escaping from the equally tempered scale

So, usually the 12 semi-tones in the western octave are 'standardised' so that they are exactly the same distance apart from each other (in terms of frequency ratio). This is called Equal Temperament and the result is the equally tempered 12 note scale.

I'm assuming that MIDI and nearly all music software uses this scale, but there are some disadvantages to it. Its good for modulation and transposition and so on, but most of the intervals become approximations to what they could be if played purely. See this table to see the percentage error.

An alternative is to tune the instruments to Just Intonation, which will optimise for 'perfect' intervals from a specific base note, but will make some related intervals between the other notes sound all wrong.

So my question is, has anyone tried this, what are the cent tuning values needed to do it with synths, is it worth the effort (is the result noticable to the ear), and so on.

i've not tried but i've been planning on trying for some time - new version of modplugtracker will have resettable tunings per note so any tunings should be possible with a bit of typing-in-numbers effort, just need to wait and have a go when its out.

i have no idea if it sounds better but i find it totally fucked that certain classical pieces are always performed with different tunings from how they were written - how can they possibly justify that? if someone composes something for a certain tuning then moving it around to others totally fucks it - wrong wrong wrong!

i look forward to having a go and will try and do something to compare tunings if i can be bothered.

If your crafty with midi you could route the midi signal so that certain midi notes have certain pitch-bend values transmitted along with it.

cubase has loads of preset different tunings available (including the carlos ones and some wacky shit) but i never use it to compose with so not experimented properly. i think most decent progs have some method of changing the tunings of the midi stuff somewhere under the hood. i'm surprised the tracker has taken so long to have it implemented because i think every note value is stored in a table somewhere and its not too hard to change them about.

max/msp imo

some resources:

http://www.microtonal.co.uk/

http://www.microtonal-synthesis.com/

the wiki is a nice hub: http://en.wikipedia.org/wiki/Microtonal_music

http://www.microtonal.org/resources.html

http://www.xs4all.nl/~huygensf/english/index.html

i don't know if it's worth the effort for you but it sounds like an interesting idea. The trouble with unequally tempered instruments is that unless you're playing in the the key of the scale certain intervals will sound wrong because they'll be out by a quarter tone or so, this can be very grating on the ear but i think it sounds really cool sometimes, you can end up with tuning that sounds like the dulcimer style sound on fenixfunk 5, noticeably out of tune but still great.

Sounds tempting, but it sounds like a lot of work in order to get anything to sound any good. However I'm building an acoustic instrument with a friend, maybe I'll suggest that we keep it fretless for microtone use, should be interesting.

  zazen said:

the 12 semi-tones in the western octave are 'standardised' so that they are exactly the same distance apart from each other (in terms of frequency ratio).

Not completely true.

  TechDiff said:

  zazen said:

the 12 semi-tones in the western octave are 'standardised' so that they are exactly the same distance apart from each other (in terms of frequency ratio).

 

Not completely true.

I thought that was right - how are they arranged then?

Can't remember exactly why but the perfect logarythmic ratios only work well for a major scale. Any accidentals and jesus christ does it soud like shit. The basic tuning scale came out of compromise so that all the notes sounded roughly OK without making some notes utterly unusable.

If memory serves.

I dont think thats true at all. the logarythmic ratio is what most intruments use today and it sounds fine to me. the ratio between c and c# is the same as between e and f. Im not really sure what your saying, but maybe an elaboration is required. I think you have it backwards that in the just intonation scale the tunings were mathematicaly pleasant in ratio..... taking thirds of strings, etc. but the accidentals had to be fudged and there were many different systems before the straitforeward equal temperment. The nice thing about just intonation is the the natural harmonics were more apparent (less than a percent more accurate) in the difference between notes (see the table in the first post), so two of the same instuments would have a more pure harmony together, i guess. The pythagorean sort of experiments with math and music.

absynth allows you to change your tuning, and has a bunch of preset tunings built in

Yes, there is a format for tunings. forget what its called. But there are a few soft-synth which load it. And somebody just released a synthedit module for it.

mm, its not difficult to do or use. as someone said before theres a cubase midi effect which will do all the necassary math, and to use it you just have to ensure you are composing in only the key you're just-intonation is set up for!

listening to the examples on the wiki page in the first post, I actually found the compromise (equal temp) slightly more pleasing, aqs i guess thats what were all used to.

never tried to write using it mind.

  zazen said:

  TechDiff said:

  zazen said:

the 12 semi-tones in the western octave are 'standardised' so that they are exactly the same distance apart from each other (in terms of frequency ratio).

