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716 Posts
Discussion Starter · #1 ·
It is my understanding that with just tuning, the pitch will get higher relative to equal temperament as you move to higher notes. So if you start with C2, and then play C4, the C4 will actually be higher than an equal tempered C4. This is confusing, because in both tempered at just tuning it seems the octave is a doubling of frequencies.

Can anyone who has a better grip of the physics of sound comment on this? Are octaves in just tuning not a doubling of frequencies?

As a side note, I have keyboards that can do just tuning. With the Yamaha the notes of a major scale are close to the same as equal temperament, but the 3rds and 6ths are flat in comparison to equal temperament. But with the Casio Privia, as you move up the octaves the notes become more and more sharp in relation to equal temperament. It would be my belief that this is a more correct rendering of just tuning. But again, this is above my pay grade and my knowledge of physics.

Distinguished SOTW Coffee Guru
40,343 Posts
Pete Thomas wrote extensively on tuning

Distinguished SOTW Coffee Guru
40,343 Posts
well, you may ask him, this was my spirit in pointing you in his direction, since searching the name of of the oldest and most active members and the word intonation may not return any precise results.

Good Luck.

Distinguished SOTW Coffee Guru
40,343 Posts
old membership :)

1,224 Posts
I did a search, but didn't find much. I'll look around at his posts.
1/ use the "Google: custom search" box, not the forum search
2/ possibly try "just intonation" or "just temperament"
3/ have a read of "how equal temperament ruined harmony: And Why You Should Care" which is a fun book.

Grafton alto | Martin Comm III tenor
29,583 Posts
Pete Thomas wrote extensively on tuning
Pete鈥檚 Old ? Dang, that means I鈥檓 old too....
So old I've forgotten about writing anything on tuning.

What I can say meanwhile (while trying to remember what and where I said stuff before about tuning) is that higher notes being relatively higher pitch (stretched octaves, have nothing to do with just or equal intonation.

The octaves should still be in tune, exactly 2:1 ratio

Stretched tuning (where e.g. C5 is more than a 2:1 ration with the lower C4) is something to do with inharmonicity of metal strings, ie perfect 2:1 octaves sound flat on some instruments due to some complicated acoustic stuff. Ask a piano tuner. Then try playing a piccolo in unison with a piano.

But it is time again, while on the subject, to show the picture of my wife's meantime tuned harpsichord withy separate D#/Eb keys etc.


716 Posts
Discussion Starter · #9 ·
Thanks for that answer Pete. I started wondering about this because I have a strobe tuner, and when I set it to just intonation and choose a root note, the tuner tells me I need to push the pitch up on the high notes. It sounds great for playing alone, but obviously not with other equal tempered instruments.

Grafton alto | Martin Comm III tenor
29,583 Posts
Thanks for that answer Pete. I started wondering about this because I have a strobe tuner, and when I set it to just intonation and choose a root note, the tuner tells me I need to push the pitch up on the high notes. It sounds great for playing alone, but obviously not with other equal tempered instruments.
In some cases you can get away with just intonation when playing with equal temper tuned instruments - it is all contextual and can depend on the timbre of the instruments you are playing with and how far away in pitch you are. e.g. I joked about the piccolo playing union with piano, but I have looked into this as a music producer.

These days I can check my recorded saxophone tuning and have the option to autotune it, however I found 9 times out of 10 it sounded better without being tuned perfectly to equal temperament. Whether my tuning "discrepancies" that sound "better" actually do sound better to everyone I don't know. Somewhere I have a comparison according of a flat G sounding better than the "in tune" G. It can also be tune to blues notes.

I do know a singer who specialises in early music who has the capability to pitch exactly how she wants, e.g. sing a scale in just intonation, meantime or equal temperament. On a session she demonstrated to me a 7th using various different tuning systems.

This is exceptional ear and memory function IMO.

