I am one of them…

If C to E is a third (3)

& C to D is a second

then C to C must logically be a first (1), or what we call a unison.

To my way of thinking, this leaves no option of a zeroth (0).

(runs!)

I am one of them…

If C to E is a third (3)

& C to D is a second

then C to C must logically be a first (1), or what we call a unison.

To my way of thinking, this leaves no option of a zeroth (0).

(runs!)

Even with this logic, it still doesn’t make sense. In this case, would a diminished unison be ½? Would a minor third be 2.5? This is one of the reasons integer notation was invented.

The englisch version of Wikipedia knows the diminished unison and tells about sources.

The german version of Wikipedia knows the “verminderte Prim” and gives an example.

I think it’s rather theoretical, but as a mathematician I would call it logical. .

Thomas

The problem with interval names, i.e., using numbers, is that we have enharmonic spelling:

C-Db is a minor second, or a second due to D being the second letter in the alphabetic sequence. Hence C-C is a unison, but so is C-C#, but that is not by virtue is its distance, but of its spelling. We could avoid all this my naming intervals discretely and uniquely (no more Aug 4/dim5). I vote for formal names: Bob, Ted, Irving, Leopold, and so forth. Rather than “a P4th above B” it would be an “Irving above B.”

I’m on to somethin’…

The convention of counting *both* ends of the range of an interval as different notes is consistent with the ancient Roman method of counting, and medieval music theory was controlled by the church, which of course worked exclusively in the Latin language. Like the ancient Greeks, the Romans did not recognize “zero” as a number at all.

This leads to the illogical situation (compared with contemporary mathematics) that an “octave” contains 8 diatonic steps, but a double-octave only contains 15, not 16.

In fact, the logical solution would be for an “octave” to contain 7, and a double-octave to contain 14. The interval from C to D would then be a “first”, not a “second”, and a “unison” would contain zero steps.

But since the term "octave has been used for at least 1000 years, nothing is likely to change any time soon!

(Note, the Catholic church still uses the term “octave” to refer to a full week of days starting and ending on days of the same name - e.g. Monday to Monday inclusive. This follows the same counting system.)

I don’t understand how a diminished unison can exist at all. If you have any other perfect interval, say a forth, fifth or octave, and increase the distance between the two notes by one half-step you get an augmented interval. The direction of the doesn’t enter into it.

I don’t see the logic behind the thinking that direction should decide the size of the interval. If you go from C upwards to a G# you get an augmented fifth, if you go from G# downwards to C you get an augmented fifth, not a diminished. But in Dorico if you go from F upwards to F# you get an augmented unison, but if you go from F# downwards to F, you get a diminished unison. That makes no sense…

In software terms, I think the Dorico way is the clearest way to describe this conundrum.

That’s only if one is determined to live a millennium or two ago, before (as Rob Tuley described) people decided that the concept of zero made every type of computation easier and more logical. Traditional interval naming assumes that 1 is the smallest conceivable integer, but we now think in terms of 0 as the reference point from which distances are measured, the measuring unit for intervals being the semitone (or whatever one prefers to call it).

We can certainly *translate* the 0-to-12 measurement into the familiar “perfect octave,” for ease of communication. But the size remains the same.

Have we stacked enough angels on the head of a pin yet?

I think we can squeeze a few more in!

Do you mean the number of perfect angels, augmented angels, or diminished angels that will stack on a pinhead?

I would assume that angles, by their definition would be perfect. Depends on if we are looking at them through the modern lens, where we use the 0 for placeholder, or the older Roman approach. I suspect that we have exhausted the limits of what musicians can bring to this debate, and it is time to being on the theologians. We all knew it would come to this, didn’t we?

Except for the fallen ones - and “diminished” seems a better name for those than “augmented.”

Hah!

According to Wikipedia, negative numbers were discussed in China in the 3rd Century C.E.; they reached India soon after and were in moderate use in the 7th C, but regarded with suspicion until the 12th C. Some Islamic mathematicians used them by the 9th C., with general use by the 12th C. In Europe, Fibonacci used them to represent debts in a publication dated 1202, but they were viewed with suspicion until the 18th C. That they are still creating difficulty within the sphere of music is not surprising: our interval system uses the integers 0 … 7 but labels them 1 … 8. In comparison with that, diminished unisons look like a minor problem.

A diminished unison is obviously an interval of i (√-1) semitones. Similar to how a double diminished unison is an interval of 2i semitones, a triple diminished unison is 3i and so on. My goodness, what do they teach in music schools nowadays?

Claude said:

“I sure hope there are no alien civilisations presently judging our species solely by studying this thread!”

Well, I am an alien, and I find this whole discussion fascinating. I also enjoy watching grass grow. (We have no plants on my planet.)

Perhaps an option to add for Dorico 6.5., when there is not much left, which is missing in the program:

“Do you want to transpose from c to ces as a diminshed unison or as an augmented unsion downwards?”

In order to entirely derail this discussion, I would just like to point out that there is a disease called Malaria tertiana which is named that way because the fever spikes every other day (Day 1, 3, 5, …) whereas Malaria quartana has spikes every third day (1, 4, 7, …).

I was unable to explain the thinking behind that naming scheme to my students until I came across this discussion. Funny that it never occurred to me that it’s the same with intervals. So thanks a lot for this insightful thread!

Which reminds me of this old party trick: the ‘entertainer’ asks the audience, “How many fingers have I got?”

They reply, “Ten! Of course.”

To which the entertainer says, “No; look…” and counts down from the little finger on their left hand: “10, 9, 8, 7, 6 plus the five on the other hand. Six and five makes eleven”.