 

Not completely true.

 

 

I thought that was right - how are they arranged then?

yeah its not completely true;

an interval is the proportion between two frequencies, right? and if you go up one octave, you double the frequency, right? (if you go up one octave from the note A3, which is 440hz, you'll find that A4 is 880hz). so the proportion of an octave is 2:1.

now, if you divide the octave into 12 equal intervals, the frequency factor (of each of those intervals) to the 12th power must be 2. so if that frequency factor would be called 'ff' --> ff^12 = 2, and subsequently, ff = 2^(1/12), which is also the twelveth root of 2. which is something like 1.0594631...

so, the frequency ratio between each semitone (each interval) is 2^(1/12) = 1.0594631. now lets take take an interval like a third; it should have an frequency ratio of 5:4, which is 1.25, right? but if you'd take ff^4, you'd get something like 1.259921...

same for a fifth: it should have a frequency ratio of 3:2 (1.5) but when you'd calculate ff^7, you'd find 1.4983071....

that's why the equal temperament isn't really equally tempered ;)

  iep said:

  zazen said:

  TechDiff said:

  zazen said:

the 12 semi-tones in the western octave are 'standardised' so that they are exactly the same distance apart from each other (in terms of frequency ratio).

 

Not completely true.

 

 

I thought that was right - how are they arranged then?

 

yeah its not completely true;

 

an interval is the proportion between two frequencies, right? and if you go up one octave, you double the frequency, right? (if you go up one octave from the note A3, which is 440hz, you'll find that A4 is 880hz). so the proportion of an octave is 2:1.

 

now, if you divide the octave into 12 equal intervals, the frequency factor (of each of those intervals) to the 12th power must be 2. so if that frequency factor would be called 'ff' --> ff^12 = 2, and subsequently, ff = 2^(1/12), which is also the twelveth root of 2. which is something like 1.0594631...

 

so, the frequency ratio between each semitone (each interval) is 2^(1/12) = 1.0594631. now lets take take an interval like a third; it should have an frequency ratio of 5:4, which is 1.25, right? but if you'd take ff^4, you'd get something like 1.259921...

same for a fifth: it should have a frequency ratio of 3:2 (1.5) but when you'd calculate ff^7, you'd find 1.4983071....

 

that's why the equal temperament isn't really equally tempered ;)

Well put. I think that originally it was slightly rounded off for the exact pupose of avoiding having hundreds of notes in one scale. The intervals where rounded down ever so slightly to maintain a tidy octave. I saw it described it using the cycle of fifths as an example. A rotation throgh every whole note through fifth intervals, so If you started on c you'd get c,g,d,a,e,b,f and back to c. However, if you calculated this same pattern you wouldnt end up at c but somewhere around C and 1/32 of a sharp. Obviously pointless because it would take you another 31 cycles to get back to a c which is an exact harmonic relation of your starting note. So instead the cycle of fifths was rounded down ever so slightly to reduce the number of notes per octave.

Can you imagine what a piano would look like? You'd need a stool with wheels to play it.

  Quote

[ A rotation throgh every whole note through fifth intervals, so If you started on c you'd get c,g,d,a,e,b,f and back to c. However, if you calculated this same pattern you wouldnt end up at c but somewhere around C and 1/32 of a sharp. Obviously pointless because it would take you another 31 cycles to get back to a c which is an exact harmonic relation of your starting note. So instead the cycle of fifths was rounded down ever so slightly to reduce the number of notes per octave.

 

Can you imagine what a piano would look like? You'd need a stool with wheels to play it.

...amm no, b to f is an augmented 4th, it would go F#, C#, G#, D#, A#, E# (F natural enharmonic) then back to C.

Anyway the concept of the equal tempered instrument came from a need for modulation, not to save space on the keyboard although the idea of a keyboard being tuned to accomodate every interval contained in the harmonic series of each note is, at best impractical, at worst idiotic. Because just intonation is an accurate representation of the harmonic series it doesn't lend itself to moving between the different keys, each of which has it's own specific harmonic series which converges as you move further and further up (i think, this can be heard on a synthesizer when you put the resonance knob up full and gradually change the cutoff), so a "line of best fit" was invented where the intervals are as accurate an average as possible. The modern piano, therefore, is one of the most out-of-tune instruments in the world, the octaves are accurate and the fifths too i believe, but some other intervals in between are either too wide or too narrow, the major 3rd being an obvious example of the former.

  oxiactionmax said:

...amm no, b to f is an augmented 4th, it would go F#, C#, G#, D#, A#, E# (F natural enharmonic) then back to C.

woops. Yep your right, my bad.