Grafton alto | Martin Comm III tenor
29,583 Posts

Distinguished SOTW Technician.
3,237 Posts
If you start with a notes of 100 the octave is 200. Use the harmonic series to get more notes.
100 200 300. The 300 is a different note to the 100 and its octave 200. To fit it in the range between the 100-200 range we can use the octave rule and half it, so then you have notes of 100, 150 & 200.
So now you take the 150 and go through its harmonics 150, 300, 450. Reduce that to our 100-200 range to get 112.5. Then have use this to find another note.
Keep doing this and you eventually get to 200 - except that it isn't 200 it's 203 or something. The scale doesn't fit. To try and make the notes fit you need to shift some from where they should be

Just intonation is an early intonation used on keyboards where most intervals are tuned pure. It leaves some intervals discordant, usually the second and seventh. It means there are 3 kinds of intervals Major tone (C-D) Minor tone (D-E) and semitone (E-F).

As far as I'm aware all temperaments tune octaves pure - however sometimes in practise pianos can sound flat with pure octaves and some will widen the interval to make it sound better.

Modern equal temperament shifts all the intervals slightly to give 12 equally spaced semitones.

716 Posts
Discussion Starter · #18 ·
All pianos have to be stretch tuned.
I do it everyday.
Thank you for clarifying this. When I use a strobe tuner on my electric Casio it shows that the octaves are right in tune to equal temperament. Then I put the keyboard on just major mode, select a root, and viola, the strobe tuner shows each octave as getting sharper in relation to equal temperament. So I am left wondering why the octaves are not the exact same ratio in equal and just temperament. I believe the stretch tuning has nothing to do with equal temperament though.

Distinguished SOTW Member, Forum Contributor 2013-
5,433 Posts
See Pythagorean Tuning.

In music theory (over 40 years ago), the professor told us that perfect tuning to do with the ancient Greek philosophers, who had a big thing for logic and numbers.

Music was, in ancient times, taught as a branch of mathematics. Oh, and everything else was, too (justice is represented as the number 4, the stable sides of the cubic solid).

Pluck a length of tightened gutstring. Then pinch off the string against the sounding board, plucking it at exactly 1/2, 1/3, 1/4 of it's length, and so on. The relationship between the ratios and the sound is the very definition of music. There. Proof that music is math. (See details below.) This is, of course, exactly represented in the music of the spheres, which contain the various planets, which revolve around earth. The math can also be shown in the direct relationship between the five perfect platonic solids (Tetrahedron, Cube, Octahedron, Dodecahedron, Icosahedron) which represent each of the elements (earth, air, fire, water, and the void).

Pythagoras was a nut. A lot of the ancient goofiness is attributed to him. He thought of air, vapors, and gases as evidence of spirits and souls. Consequently, one should not eat beans, as it would lead to corrupting souls being released at the back end of the process. (told you he was a nut)

These guys had a whole bunch of goofy ideas that go along with the growing pains of a youngster first learning to think. The need for consistency where none can be found is a prime example of the kind of notion that troubled these fellows. That, combined with a disdain for observation compared to "reason" led to scholasticism, and the belief in a golden age of the past, hampering further advances.

The Pythagorean mathematical model would require a different keyboard to be played for every key change.

Other tunings are attempts to get around that requirement.

See, e.g.:

The Musical Construction of the World-Soul in Plato's Timaeus


The god, however, gave priority and seniority to the soul [of the world], both in its coming to be and in the degree of its excellence, to be the body's mistress and to rule over it as her subject.

The components from which he made the soul and the way in which he made it were as follows: In between the being that is indivisible and always changeless, and the one that is divisible and comes to be in the corporeal realm, he mixed a third, intermediate form of being, derived from the other two. Similarly, he made a mixture of the same, and then one of the different, in between their indivisible and their corporeal, divisible counterparts. And he took the three mixtures and mixed them together to make a uniform mixture, forcing the different, which was hard to mix, into conformity with the same. Now when he had mixed these two together with being, and from the three had made a single mixture, he redivided the whole mixture into as many parts as his task required, each part remaining a mixture of the same, the different, and of being. (Timaeus, 34c-35b)

Step One:

This is how he began the division: first he took one portion away from the whole, and then he took another, twice as large, followed by a third, one and a half times as large as the second and three times as large as the first. The fourth portion he took was twice as large as the third, the sixth eight times that of the first, and the seventh twenty-seven times that of the first. (35b)

This produces the following seven quantities: 1, 2, 3, 4, 9, 8, 27.

