Decimal Notation

Definition
Diatonic scales
Neutral third scales
Absolute pitch
Outer reaches
13-limit tuning

This page shows how my decimal notation can be written on the staff, and sort of follows on from the decimal lattices page.

The idea is to write the 19 nominals on the 5-line staff, with 0 being the gap between the staves. That means conventional manuscript paper can still be used, although ideally it would have pairs of staves closer together than standard.

So, here are the nominals:

```                                        0
------------------------------------9-----
8
----------------------------7-------------
6
--------------------5---------------------
4
------------3-----------------------------
2
----1-------------------------------------
0
```

They look like this melodically

```0   1   2   3   4   5   6   7   8   9   0
s   s   s   s   s   s   s   s   s  s+q
```

With s as 3/31, 4/41 or 7/72 octaves and q as 1/31, 1/41 or 2/72 octaves. Also covering in-between temperaments. The ^ and v symbols denote upward and downward shifts of q.

For the rest of the page, I'll use round noteheads to emulate what you'd see on the page.

This is a major key starting on 0:

```                                O
---------------------------^O-----
vO
----------------------^^O---------
O
---------------wO-----------------
^O
----------^O----------------------
vO
----^^O---------------------------
O
```

Notes lined up in columns are equivalent in meantone. This gives you two ways of playing the ii chord for comma shifts. You don't need any accidentals for diatonic music. You choose the staff positions so that the chord shape is right, and 5-limit harmony (with comma shifts) results.

0 Minor

```                                    O
------------------------------^O--------vO------------------------------
vO
---------------------------------------------O--------------------------
O                            O
------------------------------------------------------------------------
^O                                     ^O
----------vO-----------------------------------------------vO-----------
vO                                                         vO
----^^O--------------------------------------------------------^^O------
O                                                                    O
```

7+3 scales naturally fall out of decimal notation. So, I notate them accordingly. That means some consonances that are characteristic of the meantone-range remain 31-equal enharmonies. I think the simplicity makes this worthwhile.

0 anti-Dorian (Mohajira). Note that the 21 note Blackjack scale is composed of exaclty 3 instances of anti-Dorian.

```                                    O
-------------------------------O------

-------------------------^O-----------
O
--------------------------------------
^O
-----------O--------------------------

-----^O-------------------------------
O
```

0 anti-Phrygian (Rast)

```                                    O
-------------------------------O------
vO
--------------------------------------
O
--------------------------------------
^O
-----------O--------------------------
vO
--------------------------------------
O
```

10 note neutral-third MOS around 3-6

```                                                   O
----------------------------------------------O------
vO
-----------------------------------^O----------------
O
-------------------------vO--------------------------
^O
----------------O------------------------------------
vO
-----^O----------------------------------------------
O
```

So, what should 0 be tuned to? I think it should be equal to C in diatonic tuning. If A is tuned to 440Hz, then the C a 5:3 below will be 264 Hz. The logarithm to base 2 of 264 is 8.04, which we can call an absolute pitch in octaves. So 0 can be written as 8.04 Oct. Then, 0v will be between 8.008 and 8.016 Oct.

We can then assign an "octave number" to each note which will usually be the integer part of the logarithm of its frequency to base 2. The first decimal place will be somehow related to the note number as well.

As 1Hz is inaudible, I think we should take 4 from the octave number. This makes the reference frequency 16Hz. It also makes us consistent with the common MIDI numbering.

So, the bottom half of the treble clef is octave 4. the top half of the bass clef is octave 3.

To show the octave on the decimal staff, I suggest writing the Arabic numeral(s) for the octave number. That's easy to remember, and will make decimal scores look distinctive. The treble clef becomes two staves for 4 and 5. The bass clef becomes two staves for 2 and 3.

Here I show how the notation can be adapted to write stranger, xenharmonic tunings.

Here's 12-equal as a subset of 72. It uses half-shifts where two / make a ^ and two \ make a v. That is / and \ show 1 step in 72-equal.

```                                                             O
-------------------------------------------------\vO--/^O------
vO
----------------------------------------\O---------------------
O
------------------------------/O-------------------------------
^O
--------------\vO--/^O-----------------------------------------
vO
-----\O--------------------------------------------------------
O
```

12-equal as a decimal scale where s=1 q=2

```                                            vO  O
-------------------------------------O---^O-------
O
-----------------------------O--------------------
O
---------------------O----------------------------
O
-------------O------------------------------------
O
-----O--------------------------------------------
O
```

12-equal using a related 10 notation where 5s+q=Oct/2 and s=q=1

```                                            vO  O
-----------------------------------------O--^O----
O
---------------------------------O----------------
O
--------------------vO---O------------------------
O  ^O
-------------O------------------------------------
O
-----O--------------------------------------------
O
```

24-equal as a subset of 72

```                                                                            \vO O
------------------------------------------------------------------\vO-O-/^O-------
vO /O
-----------------------------------------------------\O-^O------------------------
\vO O /^O
------------------------------------vO-/O-----------------------------------------
\O ^O
--------------------\vO-O-/^O-----------------------------------------------------
vO  /O
-------\O-^O----------------------------------------------------------------------
O ^/O
```

11-equal

```                                        vO   O
-------------------------------------O--^O-----
O
-----------------------------O-----------------
O
---------------------O-------------------------
O
-------------O---------------------------------
O
-----O-----------------------------------------
O
```

22-equal from 11-equal using half-shifts

```                                                   vO \O O
-------------------------------------------\O-O-/O-^O------
\O O /O
---------------------------------\O-O-/O-------------------
\O O /O
-----------------------\O-O-/O-----------------------------
\O O /O
-------------\O-O-/O---------------------------------------
\O O /O
---\O-O-/O-------------------------------------------------
O /O
```

Decatonic 22-equal (5s+q=Oct/2)

```                                                      vO O
----------------------------------------------vO-O-^O------
vO O ^O
------------------------------------vO-O-^O----------------
vO O ^O
--------------------------vO-O-^O--------------------------
vO O ^O
-------------vO-O-^O---------------------------------------
vO O ^O
---vO-O-^O-------------------------------------------------
O ^O
```

With this notation, we can write all extended 11-limit intervals. But what about the 13-limit? Well, 13:10 is roughly the wolf fourth in 31-equal. That's 0-4 in decimal notation. In 72-equal, 0-4 is 28 steps, but 13:10 is 27.25 steps. As we have a symbol for half-lowering, we can use that to write 13:10 as 0-4\. Similarly, 13:8 can be the neutral sixth 0-7^\.

The \ and / symbols could also be used to lower and raise by a syntonic comma, or equivalent intervals. So 0/ is the same as 1vvv and 0^^^ is 1\. Be sure not to mix the two definitions, and make it clear which your using.

In general, the 13-limit "half-lowering" is slightly less than half of a v. So you can ignore it in 31- or 41-equal if you don't want to step outside the tuning. The safest thing is not to bother about the 13-limit, as we have such a neat approximation to the 11-limit. But, if you want it, it's there.

17- and 19-limit are more tricky. Hopefully the cases where they become important won't fit with this kind of notation anyway, so we won't need to worry about them.