This makes me wonder though. Do electronic tuners for guitars etc take these alterations into acount or do they measure the intervals in a purely mathematic sence? What I mean is, if you tuned a piano using an electronic tuner note by note without comparison between notes. Once every note had been tuned, would the whole thing sound out of tune? Ive had it happen when Ive been tuning my guitar with a digital tuner string by string and although every string is exactly in tune, when you play a chord it sounds wrong. It only sounds right if I do it by ear. Its crazy if you think about it. Our brains are used to imperfections. Music which is bassed on a mathematicaly accurate tuning sounds out of tune, yet music bassed on an imperfect scale structure sounds in tune.

I use Indian scales.

  TechDiff said:

Ive had it happen when Ive been tuning my guitar with a digital tuner string by string and although every string is exactly in tune, when you play a chord it sounds wrong.

You need a new set of strings, methinks. String intonation gets well fucked as they get older and stretch. Also check your bridge saddles as these have to be fine-tuned to get the string length right.

http://www.fender.com/support/setup/stratsetup.php

this channel has lots of goo stuff

Necro-ing a 10-year old thread with zero replies?

Make a fresh new thread, Salv!

(This stuff is interesting so it should get the proper treatment)

i did a few tracks in pythagorean scale I think, by just directly editing a ReaJS synth to the right frequencies? it was ok i guess

i basically just did it cause i was sick of 432 hz youtube videos that just pitch down the entire song

This is a really fun field for the adventurous producer. It can potentially open up a new world for you, but maybe cause a bunch of headaches in the process. It takes a good ear to begin with and some training to even notice a difference at first. It's also important to take one of less problematic routes. That should be friendly VSTs like Native Instruments or Korg's legacy line - Native Instruments' FM8 is my favorite. A programming environment like PD or Max should also be useful, but probably necessitates some previous experience. I tried going by the *.scala and *.tun file route, or midi VSTs that use pitch bending magic, but it's a fuckload of trouble and either didn't work at all or didn't give you a direct modifying ability.

What you want to do is put your microtunable synth through a spectrum analyzer, play some very basic wave tone (saw or sine) intervals like fifths or thirds, microtune one of the keys and watch how the tones phase in and out with each other. After some practice you'll hear when they are perfectly in phase - a scale that emphasises this is the root of the pleasing sound of just intonation. Math makes this phenomenon kinda tricky in a twelve tone environment, you can't harmonize all tones perfectly, so scale tuning is always a matter of prioritization and compromise - which intervals are more important, what do you want to emphasize? 12TET (the typical western sound) tries to make the distance between all 12 tones equal across the scale to allow easy transposition. This is because deviating from that system will easily create "wolf tones" that wildly clash with certain intervals. But that makes the whole scale slightly dissonant from a pure tone perspective. There are theories that this inherent dissonance (or "edge") has lead to the energetic sound of Western music.

But this is where we can step in and experiment, and that's what's so fun about microtuning. You can choose intervals that you prefer, according to certain scales, and try to prioritize them into a harmonization system that appeals to you. You just stick to a particular scale and don't transpose it, so the wolf tones outside of the scale are naturally absent. This might seem like a lot of work for a small effect, but it's really quite a magical experience when you've come up with a personal tuning system that appeals to you personally.

In my experimentation I developed a natural minor scale, which derives from just intonation but with some changes. Most importantly it emphasises pure thirds instead of sevenths, since I love using a lot of thirds. You can see it here:

I used this scale for a track for a 432hz watmm compilation (that didn't go anywhere due to lack of submissions). What I did was simply use FM8 with the custom scale on each instrument track. I think I also replaced the 808 kick with a custom made FM8 emulation that was tuned as well. it's important to note that the 432hz system detunes the whole keyboard 8 cents (with A from 440hz to 432hz as a reference), which gives the global key a duller sound. But I think it showcases, at least to a careful ear, how good the intervals of the custom tuned scale sound together, especially in the latter minutes of the track when the pure, dry FM tones harmonize and interplay.