Step Two:

After this, he went on to fill the double and the triple intervals... (36a)

A double interval is one where the larger quantity is twice the smaller and a triple interval is one where the larger quantity is thrice the smaller. So we need to order our quantities in a way where they make these intervals. And sure enough, this works out to give us a double interval and a triple interval series:

Double interval series: 1, 2, 4, 8
Triple interval series: 1, 3, 9, 27

Step Three: cutting off still more portions from the mixture and placing these between them, in such a way that in each interval there were two middle terms... (36a)

That is, each interval is divided into three, by inserting two middle terms between the extremes. So that looks like figure one.

Step Four: exceeding the first extreme by the same fraction of the extremes by which it was exceeded by the second, and the other exceeding the first extreme by a number equal to that by which it was exceeded by the second. (36a)

The first of these rules describes the procedure for a harmonic mean and the second the procedure for a geometric mean. So these allow us to precisely place our two middle terms. This gives us figure two.

Step Five:

These connections produced intervals of 3/2, 4/3, and 9/8 within the previous intervals. (36a-b)

That is, the ratio of the larger to the smaller number in each of the nine intervals making up each of our two series will be one of these three values. That works out as shown in figure three.

In musical terms, the ratio 3/2 is called a perfect fifth. That is, when the ratio of the frequency of the higher note to the lower is 3/2, the musical interval that separates them is a perfect fifth. Similarly, 4/3 is a perfect fourth and 9/8 is a major second. So our two series can be understood as two systems of musical intervals, as shown in figure four.

This way of presenting the relationships considers the interval between each value (after the first) and the one that precedes it, so that each series is divided into three big intervals and each of these is divided into three small intervals, for a total of nine intervals making up the series. But we can also consider these series as a number of octaves (2/1 ratios) organized into scales, i.e. where we consider the ratio of each value to the value of the first member of the octave. On this way of construing them, the series consist of four major points, and within the span of each of these pairs there is a similar division consisting of four major points (i.e. sets of four points rather than sets of three intervals).

This is easy to show with the first ("double interval") series, since the major divisions correspond to octaves (2/1). It's a bit more awkward an analysis on the second ("triple interval") series, but in this case the first three points will make an octave, and we'll be left with an interval of a perfect fifth between our octaves. This looks like figure five.

By comparing the first kind of figure (showing individual intervals) to the second (showing scales), we can see how intervals are constructed from (or can be analyzed into) further intervals. Namely, an Octave is composed of a Per5 plus a Per4, and a Per5 is composed of a Per4 plus a Maj2.

Step Six:

He then proceeded to fill all the 4/3 intervals with the 9/8 interval, leaving a small portion over every time. The terms of this interval of the portion left over made a numerical ratio of 256/243. And so it was that the mixture, from which he had cut off these portions, was eventually completely used up. (36b)

Naturally enough, we'd like next to know how to analyze a Per4, and we'll have to construct the Per4 (4/3) out of Maj2s (9/8)--and that's exactly what Plato tells us. We're actually going to need two Maj2s, and even that won't be quite enough for a Per4. (E.g., the series {1, 1.12, 1.27} gives us two Maj2 [9/8] intervals, but it doesn't span far enough to get us to Per4 [4/3].) What we need to complete the Per4, having first added these two Maj2s, is exactly what Plato tells us: a 256/243 interval. (In the example just stated, that would give us {1, 1.12, 1.27, 1.33}, and that gives us a Per4, i.e. the interval between 1.33 and 1 is 4/3.) 256/243 is a musically significant ratio too, called a minor second.

