No joy with TMC2208 driver so ordered TMC2100 which are known to work silently in 3D printers.

A TMC2100 works well.  The motor is quiet.

Unfortunately, the 25:32 gears that provide the 1 second drive to the seconds wheel are far from quiet in spite of being accurately printed.  There are possible ways of improving the gears such as helical teeth but this mainly applies to machined gears.  Printed helical gears will be rough due to the layers.  Consequently, I'm looking into the possibility of just using code in the Arduino sketch.

The RTC module is capable of producing square waves of 32kHz, 8192Hz, 4096Hz, 1024Hz and 1Hz.  The stepper motor has 200 steps per revolution or 3200 steps/rev with 16x microstepping.  The TMC2100 only does "quiet" in 16x or 4x microstepping.

One thought was to apply a time correction every so often to keep the clock in time.  ie. using the 32KHz (actually 32768Hz) gives a ratio of 10.24 for 3200 microsteps per second so a correction of 24 counts per 1000 or 3 in 25. 

Another thought is to generate pulses controlled by the Arduino clock such that a number of them finish just short of the fixed time.  eg. produce 3200 pulses in just short of a second and use the 1Hz square wave to restart the sequence.  The 1Hz is a symmetrical square wave so timing can be taken from both edges resulting in two smaller gaps in the pulse train to the motor rather than one larger one each second.

Here is an Arduino sketch.

// Filename :- Giant_Wall_Clock_v2_2019-11-14
// Software timing from RTC using polling
//
#include <DS3232RTC.h>    //http://github.com/JChristensen/DS3232RTC
#include <Time.h>         //http://www.arduino.cc/playground/Code/Time  
#include <Wire.h>         //http://arduino.cc/en/Reference/Wire (included with Arduino IDE)
//
boolean lastSqWave = 0; 
boolean ledON = 0; 
int sqwPin = A6;
int stepPin = 6; // STEP pin
int enPin = 12;  // Enable pin
int ledPin = 13;  // Internal LED pin
//
void setup() {
  pinMode(stepPin, OUTPUT);
  pinMode(enPin, OUTPUT);
  pinMode(ledPin, OUTPUT);
  digitalWrite(enPin, 0);   //  enable
  pinMode(sqwPin,INPUT_PULLUP);   //  RTC timing pin
  RTC.squareWave(SQWAVE_1_HZ);    // 1Hz square wave            
}
//
void runClock(void){
  for (int i = 1600; i >= 1; i--) { //  count 1600 pulaes in half second
    digitalWrite(stepPin, 1);
    delayMicroseconds(10);     // Make STEP pulse 10μs long
    digitalWrite(stepPin, 0); 
    delayMicroseconds(300); }     //  1600×310 = 496ms
  ledON = !ledON; digitalWrite(ledPin, ledON); }  //  flash LED on & off

//
void loop(){
  boolean sqUp = ((analogRead(sqwPin) > 500));  // read logic level of 1Hz square wave
  if (sqUp != lastSqWave)  { lastSqWave = sqUp; runClock(); }  //  Call runClock on both edges of RTC square wave
}
// End

Taken the motor and gears unit off the living room wall and brought the electronics in from the cupboard for rewiring and installing replacement sketch into the Arduino Nano.  On inspecting the TMC2100 driver module connections I've found a mistake.  I have connected a pin to Gnd presumably to change from stealthChop to spreadCycle mode but I've connected Diag1 instead of Cfg1.  This would have no effect so I've been running it in stealthChop mode quite happily.  This is the silent mode with lower power but the clock has bee running quite happily on just 9v so I'll stay with this mode.

The wiring mod is simply connecting the Square Wave output from the RTC module to the Nano and an output from Nano to Step input of driver instead of a direct connection..

New sketch compiled and uploaded to the Arduino Nano.  Motor and control electronics board set up on table and all working fine.  Turned the motor current down until the PSU showed 100mA at 9v and the NEMA17 motor still produced plenty of power.  I could hear a faint hum if I put the motor to my ear.

Clock drive reinstalled and all working fire after a tiny tweak to the timing.  Clock is now virtually silent - just a faint pulsating hum only audible if sitting quietly with no TV, radio or records.  I can probably make that tiny hum quite inaudible with flexible mounting - the motor is currently fastened firmly to the partition wall (plasterboard).

Final sketch and IDE report.

// Filename :- Giant_Wall_Clock_v2_2019-11-15
// Software timing from RTC using polling
//
#include <DS3232RTC.h>    //http://github.com/JChristensen/DS3232RTC
#include <Wire.h>         //http://arduino.cc/en/Reference/Wire (included with Arduino IDE)
//
boolean lastSqWave = 0; 
boolean ledON = 0; 
int sqwPin = A6;
int dirPin = 5;   // DIRECTION pin
int stepPin = 6;  // STEP pin
int enPin = 12;   // Enable pin
int ledPin = 13;  // Internal LED pin
//
void setup() {
  pinMode(dirPin, OUTPUT);
  pinMode(stepPin, OUTPUT);
  pinMode(enPin, OUTPUT);
  pinMode(ledPin, OUTPUT);
  digitalWrite(enPin, 0);        //  enable
  digitalWrite(dirPin, 1);       //  set on test
  pinMode(sqwPin,INPUT_PULLUP);  //  RTC timing pin
  RTC.squareWave(SQWAVE_1_HZ);   // 4096Hz square wave
  }
//
//
void runClock(void){
  for (int i = 1600; i >= 1; i--) { //  count 1600 pulaes in half second
    digitalWrite(stepPin, 1);
    delayMicroseconds(10);          // Make STEP pulse 10μs long
    digitalWrite(stepPin, 0); 
    delayMicroseconds(290); }       //  1600×300 = 480ms  **** 2019-11-17
  ledON = !ledON; digitalWrite(ledPin, ledON); }  //  flash LED on & off
//
void loop(){
  boolean val = ((analogRead(sqwPin) > 500));     // read logic level of 1Hz square wave
  if (val != lastSqWave)  { lastSqWave = val; runClock(); }  //  Call runClock on rising edge of RTC square wave
}
// End

