submediant and leading tone (the rest of the major scale) you are now tuned to JUST INTONATON. Great as long as you stay in the key of E. Here is the break out if you will. Key of A now.

JUST INTONATION
A B C D E F G A
Tonic Supertonic Mediant Submediant Dominant Submediant Leading tone Tonic
HZ 440 495 550 586 660 733 825 880
Ratio 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1
Intval 0 204 386 498 702 884 1088 1200

EQUAL TEMPURAMENT
A B C D E F G A
Tonic Supertonic Mediant Submediant Dominant Submediant Leading tone Tonic
HZ 440 493 554 587 659 734 830 880
Ratio 1 1.1225 1.2599 1.3348 1.4983 1.6818 1.8877 2
Intval 0 200 400 500 700 900 1100 1200

Tlhe easy explaination is. Just Intonation is tuning using the math on frequencies. Equal Temperament is tuning using the math on all notes evenly spaced.

What you have constructed there is a just tuning. One of many, and not one that is particularly common or useful for tonal music. First problem being it only has 7 notes. There are few different ways to fill in the other 5 but none of them do a very good job of solving the second problem with that tuning: almost any other key is not just a little but a lot out of tune.

There are a lot of different just tunings with different tradeoffs depending on what you are trying to do.

A better explanation is that just tunings use frequencies that are integer ratios of each other. Equal tempered tunings do not.

[blakelock]

this is an awesome thread. i was completely unaware of just vs equal temperaments but for a while i had been thinking about "natural" scale forms based on mathematical or natural series. how about:

- somehthing like a taylor series: start with root note C, then add C/2, C/3, C/4.....this would be an infinite scale with most notes bunched up at one end.

- fibonacci sequence where each number equals the sum of the 2 previous numbers: 0, 1, 1, 2, 3, 5, 8, 13, ...you could invert each and add it to the root note. again notes would get bunched up.

- prime numbers, etc.......

and with all these, we have to think about how measure frequency. how about instead of "oscillations/sec", we chose a more "natural" time scale like "oscillations/heartbeat" or "oscillations/day"?

thought? are there simple computer tools that can generate notes of a defined frequency? then we could create these scales and see (hear) how they sound.

[Stealth]

Very interesting thread - I hadn't thought much about JI vs. ET until now.

blakelock - the Taylor series approach would make sense, insofar that human hearing (when it comes to pitch) behaves logarithmically, so they wouldn't get as bunched up as you think.

[mastershake]

found another interesting article Twelve-Tone Musical Scale -in defence of equal temperment over just intonation the article says
" Although scales with many tones have many nearly pure intervals that are consonant (ratios of small integers), they have even more intervals that are dissonant (not ratios of small integers). In contrast, the small twelve-tone equal-tempered scale has more consonant intervals (seven) than dissonant intervals (five). We observe:

The twelve-tone equal-tempered scale is the only equal-tempered scale that contains all seven of the basic consonant intervals to a good approximation — within one percent — and contains more consonant intervals than dissonant intervals.

Also, scales with many tones are too large to be really practical: a keyboard with the same range as a piano would be huge.

In summary, the twelve-tone equal-tempered scale is probably the best compromise of all possible scales, and that is why it is now standard in the Western world and common all over the world."

[Clef_de_fa]

I had an art history class during my college degree in music and at one point we talked about that.

The thing is if you wanted to play something in Db on a piano you couldn't play in any other key because the intervals between notes isn't right. So in short you had to have a piano for each key ...

So went they re-discovered what Pythagore found, well they adopted it and it opened music for a lot of people. Until they had the technology for the faster hammer recovery in a piano ... which opened the door for average player to play music only for the extrem talented player.

[FourIsEnough]

Another case for Bach's genius.

[Russell L]

Bach woulda loved my Snark.

[Chris K]

If you want just intonation, pick up a fretless.