So Plato explains a further step in the analysis here: an Octave is a Per5 plus a Per4, a Per5 is a Per4 plus a Maj2, and a Per4 is two Maj2s plus a Min2. (We can of course run with this: a Per5 is three Maj2s plus a Min2, and an Octave is five Maj2s plus two Min2s.)

Our construction of the first ("double interval") series involved two Per4 (4/3) intervals within each octave. If we now add this new division of Per4 intervals to this series, we get another layer of divisions within this series: each Per4 gets analyzed into three intervals (Maj2, Maj2, Min2) or into the four points that span these intervals. Considering all our analyzes, we now see that the series consists of three octaves, and each octave can be thought as five Maj2s plus 2 Min2s, as two Per4s plus a Maj2, or as a Per5 plus a Per4. This looks like figure six.

If we now construct this series as scales rather than individual intervals, we get three octaves, each consisting of eight notes, with the scale having the intervals: Unison, Maj2, Maj3, Per4, Per5, Maj6, Maj7, Octave. This looks like figure seven.

In modern terms, this is a major scale (although most modern instruments cannot play the mathematically precise intervals used in this system of identifying notes). In ancient terms, the key unit is a four note series called the tetrachord, and the octave is constructed by conjoining two tetrachords with a Maj2 interval separating them, which gives us the same construction (i.e. the tetrachords are the first four and the last four notes of the modern scale). In ancient music theory, there are different ways of dividing up the tetrachords, and the one Plato uses here, which corresponds to the modern major scale, is the Pythagorean scale derived from a similar mathematical procedure by Philolaus.

So Plato, following Philolaus' Pythagorean derivation of music from mathematics, constructs what is to us a major scale, merely on the basis of the ratios produced from a geometrical series analyzed into harmonic and geometric means. At least for the most part--the Min2 (256/243) interval seems to be needed simply to make up what's left over when we analyze the Per4 into two Maj2s. Note that this interval corresponds to what we would call a "semi-tone", which is the smallest interval used in modern music, corresponding to the space between two immediately adjacent notes (e.g. C to C#).

716 Posts
Discussion Starter · #20 ·
See Pythagorean Tuning. I
I have a keyboard that does Pythagorean tuning, but it seems pretty useless. With JUST INTONATION the tonic, subdominant, and dominant chords are more in tune. Here is a nice description I found of the difference.

The basic difference between "Just" and "Pythagorean" is that the pythagorean scale is built up of whole tones that are all the same size, rising evenly, but with a short semitone before the dominant, and also a short "leading tone" behind the tonic. In just intonation, there is more than one whole tone size, and of course you can also have more than 7 notes in a just scale--it is all a matter of deciding which notes you want to keep! There are different just scales: for instance a "just" major 6th could be 1.6667 (that is in an "8-limit" just intonation major scale) or 1.6875 ("16-limit"). Note that when you transpose to a new mode, the 8-limit becomes 16-limit anyway...)

The tonic, dominant and perfect 4th are identical in these two systems, and so where you feel the difference (using a "major" scale for compoarison)is at the major third, the major 6th and the major seventh (which of course means that also the minor third etc when you transpose to minor key).

The just major third is rather "flat" compared to a pythagorean--and even compared to an equal tempered major 3rd.

Pythagorean major 3rd: 1.265625:1

Just major third: 1.25:1 (5/4)

Equal tempered major 3rd: 1.25992 = (2^(1/12))^4

Conversely, the minor thirds are as follows:

Pythagorean minor 3rd: 1.185182

Just Minor 3rd: 1.2:1 (6/5)

Pythagorean major 7th: 1.8984375

Just major 7th: 1.875 (15/8)

Here is the basic construction of the pyhtagorean scale:

Tonic: 1/1

second [(3/2)^2] /2

M3: [(3/2)^4] /4

4th: 2*(2/3) (inverse of 5th)

5th: 3/2

M6: [(3/2)^3] /2

M7th: [(3/2)^5] /4

Octave: 2:1
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