Sketch uses 3896 bytes (12%) of program storage space. Maximum is 30720 bytes.
Global variables use 283 bytes (13%) of dynamic memory, leaving 1765 bytes for local variables. Maximum is 2048 bytes.
/home/gina/arduino-1.8.10/hardware/tools/avr/bin/avrdude -C/home/gina/arduino-1.8.10/hardware/tools/avr/etc/avrdude.conf -v -patmega328p -carduino -P/dev/ttyUSB0 -b115200 -D -Uflash:w:/tmp/arduino_build_979028/Giant_Wall_Clock_I2C-control_2019-11-15.ino.hex:i 

avrdude: Version 6.3-20190619
         Copyright (c) 2000-2005 Brian Dean, http://www.bdmicro.com/
         Copyright (c) 2007-2014 Joerg Wunsch

         System wide configuration file is "/home/gina/arduino-1.8.10/hardware/tools/avr/etc/avrdude.conf"
         User configuration file is "/home/gina/.avrduderc"
         User configuration file does not exist or is not a regular file, skipping

         Using Port                    : /dev/ttyUSB0
         Using Programmer              : arduino
         Overriding Baud Rate          : 115200
         AVR Part                      : ATmega328P
         Chip Erase delay              : 9000 us
         PAGEL                         : PD7
         BS2                           : PC2
         RESET disposition             : dedicated
         RETRY pulse                   : SCK
         serial program mode           : yes
         parallel program mode         : yes
         Timeout                       : 200
         StabDelay                     : 100
         CmdexeDelay                   : 25
         SyncLoops                     : 32
         ByteDelay                     : 0
         PollIndex                     : 3
         PollValue                     : 0x53
         Memory Detail                 :

                                  Block Poll               Page                       Polled
           Memory Type Mode Delay Size  Indx Paged  Size   Size #Pages MinW  MaxW   ReadBack
           ----------- ---- ----- ----- ---- ------ ------ ---- ------ ----- ----- ---------
           eeprom        65    20     4    0 no       1024    4      0  3600  3600 0xff 0xff
           flash         65     6   128    0 yes     32768  128    256  4500  4500 0xff 0xff
           lfuse          0     0     0    0 no          1    0      0  4500  4500 0x00 0x00
           hfuse          0     0     0    0 no          1    0      0  4500  4500 0x00 0x00
           efuse          0     0     0    0 no          1    0      0  4500  4500 0x00 0x00
           lock           0     0     0    0 no          1    0      0  4500  4500 0x00 0x00
           calibration    0     0     0    0 no          1    0      0     0     0 0x00 0x00
           signature      0     0     0    0 no          3    0      0     0     0 0x00 0x00

         Programmer Type : Arduino
         Description     : Arduino
         Hardware Version: 3
         Firmware Version: 4.4
         Vtarget         : 0.3 V
         Varef           : 0.3 V
         Oscillator      : 28.800 kHz
         SCK period      : 3.3 us

avrdude: AVR device initialized and ready to accept instructions

Reading | ################################################## | 100% 0.00s

avrdude: Device signature = 0x1e950f (probably m328p)
avrdude: reading input file "/tmp/arduino_build_979028/Giant_Wall_Clock_I2C-control_2019-11-15.ino.hex"
avrdude: writing flash (3896 bytes):

Writing | ################################################## | 100% 0.59s

avrdude: 3896 bytes of flash written
avrdude: verifying flash memory against /tmp/arduino_build_979028/Giant_Wall_Clock_I2C-control_2019-11-15.ino.hex:
avrdude: load data flash data from input file /tmp/arduino_build_979028/Giant_Wall_Clock_I2C-control_2019-11-15.ino.hex:
avrdude: input file /tmp/arduino_build_979028/Giant_Wall_Clock_I2C-control_2019-11-15.ino.hex contains 3896 bytes
avrdude: reading on-chip flash data:

Reading | ################################################## | 100% 0.44s

avrdude: verifying ...
avrdude: 3896 bytes of flash verified

avrdude done.  Thank you.

Now looking at another clock to go in the kitchen.  Just slightly smaller dial and hands.  May go for spur gear drive to seconds wheel rather than ratchet.  Maybe a slightly smaller gear unit would look better in the smaller space.

Now for some calculation for a pair of spur gears for the seconds drive.

1. Seconds wheel wants to turn once a minute = 60s.
2. Stepper motor driven by TMC2100 in it's quietest mode takes 3200 microsteps per revolution.
3. With a 6:1 reduction ratio (16t motor pinion and 96t seconds wheel) motor takes 10s per revolution.
4. This means 320 microsteps per second.
5. In the Arduino sketch above the 1600 pulses in a half-second becomes 160 (which agrees with the 1/10th speed of the motor).

Seconds gear drive.

Finished clock.  Apart from changing to gear drive from the motor I've cut out the ball bearing on the moving hours gear - it's sufficient to run PLA on a stainless steel bolt with washers either side of gear.  With such a low speed, wear will be insignificant.  (Bolts and washers are not shown in this model.)

Fixed hour and minute gears are attached to base plate with three countersunk bolts.  Then washer, moving hour gear, another washer and all retained by a 6mm SS bolt into the base plate.